Research ArticleInfluence of Quasiperiodic Gravitational Modulation onConvective Instability of Liquid-Liquid Polymerization Front

Saadia Assiyad1 Karam Allali1 and Mohamed Belhaq2

1Department of Mathematics University of Hassan II-Casablanca FST PO Box 146 Mohammedia Morocco2Department of Mechanics University of Hassan II-Casablanca Casablanca Morocco

Correspondence should be addressed to Karam Allali allalihotmailcom

Received 11 August 2015 Accepted 7 October 2015

Academic Editor Fazal M Mahomed

Copyright copy 2015 Saadia Assiyad et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The influence of quasiperiodic gravitational modulation on convective instability of polymerization front with liquid monomerand liquid polymer is studied The model includes the heat equation the concentration equation and the Navier-Stokes equationsunder the Boussinesq approximationThe linear stability analysis of the problem is carried out and the interface problem is derivedUsing numerical simulations the convective instability threshold is determined and the boundary of the convective instability isobtained for different amplitudes and frequencies ratio

1 Introduction

Frontal polymerization is the process of polymer productionin propagating reaction fronts [1ndash4] In the absence of vibra-tion the influence of convective instability on polymerizationfront when the monomer is liquid and the polymer is solidwas studied in [5] while the case of liquid polymer wasconsidered in [6] The influence of periodic gravitationalmodulation on the convective instability in the case of liquid-solid polymerization front was studied [7] and it was shownthat the propagation of polymerization reaction front isstrongly affected by the amplitude and the frequency of vibra-tions In particular the polymerization front can be stableor unstable depending on the values of vibration parametersThe influence of periodic vibrations on convective instabilityof reaction front was also studied in the case of liquids [8]and it was concluded that for small vibration amplitudesthe reaction front remains stable and it loses its stability forsufficiently large amplitude of vibrations

Recent works were devoted to the influence of a qua-siperiodic (QP) gravitational modulation on the convectiveinstability of reaction front For instance the influenceof the QP gravitational modulation on reaction front wasexamined in the case of porousmedia described by the Darcyequation [9] On the other hand the case of liquid-solid

polymerization front was considered in [10] using theNavier-Stokes equations instead of the Darcy equation and it wasrevealed that both the amplitudes and the frequencies ratioinfluence the stability domain of the polymerization frontSpecifically for appropriate values of vibration amplitudesand increasing values of the frequencies ratio a stabilizingeffect is observed The effect of the wave number on thereaction front was also examined showing that increasing thewave number widens the stability domain

The present work studies the effect of QP gravitationalmodulation on the convective instability of the polymeriza-tion front but in the case of liquid-liquid frontal polymer-ization This case is different from the previous one [10] inthe sense that in [10] the equation of motion is consideredonly after the reaction zone since the polymer is in the solidphase In the present work instead the equation of motion isconsidered before and after the reaction zone because boththe monomer and the polymer are liquids

The next section presents the frontal polymerizationmodel Section 3 develops the perturbation analysis whilethe interface problem is examined in Section 4 The linearstability analysis is discussed in Section 5 and numericalinvestigations are carried out in Section 6 The last sectionconcludes the work

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 484562 11 pageshttpdxdoiorg1011552015484562

2 Mathematical Problems in Engineering

2 Frontal Polymerization Model

We consider a polymerization process with a liquid reactantand a liquid product by assuming that the reaction front prop-agates in the direction opposite to the direction of gravity Inthis case the model of the frontal polymerization is given bythe system of equations

120597119879

120597119905+ (V sdot nabla) 119879 = 120581Δ119879 + 119902119882

120597120572

120597119905+ (V sdot nabla) 120572 = 119882

120597V120597119905

+ (V sdot nabla) V = minus1

120588nabla119901 + ]ΔV

+ 119892 (1 + 119887 (119905)) 120573 (119879 minus 1198790) 120574

div (V) = 0

(1)

with the following boundary conditions

119911 997888rarr +infin 119879 = 119879119894 120572 = 0 V = 0 (2)

119911 997888rarr minusinfin 119879 = 119879119887 120572 = 1 V = 0 (3)

The gradient divergence and Laplace operators aredefined by

nablaV = (120597V120597119909

120597V120597119910120597V120597119911)

div k = 120597k1

120597119909+120597k2

120597119910+120597k3

120597119911

ΔV =1205972V1205972119909

+1205972V1205972119910

+1205972V1205972119911

(4)

where the variables (119909 119910 and 119911) are the spatial coordinatessuch that minusinfin lt 119909 119910 119911 lt +infin 119879 is the temperature 120572 is theconcentration of the reaction product V is the velocity 119901 isthe pressure 120581 is the coefficient of thermal diffusivity 119902 is theadiabatic temperature heat release 120588 is an average value of thedensity ] is the coefficient of kinematic viscosity 120574 is the unitvector in the upward direction 120573 is the coefficient of thermalexpansion 119892 is the gravitational acceleration and 119887(119905) is theQP acceleration acting on the fluid which is given by 119887(119905) =1205821sin(1205901119905) + 120582

2sin(1205902119905) such that 120582

1 1205822are the amplitudes

and 1205901 1205902are the incommensurate frequencies of the QP

gravitational modulation The quantity 1198790is a mean value of

temperature 119879119894is the initial temperature and 119879

119887= 119879119894+ 119902

is the temperature of the reacted mixture We consider one-step reaction of zero order where the reaction rate is definedas follows

119882 = 119896 (119879) 120601 (120572) 120601 (120572) =

1 if 120572 lt 1

0 if 120572 = 1

(5)

The temperature dependence of the reaction rate is givenby the Arrhenius law [11]

119896 (119879) = 1198960exp(minus 119864

1198770119879) (6)

where 1198960is the preexponential factor 119864 is the activation

energy and 1198770is the universal gas constant We assume

that the two liquids are incompressible and the diffusivitycoefficient is very small comparing to the thermal diffusivitycoefficient such that the diffusivity will be neglected in theconcentration equation

To obtain the dimensionless model we introduce thedimensionless spatial variables as

1199091=1199091198881

120581

1199101=1199101198881

120581

1199111=1199111198881

120581

1199051=1199051198882

1

120581

1199011=

119901

11988821120588

1198881=

119888

radic2

V1=

V1198881

120579 =119879 minus 119879119887

119902

1198882=2119896012058111987701198792

119887

119902119864exp(minus 119864

1198770119879119887

)

(7)

where 119888 defines the stationary reaction front velocity andcan be calculated asymptotically for large Zeldovich number[12] For simplicity we keep the same notation for the othervariables and pressure System (1) with the two boundaryconditions (2)-(3) can be written in the form

120597120579

120597119905+ (V sdot nabla) 120579 = Δ120579 + 119885 exp( 120579

119885minus1 + 120575120579)120601 (120572) (8)

120597120572

120597119905+ (V sdot nabla) 120572 = 119885 exp( 120579

119885minus1 + 120575120579)120601 (120572) (9)

120597V120597119905

+ (V sdot nabla) V

= minusnabla119901 + 119875ΔV

+ 119875119877 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) (120579 + 120579

0) 120574

(10)

div (V) = 0 (11)

with the following conditions at infinity

119911 997888rarr +infin 120579 = minus1 120572 = 0 V = 0

119911 997888rarr minusinfin 120579 = 0 120572 = 1 V = 0

(12)

and 119875 = ]120581 is the Prandtl number 119877 = 1198921205731199021205812(]1198883) is the

Rayleigh number119885 = 11990211986411987701198792

119887is the Zeldovich number and

Mathematical Problems in Engineering 3

120575 and 1205790are given respectively by 120575 = 119877

0119879119887119864 and 120579

0= (119879119887minus

1198790)119902 and 120583 = 2120581120590119888

2Next we perform the linear stability analysis to tackle the

interface problem

3 Approximation of Infinitely NarrowReaction Zone

To study the interface problem analytically we use a singularperturbation analysis where the reaction zone is infinitelynarrow and the reaction term is neglected outside the zone[13] In this way the problem can be reduced to an interfaceproblem

To perform a formal asymptotic analysis 120598 = 119885minus1 is

considered as a small parameter The new independent var-iable is given by 119911

1= 119911 minus 120577(119909 119910 119905) where 120577(119909 119910 119905) defines the

location of the reaction zone Introducing new functions 1205791

1205721 V1 and 119901

1as

120579 (119909 119910 119911 119905) = 1205791(119909 119910 119911

1 119905)

120572 (119909 119910 119911 119905) = 1205721(119909 119910 119911

1 119905)

V (119909 119910 119911 119905) = V1(119909 119910 119911

1 119905)

119901 (119909 119910 119911 119905) = 1199011(119909 119910 119911

1 119905)

(13)

The system of (8)ndash(11) can be written in the form (index 1 forthe new function is omitted)120597120579

120597119905minus120597120579

1205971199111

120597120577

120597119905+ (V sdot nabla) 120579

= Δ120579 + 119885 exp( 120579

119885minus1 + 120575120579)120601 (120572)

120597120572

120597119905minus120597120572

1205971199111

120597120577

120597119905+ (V sdot nabla) 120572 = 119885 exp( 120579

119885minus1 + 120575120579)120601 (120572)

120597V120597119905

minus120597V1205971199111

120597120577

120597119905+ (V sdot nabla) V

= minusnabla119901 + 119875ΔV

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) (120579 + 120579

0) 120574

120597V119909

120597119909minus120597V119909

1205971199111

120597120577

120597119909+120597V119910

120597119910minus120597V119910

1205971199111

120597120577

120597119910+120597V119911

1205971199111

= 0

(14)

where Δ nabla and 119876 are given by

Δ =1205972

1205971199092+

1205972

1205971199102+

1205972

12059711991121

minus 21205972

1205971199091205971199111

120597120577

120597119909minus 2

1205972

1205971199101205971199111

120597120577

120597119910

+1205972

12059711991121

((120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus120597

1205971199111

(1205972120577

1205971199092+1205972120577

1205971199102)

nabla = (120597

120597119909minus

120597

1205971199111

120597120577

120597119909120597

120597119910minus

120597

1205971199111

120597120577

120597119910120597

1205971199111

)

119876 = 119875119877

(15)

To approximate the jump conditions and resolve theinterface problem we use the matched asymptotic expansionby seeking the outer solution of problem (14) in the form

120579 = 1205790+ 1205981205791+ sdot sdot sdot

120572 = 1205720+ 1205981205721+ sdot sdot sdot

V = V0+ 120598V1+ sdot sdot sdot

119901 = 1199010+ 1205981199011+ sdot sdot sdot

(16)

Introducing the stretched coordinate 120578 = 1199111120598minus1 where 120598 =

119885minus1 the inner solution can be approximated in the following

form

120579 = 1205981205791+ sdot sdot sdot

120572 = 0+ 1205981+ sdot sdot sdot

V = V0+ 120598V1+ sdot sdot sdot

119901 = 1199010+ 1205981199011+ sdot sdot sdot

120577 = 1205770+ 1205981205771+ sdot sdot sdot

(17)

Substituting the inner and outer solutions in (14) leads to

Order 120598minus2

119875(1 + (1205971205770

120597119909)

2

+ (1205971205770

120597119910)

2

)1205972V0

1205971205782= 0 (18)

Order 120598minus1

(1 + (1205971205770

120597119909)

2

+ (1205971205770

120597119910)

2

)12059721205791

1205971205782

+ exp( 1205791

1 + 1205751205791

)120601 (0) = 0

(19)

minus1205970

120597120578

1205971205770

120597119905minus1205970

120597120578(V0119909

1205971205770

120597119909+ V0119910

1205971205770

120597119910minus V0119911)

= exp( 1205791

1 + 1205751205791

)120601 (0)

(20)

minus120597V0119909

120597120578

1205971205770

120597119905minus V0119909

120597V0

120597120578

1205971205770

120597119909minus V0119910

120597V0

120597120578

1205971205770

120597119910+ V0119911

120597V0

120597120578

= 1199050

1205971199010

120597120578+ 119875(119860

1205972V1

1205971205782+ 1198650

120597V0

120597120578)

(21)

minus120597V0119909

120597120578

1205971205770

120597119909minus120597V0119910

120597120578

1205971205770

120597119910+120597V0119911

120597120578= 0 (22)

4 Mathematical Problems in Engineering

Order 1205980

120597V0

120597119905minus120597V1

120597120578

1205971205770

120597119905minus120597V0

120597120578

1205971205771

120597119905+ V0119909(120597V0

120597119909minus120597V1

120597120578

1205971205770

120597119909

minus120597V0

120597120578

1205971205771

120597119909) + V1119909

120597V0

120597120578

1205971205770

120597119909+ V1119910

120597V0

120597120578

1205971205770

120597119910

+ V0119910(120597V0

120597119910minus120597V1

120597120578

1205971205770

120597119910minus120597V0

120597120578

1205971205771

120597119910) + V0119911

120597V1

120597120578+ V1119911

sdot120597V0

120597120578= minusnabla01199010+ 1199051

1205971199010

120597120578+ 1199050

1205971199011

120597120578+ 119875(119860

1205972V2

1205971205782

+ 1199053

1205972V1

1205971205782+ 1198650

120597V1

120597120578+ 1199054

1205972V0

1205971205782+ 1198651

120597V0

120597120578+ Δ1V0)

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) 120574120579

0

(23)

120597V0119909

120597119909minus120597V1119909

120597120578

1205971205770

120597119909minus120597V0119909

120597120578

1205971205771

120597119909+120597V0119910

120597119910minus120597V1119910

120597120578

1205971205770

120597119910

minus120597V0119910

120597120578

1205971205771

120597119910+120597V1119911

120597120578= 0

(24)

Order 1205981

120597V1

120597119905minus (

120597V2

120597120578

1205971205770

120597119905+120597V1

120597120578

1205971205771

120597119905+120597V0

120597120578

1205971205772

120597119905) + V0119909(120597V1

120597119909

minus120597V1

120597120578

1205971205771

120597119909minus120597V0

120597120578

1205971205772

120597119909minus120597V2

120597120578

1205971205770

120597119909) + V1119909(120597V0

120597119909

minus120597V0

120597120578

1205971205771

120597119909minus120597V1

120597120578

1205971205770

120597119909) minus V2119909

120597V0

120597120578

1205971205770

120597119909+ V0119910(120597V1

120597119910

minus120597V1

120597120578

1205971205771

120597119910minus120597V0

120597120578

1205971205772

120597119910minus120597V2

120597120578

1205971205770

120597119910) + V1119910(120597V0

120597119910

minus120597V0

120597120578

1205971205771

120597119910minus120597V1

120597120578

1205971205770

120597119910) minus V2119910

120597V0

120597120578

1205971205770

120597119910+ V0119911

120597V2

120597120578

+ V1119911

120597V1

120597120578+ V2119911

120597V0

120597120578= 1199050

1205971199012

120597120578+ 1199051

1205971199011

120597120578minus nabla01199011+ 1199052

sdot1205971199010

120597120578+ 119875(119860

1205972V3

1205971205782+ 1199053

1205972V2

1205971205782+ 1198650

120597V2

120597120578+ 1198651

120597V1

120597120578

+ Δ1V1+ 1198652

120597V0

120597120578) + 119876 (1 + 120582

1sin (120590

1119905)

+ 1205822sin (120590

2119905)) 1205790120574

(25)

120597V1119909

120597119909minus120597V2119909

120597120578

1205971205770

120597119909minus120597V1119909

120597120578

1205971205771

120597119909minus120597V0119909

120597120578

1205971205772

120597119909+120597V1119910

120597119910

minus120597V2119910

120597120578

1205971205770

120597119910minus120597V1119910

120597120578

1205971205771

120597119910minus120597V0119910

120597120578

1205971205772

120597119910+120597V2119911

120597120578= 0

(26)

Order 1205982

120597V2119909

120597119909minus120597V3119909

120597120578

1205971205770

120597119909minus120597V2119909

120597120578

1205971205771

120597119909minus120597V1119909

120597120578

1205971205772

120597119909minus120597V0119909

120597120578

1205971205773

120597119909

+120597V2119910

120597119910minus120597V3119910

120597120578

1205971205770

120597119910minus120597V2119910

120597120578

1205971205771

120597119910minus120597V1119910

120597120578

1205971205772

120597119910

minus120597V0119910

120597120578

1205971205773

120597119910+120597V3119911

120597120578= 0

(27)

where

Δ1=

1205972

1205971199092+

1205972

1205971199102

nabla0= (

120597

120597119909120597

120597119910 0)

119865119894= minus2(

120597120577119894

120597119909

120597

120597119909+120597120577119894

120597119910

120597

120597119910) minus (

1205972120577119894

1205971199092+1205972120577119894

1205971199102) 119868

119894 = 0 1 2

1199050= (

1205971205770

1205971199091205971205770

120597119910 minus1)

119905119894= (

120597120577119894

120597119909120597120577119894

120597119910 minus 1) 119894 = 1 2

1199053= 2(

1205971205771

120597119909

1205971205770

120597119909+1205971205770

120597119910

1205971205771

120597119910)

1199054= (

1205971205771

120597119909)

2

+ (1205971205771

120597119910)

2

+1205971205770

120597119909

1205971205772

120597119909+1205971205770

120597119910

1205971205772

120597119910

119860 = (1 + (1205971205770

120597119909)

2

+ (1205971205770

120597119910)

2

)

(28)

and 119868 is the identity operator The matching conditions as120578 rarr plusmninfin are given by

V0sim V0

10038161003816100381610038161199111=plusmn0

(29)

V1sim (

120597V0

1205971199111

100381610038161003816100381610038161003816100381610038161199111=plusmn0

)120578 + V1

10038161003816100381610038161199111=plusmn0

(30)

V2sim1

2(1205972V0

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=plusmn0

)1205782+ (

120597V1

1205971199111

100381610038161003816100381610038161003816100381610038161199111=plusmn0

)120578

+ V2

10038161003816100381610038161199111=plusmn0

(31)

V3sim1

6(1205973V0

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=plusmn0

)1205783+1

2(1205972V1

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=plusmn0

)1205782

+ (120597V2

1205971199111

100381610038161003816100381610038161003816100381610038161199111=plusmn0

)120578 + V3

10038161003816100381610038161199111=plusmn0

(32)

Mathematical Problems in Engineering 5

As 120578 rarr +infin

1205791asymp 1205791

10038161003816100381610038161199111=+0

+ (1205971205790

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

)120578 0997888rarr 0 (33)

As 120578 rarr minusinfin

1205791sim 1205791

10038161003816100381610038161199111=minus0

0997888rarr 1 (34)

From (18) we obtain

1205972V0

1205971205782= 0 (35)

Consequently we can conclude that V0(120578) is a linear

function of 120578 and identically constant since the velocity isbounded Equation (29) becomes

V0

10038161003816100381610038161199111=+0

= V0

10038161003816100381610038161199111=minus0

(36)

120597V0

120597120578= 0 (37)

From (36)-(37) we deduce that the first term in theexpression of the velocity V

0is continuous at the front By

substituting (37) into (21) we find

1198751198601205972V1

1205971205782+ 1199050

1205971199010

120597120578= 0 (38)

120597V0119909

120597119909minus120597V1119909

120597120578

1205971205770

120597119909+120597V0119910

120597119910minus120597V1119910

120597120578

1205971205770

120597119910+120597V1119911

120597120578= 0 (39)

Differentiating (39) with respect to 120578 one obtains

1205972V1119909

1205971205782

1205971205770

120597119909+1205972V1119910

1205971205782

1205971205770

120597119910minus1205972V1119911

1205971205782= 0 (40)

As a result (38) is a vectorial equation with three com-ponents We multiply the first component by 120597120577

0120597119909 the

second by 1205971205770120597119910 and the third by minus1 and adding we have

119875119860(1205972V1119909

1205971205782

1205971205770

120597119909+1205972V1119910

1205971205782

1205971205770

120597119910minus1205972V1119911

1205971205782) + 119860

1205971199010

120597120578= 0 (41)

Considering the two equations (38) and (40) we obtain

1205971199010

120597120578= 0 (42)

1205972V1

1205971205782= 0 (43)

From the boundedness of the velocity and (30) the con-tinuity of the first order term of the outer expansion and ofthe first derivative of the zero-order term we can write

120597V0

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

=120597V0

1205971199111

100381610038161003816100381610038161003816100381610038161199111=minus0

V1

10038161003816100381610038161199111=+0

= V1

10038161003816100381610038161199111=minus0

(44)

Differentiating (23) once and (26) twice with respect to 120578and using the three equations (37) (42) and (43) we get

1199050

12059721199011

1205971205782+ 119875119860

1205973V2

1205971205783= 0 (45)

1205973V2119909

1205971205783

1205971205770

120597119909+1205973V2119910

1205971205783

1205971205770

120597119910minus1205973V2119911

1205971205783= 0 (46)

As above we multiply the three components of (45)respectively by 120597120577

0120597119909 120597120577

0120597119910 and minus1 and adding we get

119875119860(1205973V2119909

1205971205783

1205971205770

120597119909+1205973V2119910

1205971205783

1205971205770

120597119910minus1205973V2119911

1205971205783) + 119860

12059721199011

1205971205782= 0 (47)

From the previous equation and using (45)-(46) we have

12059721199011

1205971205782= 0

1205973V2

1205971205783= 0

(48)

Knowing that the velocity is bounded and considering(31) we obtain the jump conditions in the following forms

1205972V0

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=+0

=1205972V0

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=minus0

120597V1

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

=120597V1

1205971199111

100381610038161003816100381610038161003816100381610038161199111=minus0

V2

10038161003816100381610038161199111=+0

= V2

10038161003816100381610038161199111=minus0

(49)

Differentiating (25) twice and (27) three times withrespect to 120578 and taking into account (37) (42) (48) we obtain

1199050

12059731199012

1205971205783+ 119875119860

1205974V3

1205971205784

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) 120574

12059721205791

1205971205782= 0

1205974V3119909

1205971205784

1205971205770

120597119909+1205974V3119910

1205971205784

1205971205770

120597119910minus1205974V3119911

1205971205784= 0

(50)

119875119860(1205974V3119909

1205971205784

1205971205770

120597119909+1205974V3119910

1205971205784

1205971205770

120597119910minus1205974V3119911

1205971205784) + 119860

12059731199012

1205971205783

minus 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905))

12059721205791

1205971205782= 0

(51)

From the last equation and using (50) we can write

11986012059731199012

1205971205783minus 119876 (1 + 120582

1sin (120590

1119905) + 120582

2sin (120590

2119905))

12059721205791

1205971205782

= 0

(52)

1205740

12059721205791

1205971205782=1205974V3

1205971205784 (53)

6 Mathematical Problems in Engineering

where 1205740is defined by

1205740= (minus

1205971205770

120597119909

119877

1198602 minus1205971205770

120597119910

119877

1198602119877

1198602minus119877

119860)

sdot (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905))

(54)

Now integrating (53) with respect to 120578 using the system(32)ndash(34) lead to

1205973V0

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=minus0

minus1205973V0

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=+0

= minus1205740

1205971205790

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

(55)

As a first conclusion the velocity jump conditions acrossthe reaction front are given by (36) (44) (49) and (55)

From (20) (22) (37) we conclude that 0is a monotonic

function satisfying 0 lt 0lt 1 Therefore the reaction is of

zero order and 120601(0) equiv 1 Multiplying (19) by 120597120579

1120597120578 and

integrating the result yield

(1205971205791

120597120578)

210038161003816100381610038161003816100381610038161003816100381610038161003816+infin

minus (1205971205791

120597120578)

210038161003816100381610038161003816100381610038161003816100381610038161003816minusinfin

= 2119860minus1int

1205791|1199111=minus0

minusinfin

exp( 120591

1 + 120591120575) 119889120591

(56)

Subtracting (19) from (20) and integrating the result oneobtains

1205971205791

120597120578

100381610038161003816100381610038161003816100381610038161003816+infin

minus1205971205791

120597120578

100381610038161003816100381610038161003816100381610038161003816minusinfin

= minus119860minus1(1205971205770

120597119905+ 119904) (57)

where 119904 is given by

119904 = V0119909

1205971205770

120597119909+ V0119910

1205971205770

120597119910minus V0119911 (58)

As a second conclusion the temperature jump conditionsacross the reaction front are given by the two last equations(56)-(57) By using the matching conditions above andtruncating the expansion as

120579 asymp 1205790

1205791

10038161003816100381610038161199111=minus0

asymp 119885120579|1199111=+0

120577 asymp 1205770

V asymp V0

(59)

the jump conditions can be written as

(120597120579

1205971199111

)

2100381610038161003816100381610038161003816100381610038161003816+0

minus (120597120579

1205971199111

)

2100381610038161003816100381610038161003816100381610038161003816minus0

= 2119885(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot int

1205791|1199111=minus0

minusinfin

exp ( 120591

119885minus1 + 120591120575) 119889120591

120597120579

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

minus120597120579

1205971199111

100381610038161003816100381610038161003816100381610038161199111=minus0

= minus(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot (120597120577

120597119905+ (V119909

120597120577

120597119909+ V119910

120597120577

120597119910minus V119911)

100381610038161003816100381610038161003816100381610038161199111=+0

)

V119911

10038161003816100381610038161199111=+0

= V119911

10038161003816100381610038161199111=minus0

120597V119911

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

=120597V119911

1205971199111

100381610038161003816100381610038161003816100381610038161199111=minus0

1205972V119911

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=+0

=120597V119911

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=minus0

1205973V119911

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=minus0

minus1205973V119911

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=+0

= minus119877(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot ((1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

minus 1)

sdot (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905))

120597120579

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

(60)

4 The Interface Problem and Perturbation

To study the propagation of polymerization front with aliquid reactant and liquid product the equation of motionhas been considered before and after the reaction zone Thischanges the jump conditions and influences the stabilityconditions of the frontal polymerization process Notice thatin the case of liquid-solid polymerization front the equationof motion is considered after the reaction zone The originalsystem (2) (3) (8) (9) leads to the following interfaceproblem

In the liquid monomer 119911 gt 120577

120597120579

120597119905+ (V sdot nabla) 120579 = Δ120579 (61)

120572 = 0 (62)

Mathematical Problems in Engineering 7

120597V120597119905

+ (V sdot nabla) V

= minusnabla119901 + 119875ΔV

+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579

0) 120574

(63)

div (V) = 0 (64)

In the liquid polymer 119911 lt 120577

120597120579

120597119905+ (V sdot nabla) 120579 = Δ120579 (65)

120572 = 1 (66)

120597V120597119905

+ (V sdot nabla) V

= minusnabla119901 + 119875ΔV

+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579

0) 120574

(67)

div (V) = 0 (68)

At the interface 119911 = 120577

120579|120577minus0 = 120579|120577+0

120597120579

120597119911

10038161003816100381610038161003816100381610038161003816120577=minus0minus120597120579

120597119911

10038161003816100381610038161003816100381610038161003816120577=+0= (1 + (

120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot (120597120577

120597119905+ (V119909

120597120577

120597119909+ V119910

120597120577

120597119910minus V119911)

10038161003816100381610038161003816100381610038161003816120577

)

(120597120579

120597119911)

2100381610038161003816100381610038161003816100381610038161003816120577minus0

minus (120597120579

120597119911)

21003816100381610038161003816100381610038161003816100381610038161205771+0

= minus2119885(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot int

120579|120577

minusinfin

exp ( 120591

119885minus1 + 120575120591) 119889120591

V119911

1003816100381610038161003816120577=minus0 = V119911

1003816100381610038161003816120577=+0

120597V119911

120597119911

10038161003816100381610038161003816100381610038161003816120577=minus0=120597V119911

120597119911

10038161003816100381610038161003816100381610038161003816120577=+0

1205972V119911

1205971199112

100381610038161003816100381610038161003816100381610038161003816120577=minus0

=1205972V119911

1205971199112

100381610038161003816100381610038161003816100381610038161003816120577=+0

1205973V119911

1205971199113

100381610038161003816100381610038161003816100381610038161003816120577=minus0

minus1205973V119911

1205971199113

100381610038161003816100381610038161003816100381610038161003816120577=+0

= minus119877(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot ((1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

minus 1)

sdot (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) (

120597120579

120597119911)

10038161003816100381610038161003816100381610038161003816120577=+0

(69)

with the conditions at infinity

119911 = minusinfin 120579 = 0 V = 0

119911 = +infin 120579 = minus1 V = 0

(70)

5 Stability Analysis

The interface problem analyzed in Section 4 has a travellingwave solution in the following form

(120579 (119909 119910 119911 119905) 120572 (119909 119910 119911 119905) V)

= (120579119904(119911 minus 119906119905) 120572

119904(119911 minus 119906119905) 0)

(120579119904 (119911 minus 119906119905) 120572119904 (119911 minus 119906119905))

=

(0 1) 1199112lt 0

(exp (minus1199061199112) minus 1 0) 119911

2gt 0

(71)

where 1199112= 119911minus119906119905 and 119906 is the speed of the stationary reaction

frontThis solution is a basic and stationary solution of inter-face problem (61)ndash(70) written in the moving coordinatesThen (61) (63) (65) and (67) can be replaced by

120597120579

120597119905+ (V sdot nabla) 120579 = Δ120579 + 119906

120597120579

1205971199112

120597V120597119905

+ (Vnabla) V

= minusnabla119901 + 119875ΔV + 119906120597120579

1205971199112

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) (120579 + 120579

0) 120574

(72)

The other equations remain unchanged To study thestability of reaction front we seek the solution of the problemas follows

120579 = 120579119904+ 120579

119901 = 119901119904+ 119901

V = V119904+ V

(73)

where 120579 119901 and V are respectively small perturbation oftemperature pressure and velocity

8 Mathematical Problems in Engineering

Subtituting (73) into (64) (72) one obtains the first-orderterms as follows

In the liquid monomer 1199112gt 120585

120597120579

120597119905= Δ120579 + 119906

120597120579

1205971199112

minus V1199111205791015840

119904 (74)

120597V120597119905

= minusnabla119901 + 119875ΔV + 119906120597120579

1205971199112

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579120574

(75)

div (V) = 0 (76)

In the liquid polymer 1199112lt 120585

120597120579

120597119905= Δ120579 + 119906

120597120579

1205971199112

(77)

120597V120597119905

= minusnabla119901 + 119875ΔV + 119906120597V1205971199112

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579120574

(78)

div (V) = 0 (79)

where 120585 is defined by 120585 = 120577 minus 119906119905We note

(120579 V119911) =

(1205791 V1199111) for 119911

2lt 120585

(1205792 V1199112) for 119911

2lt 120585

(80)

and consider the perturbation as follows

120579119894= 120579119894(1199112 119905) exp (119895 (119896

1119909 + 1198962119910))

V119911119894= V119911119894(1199112 119905) exp (119895 (119896

1119909 + 1198962119910))

120585 = 1205981(119905) exp (119895 (119896

1119909 + 1198962119910))

(81)

where 119896119894 119894 = 1 2 and 120598

1denote respectively the wave

numbers (in 119909 and 119910 directions) and the amplitude of theperturbation and 119895

2= minus1 Then jump conditions (69) are

linearized by taking into account the fact that

120579|120585=plusmn0 = 120579119904(plusmn0) + 120585120579

1015840

119904(plusmn0) + 120579 (plusmn0)

120597120579

1205971199112

10038161003816100381610038161003816100381610038161003816120585=plusmn0

= 1205791015840

119904(plusmn0) + 120585120579

10158401015840

119904(plusmn0) +

120597120579

1205971199112

100381610038161003816100381610038161003816100381610038161003816120585=plusmn0

(82)

Then the higher-order terms are given as

1205792

100381610038161003816100381610038161199112=0minus 1205791

100381610038161003816100381610038161199112=0= 119906120585 (83)

1205971205792

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

minus1205971205791

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

= minus1199062120585 minus

120597120585

120597119905+ V119911

10038161003816100381610038161199112=0 (84)

1199062120585 +

1205971205792

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

= minus119885

1199061205791

100381610038161003816100381610038161199112=0 (85)

V2119911

10038161003816100381610038161199112=0= V1119911

10038161003816100381610038161199112=0 (86)

120597V1199112

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

=120597V1119911

1205971199112

100381610038161003816100381610038161003816100381610038161199112=0

(87)

1205972V2119911

12059711991122

1003816100381610038161003816100381610038161003816100381610038161199112=0

=1205972V1119911

12059711991122

1003816100381610038161003816100381610038161003816100381610038161199112=0

(88)

1205973V2119911

12059711991132

1003816100381610038161003816100381610038161003816100381610038161199112=0

=1205973V1119911

12059711991132

1003816100381610038161003816100381610038161003816100381610038161199112=0

(89)

Substituting (81) into (83)ndash(89) leads to

1205792(0 119905) minus 120579

1(0 119905) = 119906120598

1(119905)

1205791015840

2(0 119905) minus 120579

1015840

1(0 119905) = minus120598

1 (119905) 1199062minus 1205981015840

1(119905) + V

1 (0 119905)

1205981 (119905) 119906

2+ 1205791015840

2(0 119905) = minus

119885

1199061205791 (0 119905)

V(119894)2(0 119905) = V(119894)

1(0 119905) 119894 = 0 1 2 3

(90)

where

V(119894) =120597119894V120597V119894

1205791015840=120597120579119894

120597119911

1205981015840

1(119905) =

1198891205981(119905)

119889119905

(91)

After simplification (75) and (78) become

120597

120597119905ΔV119911minus 119906

120597

1205971199112

ΔV119911= 119875ΔΔV

119911+ 119876(1 + 120582

1sin (120590

1119905)

+ 1205822sin (120590

2119905) (

1205972

1205971199092+

1205972

1205971199102))120579

(92)

Substituting (71) (87) (88) into (74) (77) (92) andintroducing the variable 119908 = V10158401015840 minus 1198962V where 119896 = radic1198962

1+ 11989622

one obtains two systems of equations as follows

1199112lt 120585

120597119908

120597119905minus 1199061199081015840minus 119875 (119908

10158401015840minus 1198962119908)

= minus1198761198962(1 + 120582

1sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579

119908 = V10158401015840 minus 1198962V

120597120579

120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 0

(93)

Mathematical Problems in Engineering 9

0 5 10 15 20 25 300

50

100

150

200

250

300

350

Rc

1205901

1205902 = radic212059011205902 = radic71205901

1205902 = 0

1205902 = radic371205901

Figure 1 Critical Rayleigh number as a function of 1205901for different

frequencies ratio and for 1205902= 0 120582

1= 1205822= 5 119875 = 05 119885 = 8 and

119896 = 07

1199112gt 120585

120597119908

120597119905minus 1199061199081015840minus 119875 (119908

10158401015840minus 1198962119908)

= minus1198761198962(1 + 120582

1sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579

119908 = V10158401015840 minus 1198962V

120597120579

120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 119906 exp (minus119906119911

2) V

(94)

with the following boundary conditions

1205792(0 119905) minus 120579

1(0 119905) = 119906120598

1(119905) (95)

1205791015840

2(0 119905) minus 120579

1015840

1(0 119905) = minus119906

21205761(119905) minus 120576

1015840

1(119905) + V

1(0 119905) (96)

1205791015840

2(0 119905) +

119885

1199061205791 (0 119905) = minus119906

21205761 (119905) (97)

V(119894)1(0 119905) = V(119894)

2(0 119905) 119894 = 0 1 2 3 (98)

Combining the three boundary conditions (95)ndash(97)yields

1205791015840

1(0 119905) +

119885

1199061205791(0 119905) =

1

119906(1205792(0 119905) minus 120579

1(0 119905))

1015840

119905

minus V1(0 119905)

(99)

6 Numerical Results

The numerical simulations of the problem are performedusing the finite-difference approximation with implicitscheme The algorithm is given in Appendix

Evolution of maximum temperature versus time providesthe onset of stability of the polymerization front The con-vective instability occurs when a jump from bounded tounbounded values of maximum temperature is achieved

Figure 1 shows the critical Rayleigh number as a functionof the frequency 120590

1for different frequencies ratio and for

0 25 50 75 100 125 1500

5

10

15

20

25

Rc

1205822

1205821 = 20

1205821 = 50

1205821 = 80

1205821 = 150

Figure 2 Critical Rayleigh number as a function of 1205822for different

values of 1205821 1205902= radic2120590

1 1205901= 5 119875 = 05 119885 = 8 and 119896 = 07

0 25 50 75 100 125 150 175 2000

50

100

150

200

250

Rc

1205822

1205902 = radic371205901

1205902 = radic1712059011205902 = radic71205901

1205902 = 1radic712059011205902 = 1radic171205901

1205902 = 1radic371205901

Figure 3 Critical Rayleigh number as a function of 1205822for different

frequencies ratio 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and 120590

1= 5

1205902= 0 In the absence of gravitational modulation (120590

1= 1205902=

0) we find the same result (119877119888asymp 27) as in [8 14] It can be

observed that the frequencies ratio of the QP gravitationalmodulation has a significant influence on the convectiveinstability of the reaction front As indicated increasing thefrequencies ratio the stability domain becomes larger Inparticular this can be seen in the range of the frequency1205901belonging to the interval (0 10) For large values of the

frequency 1205901 the frequencies ratio 120590

21205901has no effect on

the convective instability It is also worthy to notice thatthe QP gravitational modulation has a stabilizing effect forsufficiently large frequencies ratio (120590

2= radic37120590

1) comparing

to the periodic modulation case (1205902= 0)

The critical Rayleigh number as a function of the ampli-tude 120582

2is shown in Figure 2 for different values of 120582

1 We can

conclude that for a fixed value of 1205822and increased values of

1205821 the polymerization front loses its stability monotonically

revealing that the stability of the reaction front can becontrolled by acting on the amplitudes of the QPmodulation

Figure 3 illustrates the critical Rayleigh number as afunction of the amplitude 120582

2for different frequencies ratio

It can be observed that increasing the frequencies ratio thestability domain becomes large for certain values of 120582

2 For

large values of 1205822(1205822ge 100) the effect of the frequencies

ratio on the convective instability boundary is insignificant

10 Mathematical Problems in Engineering

0 25 50 75 100 125 150

k = 15

k = 2

k = 1

0

50

100

150

200

250

300

350

Rc

1205822

Figure 4 Critical Rayleigh number as a function of 1205822for different

values of number wave 1205902= radic7120590

1 1205901= 10 119875 = 05 119885 = 8 and

1205821= 5

0 25 50 75 100 125 1500

255075

100125150175200

Rc

1205822

1205902 = radic21205901

1205902 = radic71205901

Figure 5 Critical Rayleigh number as a function of 1205822for two

different frequencies ratios 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and

1205901= 10

Figure 4 shows for a fixed frequencies ratio 12059021205901= radic7

the influence of the wave number on the critical Rayleighnumber versus 120582

2 The plots indicate that for large values of

the wave number the reaction front gains stabilityThe influence of 120582

2on the stability of the reaction front is

given in Figure 5 for two different frequencies ratio From thisfigure one can conclude that an increase of the frequenciesratio has a stabilizing effect

7 Conclusion

In this work we have studied the influence of the QP vibra-tions on the convective instability of liquid-liquid polymer-ization frontWe have used themodel which includes the heatequation the concentration equation and the Navier-Stokesequations under Boussinesq approximation The ZeldovichFrank-Kamenetskii method has been used assuming that thereaction occurs in a narrow zone To obtain the convec-tive instability threshold using the linear stability analysisthe reduced system of equations has been discretized andresolved using the finite-difference method with implicitscheme The results have shown that for fixed values ofamplitudes an increase of the frequencies ratio stabilizes thereaction front and for large values of the frequency 120590

1the

critical Rayleigh number tends to the unmodulated criticalvalue In addition it is observed that for a fixed frequencies

ratio and for a given amplitude 1205822an increase of the ampli-

tude1205821destabilizes the reaction front Also the reaction front

becomes more stable by increasing the wave number

Appendix

The algorithm of the considered problem uses the finite-difference approximation with implicit scheme The velocityis computed from the previous time values of the problemThe discretization in space is given by 119911

119894= 119894ℎ where ℎ is the

space step and 119894 is a positive integerThe discretization in timeis given by 119905

119894= 119894120591 where 120591 is sufficiently small time step The

variables of discretization are 119908119895119894= 119908(119911

119894 119905119895) V119895119894= V(119911

119894 119905119895)

and 120579119895119894= 120579(119911119894 119905119895) One obtains the following

In the liquid monomer 119911 lt 120577

(minus1

ℎ2) 120579119895+1

119894minus1+ (

1

120591+

2

ℎ2+ 1198962+119906

ℎ) 120579119895+1

119894

+ (minus1

ℎ2minus119906

ℎ) 120579119895+1

119894+1=1

120591120579119895

119894

(minus119875

ℎ2)119908119895+1

119894minus1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

119894

+ (minus119875

ℎ2minus119906

ℎ)119908119895+1

119894+1=1

120591119908119895

119894

minus 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895)) 120579119895

119894

(1

ℎ2) V119895+1119894minus1

+ (minus2

ℎ2minus 1198962) V119895+1119894

+ (1

ℎ2) V119895+1119894+1

= 119908119895

119894

(A1)

In the liquid polymer 119911 gt 120577

(minus1

ℎ2)119879119895+1

119894minus1+ (

1

120591+

2

ℎ2+ 1198962+119906

ℎ)119879119895+1

119894

+ (minus1

ℎ2minus119906

ℎ)119879119895+1

119894+1=1

120591119879119895

119894+ 119906 exp (minus119906119911

119894) 119881119895

119894

(minus119875

ℎ2)119882119895+1

119894minus1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119882119895+1

119894

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

119894+1=1

120591119882119895

119894

minus 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895)) 119879119895

119894

(1

ℎ2)119881119895+1

119894minus1+ (minus

2

ℎ2minus 1198962)119881119895+1

119894+ (

1

ℎ2)119881119895+1

119894+1= 119882119895

119894

(A2)

From the jump conditions we have also the followingsystems

(minus1

ℎ2) 120579119895+1

1

+ (1

120591+

2

ℎ2+ 1198962+119906

ℎ+ (

1198881

1198882

)(1

ℎ2+119906

ℎ)) 120579119895+1

0

Mathematical Problems in Engineering 11

= (minus1

120591) 120579119895

0minus (

1

1199061205911198882

)(1

ℎ2+119906

ℎ) (119879119895

0minus 120579119895

0)

minus (V0

1198882

)(1

ℎ2+119906

ℎ)

(minus119875

ℎ2)119908119895+1

1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

0

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

0= (

minus1

120591)119908119895

1

+ 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895))

sdot ((minus1198882

1198881

)1198790minus

1

1199061205911198881

(119879119895

0minus 120579119895

0) minus

V0

1198881

)

(A3)

(1

120591+

2

ℎ2+ 1198962+119906

ℎ+

1198882

ℎ21198881

)119879119895+1

0+ (

minus1

ℎ2minus119906

ℎ)119879119895+1

1

= (minus1

120591)119879119895

0minus 119906 exp (minus119906ℎ) minus ( 1

119906120591ℎ21198881

) (119879119895

0minus 120579119895

0)

minusV0

ℎ21198881

(A4)

(minus119875

ℎ2)119908119895+1

1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

0

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

0= (

minus1

120591)119908119895

0

+ 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895))

sdot ((minus1198881

1198882

)1205790minus

1

1199061205911198882

)(119879119895

0minus 120579119895

0minusV0

1198882

)

(A5)

with

1198881= (

1

ℎminus119885

119906minus

1

119906120591)

1198882= (minus

1

ℎ+

1

119906120591)

(A6)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] B B Khanukaev M A Kozhushner N S Enikolopyan and NM Chechilo ldquoTheory of the propagation of a polymerizationfrontrdquo Combustion Explosion and Shock Waves vol 10 no 1pp 17ndash21 1974

[2] J A Pojman and T-C-M Qui Nonlinear Dynamics with Pol-ymers Fundamentals Methods and Applications John Wiley ampSons 2011

[3] I D Robertson H L Hernandez S R White and J S MooreldquoRapid stiffening of a microfluidic endoskeleton via frontalpolymerizationrdquo ACS Applied Materials and Interfaces vol 6no 21 pp 18469ndash18474 2014

[4] Vi Volpert and Vl Volpert ldquoPropagation of frontal polym-erizationmdashcrystallization wavesrdquo European Journal of AppliedMathematics vol 5 no 2 pp 201ndash215 1994

[5] G Bowden M Garbey V M Ilyashenko et al ldquoEffect of con-vection on a propagating front with a solid product comparisonof theory and experimentsrdquoThe Journal of Physical Chemistry Bvol 101 no 4 pp 678ndash686 1997

[6] M Garbey A Taik and V Volpert ldquoLinear stability analysisof reaction fronts in liquidsrdquo Quarterly of Applied Mathematicsvol 54 no 2 pp 225ndash247 1996

[7] K Allali V Volpert and J A Pojman ldquoInfluence of vibrationson convective instability of polymerization frontsrdquo Journal ofEngineering Mathematics vol 41 no 1 pp 13ndash31 2001

[8] K Allali F Bikany A Taik and V Volpert ldquoInfluence ofvibrations on convective instability of reaction fronts in liquidsrdquoMathematical Modelling of Natural Phenomena vol 5 no 7 pp35ndash41 2010

[9] K Allali M Belhaq and K El Karouni ldquoInfluence of quasi-periodic gravitational modulation on convective instability ofreaction fronts in porous mediardquo Communications in NonlinearScience and Numerical Simulation vol 17 no 4 pp 1588ndash15962012

[10] KAllali S Assiyad andM Belhaq ldquoConvection of polymeriza-tion front with solid product under quasi-periodic gravitationalmodulationrdquo Nonlinear Dynamics and Systems Theory vol 14no 4 pp 323ndash334 2014

[11] M Menzinger and R Wolfgang ldquoThe meaning and use of theArrhenius activation energyrdquoAngewandte Chemie InternationalEdition in English vol 8 no 6 pp 438ndash444 1969

[12] B V Novozhilov ldquoThe rate of propagation of the front of anexothermic reaction in a condensed phaserdquo Doklady PhysicalChemistry vol 141 pp 836ndash838 1961

[13] Y B Zeldovich and D A Frank-Kamenetskii ldquoA theory ofthermal propagation of amerdquo Acta Physicochimica USSR vol 9pp 341ndash350 1938

[14] M Garbey A Taık and V Volpert ldquoInfluence of natural con-vection on stability of reaction fronts in liquidsrdquo Quarterly ofApplied Mathematics vol 56 no 1 pp 1ndash35 1998

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

2 Frontal Polymerization Model

We consider a polymerization process with a liquid reactantand a liquid product by assuming that the reaction front prop-agates in the direction opposite to the direction of gravity Inthis case the model of the frontal polymerization is given bythe system of equations

120597119879

120597119905+ (V sdot nabla) 119879 = 120581Δ119879 + 119902119882

120597120572

120597119905+ (V sdot nabla) 120572 = 119882

120597V120597119905

+ (V sdot nabla) V = minus1

120588nabla119901 + ]ΔV

+ 119892 (1 + 119887 (119905)) 120573 (119879 minus 1198790) 120574

div (V) = 0

(1)

with the following boundary conditions

119911 997888rarr +infin 119879 = 119879119894 120572 = 0 V = 0 (2)

119911 997888rarr minusinfin 119879 = 119879119887 120572 = 1 V = 0 (3)

The gradient divergence and Laplace operators aredefined by

nablaV = (120597V120597119909

120597V120597119910120597V120597119911)

div k = 120597k1

120597119909+120597k2

120597119910+120597k3

120597119911

ΔV =1205972V1205972119909

+1205972V1205972119910

+1205972V1205972119911

(4)

where the variables (119909 119910 and 119911) are the spatial coordinatessuch that minusinfin lt 119909 119910 119911 lt +infin 119879 is the temperature 120572 is theconcentration of the reaction product V is the velocity 119901 isthe pressure 120581 is the coefficient of thermal diffusivity 119902 is theadiabatic temperature heat release 120588 is an average value of thedensity ] is the coefficient of kinematic viscosity 120574 is the unitvector in the upward direction 120573 is the coefficient of thermalexpansion 119892 is the gravitational acceleration and 119887(119905) is theQP acceleration acting on the fluid which is given by 119887(119905) =1205821sin(1205901119905) + 120582

2sin(1205902119905) such that 120582

1 1205822are the amplitudes

and 1205901 1205902are the incommensurate frequencies of the QP

gravitational modulation The quantity 1198790is a mean value of

temperature 119879119894is the initial temperature and 119879

119887= 119879119894+ 119902

is the temperature of the reacted mixture We consider one-step reaction of zero order where the reaction rate is definedas follows

119882 = 119896 (119879) 120601 (120572) 120601 (120572) =

1 if 120572 lt 1

0 if 120572 = 1

(5)

The temperature dependence of the reaction rate is givenby the Arrhenius law [11]

119896 (119879) = 1198960exp(minus 119864

1198770119879) (6)

where 1198960is the preexponential factor 119864 is the activation

energy and 1198770is the universal gas constant We assume

that the two liquids are incompressible and the diffusivitycoefficient is very small comparing to the thermal diffusivitycoefficient such that the diffusivity will be neglected in theconcentration equation

To obtain the dimensionless model we introduce thedimensionless spatial variables as

1199091=1199091198881

120581

1199101=1199101198881

120581

1199111=1199111198881

120581

1199051=1199051198882

1

120581

1199011=

119901

11988821120588

1198881=

119888

radic2

V1=

V1198881

120579 =119879 minus 119879119887

119902

1198882=2119896012058111987701198792

119887

119902119864exp(minus 119864

1198770119879119887

)

(7)

where 119888 defines the stationary reaction front velocity andcan be calculated asymptotically for large Zeldovich number[12] For simplicity we keep the same notation for the othervariables and pressure System (1) with the two boundaryconditions (2)-(3) can be written in the form

120597120579

120597119905+ (V sdot nabla) 120579 = Δ120579 + 119885 exp( 120579

119885minus1 + 120575120579)120601 (120572) (8)

120597120572

120597119905+ (V sdot nabla) 120572 = 119885 exp( 120579

119885minus1 + 120575120579)120601 (120572) (9)

120597V120597119905

+ (V sdot nabla) V

= minusnabla119901 + 119875ΔV

+ 119875119877 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) (120579 + 120579

0) 120574

(10)

div (V) = 0 (11)

with the following conditions at infinity

119911 997888rarr +infin 120579 = minus1 120572 = 0 V = 0

119911 997888rarr minusinfin 120579 = 0 120572 = 1 V = 0

(12)

and 119875 = ]120581 is the Prandtl number 119877 = 1198921205731199021205812(]1198883) is the

Rayleigh number119885 = 11990211986411987701198792

119887is the Zeldovich number and

Mathematical Problems in Engineering 3

120575 and 1205790are given respectively by 120575 = 119877

0119879119887119864 and 120579

0= (119879119887minus

1198790)119902 and 120583 = 2120581120590119888

2Next we perform the linear stability analysis to tackle the

interface problem

3 Approximation of Infinitely NarrowReaction Zone

To study the interface problem analytically we use a singularperturbation analysis where the reaction zone is infinitelynarrow and the reaction term is neglected outside the zone[13] In this way the problem can be reduced to an interfaceproblem

To perform a formal asymptotic analysis 120598 = 119885minus1 is

considered as a small parameter The new independent var-iable is given by 119911

1= 119911 minus 120577(119909 119910 119905) where 120577(119909 119910 119905) defines the

location of the reaction zone Introducing new functions 1205791

1205721 V1 and 119901

1as

120579 (119909 119910 119911 119905) = 1205791(119909 119910 119911

1 119905)

120572 (119909 119910 119911 119905) = 1205721(119909 119910 119911

1 119905)

V (119909 119910 119911 119905) = V1(119909 119910 119911

1 119905)

119901 (119909 119910 119911 119905) = 1199011(119909 119910 119911

1 119905)

(13)

The system of (8)ndash(11) can be written in the form (index 1 forthe new function is omitted)120597120579

120597119905minus120597120579

1205971199111

120597120577

120597119905+ (V sdot nabla) 120579

= Δ120579 + 119885 exp( 120579

119885minus1 + 120575120579)120601 (120572)

120597120572

120597119905minus120597120572

1205971199111

120597120577

120597119905+ (V sdot nabla) 120572 = 119885 exp( 120579

119885minus1 + 120575120579)120601 (120572)

120597V120597119905

minus120597V1205971199111

120597120577

120597119905+ (V sdot nabla) V

= minusnabla119901 + 119875ΔV

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) (120579 + 120579

0) 120574

120597V119909

120597119909minus120597V119909

1205971199111

120597120577

120597119909+120597V119910

120597119910minus120597V119910

1205971199111

120597120577

120597119910+120597V119911

1205971199111

= 0

(14)

where Δ nabla and 119876 are given by

Δ =1205972

1205971199092+

1205972

1205971199102+

1205972

12059711991121

minus 21205972

1205971199091205971199111

120597120577

120597119909minus 2

1205972

1205971199101205971199111

120597120577

120597119910

+1205972

12059711991121

((120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus120597

1205971199111

(1205972120577

1205971199092+1205972120577

1205971199102)

nabla = (120597

120597119909minus

120597

1205971199111

120597120577

120597119909120597

120597119910minus

120597

1205971199111

120597120577

120597119910120597

1205971199111

)

119876 = 119875119877

(15)

To approximate the jump conditions and resolve theinterface problem we use the matched asymptotic expansionby seeking the outer solution of problem (14) in the form

120579 = 1205790+ 1205981205791+ sdot sdot sdot

120572 = 1205720+ 1205981205721+ sdot sdot sdot

V = V0+ 120598V1+ sdot sdot sdot

119901 = 1199010+ 1205981199011+ sdot sdot sdot

(16)

Introducing the stretched coordinate 120578 = 1199111120598minus1 where 120598 =

119885minus1 the inner solution can be approximated in the following

form

120579 = 1205981205791+ sdot sdot sdot

120572 = 0+ 1205981+ sdot sdot sdot

V = V0+ 120598V1+ sdot sdot sdot

119901 = 1199010+ 1205981199011+ sdot sdot sdot

120577 = 1205770+ 1205981205771+ sdot sdot sdot

(17)

Substituting the inner and outer solutions in (14) leads to

Order 120598minus2

119875(1 + (1205971205770

120597119909)

2

+ (1205971205770

120597119910)

2

)1205972V0

1205971205782= 0 (18)

Order 120598minus1

(1 + (1205971205770

120597119909)

2

+ (1205971205770

120597119910)

2

)12059721205791

1205971205782

+ exp( 1205791

1 + 1205751205791

)120601 (0) = 0

(19)

minus1205970

120597120578

1205971205770

120597119905minus1205970

120597120578(V0119909

1205971205770

120597119909+ V0119910

1205971205770

120597119910minus V0119911)

= exp( 1205791

1 + 1205751205791

)120601 (0)

(20)

minus120597V0119909

120597120578

1205971205770

120597119905minus V0119909

120597V0

120597120578

1205971205770

120597119909minus V0119910

120597V0

120597120578

1205971205770

120597119910+ V0119911

120597V0

120597120578

= 1199050

1205971199010

120597120578+ 119875(119860

1205972V1

1205971205782+ 1198650

120597V0

120597120578)

(21)

minus120597V0119909

120597120578

1205971205770

120597119909minus120597V0119910

120597120578

1205971205770

120597119910+120597V0119911

120597120578= 0 (22)

4 Mathematical Problems in Engineering

Order 1205980

120597V0

120597119905minus120597V1

120597120578

1205971205770

120597119905minus120597V0

120597120578

1205971205771

120597119905+ V0119909(120597V0

120597119909minus120597V1

120597120578

1205971205770

120597119909

minus120597V0

120597120578

1205971205771

120597119909) + V1119909

120597V0

120597120578

1205971205770

120597119909+ V1119910

120597V0

120597120578

1205971205770

120597119910

+ V0119910(120597V0

120597119910minus120597V1

120597120578

1205971205770

120597119910minus120597V0

120597120578

1205971205771

120597119910) + V0119911

120597V1

120597120578+ V1119911

sdot120597V0

120597120578= minusnabla01199010+ 1199051

1205971199010

120597120578+ 1199050

1205971199011

120597120578+ 119875(119860

1205972V2

1205971205782

+ 1199053

1205972V1

1205971205782+ 1198650

120597V1

120597120578+ 1199054

1205972V0

1205971205782+ 1198651

120597V0

120597120578+ Δ1V0)

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) 120574120579

0

(23)

120597V0119909

120597119909minus120597V1119909

120597120578

1205971205770

120597119909minus120597V0119909

120597120578

1205971205771

120597119909+120597V0119910

120597119910minus120597V1119910

120597120578

1205971205770

120597119910

minus120597V0119910

120597120578

1205971205771

120597119910+120597V1119911

120597120578= 0

(24)

Order 1205981

120597V1

120597119905minus (

120597V2

120597120578

1205971205770

120597119905+120597V1

120597120578

1205971205771

120597119905+120597V0

120597120578

1205971205772

120597119905) + V0119909(120597V1

120597119909

minus120597V1

120597120578

1205971205771

120597119909minus120597V0

120597120578

1205971205772

120597119909minus120597V2

120597120578

1205971205770

120597119909) + V1119909(120597V0

120597119909

minus120597V0

120597120578

1205971205771

120597119909minus120597V1

120597120578

1205971205770

120597119909) minus V2119909

120597V0

120597120578

1205971205770

120597119909+ V0119910(120597V1

120597119910

minus120597V1

120597120578

1205971205771

120597119910minus120597V0

120597120578

1205971205772

120597119910minus120597V2

120597120578

1205971205770

120597119910) + V1119910(120597V0

120597119910

minus120597V0

120597120578

1205971205771

120597119910minus120597V1

120597120578

1205971205770

120597119910) minus V2119910

120597V0

120597120578

1205971205770

120597119910+ V0119911

120597V2

120597120578

+ V1119911

120597V1

120597120578+ V2119911

120597V0

120597120578= 1199050

1205971199012

120597120578+ 1199051

1205971199011

120597120578minus nabla01199011+ 1199052

sdot1205971199010

120597120578+ 119875(119860

1205972V3

1205971205782+ 1199053

1205972V2

1205971205782+ 1198650

120597V2

120597120578+ 1198651

120597V1

120597120578

+ Δ1V1+ 1198652

120597V0

120597120578) + 119876 (1 + 120582

1sin (120590

1119905)

+ 1205822sin (120590

2119905)) 1205790120574

(25)

120597V1119909

120597119909minus120597V2119909

120597120578

1205971205770

120597119909minus120597V1119909

120597120578

1205971205771

120597119909minus120597V0119909

120597120578

1205971205772

120597119909+120597V1119910

120597119910

minus120597V2119910

120597120578

1205971205770

120597119910minus120597V1119910

120597120578

1205971205771

120597119910minus120597V0119910

120597120578

1205971205772

120597119910+120597V2119911

120597120578= 0

(26)

Order 1205982

120597V2119909

120597119909minus120597V3119909

120597120578

1205971205770

120597119909minus120597V2119909

120597120578

1205971205771

120597119909minus120597V1119909

120597120578

1205971205772

120597119909minus120597V0119909

120597120578

1205971205773

120597119909

+120597V2119910

120597119910minus120597V3119910

120597120578

1205971205770

120597119910minus120597V2119910

120597120578

1205971205771

120597119910minus120597V1119910

120597120578

1205971205772

120597119910

minus120597V0119910

120597120578

1205971205773

120597119910+120597V3119911

120597120578= 0

(27)

where

Δ1=

1205972

1205971199092+

1205972

1205971199102

nabla0= (

120597

120597119909120597

120597119910 0)

119865119894= minus2(

120597120577119894

120597119909

120597

120597119909+120597120577119894

120597119910

120597

120597119910) minus (

1205972120577119894

1205971199092+1205972120577119894

1205971199102) 119868

119894 = 0 1 2

1199050= (

1205971205770

1205971199091205971205770

120597119910 minus1)

119905119894= (

120597120577119894

120597119909120597120577119894

120597119910 minus 1) 119894 = 1 2

1199053= 2(

1205971205771

120597119909

1205971205770

120597119909+1205971205770

120597119910

1205971205771

120597119910)

1199054= (

1205971205771

120597119909)

2

+ (1205971205771

120597119910)

2

+1205971205770

120597119909

1205971205772

120597119909+1205971205770

120597119910

1205971205772

120597119910

119860 = (1 + (1205971205770

120597119909)

2

+ (1205971205770

120597119910)

2

)

(28)

and 119868 is the identity operator The matching conditions as120578 rarr plusmninfin are given by

V0sim V0

10038161003816100381610038161199111=plusmn0

(29)

V1sim (

120597V0

1205971199111

100381610038161003816100381610038161003816100381610038161199111=plusmn0

)120578 + V1

10038161003816100381610038161199111=plusmn0

(30)

V2sim1

2(1205972V0

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=plusmn0

)1205782+ (

120597V1

1205971199111

100381610038161003816100381610038161003816100381610038161199111=plusmn0

)120578

+ V2

10038161003816100381610038161199111=plusmn0

(31)

V3sim1

6(1205973V0

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=plusmn0

)1205783+1

2(1205972V1

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=plusmn0

)1205782

+ (120597V2

1205971199111

100381610038161003816100381610038161003816100381610038161199111=plusmn0

)120578 + V3

10038161003816100381610038161199111=plusmn0

(32)

Mathematical Problems in Engineering 5

As 120578 rarr +infin

1205791asymp 1205791

10038161003816100381610038161199111=+0

+ (1205971205790

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

)120578 0997888rarr 0 (33)

As 120578 rarr minusinfin

1205791sim 1205791

10038161003816100381610038161199111=minus0

0997888rarr 1 (34)

From (18) we obtain

1205972V0

1205971205782= 0 (35)

Consequently we can conclude that V0(120578) is a linear

function of 120578 and identically constant since the velocity isbounded Equation (29) becomes

V0

10038161003816100381610038161199111=+0

= V0

10038161003816100381610038161199111=minus0

(36)

120597V0

120597120578= 0 (37)

From (36)-(37) we deduce that the first term in theexpression of the velocity V

0is continuous at the front By

substituting (37) into (21) we find

1198751198601205972V1

1205971205782+ 1199050

1205971199010

120597120578= 0 (38)

120597V0119909

120597119909minus120597V1119909

120597120578

1205971205770

120597119909+120597V0119910

120597119910minus120597V1119910

120597120578

1205971205770

120597119910+120597V1119911

120597120578= 0 (39)

Differentiating (39) with respect to 120578 one obtains

1205972V1119909

1205971205782

1205971205770

120597119909+1205972V1119910

1205971205782

1205971205770

120597119910minus1205972V1119911

1205971205782= 0 (40)

As a result (38) is a vectorial equation with three com-ponents We multiply the first component by 120597120577

0120597119909 the

second by 1205971205770120597119910 and the third by minus1 and adding we have

119875119860(1205972V1119909

1205971205782

1205971205770

120597119909+1205972V1119910

1205971205782

1205971205770

120597119910minus1205972V1119911

1205971205782) + 119860

1205971199010

120597120578= 0 (41)

Considering the two equations (38) and (40) we obtain

1205971199010

120597120578= 0 (42)

1205972V1

1205971205782= 0 (43)

From the boundedness of the velocity and (30) the con-tinuity of the first order term of the outer expansion and ofthe first derivative of the zero-order term we can write

120597V0

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

=120597V0

1205971199111

100381610038161003816100381610038161003816100381610038161199111=minus0

V1

10038161003816100381610038161199111=+0

= V1

10038161003816100381610038161199111=minus0

(44)

Differentiating (23) once and (26) twice with respect to 120578and using the three equations (37) (42) and (43) we get

1199050

12059721199011

1205971205782+ 119875119860

1205973V2

1205971205783= 0 (45)

1205973V2119909

1205971205783

1205971205770

120597119909+1205973V2119910

1205971205783

1205971205770

120597119910minus1205973V2119911

1205971205783= 0 (46)

As above we multiply the three components of (45)respectively by 120597120577

0120597119909 120597120577

0120597119910 and minus1 and adding we get

119875119860(1205973V2119909

1205971205783

1205971205770

120597119909+1205973V2119910

1205971205783

1205971205770

120597119910minus1205973V2119911

1205971205783) + 119860

12059721199011

1205971205782= 0 (47)

From the previous equation and using (45)-(46) we have

12059721199011

1205971205782= 0

1205973V2

1205971205783= 0

(48)

Knowing that the velocity is bounded and considering(31) we obtain the jump conditions in the following forms

1205972V0

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=+0

=1205972V0

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=minus0

120597V1

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

=120597V1

1205971199111

100381610038161003816100381610038161003816100381610038161199111=minus0

V2

10038161003816100381610038161199111=+0

= V2

10038161003816100381610038161199111=minus0

(49)

Differentiating (25) twice and (27) three times withrespect to 120578 and taking into account (37) (42) (48) we obtain

1199050

12059731199012

1205971205783+ 119875119860

1205974V3

1205971205784

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) 120574

12059721205791

1205971205782= 0

1205974V3119909

1205971205784

1205971205770

120597119909+1205974V3119910

1205971205784

1205971205770

120597119910minus1205974V3119911

1205971205784= 0

(50)

119875119860(1205974V3119909

1205971205784

1205971205770

120597119909+1205974V3119910

1205971205784

1205971205770

120597119910minus1205974V3119911

1205971205784) + 119860

12059731199012

1205971205783

minus 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905))

12059721205791

1205971205782= 0

(51)

From the last equation and using (50) we can write

11986012059731199012

1205971205783minus 119876 (1 + 120582

1sin (120590

1119905) + 120582

2sin (120590

2119905))

12059721205791

1205971205782

= 0

(52)

1205740

12059721205791

1205971205782=1205974V3

1205971205784 (53)

6 Mathematical Problems in Engineering

where 1205740is defined by

1205740= (minus

1205971205770

120597119909

119877

1198602 minus1205971205770

120597119910

119877

1198602119877

1198602minus119877

119860)

sdot (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905))

(54)

Now integrating (53) with respect to 120578 using the system(32)ndash(34) lead to

1205973V0

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=minus0

minus1205973V0

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=+0

= minus1205740

1205971205790

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

(55)

As a first conclusion the velocity jump conditions acrossthe reaction front are given by (36) (44) (49) and (55)

From (20) (22) (37) we conclude that 0is a monotonic

function satisfying 0 lt 0lt 1 Therefore the reaction is of

zero order and 120601(0) equiv 1 Multiplying (19) by 120597120579

1120597120578 and

integrating the result yield

(1205971205791

120597120578)

210038161003816100381610038161003816100381610038161003816100381610038161003816+infin

minus (1205971205791

120597120578)

210038161003816100381610038161003816100381610038161003816100381610038161003816minusinfin

= 2119860minus1int

1205791|1199111=minus0

minusinfin

exp( 120591

1 + 120591120575) 119889120591

(56)

Subtracting (19) from (20) and integrating the result oneobtains

1205971205791

120597120578

100381610038161003816100381610038161003816100381610038161003816+infin

minus1205971205791

120597120578

100381610038161003816100381610038161003816100381610038161003816minusinfin

= minus119860minus1(1205971205770

120597119905+ 119904) (57)

where 119904 is given by

119904 = V0119909

1205971205770

120597119909+ V0119910

1205971205770

120597119910minus V0119911 (58)

As a second conclusion the temperature jump conditionsacross the reaction front are given by the two last equations(56)-(57) By using the matching conditions above andtruncating the expansion as

120579 asymp 1205790

1205791

10038161003816100381610038161199111=minus0

asymp 119885120579|1199111=+0

120577 asymp 1205770

V asymp V0

(59)

the jump conditions can be written as

(120597120579

1205971199111

)

2100381610038161003816100381610038161003816100381610038161003816+0

minus (120597120579

1205971199111

)

2100381610038161003816100381610038161003816100381610038161003816minus0

= 2119885(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot int

1205791|1199111=minus0

minusinfin

exp ( 120591

119885minus1 + 120591120575) 119889120591

120597120579

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

minus120597120579

1205971199111

100381610038161003816100381610038161003816100381610038161199111=minus0

= minus(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot (120597120577

120597119905+ (V119909

120597120577

120597119909+ V119910

120597120577

120597119910minus V119911)

100381610038161003816100381610038161003816100381610038161199111=+0

)

V119911

10038161003816100381610038161199111=+0

= V119911

10038161003816100381610038161199111=minus0

120597V119911

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

=120597V119911

1205971199111

100381610038161003816100381610038161003816100381610038161199111=minus0

1205972V119911

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=+0

=120597V119911

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=minus0

1205973V119911

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=minus0

minus1205973V119911

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=+0

= minus119877(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot ((1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

minus 1)

sdot (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905))

120597120579

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

(60)

4 The Interface Problem and Perturbation

To study the propagation of polymerization front with aliquid reactant and liquid product the equation of motionhas been considered before and after the reaction zone Thischanges the jump conditions and influences the stabilityconditions of the frontal polymerization process Notice thatin the case of liquid-solid polymerization front the equationof motion is considered after the reaction zone The originalsystem (2) (3) (8) (9) leads to the following interfaceproblem

In the liquid monomer 119911 gt 120577

120597120579

120597119905+ (V sdot nabla) 120579 = Δ120579 (61)

120572 = 0 (62)

Mathematical Problems in Engineering 7

120597V120597119905

+ (V sdot nabla) V

= minusnabla119901 + 119875ΔV

+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579

0) 120574

(63)

div (V) = 0 (64)

In the liquid polymer 119911 lt 120577

120597120579

120597119905+ (V sdot nabla) 120579 = Δ120579 (65)

120572 = 1 (66)

120597V120597119905

+ (V sdot nabla) V

= minusnabla119901 + 119875ΔV

+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579

0) 120574

(67)

div (V) = 0 (68)

At the interface 119911 = 120577

120579|120577minus0 = 120579|120577+0

120597120579

120597119911

10038161003816100381610038161003816100381610038161003816120577=minus0minus120597120579

120597119911

10038161003816100381610038161003816100381610038161003816120577=+0= (1 + (

120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot (120597120577

120597119905+ (V119909

120597120577

120597119909+ V119910

120597120577

120597119910minus V119911)

10038161003816100381610038161003816100381610038161003816120577

)

(120597120579

120597119911)

2100381610038161003816100381610038161003816100381610038161003816120577minus0

minus (120597120579

120597119911)

21003816100381610038161003816100381610038161003816100381610038161205771+0

= minus2119885(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot int

120579|120577

minusinfin

exp ( 120591

119885minus1 + 120575120591) 119889120591

V119911

1003816100381610038161003816120577=minus0 = V119911

1003816100381610038161003816120577=+0

120597V119911

120597119911

10038161003816100381610038161003816100381610038161003816120577=minus0=120597V119911

120597119911

10038161003816100381610038161003816100381610038161003816120577=+0

1205972V119911

1205971199112

100381610038161003816100381610038161003816100381610038161003816120577=minus0

=1205972V119911

1205971199112

100381610038161003816100381610038161003816100381610038161003816120577=+0

1205973V119911

1205971199113

100381610038161003816100381610038161003816100381610038161003816120577=minus0

minus1205973V119911

1205971199113

100381610038161003816100381610038161003816100381610038161003816120577=+0

= minus119877(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot ((1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

minus 1)

sdot (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) (

120597120579

120597119911)

10038161003816100381610038161003816100381610038161003816120577=+0

(69)

with the conditions at infinity

119911 = minusinfin 120579 = 0 V = 0

119911 = +infin 120579 = minus1 V = 0

(70)

5 Stability Analysis

The interface problem analyzed in Section 4 has a travellingwave solution in the following form

(120579 (119909 119910 119911 119905) 120572 (119909 119910 119911 119905) V)

= (120579119904(119911 minus 119906119905) 120572

119904(119911 minus 119906119905) 0)

(120579119904 (119911 minus 119906119905) 120572119904 (119911 minus 119906119905))

=

(0 1) 1199112lt 0

(exp (minus1199061199112) minus 1 0) 119911

2gt 0

(71)

where 1199112= 119911minus119906119905 and 119906 is the speed of the stationary reaction

frontThis solution is a basic and stationary solution of inter-face problem (61)ndash(70) written in the moving coordinatesThen (61) (63) (65) and (67) can be replaced by

120597120579

120597119905+ (V sdot nabla) 120579 = Δ120579 + 119906

120597120579

1205971199112

120597V120597119905

+ (Vnabla) V

= minusnabla119901 + 119875ΔV + 119906120597120579

1205971199112

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) (120579 + 120579

0) 120574

(72)

The other equations remain unchanged To study thestability of reaction front we seek the solution of the problemas follows

120579 = 120579119904+ 120579

119901 = 119901119904+ 119901

V = V119904+ V

(73)

where 120579 119901 and V are respectively small perturbation oftemperature pressure and velocity

8 Mathematical Problems in Engineering

Subtituting (73) into (64) (72) one obtains the first-orderterms as follows

In the liquid monomer 1199112gt 120585

120597120579

120597119905= Δ120579 + 119906

120597120579

1205971199112

minus V1199111205791015840

119904 (74)

120597V120597119905

= minusnabla119901 + 119875ΔV + 119906120597120579

1205971199112

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579120574

(75)

div (V) = 0 (76)

In the liquid polymer 1199112lt 120585

120597120579

120597119905= Δ120579 + 119906

120597120579

1205971199112

(77)

120597V120597119905

= minusnabla119901 + 119875ΔV + 119906120597V1205971199112

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579120574

(78)

div (V) = 0 (79)

where 120585 is defined by 120585 = 120577 minus 119906119905We note

(120579 V119911) =

(1205791 V1199111) for 119911

2lt 120585

(1205792 V1199112) for 119911

2lt 120585

(80)

and consider the perturbation as follows

120579119894= 120579119894(1199112 119905) exp (119895 (119896

1119909 + 1198962119910))

V119911119894= V119911119894(1199112 119905) exp (119895 (119896

1119909 + 1198962119910))

120585 = 1205981(119905) exp (119895 (119896

1119909 + 1198962119910))

(81)

where 119896119894 119894 = 1 2 and 120598

1denote respectively the wave

numbers (in 119909 and 119910 directions) and the amplitude of theperturbation and 119895

2= minus1 Then jump conditions (69) are

linearized by taking into account the fact that

120579|120585=plusmn0 = 120579119904(plusmn0) + 120585120579

1015840

119904(plusmn0) + 120579 (plusmn0)

120597120579

1205971199112

10038161003816100381610038161003816100381610038161003816120585=plusmn0

= 1205791015840

119904(plusmn0) + 120585120579

10158401015840

119904(plusmn0) +

120597120579

1205971199112

100381610038161003816100381610038161003816100381610038161003816120585=plusmn0

(82)

Then the higher-order terms are given as

1205792

100381610038161003816100381610038161199112=0minus 1205791

100381610038161003816100381610038161199112=0= 119906120585 (83)

1205971205792

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

minus1205971205791

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

= minus1199062120585 minus

120597120585

120597119905+ V119911

10038161003816100381610038161199112=0 (84)

1199062120585 +

1205971205792

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

= minus119885

1199061205791

100381610038161003816100381610038161199112=0 (85)

V2119911

10038161003816100381610038161199112=0= V1119911

10038161003816100381610038161199112=0 (86)

120597V1199112

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

=120597V1119911

1205971199112

100381610038161003816100381610038161003816100381610038161199112=0

(87)

1205972V2119911

12059711991122

1003816100381610038161003816100381610038161003816100381610038161199112=0

=1205972V1119911

12059711991122

1003816100381610038161003816100381610038161003816100381610038161199112=0

(88)

1205973V2119911

12059711991132

1003816100381610038161003816100381610038161003816100381610038161199112=0

=1205973V1119911

12059711991132

1003816100381610038161003816100381610038161003816100381610038161199112=0

(89)

Substituting (81) into (83)ndash(89) leads to

1205792(0 119905) minus 120579

1(0 119905) = 119906120598

1(119905)

1205791015840

2(0 119905) minus 120579

1015840

1(0 119905) = minus120598

1 (119905) 1199062minus 1205981015840

1(119905) + V

1 (0 119905)

1205981 (119905) 119906

2+ 1205791015840

2(0 119905) = minus

119885

1199061205791 (0 119905)

V(119894)2(0 119905) = V(119894)

1(0 119905) 119894 = 0 1 2 3

(90)

where

V(119894) =120597119894V120597V119894

1205791015840=120597120579119894

120597119911

1205981015840

1(119905) =

1198891205981(119905)

119889119905

(91)

After simplification (75) and (78) become

120597

120597119905ΔV119911minus 119906

120597

1205971199112

ΔV119911= 119875ΔΔV

119911+ 119876(1 + 120582

1sin (120590

1119905)

+ 1205822sin (120590

2119905) (

1205972

1205971199092+

1205972

1205971199102))120579

(92)

Substituting (71) (87) (88) into (74) (77) (92) andintroducing the variable 119908 = V10158401015840 minus 1198962V where 119896 = radic1198962

1+ 11989622

one obtains two systems of equations as follows

1199112lt 120585

120597119908

120597119905minus 1199061199081015840minus 119875 (119908

10158401015840minus 1198962119908)

= minus1198761198962(1 + 120582

1sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579

119908 = V10158401015840 minus 1198962V

120597120579

120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 0

(93)

Mathematical Problems in Engineering 9

0 5 10 15 20 25 300

50

100

150

200

250

300

350

Rc

1205901

1205902 = radic212059011205902 = radic71205901

1205902 = 0

1205902 = radic371205901

Figure 1 Critical Rayleigh number as a function of 1205901for different

frequencies ratio and for 1205902= 0 120582

1= 1205822= 5 119875 = 05 119885 = 8 and

119896 = 07

1199112gt 120585

120597119908

120597119905minus 1199061199081015840minus 119875 (119908

10158401015840minus 1198962119908)

= minus1198761198962(1 + 120582

1sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579

119908 = V10158401015840 minus 1198962V

120597120579

120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 119906 exp (minus119906119911

2) V

(94)

with the following boundary conditions

1205792(0 119905) minus 120579

1(0 119905) = 119906120598

1(119905) (95)

1205791015840

2(0 119905) minus 120579

1015840

1(0 119905) = minus119906

21205761(119905) minus 120576

1015840

1(119905) + V

1(0 119905) (96)

1205791015840

2(0 119905) +

119885

1199061205791 (0 119905) = minus119906

21205761 (119905) (97)

V(119894)1(0 119905) = V(119894)

2(0 119905) 119894 = 0 1 2 3 (98)

Combining the three boundary conditions (95)ndash(97)yields

1205791015840

1(0 119905) +

119885

1199061205791(0 119905) =

1

119906(1205792(0 119905) minus 120579

1(0 119905))

1015840

119905

minus V1(0 119905)

(99)

6 Numerical Results

The numerical simulations of the problem are performedusing the finite-difference approximation with implicitscheme The algorithm is given in Appendix

Evolution of maximum temperature versus time providesthe onset of stability of the polymerization front The con-vective instability occurs when a jump from bounded tounbounded values of maximum temperature is achieved

Figure 1 shows the critical Rayleigh number as a functionof the frequency 120590

1for different frequencies ratio and for

0 25 50 75 100 125 1500

5

10

15

20

25

Rc

1205822

1205821 = 20

1205821 = 50

1205821 = 80

1205821 = 150

Figure 2 Critical Rayleigh number as a function of 1205822for different

values of 1205821 1205902= radic2120590

1 1205901= 5 119875 = 05 119885 = 8 and 119896 = 07

0 25 50 75 100 125 150 175 2000

50

100

150

200

250

Rc

1205822

1205902 = radic371205901

1205902 = radic1712059011205902 = radic71205901

1205902 = 1radic712059011205902 = 1radic171205901

1205902 = 1radic371205901

Figure 3 Critical Rayleigh number as a function of 1205822for different

frequencies ratio 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and 120590

1= 5

1205902= 0 In the absence of gravitational modulation (120590

1= 1205902=

0) we find the same result (119877119888asymp 27) as in [8 14] It can be

observed that the frequencies ratio of the QP gravitationalmodulation has a significant influence on the convectiveinstability of the reaction front As indicated increasing thefrequencies ratio the stability domain becomes larger Inparticular this can be seen in the range of the frequency1205901belonging to the interval (0 10) For large values of the

frequency 1205901 the frequencies ratio 120590

21205901has no effect on

the convective instability It is also worthy to notice thatthe QP gravitational modulation has a stabilizing effect forsufficiently large frequencies ratio (120590

2= radic37120590

1) comparing

to the periodic modulation case (1205902= 0)

The critical Rayleigh number as a function of the ampli-tude 120582

2is shown in Figure 2 for different values of 120582

1 We can

conclude that for a fixed value of 1205822and increased values of

1205821 the polymerization front loses its stability monotonically

revealing that the stability of the reaction front can becontrolled by acting on the amplitudes of the QPmodulation

Figure 3 illustrates the critical Rayleigh number as afunction of the amplitude 120582

2for different frequencies ratio

It can be observed that increasing the frequencies ratio thestability domain becomes large for certain values of 120582

2 For

large values of 1205822(1205822ge 100) the effect of the frequencies

ratio on the convective instability boundary is insignificant

10 Mathematical Problems in Engineering

0 25 50 75 100 125 150

k = 15

k = 2

k = 1

0

50

100

150

200

250

300

350

Rc

1205822

Figure 4 Critical Rayleigh number as a function of 1205822for different

values of number wave 1205902= radic7120590

1 1205901= 10 119875 = 05 119885 = 8 and

1205821= 5

0 25 50 75 100 125 1500

255075

100125150175200

Rc

1205822

1205902 = radic21205901

1205902 = radic71205901

Figure 5 Critical Rayleigh number as a function of 1205822for two

different frequencies ratios 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and

1205901= 10

Figure 4 shows for a fixed frequencies ratio 12059021205901= radic7

the influence of the wave number on the critical Rayleighnumber versus 120582

2 The plots indicate that for large values of

the wave number the reaction front gains stabilityThe influence of 120582

2on the stability of the reaction front is

given in Figure 5 for two different frequencies ratio From thisfigure one can conclude that an increase of the frequenciesratio has a stabilizing effect

7 Conclusion

In this work we have studied the influence of the QP vibra-tions on the convective instability of liquid-liquid polymer-ization frontWe have used themodel which includes the heatequation the concentration equation and the Navier-Stokesequations under Boussinesq approximation The ZeldovichFrank-Kamenetskii method has been used assuming that thereaction occurs in a narrow zone To obtain the convec-tive instability threshold using the linear stability analysisthe reduced system of equations has been discretized andresolved using the finite-difference method with implicitscheme The results have shown that for fixed values ofamplitudes an increase of the frequencies ratio stabilizes thereaction front and for large values of the frequency 120590

1the

critical Rayleigh number tends to the unmodulated criticalvalue In addition it is observed that for a fixed frequencies

ratio and for a given amplitude 1205822an increase of the ampli-

tude1205821destabilizes the reaction front Also the reaction front

becomes more stable by increasing the wave number

Appendix

The algorithm of the considered problem uses the finite-difference approximation with implicit scheme The velocityis computed from the previous time values of the problemThe discretization in space is given by 119911

119894= 119894ℎ where ℎ is the

space step and 119894 is a positive integerThe discretization in timeis given by 119905

119894= 119894120591 where 120591 is sufficiently small time step The

variables of discretization are 119908119895119894= 119908(119911

119894 119905119895) V119895119894= V(119911

119894 119905119895)

and 120579119895119894= 120579(119911119894 119905119895) One obtains the following

In the liquid monomer 119911 lt 120577

(minus1

ℎ2) 120579119895+1

119894minus1+ (

1

120591+

2

ℎ2+ 1198962+119906

ℎ) 120579119895+1

119894

+ (minus1

ℎ2minus119906

ℎ) 120579119895+1

119894+1=1

120591120579119895

119894

(minus119875

ℎ2)119908119895+1

119894minus1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

119894

+ (minus119875

ℎ2minus119906

ℎ)119908119895+1

119894+1=1

120591119908119895

119894

minus 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895)) 120579119895

119894

(1

ℎ2) V119895+1119894minus1

+ (minus2

ℎ2minus 1198962) V119895+1119894

+ (1

ℎ2) V119895+1119894+1

= 119908119895

119894

(A1)

In the liquid polymer 119911 gt 120577

(minus1

ℎ2)119879119895+1

119894minus1+ (

1

120591+

2

ℎ2+ 1198962+119906

ℎ)119879119895+1

119894

+ (minus1

ℎ2minus119906

ℎ)119879119895+1

119894+1=1

120591119879119895

119894+ 119906 exp (minus119906119911

119894) 119881119895

119894

(minus119875

ℎ2)119882119895+1

119894minus1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119882119895+1

119894

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

119894+1=1

120591119882119895

119894

minus 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895)) 119879119895

119894

(1

ℎ2)119881119895+1

119894minus1+ (minus

2

ℎ2minus 1198962)119881119895+1

119894+ (

1

ℎ2)119881119895+1

119894+1= 119882119895

119894

(A2)

From the jump conditions we have also the followingsystems

(minus1

ℎ2) 120579119895+1

1

+ (1

120591+

2

ℎ2+ 1198962+119906

ℎ+ (

1198881

1198882

)(1

ℎ2+119906

ℎ)) 120579119895+1

0

Mathematical Problems in Engineering 11

= (minus1

120591) 120579119895

0minus (

1

1199061205911198882

)(1

ℎ2+119906

ℎ) (119879119895

0minus 120579119895

0)

minus (V0

1198882

)(1

ℎ2+119906

ℎ)

(minus119875

ℎ2)119908119895+1

1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

0

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

0= (

minus1

120591)119908119895

1

+ 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895))

sdot ((minus1198882

1198881

)1198790minus

1

1199061205911198881

(119879119895

0minus 120579119895

0) minus

V0

1198881

)

(A3)

(1

120591+

2

ℎ2+ 1198962+119906

ℎ+

1198882

ℎ21198881

)119879119895+1

0+ (

minus1

ℎ2minus119906

ℎ)119879119895+1

1

= (minus1

120591)119879119895

0minus 119906 exp (minus119906ℎ) minus ( 1

119906120591ℎ21198881

) (119879119895

0minus 120579119895

0)

minusV0

ℎ21198881

(A4)

(minus119875

ℎ2)119908119895+1

1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

0

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

0= (

minus1

120591)119908119895

0

+ 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895))

sdot ((minus1198881

1198882

)1205790minus

1

1199061205911198882

)(119879119895

0minus 120579119895

0minusV0

1198882

)

(A5)

with

1198881= (

1

ℎminus119885

119906minus

1

119906120591)

1198882= (minus

1

ℎ+

1

119906120591)

(A6)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] B B Khanukaev M A Kozhushner N S Enikolopyan and NM Chechilo ldquoTheory of the propagation of a polymerizationfrontrdquo Combustion Explosion and Shock Waves vol 10 no 1pp 17ndash21 1974

[2] J A Pojman and T-C-M Qui Nonlinear Dynamics with Pol-ymers Fundamentals Methods and Applications John Wiley ampSons 2011

[3] I D Robertson H L Hernandez S R White and J S MooreldquoRapid stiffening of a microfluidic endoskeleton via frontalpolymerizationrdquo ACS Applied Materials and Interfaces vol 6no 21 pp 18469ndash18474 2014

[4] Vi Volpert and Vl Volpert ldquoPropagation of frontal polym-erizationmdashcrystallization wavesrdquo European Journal of AppliedMathematics vol 5 no 2 pp 201ndash215 1994

[5] G Bowden M Garbey V M Ilyashenko et al ldquoEffect of con-vection on a propagating front with a solid product comparisonof theory and experimentsrdquoThe Journal of Physical Chemistry Bvol 101 no 4 pp 678ndash686 1997

[6] M Garbey A Taik and V Volpert ldquoLinear stability analysisof reaction fronts in liquidsrdquo Quarterly of Applied Mathematicsvol 54 no 2 pp 225ndash247 1996

[7] K Allali V Volpert and J A Pojman ldquoInfluence of vibrationson convective instability of polymerization frontsrdquo Journal ofEngineering Mathematics vol 41 no 1 pp 13ndash31 2001

[8] K Allali F Bikany A Taik and V Volpert ldquoInfluence ofvibrations on convective instability of reaction fronts in liquidsrdquoMathematical Modelling of Natural Phenomena vol 5 no 7 pp35ndash41 2010

[9] K Allali M Belhaq and K El Karouni ldquoInfluence of quasi-periodic gravitational modulation on convective instability ofreaction fronts in porous mediardquo Communications in NonlinearScience and Numerical Simulation vol 17 no 4 pp 1588ndash15962012

[10] KAllali S Assiyad andM Belhaq ldquoConvection of polymeriza-tion front with solid product under quasi-periodic gravitationalmodulationrdquo Nonlinear Dynamics and Systems Theory vol 14no 4 pp 323ndash334 2014

[11] M Menzinger and R Wolfgang ldquoThe meaning and use of theArrhenius activation energyrdquoAngewandte Chemie InternationalEdition in English vol 8 no 6 pp 438ndash444 1969

[12] B V Novozhilov ldquoThe rate of propagation of the front of anexothermic reaction in a condensed phaserdquo Doklady PhysicalChemistry vol 141 pp 836ndash838 1961

[13] Y B Zeldovich and D A Frank-Kamenetskii ldquoA theory ofthermal propagation of amerdquo Acta Physicochimica USSR vol 9pp 341ndash350 1938

[14] M Garbey A Taık and V Volpert ldquoInfluence of natural con-vection on stability of reaction fronts in liquidsrdquo Quarterly ofApplied Mathematics vol 56 no 1 pp 1ndash35 1998

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

120575 and 1205790are given respectively by 120575 = 119877

0119879119887119864 and 120579

0= (119879119887minus

1198790)119902 and 120583 = 2120581120590119888

2Next we perform the linear stability analysis to tackle the

interface problem

3 Approximation of Infinitely NarrowReaction Zone

To study the interface problem analytically we use a singularperturbation analysis where the reaction zone is infinitelynarrow and the reaction term is neglected outside the zone[13] In this way the problem can be reduced to an interfaceproblem

To perform a formal asymptotic analysis 120598 = 119885minus1 is

considered as a small parameter The new independent var-iable is given by 119911

1= 119911 minus 120577(119909 119910 119905) where 120577(119909 119910 119905) defines the

location of the reaction zone Introducing new functions 1205791

1205721 V1 and 119901

1as

120579 (119909 119910 119911 119905) = 1205791(119909 119910 119911

1 119905)

120572 (119909 119910 119911 119905) = 1205721(119909 119910 119911

1 119905)

V (119909 119910 119911 119905) = V1(119909 119910 119911

1 119905)

119901 (119909 119910 119911 119905) = 1199011(119909 119910 119911

1 119905)

(13)

The system of (8)ndash(11) can be written in the form (index 1 forthe new function is omitted)120597120579

120597119905minus120597120579

1205971199111

120597120577

120597119905+ (V sdot nabla) 120579

= Δ120579 + 119885 exp( 120579

119885minus1 + 120575120579)120601 (120572)

120597120572

120597119905minus120597120572

1205971199111

120597120577

120597119905+ (V sdot nabla) 120572 = 119885 exp( 120579

119885minus1 + 120575120579)120601 (120572)

120597V120597119905

minus120597V1205971199111

120597120577

120597119905+ (V sdot nabla) V

= minusnabla119901 + 119875ΔV

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) (120579 + 120579

0) 120574

120597V119909

120597119909minus120597V119909

1205971199111

120597120577

120597119909+120597V119910

120597119910minus120597V119910

1205971199111

120597120577

120597119910+120597V119911

1205971199111

= 0

(14)

where Δ nabla and 119876 are given by

Δ =1205972

1205971199092+

1205972

1205971199102+

1205972

12059711991121

minus 21205972

1205971199091205971199111

120597120577

120597119909minus 2

1205972

1205971199101205971199111

120597120577

120597119910

+1205972

12059711991121

((120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus120597

1205971199111

(1205972120577

1205971199092+1205972120577

1205971199102)

nabla = (120597

120597119909minus

120597

1205971199111

120597120577

120597119909120597

120597119910minus

120597

1205971199111

120597120577

120597119910120597

1205971199111

)

119876 = 119875119877

(15)

To approximate the jump conditions and resolve theinterface problem we use the matched asymptotic expansionby seeking the outer solution of problem (14) in the form

120579 = 1205790+ 1205981205791+ sdot sdot sdot

120572 = 1205720+ 1205981205721+ sdot sdot sdot

V = V0+ 120598V1+ sdot sdot sdot

119901 = 1199010+ 1205981199011+ sdot sdot sdot

(16)

Introducing the stretched coordinate 120578 = 1199111120598minus1 where 120598 =

119885minus1 the inner solution can be approximated in the following

form

120579 = 1205981205791+ sdot sdot sdot

120572 = 0+ 1205981+ sdot sdot sdot

V = V0+ 120598V1+ sdot sdot sdot

119901 = 1199010+ 1205981199011+ sdot sdot sdot

120577 = 1205770+ 1205981205771+ sdot sdot sdot

(17)

Substituting the inner and outer solutions in (14) leads to

Order 120598minus2

119875(1 + (1205971205770

120597119909)

2

+ (1205971205770

120597119910)

2

)1205972V0

1205971205782= 0 (18)

Order 120598minus1

(1 + (1205971205770

120597119909)

2

+ (1205971205770

120597119910)

2

)12059721205791

1205971205782

+ exp( 1205791

1 + 1205751205791

)120601 (0) = 0

(19)

minus1205970

120597120578

1205971205770

120597119905minus1205970

120597120578(V0119909

1205971205770

120597119909+ V0119910

1205971205770

120597119910minus V0119911)

= exp( 1205791

1 + 1205751205791

)120601 (0)

(20)

minus120597V0119909

120597120578

1205971205770

120597119905minus V0119909

120597V0

120597120578

1205971205770

120597119909minus V0119910

120597V0

120597120578

1205971205770

120597119910+ V0119911

120597V0

120597120578

= 1199050

1205971199010

120597120578+ 119875(119860

1205972V1

1205971205782+ 1198650

120597V0

120597120578)

(21)

minus120597V0119909

120597120578

1205971205770

120597119909minus120597V0119910

120597120578

1205971205770

120597119910+120597V0119911

120597120578= 0 (22)

4 Mathematical Problems in Engineering

Order 1205980

120597V0

120597119905minus120597V1

120597120578

1205971205770

120597119905minus120597V0

120597120578

1205971205771

120597119905+ V0119909(120597V0

120597119909minus120597V1

120597120578

1205971205770

120597119909

minus120597V0

120597120578

1205971205771

120597119909) + V1119909

120597V0

120597120578

1205971205770

120597119909+ V1119910

120597V0

120597120578

1205971205770

120597119910

+ V0119910(120597V0

120597119910minus120597V1

120597120578

1205971205770

120597119910minus120597V0

120597120578

1205971205771

120597119910) + V0119911

120597V1

120597120578+ V1119911

sdot120597V0

120597120578= minusnabla01199010+ 1199051

1205971199010

120597120578+ 1199050

1205971199011

120597120578+ 119875(119860

1205972V2

1205971205782

+ 1199053

1205972V1

1205971205782+ 1198650

120597V1

120597120578+ 1199054

1205972V0

1205971205782+ 1198651

120597V0

120597120578+ Δ1V0)

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) 120574120579

0

(23)

120597V0119909

120597119909minus120597V1119909

120597120578

1205971205770

120597119909minus120597V0119909

120597120578

1205971205771

120597119909+120597V0119910

120597119910minus120597V1119910

120597120578

1205971205770

120597119910

minus120597V0119910

120597120578

1205971205771

120597119910+120597V1119911

120597120578= 0

(24)

Order 1205981

120597V1

120597119905minus (

120597V2

120597120578

1205971205770

120597119905+120597V1

120597120578

1205971205771

120597119905+120597V0

120597120578

1205971205772

120597119905) + V0119909(120597V1

120597119909

minus120597V1

120597120578

1205971205771

120597119909minus120597V0

120597120578

1205971205772

120597119909minus120597V2

120597120578

1205971205770

120597119909) + V1119909(120597V0

120597119909

minus120597V0

120597120578

1205971205771

120597119909minus120597V1

120597120578

1205971205770

120597119909) minus V2119909

120597V0

120597120578

1205971205770

120597119909+ V0119910(120597V1

120597119910

minus120597V1

120597120578

1205971205771

120597119910minus120597V0

120597120578

1205971205772

120597119910minus120597V2

120597120578

1205971205770

120597119910) + V1119910(120597V0

120597119910

minus120597V0

120597120578

1205971205771

120597119910minus120597V1

120597120578

1205971205770

120597119910) minus V2119910

120597V0

120597120578

1205971205770

120597119910+ V0119911

120597V2

120597120578

+ V1119911

120597V1

120597120578+ V2119911

120597V0

120597120578= 1199050

1205971199012

120597120578+ 1199051

1205971199011

120597120578minus nabla01199011+ 1199052

sdot1205971199010

120597120578+ 119875(119860

1205972V3

1205971205782+ 1199053

1205972V2

1205971205782+ 1198650

120597V2

120597120578+ 1198651

120597V1

120597120578

+ Δ1V1+ 1198652

120597V0

120597120578) + 119876 (1 + 120582

1sin (120590

1119905)

+ 1205822sin (120590

2119905)) 1205790120574

(25)

120597V1119909

120597119909minus120597V2119909

120597120578

1205971205770

120597119909minus120597V1119909

120597120578

1205971205771

120597119909minus120597V0119909

120597120578

1205971205772

120597119909+120597V1119910

120597119910

minus120597V2119910

120597120578

1205971205770

120597119910minus120597V1119910

120597120578

1205971205771

120597119910minus120597V0119910

120597120578

1205971205772

120597119910+120597V2119911

120597120578= 0

(26)

Order 1205982

120597V2119909

120597119909minus120597V3119909

120597120578

1205971205770

120597119909minus120597V2119909

120597120578

1205971205771

120597119909minus120597V1119909

120597120578

1205971205772

120597119909minus120597V0119909

120597120578

1205971205773

120597119909

+120597V2119910

120597119910minus120597V3119910

120597120578

1205971205770

120597119910minus120597V2119910

120597120578

1205971205771

120597119910minus120597V1119910

120597120578

1205971205772

120597119910

minus120597V0119910

120597120578

1205971205773

120597119910+120597V3119911

120597120578= 0

(27)

where

Δ1=

1205972

1205971199092+

1205972

1205971199102

nabla0= (

120597

120597119909120597

120597119910 0)

119865119894= minus2(

120597120577119894

120597119909

120597

120597119909+120597120577119894

120597119910

120597

120597119910) minus (

1205972120577119894

1205971199092+1205972120577119894

1205971199102) 119868

119894 = 0 1 2

1199050= (

1205971205770

1205971199091205971205770

120597119910 minus1)

119905119894= (

120597120577119894

120597119909120597120577119894

120597119910 minus 1) 119894 = 1 2

1199053= 2(

1205971205771

120597119909

1205971205770

120597119909+1205971205770

120597119910

1205971205771

120597119910)

1199054= (

1205971205771

120597119909)

2

+ (1205971205771

120597119910)

2

+1205971205770

120597119909

1205971205772

120597119909+1205971205770

120597119910

1205971205772

120597119910

119860 = (1 + (1205971205770

120597119909)

2

+ (1205971205770

120597119910)

2

)

(28)

and 119868 is the identity operator The matching conditions as120578 rarr plusmninfin are given by

V0sim V0

10038161003816100381610038161199111=plusmn0

(29)

V1sim (

120597V0

1205971199111

100381610038161003816100381610038161003816100381610038161199111=plusmn0

)120578 + V1

10038161003816100381610038161199111=plusmn0

(30)

V2sim1

2(1205972V0

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=plusmn0

)1205782+ (

120597V1

1205971199111

100381610038161003816100381610038161003816100381610038161199111=plusmn0

)120578

+ V2

10038161003816100381610038161199111=plusmn0

(31)

V3sim1

6(1205973V0

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=plusmn0

)1205783+1

2(1205972V1

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=plusmn0

)1205782

+ (120597V2

1205971199111

100381610038161003816100381610038161003816100381610038161199111=plusmn0

)120578 + V3

10038161003816100381610038161199111=plusmn0

(32)

Mathematical Problems in Engineering 5

As 120578 rarr +infin

1205791asymp 1205791

10038161003816100381610038161199111=+0

+ (1205971205790

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

)120578 0997888rarr 0 (33)

As 120578 rarr minusinfin

1205791sim 1205791

10038161003816100381610038161199111=minus0

0997888rarr 1 (34)

From (18) we obtain

1205972V0

1205971205782= 0 (35)

Consequently we can conclude that V0(120578) is a linear

function of 120578 and identically constant since the velocity isbounded Equation (29) becomes

V0

10038161003816100381610038161199111=+0

= V0

10038161003816100381610038161199111=minus0

(36)

120597V0

120597120578= 0 (37)

From (36)-(37) we deduce that the first term in theexpression of the velocity V

0is continuous at the front By

substituting (37) into (21) we find

1198751198601205972V1

1205971205782+ 1199050

1205971199010

120597120578= 0 (38)

120597V0119909

120597119909minus120597V1119909

120597120578

1205971205770

120597119909+120597V0119910

120597119910minus120597V1119910

120597120578

1205971205770

120597119910+120597V1119911

120597120578= 0 (39)

Differentiating (39) with respect to 120578 one obtains

1205972V1119909

1205971205782

1205971205770

120597119909+1205972V1119910

1205971205782

1205971205770

120597119910minus1205972V1119911

1205971205782= 0 (40)

As a result (38) is a vectorial equation with three com-ponents We multiply the first component by 120597120577

0120597119909 the

second by 1205971205770120597119910 and the third by minus1 and adding we have

119875119860(1205972V1119909

1205971205782

1205971205770

120597119909+1205972V1119910

1205971205782

1205971205770

120597119910minus1205972V1119911

1205971205782) + 119860

1205971199010

120597120578= 0 (41)

Considering the two equations (38) and (40) we obtain

1205971199010

120597120578= 0 (42)

1205972V1

1205971205782= 0 (43)

From the boundedness of the velocity and (30) the con-tinuity of the first order term of the outer expansion and ofthe first derivative of the zero-order term we can write

120597V0

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

=120597V0

1205971199111

100381610038161003816100381610038161003816100381610038161199111=minus0

V1

10038161003816100381610038161199111=+0

= V1

10038161003816100381610038161199111=minus0

(44)

Differentiating (23) once and (26) twice with respect to 120578and using the three equations (37) (42) and (43) we get

1199050

12059721199011

1205971205782+ 119875119860

1205973V2

1205971205783= 0 (45)

1205973V2119909

1205971205783

1205971205770

120597119909+1205973V2119910

1205971205783

1205971205770

120597119910minus1205973V2119911

1205971205783= 0 (46)

As above we multiply the three components of (45)respectively by 120597120577

0120597119909 120597120577

0120597119910 and minus1 and adding we get

119875119860(1205973V2119909

1205971205783

1205971205770

120597119909+1205973V2119910

1205971205783

1205971205770

120597119910minus1205973V2119911

1205971205783) + 119860

12059721199011

1205971205782= 0 (47)

From the previous equation and using (45)-(46) we have

12059721199011

1205971205782= 0

1205973V2

1205971205783= 0

(48)

Knowing that the velocity is bounded and considering(31) we obtain the jump conditions in the following forms

1205972V0

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=+0

=1205972V0

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=minus0

120597V1

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

=120597V1

1205971199111

100381610038161003816100381610038161003816100381610038161199111=minus0

V2

10038161003816100381610038161199111=+0

= V2

10038161003816100381610038161199111=minus0

(49)

Differentiating (25) twice and (27) three times withrespect to 120578 and taking into account (37) (42) (48) we obtain

1199050

12059731199012

1205971205783+ 119875119860

1205974V3

1205971205784

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) 120574

12059721205791

1205971205782= 0

1205974V3119909

1205971205784

1205971205770

120597119909+1205974V3119910

1205971205784

1205971205770

120597119910minus1205974V3119911

1205971205784= 0

(50)

119875119860(1205974V3119909

1205971205784

1205971205770

120597119909+1205974V3119910

1205971205784

1205971205770

120597119910minus1205974V3119911

1205971205784) + 119860

12059731199012

1205971205783

minus 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905))

12059721205791

1205971205782= 0

(51)

From the last equation and using (50) we can write

11986012059731199012

1205971205783minus 119876 (1 + 120582

1sin (120590

1119905) + 120582

2sin (120590

2119905))

12059721205791

1205971205782

= 0

(52)

1205740

12059721205791

1205971205782=1205974V3

1205971205784 (53)

6 Mathematical Problems in Engineering

where 1205740is defined by

1205740= (minus

1205971205770

120597119909

119877

1198602 minus1205971205770

120597119910

119877

1198602119877

1198602minus119877

119860)

sdot (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905))

(54)

Now integrating (53) with respect to 120578 using the system(32)ndash(34) lead to

1205973V0

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=minus0

minus1205973V0

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=+0

= minus1205740

1205971205790

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

(55)

As a first conclusion the velocity jump conditions acrossthe reaction front are given by (36) (44) (49) and (55)

From (20) (22) (37) we conclude that 0is a monotonic

function satisfying 0 lt 0lt 1 Therefore the reaction is of

zero order and 120601(0) equiv 1 Multiplying (19) by 120597120579

1120597120578 and

integrating the result yield

(1205971205791

120597120578)

210038161003816100381610038161003816100381610038161003816100381610038161003816+infin

minus (1205971205791

120597120578)

210038161003816100381610038161003816100381610038161003816100381610038161003816minusinfin

= 2119860minus1int

1205791|1199111=minus0

minusinfin

exp( 120591

1 + 120591120575) 119889120591

(56)

Subtracting (19) from (20) and integrating the result oneobtains

1205971205791

120597120578

100381610038161003816100381610038161003816100381610038161003816+infin

minus1205971205791

120597120578

100381610038161003816100381610038161003816100381610038161003816minusinfin

= minus119860minus1(1205971205770

120597119905+ 119904) (57)

where 119904 is given by

119904 = V0119909

1205971205770

120597119909+ V0119910

1205971205770

120597119910minus V0119911 (58)

As a second conclusion the temperature jump conditionsacross the reaction front are given by the two last equations(56)-(57) By using the matching conditions above andtruncating the expansion as

120579 asymp 1205790

1205791

10038161003816100381610038161199111=minus0

asymp 119885120579|1199111=+0

120577 asymp 1205770

V asymp V0

(59)

the jump conditions can be written as

(120597120579

1205971199111

)

2100381610038161003816100381610038161003816100381610038161003816+0

minus (120597120579

1205971199111

)

2100381610038161003816100381610038161003816100381610038161003816minus0

= 2119885(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot int

1205791|1199111=minus0

minusinfin

exp ( 120591

119885minus1 + 120591120575) 119889120591

120597120579

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

minus120597120579

1205971199111

100381610038161003816100381610038161003816100381610038161199111=minus0

= minus(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot (120597120577

120597119905+ (V119909

120597120577

120597119909+ V119910

120597120577

120597119910minus V119911)

100381610038161003816100381610038161003816100381610038161199111=+0

)

V119911

10038161003816100381610038161199111=+0

= V119911

10038161003816100381610038161199111=minus0

120597V119911

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

=120597V119911

1205971199111

100381610038161003816100381610038161003816100381610038161199111=minus0

1205972V119911

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=+0

=120597V119911

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=minus0

1205973V119911

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=minus0

minus1205973V119911

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=+0

= minus119877(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot ((1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

minus 1)

sdot (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905))

120597120579

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

(60)

4 The Interface Problem and Perturbation

To study the propagation of polymerization front with aliquid reactant and liquid product the equation of motionhas been considered before and after the reaction zone Thischanges the jump conditions and influences the stabilityconditions of the frontal polymerization process Notice thatin the case of liquid-solid polymerization front the equationof motion is considered after the reaction zone The originalsystem (2) (3) (8) (9) leads to the following interfaceproblem

In the liquid monomer 119911 gt 120577

120597120579

120597119905+ (V sdot nabla) 120579 = Δ120579 (61)

120572 = 0 (62)

Mathematical Problems in Engineering 7

120597V120597119905

+ (V sdot nabla) V

= minusnabla119901 + 119875ΔV

+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579

0) 120574

(63)

div (V) = 0 (64)

In the liquid polymer 119911 lt 120577

120597120579

120597119905+ (V sdot nabla) 120579 = Δ120579 (65)

120572 = 1 (66)

120597V120597119905

+ (V sdot nabla) V

= minusnabla119901 + 119875ΔV

+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579

0) 120574

(67)

div (V) = 0 (68)

At the interface 119911 = 120577

120579|120577minus0 = 120579|120577+0

120597120579

120597119911

10038161003816100381610038161003816100381610038161003816120577=minus0minus120597120579

120597119911

10038161003816100381610038161003816100381610038161003816120577=+0= (1 + (

120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot (120597120577

120597119905+ (V119909

120597120577

120597119909+ V119910

120597120577

120597119910minus V119911)

10038161003816100381610038161003816100381610038161003816120577

)

(120597120579

120597119911)

2100381610038161003816100381610038161003816100381610038161003816120577minus0

minus (120597120579

120597119911)

21003816100381610038161003816100381610038161003816100381610038161205771+0

= minus2119885(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot int

120579|120577

minusinfin

exp ( 120591

119885minus1 + 120575120591) 119889120591

V119911

1003816100381610038161003816120577=minus0 = V119911

1003816100381610038161003816120577=+0

120597V119911

120597119911

10038161003816100381610038161003816100381610038161003816120577=minus0=120597V119911

120597119911

10038161003816100381610038161003816100381610038161003816120577=+0

1205972V119911

1205971199112

100381610038161003816100381610038161003816100381610038161003816120577=minus0

=1205972V119911

1205971199112

100381610038161003816100381610038161003816100381610038161003816120577=+0

1205973V119911

1205971199113

100381610038161003816100381610038161003816100381610038161003816120577=minus0

minus1205973V119911

1205971199113

100381610038161003816100381610038161003816100381610038161003816120577=+0

= minus119877(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot ((1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

minus 1)

sdot (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) (

120597120579

120597119911)

10038161003816100381610038161003816100381610038161003816120577=+0

(69)

with the conditions at infinity

119911 = minusinfin 120579 = 0 V = 0

119911 = +infin 120579 = minus1 V = 0

(70)

5 Stability Analysis

The interface problem analyzed in Section 4 has a travellingwave solution in the following form

(120579 (119909 119910 119911 119905) 120572 (119909 119910 119911 119905) V)

= (120579119904(119911 minus 119906119905) 120572

119904(119911 minus 119906119905) 0)

(120579119904 (119911 minus 119906119905) 120572119904 (119911 minus 119906119905))

=

(0 1) 1199112lt 0

(exp (minus1199061199112) minus 1 0) 119911

2gt 0

(71)

where 1199112= 119911minus119906119905 and 119906 is the speed of the stationary reaction

frontThis solution is a basic and stationary solution of inter-face problem (61)ndash(70) written in the moving coordinatesThen (61) (63) (65) and (67) can be replaced by

120597120579

120597119905+ (V sdot nabla) 120579 = Δ120579 + 119906

120597120579

1205971199112

120597V120597119905

+ (Vnabla) V

= minusnabla119901 + 119875ΔV + 119906120597120579

1205971199112

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) (120579 + 120579

0) 120574

(72)

The other equations remain unchanged To study thestability of reaction front we seek the solution of the problemas follows

120579 = 120579119904+ 120579

119901 = 119901119904+ 119901

V = V119904+ V

(73)

where 120579 119901 and V are respectively small perturbation oftemperature pressure and velocity

8 Mathematical Problems in Engineering

Subtituting (73) into (64) (72) one obtains the first-orderterms as follows

In the liquid monomer 1199112gt 120585

120597120579

120597119905= Δ120579 + 119906

120597120579

1205971199112

minus V1199111205791015840

119904 (74)

120597V120597119905

= minusnabla119901 + 119875ΔV + 119906120597120579

1205971199112

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579120574

(75)

div (V) = 0 (76)

In the liquid polymer 1199112lt 120585

120597120579

120597119905= Δ120579 + 119906

120597120579

1205971199112

(77)

120597V120597119905

= minusnabla119901 + 119875ΔV + 119906120597V1205971199112

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579120574

(78)

div (V) = 0 (79)

where 120585 is defined by 120585 = 120577 minus 119906119905We note

(120579 V119911) =

(1205791 V1199111) for 119911

2lt 120585

(1205792 V1199112) for 119911

2lt 120585

(80)

and consider the perturbation as follows

120579119894= 120579119894(1199112 119905) exp (119895 (119896

1119909 + 1198962119910))

V119911119894= V119911119894(1199112 119905) exp (119895 (119896

1119909 + 1198962119910))

120585 = 1205981(119905) exp (119895 (119896

1119909 + 1198962119910))

(81)

where 119896119894 119894 = 1 2 and 120598

1denote respectively the wave

numbers (in 119909 and 119910 directions) and the amplitude of theperturbation and 119895

2= minus1 Then jump conditions (69) are

linearized by taking into account the fact that

120579|120585=plusmn0 = 120579119904(plusmn0) + 120585120579

1015840

119904(plusmn0) + 120579 (plusmn0)

120597120579

1205971199112

10038161003816100381610038161003816100381610038161003816120585=plusmn0

= 1205791015840

119904(plusmn0) + 120585120579

10158401015840

119904(plusmn0) +

120597120579

1205971199112

100381610038161003816100381610038161003816100381610038161003816120585=plusmn0

(82)

Then the higher-order terms are given as

1205792

100381610038161003816100381610038161199112=0minus 1205791

100381610038161003816100381610038161199112=0= 119906120585 (83)

1205971205792

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

minus1205971205791

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

= minus1199062120585 minus

120597120585

120597119905+ V119911

10038161003816100381610038161199112=0 (84)

1199062120585 +

1205971205792

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

= minus119885

1199061205791

100381610038161003816100381610038161199112=0 (85)

V2119911

10038161003816100381610038161199112=0= V1119911

10038161003816100381610038161199112=0 (86)

120597V1199112

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

=120597V1119911

1205971199112

100381610038161003816100381610038161003816100381610038161199112=0

(87)

1205972V2119911

12059711991122

1003816100381610038161003816100381610038161003816100381610038161199112=0

=1205972V1119911

12059711991122

1003816100381610038161003816100381610038161003816100381610038161199112=0

(88)

1205973V2119911

12059711991132

1003816100381610038161003816100381610038161003816100381610038161199112=0

=1205973V1119911

12059711991132

1003816100381610038161003816100381610038161003816100381610038161199112=0

(89)

Substituting (81) into (83)ndash(89) leads to

1205792(0 119905) minus 120579

1(0 119905) = 119906120598

1(119905)

1205791015840

2(0 119905) minus 120579

1015840

1(0 119905) = minus120598

1 (119905) 1199062minus 1205981015840

1(119905) + V

1 (0 119905)

1205981 (119905) 119906

2+ 1205791015840

2(0 119905) = minus

119885

1199061205791 (0 119905)

V(119894)2(0 119905) = V(119894)

1(0 119905) 119894 = 0 1 2 3

(90)

where

V(119894) =120597119894V120597V119894

1205791015840=120597120579119894

120597119911

1205981015840

1(119905) =

1198891205981(119905)

119889119905

(91)

After simplification (75) and (78) become

120597

120597119905ΔV119911minus 119906

120597

1205971199112

ΔV119911= 119875ΔΔV

119911+ 119876(1 + 120582

1sin (120590

1119905)

+ 1205822sin (120590

2119905) (

1205972

1205971199092+

1205972

1205971199102))120579

(92)

Substituting (71) (87) (88) into (74) (77) (92) andintroducing the variable 119908 = V10158401015840 minus 1198962V where 119896 = radic1198962

1+ 11989622

one obtains two systems of equations as follows

1199112lt 120585

120597119908

120597119905minus 1199061199081015840minus 119875 (119908

10158401015840minus 1198962119908)

= minus1198761198962(1 + 120582

1sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579

119908 = V10158401015840 minus 1198962V

120597120579

120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 0

(93)

Mathematical Problems in Engineering 9

0 5 10 15 20 25 300

50

100

150

200

250

300

350

Rc

1205901

1205902 = radic212059011205902 = radic71205901

1205902 = 0

1205902 = radic371205901

Figure 1 Critical Rayleigh number as a function of 1205901for different

frequencies ratio and for 1205902= 0 120582

1= 1205822= 5 119875 = 05 119885 = 8 and

119896 = 07

1199112gt 120585

120597119908

120597119905minus 1199061199081015840minus 119875 (119908

10158401015840minus 1198962119908)

= minus1198761198962(1 + 120582

1sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579

119908 = V10158401015840 minus 1198962V

120597120579

120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 119906 exp (minus119906119911

2) V

(94)

with the following boundary conditions

1205792(0 119905) minus 120579

1(0 119905) = 119906120598

1(119905) (95)

1205791015840

2(0 119905) minus 120579

1015840

1(0 119905) = minus119906

21205761(119905) minus 120576

1015840

1(119905) + V

1(0 119905) (96)

1205791015840

2(0 119905) +

119885

1199061205791 (0 119905) = minus119906

21205761 (119905) (97)

V(119894)1(0 119905) = V(119894)

2(0 119905) 119894 = 0 1 2 3 (98)

Combining the three boundary conditions (95)ndash(97)yields

1205791015840

1(0 119905) +

119885

1199061205791(0 119905) =

1

119906(1205792(0 119905) minus 120579

1(0 119905))

1015840

119905

minus V1(0 119905)

(99)

6 Numerical Results

The numerical simulations of the problem are performedusing the finite-difference approximation with implicitscheme The algorithm is given in Appendix

Evolution of maximum temperature versus time providesthe onset of stability of the polymerization front The con-vective instability occurs when a jump from bounded tounbounded values of maximum temperature is achieved

Figure 1 shows the critical Rayleigh number as a functionof the frequency 120590

1for different frequencies ratio and for

0 25 50 75 100 125 1500

5

10

15

20

25

Rc

1205822

1205821 = 20

1205821 = 50

1205821 = 80

1205821 = 150

Figure 2 Critical Rayleigh number as a function of 1205822for different

values of 1205821 1205902= radic2120590

1 1205901= 5 119875 = 05 119885 = 8 and 119896 = 07

0 25 50 75 100 125 150 175 2000

50

100

150

200

250

Rc

1205822

1205902 = radic371205901

1205902 = radic1712059011205902 = radic71205901

1205902 = 1radic712059011205902 = 1radic171205901

1205902 = 1radic371205901

Figure 3 Critical Rayleigh number as a function of 1205822for different

frequencies ratio 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and 120590

1= 5

1205902= 0 In the absence of gravitational modulation (120590

1= 1205902=

0) we find the same result (119877119888asymp 27) as in [8 14] It can be

observed that the frequencies ratio of the QP gravitationalmodulation has a significant influence on the convectiveinstability of the reaction front As indicated increasing thefrequencies ratio the stability domain becomes larger Inparticular this can be seen in the range of the frequency1205901belonging to the interval (0 10) For large values of the

frequency 1205901 the frequencies ratio 120590

21205901has no effect on

the convective instability It is also worthy to notice thatthe QP gravitational modulation has a stabilizing effect forsufficiently large frequencies ratio (120590

2= radic37120590

1) comparing

to the periodic modulation case (1205902= 0)

The critical Rayleigh number as a function of the ampli-tude 120582

2is shown in Figure 2 for different values of 120582

1 We can

conclude that for a fixed value of 1205822and increased values of

1205821 the polymerization front loses its stability monotonically

revealing that the stability of the reaction front can becontrolled by acting on the amplitudes of the QPmodulation

Figure 3 illustrates the critical Rayleigh number as afunction of the amplitude 120582

2for different frequencies ratio

It can be observed that increasing the frequencies ratio thestability domain becomes large for certain values of 120582

2 For

large values of 1205822(1205822ge 100) the effect of the frequencies

ratio on the convective instability boundary is insignificant

10 Mathematical Problems in Engineering

0 25 50 75 100 125 150

k = 15

k = 2

k = 1

0

50

100

150

200

250

300

350

Rc

1205822

Figure 4 Critical Rayleigh number as a function of 1205822for different

values of number wave 1205902= radic7120590

1 1205901= 10 119875 = 05 119885 = 8 and

1205821= 5

0 25 50 75 100 125 1500

255075

100125150175200

Rc

1205822

1205902 = radic21205901

1205902 = radic71205901

Figure 5 Critical Rayleigh number as a function of 1205822for two

different frequencies ratios 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and

1205901= 10

Figure 4 shows for a fixed frequencies ratio 12059021205901= radic7

the influence of the wave number on the critical Rayleighnumber versus 120582

2 The plots indicate that for large values of

the wave number the reaction front gains stabilityThe influence of 120582

2on the stability of the reaction front is

given in Figure 5 for two different frequencies ratio From thisfigure one can conclude that an increase of the frequenciesratio has a stabilizing effect

7 Conclusion

In this work we have studied the influence of the QP vibra-tions on the convective instability of liquid-liquid polymer-ization frontWe have used themodel which includes the heatequation the concentration equation and the Navier-Stokesequations under Boussinesq approximation The ZeldovichFrank-Kamenetskii method has been used assuming that thereaction occurs in a narrow zone To obtain the convec-tive instability threshold using the linear stability analysisthe reduced system of equations has been discretized andresolved using the finite-difference method with implicitscheme The results have shown that for fixed values ofamplitudes an increase of the frequencies ratio stabilizes thereaction front and for large values of the frequency 120590

1the

critical Rayleigh number tends to the unmodulated criticalvalue In addition it is observed that for a fixed frequencies

ratio and for a given amplitude 1205822an increase of the ampli-

tude1205821destabilizes the reaction front Also the reaction front

becomes more stable by increasing the wave number

Appendix

The algorithm of the considered problem uses the finite-difference approximation with implicit scheme The velocityis computed from the previous time values of the problemThe discretization in space is given by 119911

119894= 119894ℎ where ℎ is the

space step and 119894 is a positive integerThe discretization in timeis given by 119905

119894= 119894120591 where 120591 is sufficiently small time step The

variables of discretization are 119908119895119894= 119908(119911

119894 119905119895) V119895119894= V(119911

119894 119905119895)

and 120579119895119894= 120579(119911119894 119905119895) One obtains the following

In the liquid monomer 119911 lt 120577

(minus1

ℎ2) 120579119895+1

119894minus1+ (

1

120591+

2

ℎ2+ 1198962+119906

ℎ) 120579119895+1

119894

+ (minus1

ℎ2minus119906

ℎ) 120579119895+1

119894+1=1

120591120579119895

119894

(minus119875

ℎ2)119908119895+1

119894minus1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

119894

+ (minus119875

ℎ2minus119906

ℎ)119908119895+1

119894+1=1

120591119908119895

119894

minus 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895)) 120579119895

119894

(1

ℎ2) V119895+1119894minus1

+ (minus2

ℎ2minus 1198962) V119895+1119894

+ (1

ℎ2) V119895+1119894+1

= 119908119895

119894

(A1)

In the liquid polymer 119911 gt 120577

(minus1

ℎ2)119879119895+1

119894minus1+ (

1

120591+

2

ℎ2+ 1198962+119906

ℎ)119879119895+1

119894

+ (minus1

ℎ2minus119906

ℎ)119879119895+1

119894+1=1

120591119879119895

119894+ 119906 exp (minus119906119911

119894) 119881119895

119894

(minus119875

ℎ2)119882119895+1

119894minus1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119882119895+1

119894

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

119894+1=1

120591119882119895

119894

minus 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895)) 119879119895

119894

(1

ℎ2)119881119895+1

119894minus1+ (minus

2

ℎ2minus 1198962)119881119895+1

119894+ (

1

ℎ2)119881119895+1

119894+1= 119882119895

119894

(A2)

From the jump conditions we have also the followingsystems

(minus1

ℎ2) 120579119895+1

1

+ (1

120591+

2

ℎ2+ 1198962+119906

ℎ+ (

1198881

1198882

)(1

ℎ2+119906

ℎ)) 120579119895+1

0

Mathematical Problems in Engineering 11

= (minus1

120591) 120579119895

0minus (

1

1199061205911198882

)(1

ℎ2+119906

ℎ) (119879119895

0minus 120579119895

0)

minus (V0

1198882

)(1

ℎ2+119906

ℎ)

(minus119875

ℎ2)119908119895+1

1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

0

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

0= (

minus1

120591)119908119895

1

+ 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895))

sdot ((minus1198882

1198881

)1198790minus

1

1199061205911198881

(119879119895

0minus 120579119895

0) minus

V0

1198881

)

(A3)

(1

120591+

2

ℎ2+ 1198962+119906

ℎ+

1198882

ℎ21198881

)119879119895+1

0+ (

minus1

ℎ2minus119906

ℎ)119879119895+1

1

= (minus1

120591)119879119895

0minus 119906 exp (minus119906ℎ) minus ( 1

119906120591ℎ21198881

) (119879119895

0minus 120579119895

0)

minusV0

ℎ21198881

(A4)

(minus119875

ℎ2)119908119895+1

1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

0

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

0= (

minus1

120591)119908119895

0

+ 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895))

sdot ((minus1198881

1198882

)1205790minus

1

1199061205911198882

)(119879119895

0minus 120579119895

0minusV0

1198882

)

(A5)

with

1198881= (

1

ℎminus119885

119906minus

1

119906120591)

1198882= (minus

1

ℎ+

1

119906120591)

(A6)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] B B Khanukaev M A Kozhushner N S Enikolopyan and NM Chechilo ldquoTheory of the propagation of a polymerizationfrontrdquo Combustion Explosion and Shock Waves vol 10 no 1pp 17ndash21 1974

[2] J A Pojman and T-C-M Qui Nonlinear Dynamics with Pol-ymers Fundamentals Methods and Applications John Wiley ampSons 2011

[3] I D Robertson H L Hernandez S R White and J S MooreldquoRapid stiffening of a microfluidic endoskeleton via frontalpolymerizationrdquo ACS Applied Materials and Interfaces vol 6no 21 pp 18469ndash18474 2014

[4] Vi Volpert and Vl Volpert ldquoPropagation of frontal polym-erizationmdashcrystallization wavesrdquo European Journal of AppliedMathematics vol 5 no 2 pp 201ndash215 1994

[5] G Bowden M Garbey V M Ilyashenko et al ldquoEffect of con-vection on a propagating front with a solid product comparisonof theory and experimentsrdquoThe Journal of Physical Chemistry Bvol 101 no 4 pp 678ndash686 1997

[6] M Garbey A Taik and V Volpert ldquoLinear stability analysisof reaction fronts in liquidsrdquo Quarterly of Applied Mathematicsvol 54 no 2 pp 225ndash247 1996

[7] K Allali V Volpert and J A Pojman ldquoInfluence of vibrationson convective instability of polymerization frontsrdquo Journal ofEngineering Mathematics vol 41 no 1 pp 13ndash31 2001

[8] K Allali F Bikany A Taik and V Volpert ldquoInfluence ofvibrations on convective instability of reaction fronts in liquidsrdquoMathematical Modelling of Natural Phenomena vol 5 no 7 pp35ndash41 2010

[9] K Allali M Belhaq and K El Karouni ldquoInfluence of quasi-periodic gravitational modulation on convective instability ofreaction fronts in porous mediardquo Communications in NonlinearScience and Numerical Simulation vol 17 no 4 pp 1588ndash15962012

[10] KAllali S Assiyad andM Belhaq ldquoConvection of polymeriza-tion front with solid product under quasi-periodic gravitationalmodulationrdquo Nonlinear Dynamics and Systems Theory vol 14no 4 pp 323ndash334 2014

[11] M Menzinger and R Wolfgang ldquoThe meaning and use of theArrhenius activation energyrdquoAngewandte Chemie InternationalEdition in English vol 8 no 6 pp 438ndash444 1969

[12] B V Novozhilov ldquoThe rate of propagation of the front of anexothermic reaction in a condensed phaserdquo Doklady PhysicalChemistry vol 141 pp 836ndash838 1961

[13] Y B Zeldovich and D A Frank-Kamenetskii ldquoA theory ofthermal propagation of amerdquo Acta Physicochimica USSR vol 9pp 341ndash350 1938

[14] M Garbey A Taık and V Volpert ldquoInfluence of natural con-vection on stability of reaction fronts in liquidsrdquo Quarterly ofApplied Mathematics vol 56 no 1 pp 1ndash35 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Order 1205980

120597V0

120597119905minus120597V1

120597120578

1205971205770

120597119905minus120597V0

120597120578

1205971205771

120597119905+ V0119909(120597V0

120597119909minus120597V1

120597120578

1205971205770

120597119909

minus120597V0

120597120578

1205971205771

120597119909) + V1119909

120597V0

120597120578

1205971205770

120597119909+ V1119910

120597V0

120597120578

1205971205770

120597119910

+ V0119910(120597V0

120597119910minus120597V1

120597120578

1205971205770

120597119910minus120597V0

120597120578

1205971205771

120597119910) + V0119911

120597V1

120597120578+ V1119911

sdot120597V0

120597120578= minusnabla01199010+ 1199051

1205971199010

120597120578+ 1199050

1205971199011

120597120578+ 119875(119860

1205972V2

1205971205782

+ 1199053

1205972V1

1205971205782+ 1198650

120597V1

120597120578+ 1199054

1205972V0

1205971205782+ 1198651

120597V0

120597120578+ Δ1V0)

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) 120574120579

0

(23)

120597V0119909

120597119909minus120597V1119909

120597120578

1205971205770

120597119909minus120597V0119909

120597120578

1205971205771

120597119909+120597V0119910

120597119910minus120597V1119910

120597120578

1205971205770

120597119910

minus120597V0119910

120597120578

1205971205771

120597119910+120597V1119911

120597120578= 0

(24)

Order 1205981

120597V1

120597119905minus (

120597V2

120597120578

1205971205770

120597119905+120597V1

120597120578

1205971205771

120597119905+120597V0

120597120578

1205971205772

120597119905) + V0119909(120597V1

120597119909

minus120597V1

120597120578

1205971205771

120597119909minus120597V0

120597120578

1205971205772

120597119909minus120597V2

120597120578

1205971205770

120597119909) + V1119909(120597V0

120597119909

minus120597V0

120597120578

1205971205771

120597119909minus120597V1

120597120578

1205971205770

120597119909) minus V2119909

120597V0

120597120578

1205971205770

120597119909+ V0119910(120597V1

120597119910

minus120597V1

120597120578

1205971205771

120597119910minus120597V0

120597120578

1205971205772

120597119910minus120597V2

120597120578

1205971205770

120597119910) + V1119910(120597V0

120597119910

minus120597V0

120597120578

1205971205771

120597119910minus120597V1

120597120578

1205971205770

120597119910) minus V2119910

120597V0

120597120578

1205971205770

120597119910+ V0119911

120597V2

120597120578

+ V1119911

120597V1

120597120578+ V2119911

120597V0

120597120578= 1199050

1205971199012

120597120578+ 1199051

1205971199011

120597120578minus nabla01199011+ 1199052

sdot1205971199010

120597120578+ 119875(119860

1205972V3

1205971205782+ 1199053

1205972V2

1205971205782+ 1198650

120597V2

120597120578+ 1198651

120597V1

120597120578

+ Δ1V1+ 1198652

120597V0

120597120578) + 119876 (1 + 120582

1sin (120590

1119905)

+ 1205822sin (120590

2119905)) 1205790120574

(25)

120597V1119909

120597119909minus120597V2119909

120597120578

1205971205770

120597119909minus120597V1119909

120597120578

1205971205771

120597119909minus120597V0119909

120597120578

1205971205772

120597119909+120597V1119910

120597119910

minus120597V2119910

120597120578

1205971205770

120597119910minus120597V1119910

120597120578

1205971205771

120597119910minus120597V0119910

120597120578

1205971205772

120597119910+120597V2119911

120597120578= 0

(26)

Order 1205982

120597V2119909

120597119909minus120597V3119909

120597120578

1205971205770

120597119909minus120597V2119909

120597120578

1205971205771

120597119909minus120597V1119909

120597120578

1205971205772

120597119909minus120597V0119909

120597120578

1205971205773

120597119909

+120597V2119910

120597119910minus120597V3119910

120597120578

1205971205770

120597119910minus120597V2119910

120597120578

1205971205771

120597119910minus120597V1119910

120597120578

1205971205772

120597119910

minus120597V0119910

120597120578

1205971205773

120597119910+120597V3119911

120597120578= 0

(27)

where

Δ1=

1205972

1205971199092+

1205972

1205971199102

nabla0= (

120597

120597119909120597

120597119910 0)

119865119894= minus2(

120597120577119894

120597119909

120597

120597119909+120597120577119894

120597119910

120597

120597119910) minus (

1205972120577119894

1205971199092+1205972120577119894

1205971199102) 119868

119894 = 0 1 2

1199050= (

1205971205770

1205971199091205971205770

120597119910 minus1)

119905119894= (

120597120577119894

120597119909120597120577119894

120597119910 minus 1) 119894 = 1 2

1199053= 2(

1205971205771

120597119909

1205971205770

120597119909+1205971205770

120597119910

1205971205771

120597119910)

1199054= (

1205971205771

120597119909)

2

+ (1205971205771

120597119910)

2

+1205971205770

120597119909

1205971205772

120597119909+1205971205770

120597119910

1205971205772

120597119910

119860 = (1 + (1205971205770

120597119909)

2

+ (1205971205770

120597119910)

2

)

(28)

and 119868 is the identity operator The matching conditions as120578 rarr plusmninfin are given by

V0sim V0

10038161003816100381610038161199111=plusmn0

(29)

V1sim (

120597V0

1205971199111

100381610038161003816100381610038161003816100381610038161199111=plusmn0

)120578 + V1

10038161003816100381610038161199111=plusmn0

(30)

V2sim1

2(1205972V0

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=plusmn0

)1205782+ (

120597V1

1205971199111

100381610038161003816100381610038161003816100381610038161199111=plusmn0

)120578

+ V2

10038161003816100381610038161199111=plusmn0

(31)

V3sim1

6(1205973V0

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=plusmn0

)1205783+1

2(1205972V1

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=plusmn0

)1205782

+ (120597V2

1205971199111

100381610038161003816100381610038161003816100381610038161199111=plusmn0

)120578 + V3

10038161003816100381610038161199111=plusmn0

(32)

Mathematical Problems in Engineering 5

As 120578 rarr +infin

1205791asymp 1205791

10038161003816100381610038161199111=+0

+ (1205971205790

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

)120578 0997888rarr 0 (33)

As 120578 rarr minusinfin

1205791sim 1205791

10038161003816100381610038161199111=minus0

0997888rarr 1 (34)

From (18) we obtain

1205972V0

1205971205782= 0 (35)

Consequently we can conclude that V0(120578) is a linear

function of 120578 and identically constant since the velocity isbounded Equation (29) becomes

V0

10038161003816100381610038161199111=+0

= V0

10038161003816100381610038161199111=minus0

(36)

120597V0

120597120578= 0 (37)

From (36)-(37) we deduce that the first term in theexpression of the velocity V

0is continuous at the front By

substituting (37) into (21) we find

1198751198601205972V1

1205971205782+ 1199050

1205971199010

120597120578= 0 (38)

120597V0119909

120597119909minus120597V1119909

120597120578

1205971205770

120597119909+120597V0119910

120597119910minus120597V1119910

120597120578

1205971205770

120597119910+120597V1119911

120597120578= 0 (39)

Differentiating (39) with respect to 120578 one obtains

1205972V1119909

1205971205782

1205971205770

120597119909+1205972V1119910

1205971205782

1205971205770

120597119910minus1205972V1119911

1205971205782= 0 (40)

As a result (38) is a vectorial equation with three com-ponents We multiply the first component by 120597120577

0120597119909 the

second by 1205971205770120597119910 and the third by minus1 and adding we have

119875119860(1205972V1119909

1205971205782

1205971205770

120597119909+1205972V1119910

1205971205782

1205971205770

120597119910minus1205972V1119911

1205971205782) + 119860

1205971199010

120597120578= 0 (41)

Considering the two equations (38) and (40) we obtain

1205971199010

120597120578= 0 (42)

1205972V1

1205971205782= 0 (43)

From the boundedness of the velocity and (30) the con-tinuity of the first order term of the outer expansion and ofthe first derivative of the zero-order term we can write

120597V0

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

=120597V0

1205971199111

100381610038161003816100381610038161003816100381610038161199111=minus0

V1

10038161003816100381610038161199111=+0

= V1

10038161003816100381610038161199111=minus0

(44)

Differentiating (23) once and (26) twice with respect to 120578and using the three equations (37) (42) and (43) we get

1199050

12059721199011

1205971205782+ 119875119860

1205973V2

1205971205783= 0 (45)

1205973V2119909

1205971205783

1205971205770

120597119909+1205973V2119910

1205971205783

1205971205770

120597119910minus1205973V2119911

1205971205783= 0 (46)

As above we multiply the three components of (45)respectively by 120597120577

0120597119909 120597120577

0120597119910 and minus1 and adding we get

119875119860(1205973V2119909

1205971205783

1205971205770

120597119909+1205973V2119910

1205971205783

1205971205770

120597119910minus1205973V2119911

1205971205783) + 119860

12059721199011

1205971205782= 0 (47)

From the previous equation and using (45)-(46) we have

12059721199011

1205971205782= 0

1205973V2

1205971205783= 0

(48)

Knowing that the velocity is bounded and considering(31) we obtain the jump conditions in the following forms

1205972V0

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=+0

=1205972V0

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=minus0

120597V1

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

=120597V1

1205971199111

100381610038161003816100381610038161003816100381610038161199111=minus0

V2

10038161003816100381610038161199111=+0

= V2

10038161003816100381610038161199111=minus0

(49)

Differentiating (25) twice and (27) three times withrespect to 120578 and taking into account (37) (42) (48) we obtain

1199050

12059731199012

1205971205783+ 119875119860

1205974V3

1205971205784

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) 120574

12059721205791

1205971205782= 0

1205974V3119909

1205971205784

1205971205770

120597119909+1205974V3119910

1205971205784

1205971205770

120597119910minus1205974V3119911

1205971205784= 0

(50)

119875119860(1205974V3119909

1205971205784

1205971205770

120597119909+1205974V3119910

1205971205784

1205971205770

120597119910minus1205974V3119911

1205971205784) + 119860

12059731199012

1205971205783

minus 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905))

12059721205791

1205971205782= 0

(51)

From the last equation and using (50) we can write

11986012059731199012

1205971205783minus 119876 (1 + 120582

1sin (120590

1119905) + 120582

2sin (120590

2119905))

12059721205791

1205971205782

= 0

(52)

1205740

12059721205791

1205971205782=1205974V3

1205971205784 (53)

6 Mathematical Problems in Engineering

where 1205740is defined by

1205740= (minus

1205971205770

120597119909

119877

1198602 minus1205971205770

120597119910

119877

1198602119877

1198602minus119877

119860)

sdot (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905))

(54)

Now integrating (53) with respect to 120578 using the system(32)ndash(34) lead to

1205973V0

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=minus0

minus1205973V0

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=+0

= minus1205740

1205971205790

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

(55)

As a first conclusion the velocity jump conditions acrossthe reaction front are given by (36) (44) (49) and (55)

From (20) (22) (37) we conclude that 0is a monotonic

function satisfying 0 lt 0lt 1 Therefore the reaction is of

zero order and 120601(0) equiv 1 Multiplying (19) by 120597120579

1120597120578 and

integrating the result yield

(1205971205791

120597120578)

210038161003816100381610038161003816100381610038161003816100381610038161003816+infin

minus (1205971205791

120597120578)

210038161003816100381610038161003816100381610038161003816100381610038161003816minusinfin

= 2119860minus1int

1205791|1199111=minus0

minusinfin

exp( 120591

1 + 120591120575) 119889120591

(56)

Subtracting (19) from (20) and integrating the result oneobtains

1205971205791

120597120578

100381610038161003816100381610038161003816100381610038161003816+infin

minus1205971205791

120597120578

100381610038161003816100381610038161003816100381610038161003816minusinfin

= minus119860minus1(1205971205770

120597119905+ 119904) (57)

where 119904 is given by

119904 = V0119909

1205971205770

120597119909+ V0119910

1205971205770

120597119910minus V0119911 (58)

As a second conclusion the temperature jump conditionsacross the reaction front are given by the two last equations(56)-(57) By using the matching conditions above andtruncating the expansion as

120579 asymp 1205790

1205791

10038161003816100381610038161199111=minus0

asymp 119885120579|1199111=+0

120577 asymp 1205770

V asymp V0

(59)

the jump conditions can be written as

(120597120579

1205971199111

)

2100381610038161003816100381610038161003816100381610038161003816+0

minus (120597120579

1205971199111

)

2100381610038161003816100381610038161003816100381610038161003816minus0

= 2119885(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot int

1205791|1199111=minus0

minusinfin

exp ( 120591

119885minus1 + 120591120575) 119889120591

120597120579

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

minus120597120579

1205971199111

100381610038161003816100381610038161003816100381610038161199111=minus0

= minus(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot (120597120577

120597119905+ (V119909

120597120577

120597119909+ V119910

120597120577

120597119910minus V119911)

100381610038161003816100381610038161003816100381610038161199111=+0

)

V119911

10038161003816100381610038161199111=+0

= V119911

10038161003816100381610038161199111=minus0

120597V119911

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

=120597V119911

1205971199111

100381610038161003816100381610038161003816100381610038161199111=minus0

1205972V119911

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=+0

=120597V119911

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=minus0

1205973V119911

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=minus0

minus1205973V119911

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=+0

= minus119877(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot ((1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

minus 1)

sdot (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905))

120597120579

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

(60)

4 The Interface Problem and Perturbation

To study the propagation of polymerization front with aliquid reactant and liquid product the equation of motionhas been considered before and after the reaction zone Thischanges the jump conditions and influences the stabilityconditions of the frontal polymerization process Notice thatin the case of liquid-solid polymerization front the equationof motion is considered after the reaction zone The originalsystem (2) (3) (8) (9) leads to the following interfaceproblem

In the liquid monomer 119911 gt 120577

120597120579

120597119905+ (V sdot nabla) 120579 = Δ120579 (61)

120572 = 0 (62)

Mathematical Problems in Engineering 7

120597V120597119905

+ (V sdot nabla) V

= minusnabla119901 + 119875ΔV

+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579

0) 120574

(63)

div (V) = 0 (64)

In the liquid polymer 119911 lt 120577

120597120579

120597119905+ (V sdot nabla) 120579 = Δ120579 (65)

120572 = 1 (66)

120597V120597119905

+ (V sdot nabla) V

= minusnabla119901 + 119875ΔV

+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579

0) 120574

(67)

div (V) = 0 (68)

At the interface 119911 = 120577

120579|120577minus0 = 120579|120577+0

120597120579

120597119911

10038161003816100381610038161003816100381610038161003816120577=minus0minus120597120579

120597119911

10038161003816100381610038161003816100381610038161003816120577=+0= (1 + (

120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot (120597120577

120597119905+ (V119909

120597120577

120597119909+ V119910

120597120577

120597119910minus V119911)

10038161003816100381610038161003816100381610038161003816120577

)

(120597120579

120597119911)

2100381610038161003816100381610038161003816100381610038161003816120577minus0

minus (120597120579

120597119911)

21003816100381610038161003816100381610038161003816100381610038161205771+0

= minus2119885(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot int

120579|120577

minusinfin

exp ( 120591

119885minus1 + 120575120591) 119889120591

V119911

1003816100381610038161003816120577=minus0 = V119911

1003816100381610038161003816120577=+0

120597V119911

120597119911

10038161003816100381610038161003816100381610038161003816120577=minus0=120597V119911

120597119911

10038161003816100381610038161003816100381610038161003816120577=+0

1205972V119911

1205971199112

100381610038161003816100381610038161003816100381610038161003816120577=minus0

=1205972V119911

1205971199112

100381610038161003816100381610038161003816100381610038161003816120577=+0

1205973V119911

1205971199113

100381610038161003816100381610038161003816100381610038161003816120577=minus0

minus1205973V119911

1205971199113

100381610038161003816100381610038161003816100381610038161003816120577=+0

= minus119877(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot ((1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

minus 1)

sdot (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) (

120597120579

120597119911)

10038161003816100381610038161003816100381610038161003816120577=+0

(69)

with the conditions at infinity

119911 = minusinfin 120579 = 0 V = 0

119911 = +infin 120579 = minus1 V = 0

(70)

5 Stability Analysis

The interface problem analyzed in Section 4 has a travellingwave solution in the following form

(120579 (119909 119910 119911 119905) 120572 (119909 119910 119911 119905) V)

= (120579119904(119911 minus 119906119905) 120572

119904(119911 minus 119906119905) 0)

(120579119904 (119911 minus 119906119905) 120572119904 (119911 minus 119906119905))

=

(0 1) 1199112lt 0

(exp (minus1199061199112) minus 1 0) 119911

2gt 0

(71)

where 1199112= 119911minus119906119905 and 119906 is the speed of the stationary reaction

frontThis solution is a basic and stationary solution of inter-face problem (61)ndash(70) written in the moving coordinatesThen (61) (63) (65) and (67) can be replaced by

120597120579

120597119905+ (V sdot nabla) 120579 = Δ120579 + 119906

120597120579

1205971199112

120597V120597119905

+ (Vnabla) V

= minusnabla119901 + 119875ΔV + 119906120597120579

1205971199112

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) (120579 + 120579

0) 120574

(72)

The other equations remain unchanged To study thestability of reaction front we seek the solution of the problemas follows

120579 = 120579119904+ 120579

119901 = 119901119904+ 119901

V = V119904+ V

(73)

where 120579 119901 and V are respectively small perturbation oftemperature pressure and velocity

8 Mathematical Problems in Engineering

Subtituting (73) into (64) (72) one obtains the first-orderterms as follows

In the liquid monomer 1199112gt 120585

120597120579

120597119905= Δ120579 + 119906

120597120579

1205971199112

minus V1199111205791015840

119904 (74)

120597V120597119905

= minusnabla119901 + 119875ΔV + 119906120597120579

1205971199112

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579120574

(75)

div (V) = 0 (76)

In the liquid polymer 1199112lt 120585

120597120579

120597119905= Δ120579 + 119906

120597120579

1205971199112

(77)

120597V120597119905

= minusnabla119901 + 119875ΔV + 119906120597V1205971199112

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579120574

(78)

div (V) = 0 (79)

where 120585 is defined by 120585 = 120577 minus 119906119905We note

(120579 V119911) =

(1205791 V1199111) for 119911

2lt 120585

(1205792 V1199112) for 119911

2lt 120585

(80)

and consider the perturbation as follows

120579119894= 120579119894(1199112 119905) exp (119895 (119896

1119909 + 1198962119910))

V119911119894= V119911119894(1199112 119905) exp (119895 (119896

1119909 + 1198962119910))

120585 = 1205981(119905) exp (119895 (119896

1119909 + 1198962119910))

(81)

where 119896119894 119894 = 1 2 and 120598

1denote respectively the wave

numbers (in 119909 and 119910 directions) and the amplitude of theperturbation and 119895

2= minus1 Then jump conditions (69) are

linearized by taking into account the fact that

120579|120585=plusmn0 = 120579119904(plusmn0) + 120585120579

1015840

119904(plusmn0) + 120579 (plusmn0)

120597120579

1205971199112

10038161003816100381610038161003816100381610038161003816120585=plusmn0

= 1205791015840

119904(plusmn0) + 120585120579

10158401015840

119904(plusmn0) +

120597120579

1205971199112

100381610038161003816100381610038161003816100381610038161003816120585=plusmn0

(82)

Then the higher-order terms are given as

1205792

100381610038161003816100381610038161199112=0minus 1205791

100381610038161003816100381610038161199112=0= 119906120585 (83)

1205971205792

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

minus1205971205791

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

= minus1199062120585 minus

120597120585

120597119905+ V119911

10038161003816100381610038161199112=0 (84)

1199062120585 +

1205971205792

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

= minus119885

1199061205791

100381610038161003816100381610038161199112=0 (85)

V2119911

10038161003816100381610038161199112=0= V1119911

10038161003816100381610038161199112=0 (86)

120597V1199112

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

=120597V1119911

1205971199112

100381610038161003816100381610038161003816100381610038161199112=0

(87)

1205972V2119911

12059711991122

1003816100381610038161003816100381610038161003816100381610038161199112=0

=1205972V1119911

12059711991122

1003816100381610038161003816100381610038161003816100381610038161199112=0

(88)

1205973V2119911

12059711991132

1003816100381610038161003816100381610038161003816100381610038161199112=0

=1205973V1119911

12059711991132

1003816100381610038161003816100381610038161003816100381610038161199112=0

(89)

Substituting (81) into (83)ndash(89) leads to

1205792(0 119905) minus 120579

1(0 119905) = 119906120598

1(119905)

1205791015840

2(0 119905) minus 120579

1015840

1(0 119905) = minus120598

1 (119905) 1199062minus 1205981015840

1(119905) + V

1 (0 119905)

1205981 (119905) 119906

2+ 1205791015840

2(0 119905) = minus

119885

1199061205791 (0 119905)

V(119894)2(0 119905) = V(119894)

1(0 119905) 119894 = 0 1 2 3

(90)

where

V(119894) =120597119894V120597V119894

1205791015840=120597120579119894

120597119911

1205981015840

1(119905) =

1198891205981(119905)

119889119905

(91)

After simplification (75) and (78) become

120597

120597119905ΔV119911minus 119906

120597

1205971199112

ΔV119911= 119875ΔΔV

119911+ 119876(1 + 120582

1sin (120590

1119905)

+ 1205822sin (120590

2119905) (

1205972

1205971199092+

1205972

1205971199102))120579

(92)

Substituting (71) (87) (88) into (74) (77) (92) andintroducing the variable 119908 = V10158401015840 minus 1198962V where 119896 = radic1198962

1+ 11989622

one obtains two systems of equations as follows

1199112lt 120585

120597119908

120597119905minus 1199061199081015840minus 119875 (119908

10158401015840minus 1198962119908)

= minus1198761198962(1 + 120582

1sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579

119908 = V10158401015840 minus 1198962V

120597120579

120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 0

(93)

Mathematical Problems in Engineering 9

0 5 10 15 20 25 300

50

100

150

200

250

300

350

Rc

1205901

1205902 = radic212059011205902 = radic71205901

1205902 = 0

1205902 = radic371205901

Figure 1 Critical Rayleigh number as a function of 1205901for different

frequencies ratio and for 1205902= 0 120582

1= 1205822= 5 119875 = 05 119885 = 8 and

119896 = 07

1199112gt 120585

120597119908

120597119905minus 1199061199081015840minus 119875 (119908

10158401015840minus 1198962119908)

= minus1198761198962(1 + 120582

1sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579

119908 = V10158401015840 minus 1198962V

120597120579

120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 119906 exp (minus119906119911

2) V

(94)

with the following boundary conditions

1205792(0 119905) minus 120579

1(0 119905) = 119906120598

1(119905) (95)

1205791015840

2(0 119905) minus 120579

1015840

1(0 119905) = minus119906

21205761(119905) minus 120576

1015840

1(119905) + V

1(0 119905) (96)

1205791015840

2(0 119905) +

119885

1199061205791 (0 119905) = minus119906

21205761 (119905) (97)

V(119894)1(0 119905) = V(119894)

2(0 119905) 119894 = 0 1 2 3 (98)

Combining the three boundary conditions (95)ndash(97)yields

1205791015840

1(0 119905) +

119885

1199061205791(0 119905) =

1

119906(1205792(0 119905) minus 120579

1(0 119905))

1015840

119905

minus V1(0 119905)

(99)

6 Numerical Results

The numerical simulations of the problem are performedusing the finite-difference approximation with implicitscheme The algorithm is given in Appendix

Evolution of maximum temperature versus time providesthe onset of stability of the polymerization front The con-vective instability occurs when a jump from bounded tounbounded values of maximum temperature is achieved

Figure 1 shows the critical Rayleigh number as a functionof the frequency 120590

1for different frequencies ratio and for

0 25 50 75 100 125 1500

5

10

15

20

25

Rc

1205822

1205821 = 20

1205821 = 50

1205821 = 80

1205821 = 150

Figure 2 Critical Rayleigh number as a function of 1205822for different

values of 1205821 1205902= radic2120590

1 1205901= 5 119875 = 05 119885 = 8 and 119896 = 07

0 25 50 75 100 125 150 175 2000

50

100

150

200

250

Rc

1205822

1205902 = radic371205901

1205902 = radic1712059011205902 = radic71205901

1205902 = 1radic712059011205902 = 1radic171205901

1205902 = 1radic371205901

Figure 3 Critical Rayleigh number as a function of 1205822for different

frequencies ratio 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and 120590

1= 5

1205902= 0 In the absence of gravitational modulation (120590

1= 1205902=

0) we find the same result (119877119888asymp 27) as in [8 14] It can be

observed that the frequencies ratio of the QP gravitationalmodulation has a significant influence on the convectiveinstability of the reaction front As indicated increasing thefrequencies ratio the stability domain becomes larger Inparticular this can be seen in the range of the frequency1205901belonging to the interval (0 10) For large values of the

frequency 1205901 the frequencies ratio 120590

21205901has no effect on

the convective instability It is also worthy to notice thatthe QP gravitational modulation has a stabilizing effect forsufficiently large frequencies ratio (120590

2= radic37120590

1) comparing

to the periodic modulation case (1205902= 0)

The critical Rayleigh number as a function of the ampli-tude 120582

2is shown in Figure 2 for different values of 120582

1 We can

conclude that for a fixed value of 1205822and increased values of

1205821 the polymerization front loses its stability monotonically

revealing that the stability of the reaction front can becontrolled by acting on the amplitudes of the QPmodulation

Figure 3 illustrates the critical Rayleigh number as afunction of the amplitude 120582

2for different frequencies ratio

It can be observed that increasing the frequencies ratio thestability domain becomes large for certain values of 120582

2 For

large values of 1205822(1205822ge 100) the effect of the frequencies

ratio on the convective instability boundary is insignificant

10 Mathematical Problems in Engineering

0 25 50 75 100 125 150

k = 15

k = 2

k = 1

0

50

100

150

200

250

300

350

Rc

1205822

Figure 4 Critical Rayleigh number as a function of 1205822for different

values of number wave 1205902= radic7120590

1 1205901= 10 119875 = 05 119885 = 8 and

1205821= 5

0 25 50 75 100 125 1500

255075

100125150175200

Rc

1205822

1205902 = radic21205901

1205902 = radic71205901

Figure 5 Critical Rayleigh number as a function of 1205822for two

different frequencies ratios 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and

1205901= 10

Figure 4 shows for a fixed frequencies ratio 12059021205901= radic7

the influence of the wave number on the critical Rayleighnumber versus 120582

2 The plots indicate that for large values of

the wave number the reaction front gains stabilityThe influence of 120582

2on the stability of the reaction front is

given in Figure 5 for two different frequencies ratio From thisfigure one can conclude that an increase of the frequenciesratio has a stabilizing effect

7 Conclusion

In this work we have studied the influence of the QP vibra-tions on the convective instability of liquid-liquid polymer-ization frontWe have used themodel which includes the heatequation the concentration equation and the Navier-Stokesequations under Boussinesq approximation The ZeldovichFrank-Kamenetskii method has been used assuming that thereaction occurs in a narrow zone To obtain the convec-tive instability threshold using the linear stability analysisthe reduced system of equations has been discretized andresolved using the finite-difference method with implicitscheme The results have shown that for fixed values ofamplitudes an increase of the frequencies ratio stabilizes thereaction front and for large values of the frequency 120590

1the

critical Rayleigh number tends to the unmodulated criticalvalue In addition it is observed that for a fixed frequencies

ratio and for a given amplitude 1205822an increase of the ampli-

tude1205821destabilizes the reaction front Also the reaction front

becomes more stable by increasing the wave number

Appendix

The algorithm of the considered problem uses the finite-difference approximation with implicit scheme The velocityis computed from the previous time values of the problemThe discretization in space is given by 119911

119894= 119894ℎ where ℎ is the

space step and 119894 is a positive integerThe discretization in timeis given by 119905

119894= 119894120591 where 120591 is sufficiently small time step The

variables of discretization are 119908119895119894= 119908(119911

119894 119905119895) V119895119894= V(119911

119894 119905119895)

and 120579119895119894= 120579(119911119894 119905119895) One obtains the following

In the liquid monomer 119911 lt 120577

(minus1

ℎ2) 120579119895+1

119894minus1+ (

1

120591+

2

ℎ2+ 1198962+119906

ℎ) 120579119895+1

119894

+ (minus1

ℎ2minus119906

ℎ) 120579119895+1

119894+1=1

120591120579119895

119894

(minus119875

ℎ2)119908119895+1

119894minus1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

119894

+ (minus119875

ℎ2minus119906

ℎ)119908119895+1

119894+1=1

120591119908119895

119894

minus 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895)) 120579119895

119894

(1

ℎ2) V119895+1119894minus1

+ (minus2

ℎ2minus 1198962) V119895+1119894

+ (1

ℎ2) V119895+1119894+1

= 119908119895

119894

(A1)

In the liquid polymer 119911 gt 120577

(minus1

ℎ2)119879119895+1

119894minus1+ (

1

120591+

2

ℎ2+ 1198962+119906

ℎ)119879119895+1

119894

+ (minus1

ℎ2minus119906

ℎ)119879119895+1

119894+1=1

120591119879119895

119894+ 119906 exp (minus119906119911

119894) 119881119895

119894

(minus119875

ℎ2)119882119895+1

119894minus1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119882119895+1

119894

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

119894+1=1

120591119882119895

119894

minus 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895)) 119879119895

119894

(1

ℎ2)119881119895+1

119894minus1+ (minus

2

ℎ2minus 1198962)119881119895+1

119894+ (

1

ℎ2)119881119895+1

119894+1= 119882119895

119894

(A2)

From the jump conditions we have also the followingsystems

(minus1

ℎ2) 120579119895+1

1

+ (1

120591+

2

ℎ2+ 1198962+119906

ℎ+ (

1198881

1198882

)(1

ℎ2+119906

ℎ)) 120579119895+1

0

Mathematical Problems in Engineering 11

= (minus1

120591) 120579119895

0minus (

1

1199061205911198882

)(1

ℎ2+119906

ℎ) (119879119895

0minus 120579119895

0)

minus (V0

1198882

)(1

ℎ2+119906

ℎ)

(minus119875

ℎ2)119908119895+1

1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

0

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

0= (

minus1

120591)119908119895

1

+ 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895))

sdot ((minus1198882

1198881

)1198790minus

1

1199061205911198881

(119879119895

0minus 120579119895

0) minus

V0

1198881

)

(A3)

(1

120591+

2

ℎ2+ 1198962+119906

ℎ+

1198882

ℎ21198881

)119879119895+1

0+ (

minus1

ℎ2minus119906

ℎ)119879119895+1

1

= (minus1

120591)119879119895

0minus 119906 exp (minus119906ℎ) minus ( 1

119906120591ℎ21198881

) (119879119895

0minus 120579119895

0)

minusV0

ℎ21198881

(A4)

(minus119875

ℎ2)119908119895+1

1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

0

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

0= (

minus1

120591)119908119895

0

+ 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895))

sdot ((minus1198881

1198882

)1205790minus

1

1199061205911198882

)(119879119895

0minus 120579119895

0minusV0

1198882

)

(A5)

with

1198881= (

1

ℎminus119885

119906minus

1

119906120591)

1198882= (minus

1

ℎ+

1

119906120591)

(A6)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] B B Khanukaev M A Kozhushner N S Enikolopyan and NM Chechilo ldquoTheory of the propagation of a polymerizationfrontrdquo Combustion Explosion and Shock Waves vol 10 no 1pp 17ndash21 1974

[2] J A Pojman and T-C-M Qui Nonlinear Dynamics with Pol-ymers Fundamentals Methods and Applications John Wiley ampSons 2011

[3] I D Robertson H L Hernandez S R White and J S MooreldquoRapid stiffening of a microfluidic endoskeleton via frontalpolymerizationrdquo ACS Applied Materials and Interfaces vol 6no 21 pp 18469ndash18474 2014

[4] Vi Volpert and Vl Volpert ldquoPropagation of frontal polym-erizationmdashcrystallization wavesrdquo European Journal of AppliedMathematics vol 5 no 2 pp 201ndash215 1994

[5] G Bowden M Garbey V M Ilyashenko et al ldquoEffect of con-vection on a propagating front with a solid product comparisonof theory and experimentsrdquoThe Journal of Physical Chemistry Bvol 101 no 4 pp 678ndash686 1997

[6] M Garbey A Taik and V Volpert ldquoLinear stability analysisof reaction fronts in liquidsrdquo Quarterly of Applied Mathematicsvol 54 no 2 pp 225ndash247 1996

[7] K Allali V Volpert and J A Pojman ldquoInfluence of vibrationson convective instability of polymerization frontsrdquo Journal ofEngineering Mathematics vol 41 no 1 pp 13ndash31 2001

[8] K Allali F Bikany A Taik and V Volpert ldquoInfluence ofvibrations on convective instability of reaction fronts in liquidsrdquoMathematical Modelling of Natural Phenomena vol 5 no 7 pp35ndash41 2010

[9] K Allali M Belhaq and K El Karouni ldquoInfluence of quasi-periodic gravitational modulation on convective instability ofreaction fronts in porous mediardquo Communications in NonlinearScience and Numerical Simulation vol 17 no 4 pp 1588ndash15962012

[10] KAllali S Assiyad andM Belhaq ldquoConvection of polymeriza-tion front with solid product under quasi-periodic gravitationalmodulationrdquo Nonlinear Dynamics and Systems Theory vol 14no 4 pp 323ndash334 2014

[11] M Menzinger and R Wolfgang ldquoThe meaning and use of theArrhenius activation energyrdquoAngewandte Chemie InternationalEdition in English vol 8 no 6 pp 438ndash444 1969

[12] B V Novozhilov ldquoThe rate of propagation of the front of anexothermic reaction in a condensed phaserdquo Doklady PhysicalChemistry vol 141 pp 836ndash838 1961

[13] Y B Zeldovich and D A Frank-Kamenetskii ldquoA theory ofthermal propagation of amerdquo Acta Physicochimica USSR vol 9pp 341ndash350 1938

[14] M Garbey A Taık and V Volpert ldquoInfluence of natural con-vection on stability of reaction fronts in liquidsrdquo Quarterly ofApplied Mathematics vol 56 no 1 pp 1ndash35 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

As 120578 rarr +infin

1205791asymp 1205791

10038161003816100381610038161199111=+0

+ (1205971205790

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

)120578 0997888rarr 0 (33)

As 120578 rarr minusinfin

1205791sim 1205791

10038161003816100381610038161199111=minus0

0997888rarr 1 (34)

From (18) we obtain

1205972V0

1205971205782= 0 (35)

Consequently we can conclude that V0(120578) is a linear

function of 120578 and identically constant since the velocity isbounded Equation (29) becomes

V0

10038161003816100381610038161199111=+0

= V0

10038161003816100381610038161199111=minus0

(36)

120597V0

120597120578= 0 (37)

From (36)-(37) we deduce that the first term in theexpression of the velocity V

0is continuous at the front By

substituting (37) into (21) we find

1198751198601205972V1

1205971205782+ 1199050

1205971199010

120597120578= 0 (38)

120597V0119909

120597119909minus120597V1119909

120597120578

1205971205770

120597119909+120597V0119910

120597119910minus120597V1119910

120597120578

1205971205770

120597119910+120597V1119911

120597120578= 0 (39)

Differentiating (39) with respect to 120578 one obtains

1205972V1119909

1205971205782

1205971205770

120597119909+1205972V1119910

1205971205782

1205971205770

120597119910minus1205972V1119911

1205971205782= 0 (40)

As a result (38) is a vectorial equation with three com-ponents We multiply the first component by 120597120577

0120597119909 the

second by 1205971205770120597119910 and the third by minus1 and adding we have

119875119860(1205972V1119909

1205971205782

1205971205770

120597119909+1205972V1119910

1205971205782

1205971205770

120597119910minus1205972V1119911

1205971205782) + 119860

1205971199010

120597120578= 0 (41)

Considering the two equations (38) and (40) we obtain

1205971199010

120597120578= 0 (42)

1205972V1

1205971205782= 0 (43)

From the boundedness of the velocity and (30) the con-tinuity of the first order term of the outer expansion and ofthe first derivative of the zero-order term we can write

120597V0

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

=120597V0

1205971199111

100381610038161003816100381610038161003816100381610038161199111=minus0

V1

10038161003816100381610038161199111=+0

= V1

10038161003816100381610038161199111=minus0

(44)

Differentiating (23) once and (26) twice with respect to 120578and using the three equations (37) (42) and (43) we get

1199050

12059721199011

1205971205782+ 119875119860

1205973V2

1205971205783= 0 (45)

1205973V2119909

1205971205783

1205971205770

120597119909+1205973V2119910

1205971205783

1205971205770

120597119910minus1205973V2119911

1205971205783= 0 (46)

As above we multiply the three components of (45)respectively by 120597120577

0120597119909 120597120577

0120597119910 and minus1 and adding we get

119875119860(1205973V2119909

1205971205783

1205971205770

120597119909+1205973V2119910

1205971205783

1205971205770

120597119910minus1205973V2119911

1205971205783) + 119860

12059721199011

1205971205782= 0 (47)

From the previous equation and using (45)-(46) we have

12059721199011

1205971205782= 0

1205973V2

1205971205783= 0

(48)

Knowing that the velocity is bounded and considering(31) we obtain the jump conditions in the following forms

1205972V0

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=+0

=1205972V0

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=minus0

120597V1

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

=120597V1

1205971199111

100381610038161003816100381610038161003816100381610038161199111=minus0

V2

10038161003816100381610038161199111=+0

= V2

10038161003816100381610038161199111=minus0

(49)

Differentiating (25) twice and (27) three times withrespect to 120578 and taking into account (37) (42) (48) we obtain

1199050

12059731199012

1205971205783+ 119875119860

1205974V3

1205971205784

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) 120574

12059721205791

1205971205782= 0

1205974V3119909

1205971205784

1205971205770

120597119909+1205974V3119910

1205971205784

1205971205770

120597119910minus1205974V3119911

1205971205784= 0

(50)

119875119860(1205974V3119909

1205971205784

1205971205770

120597119909+1205974V3119910

1205971205784

1205971205770

120597119910minus1205974V3119911

1205971205784) + 119860

12059731199012

1205971205783

minus 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905))

12059721205791

1205971205782= 0

(51)

From the last equation and using (50) we can write

11986012059731199012

1205971205783minus 119876 (1 + 120582

1sin (120590

1119905) + 120582

2sin (120590

2119905))

12059721205791

1205971205782

= 0

(52)

1205740

12059721205791

1205971205782=1205974V3

1205971205784 (53)

6 Mathematical Problems in Engineering

where 1205740is defined by

1205740= (minus

1205971205770

120597119909

119877

1198602 minus1205971205770

120597119910

119877

1198602119877

1198602minus119877

119860)

sdot (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905))

(54)

Now integrating (53) with respect to 120578 using the system(32)ndash(34) lead to

1205973V0

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=minus0

minus1205973V0

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=+0

= minus1205740

1205971205790

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

(55)

As a first conclusion the velocity jump conditions acrossthe reaction front are given by (36) (44) (49) and (55)

From (20) (22) (37) we conclude that 0is a monotonic

function satisfying 0 lt 0lt 1 Therefore the reaction is of

zero order and 120601(0) equiv 1 Multiplying (19) by 120597120579

1120597120578 and

integrating the result yield

(1205971205791

120597120578)

210038161003816100381610038161003816100381610038161003816100381610038161003816+infin

minus (1205971205791

120597120578)

210038161003816100381610038161003816100381610038161003816100381610038161003816minusinfin

= 2119860minus1int

1205791|1199111=minus0

minusinfin

exp( 120591

1 + 120591120575) 119889120591

(56)

Subtracting (19) from (20) and integrating the result oneobtains

1205971205791

120597120578

100381610038161003816100381610038161003816100381610038161003816+infin

minus1205971205791

120597120578

100381610038161003816100381610038161003816100381610038161003816minusinfin

= minus119860minus1(1205971205770

120597119905+ 119904) (57)

where 119904 is given by

119904 = V0119909

1205971205770

120597119909+ V0119910

1205971205770

120597119910minus V0119911 (58)

As a second conclusion the temperature jump conditionsacross the reaction front are given by the two last equations(56)-(57) By using the matching conditions above andtruncating the expansion as

120579 asymp 1205790

1205791

10038161003816100381610038161199111=minus0

asymp 119885120579|1199111=+0

120577 asymp 1205770

V asymp V0

(59)

the jump conditions can be written as

(120597120579

1205971199111

)

2100381610038161003816100381610038161003816100381610038161003816+0

minus (120597120579

1205971199111

)

2100381610038161003816100381610038161003816100381610038161003816minus0

= 2119885(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot int

1205791|1199111=minus0

minusinfin

exp ( 120591

119885minus1 + 120591120575) 119889120591

120597120579

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

minus120597120579

1205971199111

100381610038161003816100381610038161003816100381610038161199111=minus0

= minus(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot (120597120577

120597119905+ (V119909

120597120577

120597119909+ V119910

120597120577

120597119910minus V119911)

100381610038161003816100381610038161003816100381610038161199111=+0

)

V119911

10038161003816100381610038161199111=+0

= V119911

10038161003816100381610038161199111=minus0

120597V119911

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

=120597V119911

1205971199111

100381610038161003816100381610038161003816100381610038161199111=minus0

1205972V119911

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=+0

=120597V119911

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=minus0

1205973V119911

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=minus0

minus1205973V119911

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=+0

= minus119877(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot ((1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

minus 1)

sdot (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905))

120597120579

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

(60)

4 The Interface Problem and Perturbation

To study the propagation of polymerization front with aliquid reactant and liquid product the equation of motionhas been considered before and after the reaction zone Thischanges the jump conditions and influences the stabilityconditions of the frontal polymerization process Notice thatin the case of liquid-solid polymerization front the equationof motion is considered after the reaction zone The originalsystem (2) (3) (8) (9) leads to the following interfaceproblem

In the liquid monomer 119911 gt 120577

120597120579

120597119905+ (V sdot nabla) 120579 = Δ120579 (61)

120572 = 0 (62)

Mathematical Problems in Engineering 7

120597V120597119905

+ (V sdot nabla) V

= minusnabla119901 + 119875ΔV

+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579

0) 120574

(63)

div (V) = 0 (64)

In the liquid polymer 119911 lt 120577

120597120579

120597119905+ (V sdot nabla) 120579 = Δ120579 (65)

120572 = 1 (66)

120597V120597119905

+ (V sdot nabla) V

= minusnabla119901 + 119875ΔV

+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579

0) 120574

(67)

div (V) = 0 (68)

At the interface 119911 = 120577

120579|120577minus0 = 120579|120577+0

120597120579

120597119911

10038161003816100381610038161003816100381610038161003816120577=minus0minus120597120579

120597119911

10038161003816100381610038161003816100381610038161003816120577=+0= (1 + (

120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot (120597120577

120597119905+ (V119909

120597120577

120597119909+ V119910

120597120577

120597119910minus V119911)

10038161003816100381610038161003816100381610038161003816120577

)

(120597120579

120597119911)

2100381610038161003816100381610038161003816100381610038161003816120577minus0

minus (120597120579

120597119911)

21003816100381610038161003816100381610038161003816100381610038161205771+0

= minus2119885(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot int

120579|120577

minusinfin

exp ( 120591

119885minus1 + 120575120591) 119889120591

V119911

1003816100381610038161003816120577=minus0 = V119911

1003816100381610038161003816120577=+0

120597V119911

120597119911

10038161003816100381610038161003816100381610038161003816120577=minus0=120597V119911

120597119911

10038161003816100381610038161003816100381610038161003816120577=+0

1205972V119911

1205971199112

100381610038161003816100381610038161003816100381610038161003816120577=minus0

=1205972V119911

1205971199112

100381610038161003816100381610038161003816100381610038161003816120577=+0

1205973V119911

1205971199113

100381610038161003816100381610038161003816100381610038161003816120577=minus0

minus1205973V119911

1205971199113

100381610038161003816100381610038161003816100381610038161003816120577=+0

= minus119877(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot ((1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

minus 1)

sdot (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) (

120597120579

120597119911)

10038161003816100381610038161003816100381610038161003816120577=+0

(69)

with the conditions at infinity

119911 = minusinfin 120579 = 0 V = 0

119911 = +infin 120579 = minus1 V = 0

(70)

5 Stability Analysis

The interface problem analyzed in Section 4 has a travellingwave solution in the following form

(120579 (119909 119910 119911 119905) 120572 (119909 119910 119911 119905) V)

= (120579119904(119911 minus 119906119905) 120572

119904(119911 minus 119906119905) 0)

(120579119904 (119911 minus 119906119905) 120572119904 (119911 minus 119906119905))

=

(0 1) 1199112lt 0

(exp (minus1199061199112) minus 1 0) 119911

2gt 0

(71)

where 1199112= 119911minus119906119905 and 119906 is the speed of the stationary reaction

frontThis solution is a basic and stationary solution of inter-face problem (61)ndash(70) written in the moving coordinatesThen (61) (63) (65) and (67) can be replaced by

120597120579

120597119905+ (V sdot nabla) 120579 = Δ120579 + 119906

120597120579

1205971199112

120597V120597119905

+ (Vnabla) V

= minusnabla119901 + 119875ΔV + 119906120597120579

1205971199112

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) (120579 + 120579

0) 120574

(72)

The other equations remain unchanged To study thestability of reaction front we seek the solution of the problemas follows

120579 = 120579119904+ 120579

119901 = 119901119904+ 119901

V = V119904+ V

(73)

where 120579 119901 and V are respectively small perturbation oftemperature pressure and velocity

8 Mathematical Problems in Engineering

Subtituting (73) into (64) (72) one obtains the first-orderterms as follows

In the liquid monomer 1199112gt 120585

120597120579

120597119905= Δ120579 + 119906

120597120579

1205971199112

minus V1199111205791015840

119904 (74)

120597V120597119905

= minusnabla119901 + 119875ΔV + 119906120597120579

1205971199112

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579120574

(75)

div (V) = 0 (76)

In the liquid polymer 1199112lt 120585

120597120579

120597119905= Δ120579 + 119906

120597120579

1205971199112

(77)

120597V120597119905

= minusnabla119901 + 119875ΔV + 119906120597V1205971199112

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579120574

(78)

div (V) = 0 (79)

where 120585 is defined by 120585 = 120577 minus 119906119905We note

(120579 V119911) =

(1205791 V1199111) for 119911

2lt 120585

(1205792 V1199112) for 119911

2lt 120585

(80)

and consider the perturbation as follows

120579119894= 120579119894(1199112 119905) exp (119895 (119896

1119909 + 1198962119910))

V119911119894= V119911119894(1199112 119905) exp (119895 (119896

1119909 + 1198962119910))

120585 = 1205981(119905) exp (119895 (119896

1119909 + 1198962119910))

(81)

where 119896119894 119894 = 1 2 and 120598

1denote respectively the wave

numbers (in 119909 and 119910 directions) and the amplitude of theperturbation and 119895

2= minus1 Then jump conditions (69) are

linearized by taking into account the fact that

120579|120585=plusmn0 = 120579119904(plusmn0) + 120585120579

1015840

119904(plusmn0) + 120579 (plusmn0)

120597120579

1205971199112

10038161003816100381610038161003816100381610038161003816120585=plusmn0

= 1205791015840

119904(plusmn0) + 120585120579

10158401015840

119904(plusmn0) +

120597120579

1205971199112

100381610038161003816100381610038161003816100381610038161003816120585=plusmn0

(82)

Then the higher-order terms are given as

1205792

100381610038161003816100381610038161199112=0minus 1205791

100381610038161003816100381610038161199112=0= 119906120585 (83)

1205971205792

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

minus1205971205791

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

= minus1199062120585 minus

120597120585

120597119905+ V119911

10038161003816100381610038161199112=0 (84)

1199062120585 +

1205971205792

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

= minus119885

1199061205791

100381610038161003816100381610038161199112=0 (85)

V2119911

10038161003816100381610038161199112=0= V1119911

10038161003816100381610038161199112=0 (86)

120597V1199112

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

=120597V1119911

1205971199112

100381610038161003816100381610038161003816100381610038161199112=0

(87)

1205972V2119911

12059711991122

1003816100381610038161003816100381610038161003816100381610038161199112=0

=1205972V1119911

12059711991122

1003816100381610038161003816100381610038161003816100381610038161199112=0

(88)

1205973V2119911

12059711991132

1003816100381610038161003816100381610038161003816100381610038161199112=0

=1205973V1119911

12059711991132

1003816100381610038161003816100381610038161003816100381610038161199112=0

(89)

Substituting (81) into (83)ndash(89) leads to

1205792(0 119905) minus 120579

1(0 119905) = 119906120598

1(119905)

1205791015840

2(0 119905) minus 120579

1015840

1(0 119905) = minus120598

1 (119905) 1199062minus 1205981015840

1(119905) + V

1 (0 119905)

1205981 (119905) 119906

2+ 1205791015840

2(0 119905) = minus

119885

1199061205791 (0 119905)

V(119894)2(0 119905) = V(119894)

1(0 119905) 119894 = 0 1 2 3

(90)

where

V(119894) =120597119894V120597V119894

1205791015840=120597120579119894

120597119911

1205981015840

1(119905) =

1198891205981(119905)

119889119905

(91)

After simplification (75) and (78) become

120597

120597119905ΔV119911minus 119906

120597

1205971199112

ΔV119911= 119875ΔΔV

119911+ 119876(1 + 120582

1sin (120590

1119905)

+ 1205822sin (120590

2119905) (

1205972

1205971199092+

1205972

1205971199102))120579

(92)

Substituting (71) (87) (88) into (74) (77) (92) andintroducing the variable 119908 = V10158401015840 minus 1198962V where 119896 = radic1198962

1+ 11989622

one obtains two systems of equations as follows

1199112lt 120585

120597119908

120597119905minus 1199061199081015840minus 119875 (119908

10158401015840minus 1198962119908)

= minus1198761198962(1 + 120582

1sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579

119908 = V10158401015840 minus 1198962V

120597120579

120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 0

(93)

Mathematical Problems in Engineering 9

0 5 10 15 20 25 300

50

100

150

200

250

300

350

Rc

1205901

1205902 = radic212059011205902 = radic71205901

1205902 = 0

1205902 = radic371205901

Figure 1 Critical Rayleigh number as a function of 1205901for different

frequencies ratio and for 1205902= 0 120582

1= 1205822= 5 119875 = 05 119885 = 8 and

119896 = 07

1199112gt 120585

120597119908

120597119905minus 1199061199081015840minus 119875 (119908

10158401015840minus 1198962119908)

= minus1198761198962(1 + 120582

1sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579

119908 = V10158401015840 minus 1198962V

120597120579

120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 119906 exp (minus119906119911

2) V

(94)

with the following boundary conditions

1205792(0 119905) minus 120579

1(0 119905) = 119906120598

1(119905) (95)

1205791015840

2(0 119905) minus 120579

1015840

1(0 119905) = minus119906

21205761(119905) minus 120576

1015840

1(119905) + V

1(0 119905) (96)

1205791015840

2(0 119905) +

119885

1199061205791 (0 119905) = minus119906

21205761 (119905) (97)

V(119894)1(0 119905) = V(119894)

2(0 119905) 119894 = 0 1 2 3 (98)

Combining the three boundary conditions (95)ndash(97)yields

1205791015840

1(0 119905) +

119885

1199061205791(0 119905) =

1

119906(1205792(0 119905) minus 120579

1(0 119905))

1015840

119905

minus V1(0 119905)

(99)

6 Numerical Results

The numerical simulations of the problem are performedusing the finite-difference approximation with implicitscheme The algorithm is given in Appendix

Evolution of maximum temperature versus time providesthe onset of stability of the polymerization front The con-vective instability occurs when a jump from bounded tounbounded values of maximum temperature is achieved

Figure 1 shows the critical Rayleigh number as a functionof the frequency 120590

1for different frequencies ratio and for

0 25 50 75 100 125 1500

5

10

15

20

25

Rc

1205822

1205821 = 20

1205821 = 50

1205821 = 80

1205821 = 150

Figure 2 Critical Rayleigh number as a function of 1205822for different

values of 1205821 1205902= radic2120590

1 1205901= 5 119875 = 05 119885 = 8 and 119896 = 07

0 25 50 75 100 125 150 175 2000

50

100

150

200

250

Rc

1205822

1205902 = radic371205901

1205902 = radic1712059011205902 = radic71205901

1205902 = 1radic712059011205902 = 1radic171205901

1205902 = 1radic371205901

Figure 3 Critical Rayleigh number as a function of 1205822for different

frequencies ratio 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and 120590

1= 5

1205902= 0 In the absence of gravitational modulation (120590

1= 1205902=

0) we find the same result (119877119888asymp 27) as in [8 14] It can be

observed that the frequencies ratio of the QP gravitationalmodulation has a significant influence on the convectiveinstability of the reaction front As indicated increasing thefrequencies ratio the stability domain becomes larger Inparticular this can be seen in the range of the frequency1205901belonging to the interval (0 10) For large values of the

frequency 1205901 the frequencies ratio 120590

21205901has no effect on

the convective instability It is also worthy to notice thatthe QP gravitational modulation has a stabilizing effect forsufficiently large frequencies ratio (120590

2= radic37120590

1) comparing

to the periodic modulation case (1205902= 0)

The critical Rayleigh number as a function of the ampli-tude 120582

2is shown in Figure 2 for different values of 120582

1 We can

conclude that for a fixed value of 1205822and increased values of

1205821 the polymerization front loses its stability monotonically

revealing that the stability of the reaction front can becontrolled by acting on the amplitudes of the QPmodulation

Figure 3 illustrates the critical Rayleigh number as afunction of the amplitude 120582

2for different frequencies ratio

It can be observed that increasing the frequencies ratio thestability domain becomes large for certain values of 120582

2 For

large values of 1205822(1205822ge 100) the effect of the frequencies

ratio on the convective instability boundary is insignificant

10 Mathematical Problems in Engineering

0 25 50 75 100 125 150

k = 15

k = 2

k = 1

0

50

100

150

200

250

300

350

Rc

1205822

Figure 4 Critical Rayleigh number as a function of 1205822for different

values of number wave 1205902= radic7120590

1 1205901= 10 119875 = 05 119885 = 8 and

1205821= 5

0 25 50 75 100 125 1500

255075

100125150175200

Rc

1205822

1205902 = radic21205901

1205902 = radic71205901

Figure 5 Critical Rayleigh number as a function of 1205822for two

different frequencies ratios 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and

1205901= 10

Figure 4 shows for a fixed frequencies ratio 12059021205901= radic7

the influence of the wave number on the critical Rayleighnumber versus 120582

2 The plots indicate that for large values of

the wave number the reaction front gains stabilityThe influence of 120582

2on the stability of the reaction front is

given in Figure 5 for two different frequencies ratio From thisfigure one can conclude that an increase of the frequenciesratio has a stabilizing effect

7 Conclusion

In this work we have studied the influence of the QP vibra-tions on the convective instability of liquid-liquid polymer-ization frontWe have used themodel which includes the heatequation the concentration equation and the Navier-Stokesequations under Boussinesq approximation The ZeldovichFrank-Kamenetskii method has been used assuming that thereaction occurs in a narrow zone To obtain the convec-tive instability threshold using the linear stability analysisthe reduced system of equations has been discretized andresolved using the finite-difference method with implicitscheme The results have shown that for fixed values ofamplitudes an increase of the frequencies ratio stabilizes thereaction front and for large values of the frequency 120590

1the

critical Rayleigh number tends to the unmodulated criticalvalue In addition it is observed that for a fixed frequencies

ratio and for a given amplitude 1205822an increase of the ampli-

tude1205821destabilizes the reaction front Also the reaction front

becomes more stable by increasing the wave number

Appendix

The algorithm of the considered problem uses the finite-difference approximation with implicit scheme The velocityis computed from the previous time values of the problemThe discretization in space is given by 119911

119894= 119894ℎ where ℎ is the

space step and 119894 is a positive integerThe discretization in timeis given by 119905

119894= 119894120591 where 120591 is sufficiently small time step The

variables of discretization are 119908119895119894= 119908(119911

119894 119905119895) V119895119894= V(119911

119894 119905119895)

and 120579119895119894= 120579(119911119894 119905119895) One obtains the following

In the liquid monomer 119911 lt 120577

(minus1

ℎ2) 120579119895+1

119894minus1+ (

1

120591+

2

ℎ2+ 1198962+119906

ℎ) 120579119895+1

119894

+ (minus1

ℎ2minus119906

ℎ) 120579119895+1

119894+1=1

120591120579119895

119894

(minus119875

ℎ2)119908119895+1

119894minus1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

119894

+ (minus119875

ℎ2minus119906

ℎ)119908119895+1

119894+1=1

120591119908119895

119894

minus 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895)) 120579119895

119894

(1

ℎ2) V119895+1119894minus1

+ (minus2

ℎ2minus 1198962) V119895+1119894

+ (1

ℎ2) V119895+1119894+1

= 119908119895

119894

(A1)

In the liquid polymer 119911 gt 120577

(minus1

ℎ2)119879119895+1

119894minus1+ (

1

120591+

2

ℎ2+ 1198962+119906

ℎ)119879119895+1

119894

+ (minus1

ℎ2minus119906

ℎ)119879119895+1

119894+1=1

120591119879119895

119894+ 119906 exp (minus119906119911

119894) 119881119895

119894

(minus119875

ℎ2)119882119895+1

119894minus1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119882119895+1

119894

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

119894+1=1

120591119882119895

119894

minus 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895)) 119879119895

119894

(1

ℎ2)119881119895+1

119894minus1+ (minus

2

ℎ2minus 1198962)119881119895+1

119894+ (

1

ℎ2)119881119895+1

119894+1= 119882119895

119894

(A2)

From the jump conditions we have also the followingsystems

(minus1

ℎ2) 120579119895+1

1

+ (1

120591+

2

ℎ2+ 1198962+119906

ℎ+ (

1198881

1198882

)(1

ℎ2+119906

ℎ)) 120579119895+1

0

Mathematical Problems in Engineering 11

= (minus1

120591) 120579119895

0minus (

1

1199061205911198882

)(1

ℎ2+119906

ℎ) (119879119895

0minus 120579119895

0)

minus (V0

1198882

)(1

ℎ2+119906

ℎ)

(minus119875

ℎ2)119908119895+1

1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

0

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

0= (

minus1

120591)119908119895

1

+ 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895))

sdot ((minus1198882

1198881

)1198790minus

1

1199061205911198881

(119879119895

0minus 120579119895

0) minus

V0

1198881

)

(A3)

(1

120591+

2

ℎ2+ 1198962+119906

ℎ+

1198882

ℎ21198881

)119879119895+1

0+ (

minus1

ℎ2minus119906

ℎ)119879119895+1

1

= (minus1

120591)119879119895

0minus 119906 exp (minus119906ℎ) minus ( 1

119906120591ℎ21198881

) (119879119895

0minus 120579119895

0)

minusV0

ℎ21198881

(A4)

(minus119875

ℎ2)119908119895+1

1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

0

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

0= (

minus1

120591)119908119895

0

+ 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895))

sdot ((minus1198881

1198882

)1205790minus

1

1199061205911198882

)(119879119895

0minus 120579119895

0minusV0

1198882

)

(A5)

with

1198881= (

1

ℎminus119885

119906minus

1

119906120591)

1198882= (minus

1

ℎ+

1

119906120591)

(A6)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] B B Khanukaev M A Kozhushner N S Enikolopyan and NM Chechilo ldquoTheory of the propagation of a polymerizationfrontrdquo Combustion Explosion and Shock Waves vol 10 no 1pp 17ndash21 1974

[2] J A Pojman and T-C-M Qui Nonlinear Dynamics with Pol-ymers Fundamentals Methods and Applications John Wiley ampSons 2011

[3] I D Robertson H L Hernandez S R White and J S MooreldquoRapid stiffening of a microfluidic endoskeleton via frontalpolymerizationrdquo ACS Applied Materials and Interfaces vol 6no 21 pp 18469ndash18474 2014

[4] Vi Volpert and Vl Volpert ldquoPropagation of frontal polym-erizationmdashcrystallization wavesrdquo European Journal of AppliedMathematics vol 5 no 2 pp 201ndash215 1994

[5] G Bowden M Garbey V M Ilyashenko et al ldquoEffect of con-vection on a propagating front with a solid product comparisonof theory and experimentsrdquoThe Journal of Physical Chemistry Bvol 101 no 4 pp 678ndash686 1997

[6] M Garbey A Taik and V Volpert ldquoLinear stability analysisof reaction fronts in liquidsrdquo Quarterly of Applied Mathematicsvol 54 no 2 pp 225ndash247 1996

[7] K Allali V Volpert and J A Pojman ldquoInfluence of vibrationson convective instability of polymerization frontsrdquo Journal ofEngineering Mathematics vol 41 no 1 pp 13ndash31 2001

[8] K Allali F Bikany A Taik and V Volpert ldquoInfluence ofvibrations on convective instability of reaction fronts in liquidsrdquoMathematical Modelling of Natural Phenomena vol 5 no 7 pp35ndash41 2010

[9] K Allali M Belhaq and K El Karouni ldquoInfluence of quasi-periodic gravitational modulation on convective instability ofreaction fronts in porous mediardquo Communications in NonlinearScience and Numerical Simulation vol 17 no 4 pp 1588ndash15962012

[10] KAllali S Assiyad andM Belhaq ldquoConvection of polymeriza-tion front with solid product under quasi-periodic gravitationalmodulationrdquo Nonlinear Dynamics and Systems Theory vol 14no 4 pp 323ndash334 2014

[11] M Menzinger and R Wolfgang ldquoThe meaning and use of theArrhenius activation energyrdquoAngewandte Chemie InternationalEdition in English vol 8 no 6 pp 438ndash444 1969

[12] B V Novozhilov ldquoThe rate of propagation of the front of anexothermic reaction in a condensed phaserdquo Doklady PhysicalChemistry vol 141 pp 836ndash838 1961

[13] Y B Zeldovich and D A Frank-Kamenetskii ldquoA theory ofthermal propagation of amerdquo Acta Physicochimica USSR vol 9pp 341ndash350 1938

[14] M Garbey A Taık and V Volpert ldquoInfluence of natural con-vection on stability of reaction fronts in liquidsrdquo Quarterly ofApplied Mathematics vol 56 no 1 pp 1ndash35 1998

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

where 1205740is defined by

1205740= (minus

1205971205770

120597119909

119877

1198602 minus1205971205770

120597119910

119877

1198602119877

1198602minus119877

119860)

sdot (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905))

(54)

Now integrating (53) with respect to 120578 using the system(32)ndash(34) lead to

1205973V0

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=minus0

minus1205973V0

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=+0

= minus1205740

1205971205790

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

(55)

As a first conclusion the velocity jump conditions acrossthe reaction front are given by (36) (44) (49) and (55)

From (20) (22) (37) we conclude that 0is a monotonic

function satisfying 0 lt 0lt 1 Therefore the reaction is of

zero order and 120601(0) equiv 1 Multiplying (19) by 120597120579

1120597120578 and

integrating the result yield

(1205971205791

120597120578)

210038161003816100381610038161003816100381610038161003816100381610038161003816+infin

minus (1205971205791

120597120578)

210038161003816100381610038161003816100381610038161003816100381610038161003816minusinfin

= 2119860minus1int

1205791|1199111=minus0

minusinfin

exp( 120591

1 + 120591120575) 119889120591

(56)

Subtracting (19) from (20) and integrating the result oneobtains

1205971205791

120597120578

100381610038161003816100381610038161003816100381610038161003816+infin

minus1205971205791

120597120578

100381610038161003816100381610038161003816100381610038161003816minusinfin

= minus119860minus1(1205971205770

120597119905+ 119904) (57)

where 119904 is given by

119904 = V0119909

1205971205770

120597119909+ V0119910

1205971205770

120597119910minus V0119911 (58)

As a second conclusion the temperature jump conditionsacross the reaction front are given by the two last equations(56)-(57) By using the matching conditions above andtruncating the expansion as

120579 asymp 1205790

1205791

10038161003816100381610038161199111=minus0

asymp 119885120579|1199111=+0

120577 asymp 1205770

V asymp V0

(59)

the jump conditions can be written as

(120597120579

1205971199111

)

2100381610038161003816100381610038161003816100381610038161003816+0

minus (120597120579

1205971199111

)

2100381610038161003816100381610038161003816100381610038161003816minus0

= 2119885(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot int

1205791|1199111=minus0

minusinfin

exp ( 120591

119885minus1 + 120591120575) 119889120591

120597120579

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

minus120597120579

1205971199111

100381610038161003816100381610038161003816100381610038161199111=minus0

= minus(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot (120597120577

120597119905+ (V119909

120597120577

120597119909+ V119910

120597120577

120597119910minus V119911)

100381610038161003816100381610038161003816100381610038161199111=+0

)

V119911

10038161003816100381610038161199111=+0

= V119911

10038161003816100381610038161199111=minus0

120597V119911

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

=120597V119911

1205971199111

100381610038161003816100381610038161003816100381610038161199111=minus0

1205972V119911

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=+0

=120597V119911

12059711991121

1003816100381610038161003816100381610038161003816100381610038161199111=minus0

1205973V119911

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=minus0

minus1205973V119911

12059711991131

1003816100381610038161003816100381610038161003816100381610038161199111=+0

= minus119877(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot ((1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

minus 1)

sdot (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905))

120597120579

1205971199111

100381610038161003816100381610038161003816100381610038161199111=+0

(60)

4 The Interface Problem and Perturbation

To study the propagation of polymerization front with aliquid reactant and liquid product the equation of motionhas been considered before and after the reaction zone Thischanges the jump conditions and influences the stabilityconditions of the frontal polymerization process Notice thatin the case of liquid-solid polymerization front the equationof motion is considered after the reaction zone The originalsystem (2) (3) (8) (9) leads to the following interfaceproblem

In the liquid monomer 119911 gt 120577

120597120579

120597119905+ (V sdot nabla) 120579 = Δ120579 (61)

120572 = 0 (62)

Mathematical Problems in Engineering 7

120597V120597119905

+ (V sdot nabla) V

= minusnabla119901 + 119875ΔV

+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579

0) 120574

(63)

div (V) = 0 (64)

In the liquid polymer 119911 lt 120577

120597120579

120597119905+ (V sdot nabla) 120579 = Δ120579 (65)

120572 = 1 (66)

120597V120597119905

+ (V sdot nabla) V

= minusnabla119901 + 119875ΔV

+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579

0) 120574

(67)

div (V) = 0 (68)

At the interface 119911 = 120577

120579|120577minus0 = 120579|120577+0

120597120579

120597119911

10038161003816100381610038161003816100381610038161003816120577=minus0minus120597120579

120597119911

10038161003816100381610038161003816100381610038161003816120577=+0= (1 + (

120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot (120597120577

120597119905+ (V119909

120597120577

120597119909+ V119910

120597120577

120597119910minus V119911)

10038161003816100381610038161003816100381610038161003816120577

)

(120597120579

120597119911)

2100381610038161003816100381610038161003816100381610038161003816120577minus0

minus (120597120579

120597119911)

21003816100381610038161003816100381610038161003816100381610038161205771+0

= minus2119885(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot int

120579|120577

minusinfin

exp ( 120591

119885minus1 + 120575120591) 119889120591

V119911

1003816100381610038161003816120577=minus0 = V119911

1003816100381610038161003816120577=+0

120597V119911

120597119911

10038161003816100381610038161003816100381610038161003816120577=minus0=120597V119911

120597119911

10038161003816100381610038161003816100381610038161003816120577=+0

1205972V119911

1205971199112

100381610038161003816100381610038161003816100381610038161003816120577=minus0

=1205972V119911

1205971199112

100381610038161003816100381610038161003816100381610038161003816120577=+0

1205973V119911

1205971199113

100381610038161003816100381610038161003816100381610038161003816120577=minus0

minus1205973V119911

1205971199113

100381610038161003816100381610038161003816100381610038161003816120577=+0

= minus119877(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot ((1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

minus 1)

sdot (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) (

120597120579

120597119911)

10038161003816100381610038161003816100381610038161003816120577=+0

(69)

with the conditions at infinity

119911 = minusinfin 120579 = 0 V = 0

119911 = +infin 120579 = minus1 V = 0

(70)

5 Stability Analysis

The interface problem analyzed in Section 4 has a travellingwave solution in the following form

(120579 (119909 119910 119911 119905) 120572 (119909 119910 119911 119905) V)

= (120579119904(119911 minus 119906119905) 120572

119904(119911 minus 119906119905) 0)

(120579119904 (119911 minus 119906119905) 120572119904 (119911 minus 119906119905))

=

(0 1) 1199112lt 0

(exp (minus1199061199112) minus 1 0) 119911

2gt 0

(71)

where 1199112= 119911minus119906119905 and 119906 is the speed of the stationary reaction

frontThis solution is a basic and stationary solution of inter-face problem (61)ndash(70) written in the moving coordinatesThen (61) (63) (65) and (67) can be replaced by

120597120579

120597119905+ (V sdot nabla) 120579 = Δ120579 + 119906

120597120579

1205971199112

120597V120597119905

+ (Vnabla) V

= minusnabla119901 + 119875ΔV + 119906120597120579

1205971199112

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) (120579 + 120579

0) 120574

(72)

The other equations remain unchanged To study thestability of reaction front we seek the solution of the problemas follows

120579 = 120579119904+ 120579

119901 = 119901119904+ 119901

V = V119904+ V

(73)

where 120579 119901 and V are respectively small perturbation oftemperature pressure and velocity

8 Mathematical Problems in Engineering

Subtituting (73) into (64) (72) one obtains the first-orderterms as follows

In the liquid monomer 1199112gt 120585

120597120579

120597119905= Δ120579 + 119906

120597120579

1205971199112

minus V1199111205791015840

119904 (74)

120597V120597119905

= minusnabla119901 + 119875ΔV + 119906120597120579

1205971199112

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579120574

(75)

div (V) = 0 (76)

In the liquid polymer 1199112lt 120585

120597120579

120597119905= Δ120579 + 119906

120597120579

1205971199112

(77)

120597V120597119905

= minusnabla119901 + 119875ΔV + 119906120597V1205971199112

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579120574

(78)

div (V) = 0 (79)

where 120585 is defined by 120585 = 120577 minus 119906119905We note

(120579 V119911) =

(1205791 V1199111) for 119911

2lt 120585

(1205792 V1199112) for 119911

2lt 120585

(80)

and consider the perturbation as follows

120579119894= 120579119894(1199112 119905) exp (119895 (119896

1119909 + 1198962119910))

V119911119894= V119911119894(1199112 119905) exp (119895 (119896

1119909 + 1198962119910))

120585 = 1205981(119905) exp (119895 (119896

1119909 + 1198962119910))

(81)

where 119896119894 119894 = 1 2 and 120598

1denote respectively the wave

numbers (in 119909 and 119910 directions) and the amplitude of theperturbation and 119895

2= minus1 Then jump conditions (69) are

linearized by taking into account the fact that

120579|120585=plusmn0 = 120579119904(plusmn0) + 120585120579

1015840

119904(plusmn0) + 120579 (plusmn0)

120597120579

1205971199112

10038161003816100381610038161003816100381610038161003816120585=plusmn0

= 1205791015840

119904(plusmn0) + 120585120579

10158401015840

119904(plusmn0) +

120597120579

1205971199112

100381610038161003816100381610038161003816100381610038161003816120585=plusmn0

(82)

Then the higher-order terms are given as

1205792

100381610038161003816100381610038161199112=0minus 1205791

100381610038161003816100381610038161199112=0= 119906120585 (83)

1205971205792

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

minus1205971205791

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

= minus1199062120585 minus

120597120585

120597119905+ V119911

10038161003816100381610038161199112=0 (84)

1199062120585 +

1205971205792

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

= minus119885

1199061205791

100381610038161003816100381610038161199112=0 (85)

V2119911

10038161003816100381610038161199112=0= V1119911

10038161003816100381610038161199112=0 (86)

120597V1199112

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

=120597V1119911

1205971199112

100381610038161003816100381610038161003816100381610038161199112=0

(87)

1205972V2119911

12059711991122

1003816100381610038161003816100381610038161003816100381610038161199112=0

=1205972V1119911

12059711991122

1003816100381610038161003816100381610038161003816100381610038161199112=0

(88)

1205973V2119911

12059711991132

1003816100381610038161003816100381610038161003816100381610038161199112=0

=1205973V1119911

12059711991132

1003816100381610038161003816100381610038161003816100381610038161199112=0

(89)

Substituting (81) into (83)ndash(89) leads to

1205792(0 119905) minus 120579

1(0 119905) = 119906120598

1(119905)

1205791015840

2(0 119905) minus 120579

1015840

1(0 119905) = minus120598

1 (119905) 1199062minus 1205981015840

1(119905) + V

1 (0 119905)

1205981 (119905) 119906

2+ 1205791015840

2(0 119905) = minus

119885

1199061205791 (0 119905)

V(119894)2(0 119905) = V(119894)

1(0 119905) 119894 = 0 1 2 3

(90)

where

V(119894) =120597119894V120597V119894

1205791015840=120597120579119894

120597119911

1205981015840

1(119905) =

1198891205981(119905)

119889119905

(91)

After simplification (75) and (78) become

120597

120597119905ΔV119911minus 119906

120597

1205971199112

ΔV119911= 119875ΔΔV

119911+ 119876(1 + 120582

1sin (120590

1119905)

+ 1205822sin (120590

2119905) (

1205972

1205971199092+

1205972

1205971199102))120579

(92)

Substituting (71) (87) (88) into (74) (77) (92) andintroducing the variable 119908 = V10158401015840 minus 1198962V where 119896 = radic1198962

1+ 11989622

one obtains two systems of equations as follows

1199112lt 120585

120597119908

120597119905minus 1199061199081015840minus 119875 (119908

10158401015840minus 1198962119908)

= minus1198761198962(1 + 120582

1sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579

119908 = V10158401015840 minus 1198962V

120597120579

120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 0

(93)

Mathematical Problems in Engineering 9

0 5 10 15 20 25 300

50

100

150

200

250

300

350

Rc

1205901

1205902 = radic212059011205902 = radic71205901

1205902 = 0

1205902 = radic371205901

Figure 1 Critical Rayleigh number as a function of 1205901for different

frequencies ratio and for 1205902= 0 120582

1= 1205822= 5 119875 = 05 119885 = 8 and

119896 = 07

1199112gt 120585

120597119908

120597119905minus 1199061199081015840minus 119875 (119908

10158401015840minus 1198962119908)

= minus1198761198962(1 + 120582

1sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579

119908 = V10158401015840 minus 1198962V

120597120579

120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 119906 exp (minus119906119911

2) V

(94)

with the following boundary conditions

1205792(0 119905) minus 120579

1(0 119905) = 119906120598

1(119905) (95)

1205791015840

2(0 119905) minus 120579

1015840

1(0 119905) = minus119906

21205761(119905) minus 120576

1015840

1(119905) + V

1(0 119905) (96)

1205791015840

2(0 119905) +

119885

1199061205791 (0 119905) = minus119906

21205761 (119905) (97)

V(119894)1(0 119905) = V(119894)

2(0 119905) 119894 = 0 1 2 3 (98)

Combining the three boundary conditions (95)ndash(97)yields

1205791015840

1(0 119905) +

119885

1199061205791(0 119905) =

1

119906(1205792(0 119905) minus 120579

1(0 119905))

1015840

119905

minus V1(0 119905)

(99)

6 Numerical Results

The numerical simulations of the problem are performedusing the finite-difference approximation with implicitscheme The algorithm is given in Appendix

Evolution of maximum temperature versus time providesthe onset of stability of the polymerization front The con-vective instability occurs when a jump from bounded tounbounded values of maximum temperature is achieved

Figure 1 shows the critical Rayleigh number as a functionof the frequency 120590

1for different frequencies ratio and for

0 25 50 75 100 125 1500

5

10

15

20

25

Rc

1205822

1205821 = 20

1205821 = 50

1205821 = 80

1205821 = 150

Figure 2 Critical Rayleigh number as a function of 1205822for different

values of 1205821 1205902= radic2120590

1 1205901= 5 119875 = 05 119885 = 8 and 119896 = 07

0 25 50 75 100 125 150 175 2000

50

100

150

200

250

Rc

1205822

1205902 = radic371205901

1205902 = radic1712059011205902 = radic71205901

1205902 = 1radic712059011205902 = 1radic171205901

1205902 = 1radic371205901

Figure 3 Critical Rayleigh number as a function of 1205822for different

frequencies ratio 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and 120590

1= 5

1205902= 0 In the absence of gravitational modulation (120590

1= 1205902=

0) we find the same result (119877119888asymp 27) as in [8 14] It can be

observed that the frequencies ratio of the QP gravitationalmodulation has a significant influence on the convectiveinstability of the reaction front As indicated increasing thefrequencies ratio the stability domain becomes larger Inparticular this can be seen in the range of the frequency1205901belonging to the interval (0 10) For large values of the

frequency 1205901 the frequencies ratio 120590

21205901has no effect on

the convective instability It is also worthy to notice thatthe QP gravitational modulation has a stabilizing effect forsufficiently large frequencies ratio (120590

2= radic37120590

1) comparing

to the periodic modulation case (1205902= 0)

The critical Rayleigh number as a function of the ampli-tude 120582

2is shown in Figure 2 for different values of 120582

1 We can

conclude that for a fixed value of 1205822and increased values of

1205821 the polymerization front loses its stability monotonically

revealing that the stability of the reaction front can becontrolled by acting on the amplitudes of the QPmodulation

Figure 3 illustrates the critical Rayleigh number as afunction of the amplitude 120582

2for different frequencies ratio

It can be observed that increasing the frequencies ratio thestability domain becomes large for certain values of 120582

2 For

large values of 1205822(1205822ge 100) the effect of the frequencies

ratio on the convective instability boundary is insignificant

10 Mathematical Problems in Engineering

0 25 50 75 100 125 150

k = 15

k = 2

k = 1

0

50

100

150

200

250

300

350

Rc

1205822

Figure 4 Critical Rayleigh number as a function of 1205822for different

values of number wave 1205902= radic7120590

1 1205901= 10 119875 = 05 119885 = 8 and

1205821= 5

0 25 50 75 100 125 1500

255075

100125150175200

Rc

1205822

1205902 = radic21205901

1205902 = radic71205901

Figure 5 Critical Rayleigh number as a function of 1205822for two

different frequencies ratios 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and

1205901= 10

Figure 4 shows for a fixed frequencies ratio 12059021205901= radic7

the influence of the wave number on the critical Rayleighnumber versus 120582

2 The plots indicate that for large values of

the wave number the reaction front gains stabilityThe influence of 120582

2on the stability of the reaction front is

given in Figure 5 for two different frequencies ratio From thisfigure one can conclude that an increase of the frequenciesratio has a stabilizing effect

7 Conclusion

In this work we have studied the influence of the QP vibra-tions on the convective instability of liquid-liquid polymer-ization frontWe have used themodel which includes the heatequation the concentration equation and the Navier-Stokesequations under Boussinesq approximation The ZeldovichFrank-Kamenetskii method has been used assuming that thereaction occurs in a narrow zone To obtain the convec-tive instability threshold using the linear stability analysisthe reduced system of equations has been discretized andresolved using the finite-difference method with implicitscheme The results have shown that for fixed values ofamplitudes an increase of the frequencies ratio stabilizes thereaction front and for large values of the frequency 120590

1the

critical Rayleigh number tends to the unmodulated criticalvalue In addition it is observed that for a fixed frequencies

ratio and for a given amplitude 1205822an increase of the ampli-

tude1205821destabilizes the reaction front Also the reaction front

becomes more stable by increasing the wave number

Appendix

The algorithm of the considered problem uses the finite-difference approximation with implicit scheme The velocityis computed from the previous time values of the problemThe discretization in space is given by 119911

119894= 119894ℎ where ℎ is the

space step and 119894 is a positive integerThe discretization in timeis given by 119905

119894= 119894120591 where 120591 is sufficiently small time step The

variables of discretization are 119908119895119894= 119908(119911

119894 119905119895) V119895119894= V(119911

119894 119905119895)

and 120579119895119894= 120579(119911119894 119905119895) One obtains the following

In the liquid monomer 119911 lt 120577

(minus1

ℎ2) 120579119895+1

119894minus1+ (

1

120591+

2

ℎ2+ 1198962+119906

ℎ) 120579119895+1

119894

+ (minus1

ℎ2minus119906

ℎ) 120579119895+1

119894+1=1

120591120579119895

119894

(minus119875

ℎ2)119908119895+1

119894minus1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

119894

+ (minus119875

ℎ2minus119906

ℎ)119908119895+1

119894+1=1

120591119908119895

119894

minus 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895)) 120579119895

119894

(1

ℎ2) V119895+1119894minus1

+ (minus2

ℎ2minus 1198962) V119895+1119894

+ (1

ℎ2) V119895+1119894+1

= 119908119895

119894

(A1)

In the liquid polymer 119911 gt 120577

(minus1

ℎ2)119879119895+1

119894minus1+ (

1

120591+

2

ℎ2+ 1198962+119906

ℎ)119879119895+1

119894

+ (minus1

ℎ2minus119906

ℎ)119879119895+1

119894+1=1

120591119879119895

119894+ 119906 exp (minus119906119911

119894) 119881119895

119894

(minus119875

ℎ2)119882119895+1

119894minus1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119882119895+1

119894

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

119894+1=1

120591119882119895

119894

minus 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895)) 119879119895

119894

(1

ℎ2)119881119895+1

119894minus1+ (minus

2

ℎ2minus 1198962)119881119895+1

119894+ (

1

ℎ2)119881119895+1

119894+1= 119882119895

119894

(A2)

From the jump conditions we have also the followingsystems

(minus1

ℎ2) 120579119895+1

1

+ (1

120591+

2

ℎ2+ 1198962+119906

ℎ+ (

1198881

1198882

)(1

ℎ2+119906

ℎ)) 120579119895+1

0

Mathematical Problems in Engineering 11

= (minus1

120591) 120579119895

0minus (

1

1199061205911198882

)(1

ℎ2+119906

ℎ) (119879119895

0minus 120579119895

0)

minus (V0

1198882

)(1

ℎ2+119906

ℎ)

(minus119875

ℎ2)119908119895+1

1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

0

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

0= (

minus1

120591)119908119895

1

+ 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895))

sdot ((minus1198882

1198881

)1198790minus

1

1199061205911198881

(119879119895

0minus 120579119895

0) minus

V0

1198881

)

(A3)

(1

120591+

2

ℎ2+ 1198962+119906

ℎ+

1198882

ℎ21198881

)119879119895+1

0+ (

minus1

ℎ2minus119906

ℎ)119879119895+1

1

= (minus1

120591)119879119895

0minus 119906 exp (minus119906ℎ) minus ( 1

119906120591ℎ21198881

) (119879119895

0minus 120579119895

0)

minusV0

ℎ21198881

(A4)

(minus119875

ℎ2)119908119895+1

1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

0

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

0= (

minus1

120591)119908119895

0

+ 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895))

sdot ((minus1198881

1198882

)1205790minus

1

1199061205911198882

)(119879119895

0minus 120579119895

0minusV0

1198882

)

(A5)

with

1198881= (

1

ℎminus119885

119906minus

1

119906120591)

1198882= (minus

1

ℎ+

1

119906120591)

(A6)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] B B Khanukaev M A Kozhushner N S Enikolopyan and NM Chechilo ldquoTheory of the propagation of a polymerizationfrontrdquo Combustion Explosion and Shock Waves vol 10 no 1pp 17ndash21 1974

[2] J A Pojman and T-C-M Qui Nonlinear Dynamics with Pol-ymers Fundamentals Methods and Applications John Wiley ampSons 2011

[3] I D Robertson H L Hernandez S R White and J S MooreldquoRapid stiffening of a microfluidic endoskeleton via frontalpolymerizationrdquo ACS Applied Materials and Interfaces vol 6no 21 pp 18469ndash18474 2014

[4] Vi Volpert and Vl Volpert ldquoPropagation of frontal polym-erizationmdashcrystallization wavesrdquo European Journal of AppliedMathematics vol 5 no 2 pp 201ndash215 1994

[5] G Bowden M Garbey V M Ilyashenko et al ldquoEffect of con-vection on a propagating front with a solid product comparisonof theory and experimentsrdquoThe Journal of Physical Chemistry Bvol 101 no 4 pp 678ndash686 1997

[6] M Garbey A Taik and V Volpert ldquoLinear stability analysisof reaction fronts in liquidsrdquo Quarterly of Applied Mathematicsvol 54 no 2 pp 225ndash247 1996

[7] K Allali V Volpert and J A Pojman ldquoInfluence of vibrationson convective instability of polymerization frontsrdquo Journal ofEngineering Mathematics vol 41 no 1 pp 13ndash31 2001

[8] K Allali F Bikany A Taik and V Volpert ldquoInfluence ofvibrations on convective instability of reaction fronts in liquidsrdquoMathematical Modelling of Natural Phenomena vol 5 no 7 pp35ndash41 2010

[9] K Allali M Belhaq and K El Karouni ldquoInfluence of quasi-periodic gravitational modulation on convective instability ofreaction fronts in porous mediardquo Communications in NonlinearScience and Numerical Simulation vol 17 no 4 pp 1588ndash15962012

[10] KAllali S Assiyad andM Belhaq ldquoConvection of polymeriza-tion front with solid product under quasi-periodic gravitationalmodulationrdquo Nonlinear Dynamics and Systems Theory vol 14no 4 pp 323ndash334 2014

[11] M Menzinger and R Wolfgang ldquoThe meaning and use of theArrhenius activation energyrdquoAngewandte Chemie InternationalEdition in English vol 8 no 6 pp 438ndash444 1969

[12] B V Novozhilov ldquoThe rate of propagation of the front of anexothermic reaction in a condensed phaserdquo Doklady PhysicalChemistry vol 141 pp 836ndash838 1961

[13] Y B Zeldovich and D A Frank-Kamenetskii ldquoA theory ofthermal propagation of amerdquo Acta Physicochimica USSR vol 9pp 341ndash350 1938

[14] M Garbey A Taık and V Volpert ldquoInfluence of natural con-vection on stability of reaction fronts in liquidsrdquo Quarterly ofApplied Mathematics vol 56 no 1 pp 1ndash35 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

120597V120597119905

+ (V sdot nabla) V

= minusnabla119901 + 119875ΔV

+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579

0) 120574

(63)

div (V) = 0 (64)

In the liquid polymer 119911 lt 120577

120597120579

120597119905+ (V sdot nabla) 120579 = Δ120579 (65)

120572 = 1 (66)

120597V120597119905

+ (V sdot nabla) V

= minusnabla119901 + 119875ΔV

+ 119876 (1 + 1205821sin1205901119905 + 1205822sin1205902119905) (120579 + 120579

0) 120574

(67)

div (V) = 0 (68)

At the interface 119911 = 120577

120579|120577minus0 = 120579|120577+0

120597120579

120597119911

10038161003816100381610038161003816100381610038161003816120577=minus0minus120597120579

120597119911

10038161003816100381610038161003816100381610038161003816120577=+0= (1 + (

120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot (120597120577

120597119905+ (V119909

120597120577

120597119909+ V119910

120597120577

120597119910minus V119911)

10038161003816100381610038161003816100381610038161003816120577

)

(120597120579

120597119911)

2100381610038161003816100381610038161003816100381610038161003816120577minus0

minus (120597120579

120597119911)

21003816100381610038161003816100381610038161003816100381610038161205771+0

= minus2119885(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot int

120579|120577

minusinfin

exp ( 120591

119885minus1 + 120575120591) 119889120591

V119911

1003816100381610038161003816120577=minus0 = V119911

1003816100381610038161003816120577=+0

120597V119911

120597119911

10038161003816100381610038161003816100381610038161003816120577=minus0=120597V119911

120597119911

10038161003816100381610038161003816100381610038161003816120577=+0

1205972V119911

1205971199112

100381610038161003816100381610038161003816100381610038161003816120577=minus0

=1205972V119911

1205971199112

100381610038161003816100381610038161003816100381610038161003816120577=+0

1205973V119911

1205971199113

100381610038161003816100381610038161003816100381610038161003816120577=minus0

minus1205973V119911

1205971199113

100381610038161003816100381610038161003816100381610038161003816120577=+0

= minus119877(1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

sdot ((1 + (120597120577

120597119909)

2

+ (120597120577

120597119910)

2

)

minus1

minus 1)

sdot (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) (

120597120579

120597119911)

10038161003816100381610038161003816100381610038161003816120577=+0

(69)

with the conditions at infinity

119911 = minusinfin 120579 = 0 V = 0

119911 = +infin 120579 = minus1 V = 0

(70)

5 Stability Analysis

The interface problem analyzed in Section 4 has a travellingwave solution in the following form

(120579 (119909 119910 119911 119905) 120572 (119909 119910 119911 119905) V)

= (120579119904(119911 minus 119906119905) 120572

119904(119911 minus 119906119905) 0)

(120579119904 (119911 minus 119906119905) 120572119904 (119911 minus 119906119905))

=

(0 1) 1199112lt 0

(exp (minus1199061199112) minus 1 0) 119911

2gt 0

(71)

where 1199112= 119911minus119906119905 and 119906 is the speed of the stationary reaction

frontThis solution is a basic and stationary solution of inter-face problem (61)ndash(70) written in the moving coordinatesThen (61) (63) (65) and (67) can be replaced by

120597120579

120597119905+ (V sdot nabla) 120579 = Δ120579 + 119906

120597120579

1205971199112

120597V120597119905

+ (Vnabla) V

= minusnabla119901 + 119875ΔV + 119906120597120579

1205971199112

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) (120579 + 120579

0) 120574

(72)

The other equations remain unchanged To study thestability of reaction front we seek the solution of the problemas follows

120579 = 120579119904+ 120579

119901 = 119901119904+ 119901

V = V119904+ V

(73)

where 120579 119901 and V are respectively small perturbation oftemperature pressure and velocity

8 Mathematical Problems in Engineering

Subtituting (73) into (64) (72) one obtains the first-orderterms as follows

In the liquid monomer 1199112gt 120585

120597120579

120597119905= Δ120579 + 119906

120597120579

1205971199112

minus V1199111205791015840

119904 (74)

120597V120597119905

= minusnabla119901 + 119875ΔV + 119906120597120579

1205971199112

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579120574

(75)

div (V) = 0 (76)

In the liquid polymer 1199112lt 120585

120597120579

120597119905= Δ120579 + 119906

120597120579

1205971199112

(77)

120597V120597119905

= minusnabla119901 + 119875ΔV + 119906120597V1205971199112

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579120574

(78)

div (V) = 0 (79)

where 120585 is defined by 120585 = 120577 minus 119906119905We note

(120579 V119911) =

(1205791 V1199111) for 119911

2lt 120585

(1205792 V1199112) for 119911

2lt 120585

(80)

and consider the perturbation as follows

120579119894= 120579119894(1199112 119905) exp (119895 (119896

1119909 + 1198962119910))

V119911119894= V119911119894(1199112 119905) exp (119895 (119896

1119909 + 1198962119910))

120585 = 1205981(119905) exp (119895 (119896

1119909 + 1198962119910))

(81)

where 119896119894 119894 = 1 2 and 120598

1denote respectively the wave

numbers (in 119909 and 119910 directions) and the amplitude of theperturbation and 119895

2= minus1 Then jump conditions (69) are

linearized by taking into account the fact that

120579|120585=plusmn0 = 120579119904(plusmn0) + 120585120579

1015840

119904(plusmn0) + 120579 (plusmn0)

120597120579

1205971199112

10038161003816100381610038161003816100381610038161003816120585=plusmn0

= 1205791015840

119904(plusmn0) + 120585120579

10158401015840

119904(plusmn0) +

120597120579

1205971199112

100381610038161003816100381610038161003816100381610038161003816120585=plusmn0

(82)

Then the higher-order terms are given as

1205792

100381610038161003816100381610038161199112=0minus 1205791

100381610038161003816100381610038161199112=0= 119906120585 (83)

1205971205792

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

minus1205971205791

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

= minus1199062120585 minus

120597120585

120597119905+ V119911

10038161003816100381610038161199112=0 (84)

1199062120585 +

1205971205792

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

= minus119885

1199061205791

100381610038161003816100381610038161199112=0 (85)

V2119911

10038161003816100381610038161199112=0= V1119911

10038161003816100381610038161199112=0 (86)

120597V1199112

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

=120597V1119911

1205971199112

100381610038161003816100381610038161003816100381610038161199112=0

(87)

1205972V2119911

12059711991122

1003816100381610038161003816100381610038161003816100381610038161199112=0

=1205972V1119911

12059711991122

1003816100381610038161003816100381610038161003816100381610038161199112=0

(88)

1205973V2119911

12059711991132

1003816100381610038161003816100381610038161003816100381610038161199112=0

=1205973V1119911

12059711991132

1003816100381610038161003816100381610038161003816100381610038161199112=0

(89)

Substituting (81) into (83)ndash(89) leads to

1205792(0 119905) minus 120579

1(0 119905) = 119906120598

1(119905)

1205791015840

2(0 119905) minus 120579

1015840

1(0 119905) = minus120598

1 (119905) 1199062minus 1205981015840

1(119905) + V

1 (0 119905)

1205981 (119905) 119906

2+ 1205791015840

2(0 119905) = minus

119885

1199061205791 (0 119905)

V(119894)2(0 119905) = V(119894)

1(0 119905) 119894 = 0 1 2 3

(90)

where

V(119894) =120597119894V120597V119894

1205791015840=120597120579119894

120597119911

1205981015840

1(119905) =

1198891205981(119905)

119889119905

(91)

After simplification (75) and (78) become

120597

120597119905ΔV119911minus 119906

120597

1205971199112

ΔV119911= 119875ΔΔV

119911+ 119876(1 + 120582

1sin (120590

1119905)

+ 1205822sin (120590

2119905) (

1205972

1205971199092+

1205972

1205971199102))120579

(92)

Substituting (71) (87) (88) into (74) (77) (92) andintroducing the variable 119908 = V10158401015840 minus 1198962V where 119896 = radic1198962

1+ 11989622

one obtains two systems of equations as follows

1199112lt 120585

120597119908

120597119905minus 1199061199081015840minus 119875 (119908

10158401015840minus 1198962119908)

= minus1198761198962(1 + 120582

1sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579

119908 = V10158401015840 minus 1198962V

120597120579

120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 0

(93)

Mathematical Problems in Engineering 9

0 5 10 15 20 25 300

50

100

150

200

250

300

350

Rc

1205901

1205902 = radic212059011205902 = radic71205901

1205902 = 0

1205902 = radic371205901

Figure 1 Critical Rayleigh number as a function of 1205901for different

frequencies ratio and for 1205902= 0 120582

1= 1205822= 5 119875 = 05 119885 = 8 and

119896 = 07

1199112gt 120585

120597119908

120597119905minus 1199061199081015840minus 119875 (119908

10158401015840minus 1198962119908)

= minus1198761198962(1 + 120582

1sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579

119908 = V10158401015840 minus 1198962V

120597120579

120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 119906 exp (minus119906119911

2) V

(94)

with the following boundary conditions

1205792(0 119905) minus 120579

1(0 119905) = 119906120598

1(119905) (95)

1205791015840

2(0 119905) minus 120579

1015840

1(0 119905) = minus119906

21205761(119905) minus 120576

1015840

1(119905) + V

1(0 119905) (96)

1205791015840

2(0 119905) +

119885

1199061205791 (0 119905) = minus119906

21205761 (119905) (97)

V(119894)1(0 119905) = V(119894)

2(0 119905) 119894 = 0 1 2 3 (98)

Combining the three boundary conditions (95)ndash(97)yields

1205791015840

1(0 119905) +

119885

1199061205791(0 119905) =

1

119906(1205792(0 119905) minus 120579

1(0 119905))

1015840

119905

minus V1(0 119905)

(99)

6 Numerical Results

The numerical simulations of the problem are performedusing the finite-difference approximation with implicitscheme The algorithm is given in Appendix

Evolution of maximum temperature versus time providesthe onset of stability of the polymerization front The con-vective instability occurs when a jump from bounded tounbounded values of maximum temperature is achieved

Figure 1 shows the critical Rayleigh number as a functionof the frequency 120590

1for different frequencies ratio and for

0 25 50 75 100 125 1500

5

10

15

20

25

Rc

1205822

1205821 = 20

1205821 = 50

1205821 = 80

1205821 = 150

Figure 2 Critical Rayleigh number as a function of 1205822for different

values of 1205821 1205902= radic2120590

1 1205901= 5 119875 = 05 119885 = 8 and 119896 = 07

0 25 50 75 100 125 150 175 2000

50

100

150

200

250

Rc

1205822

1205902 = radic371205901

1205902 = radic1712059011205902 = radic71205901

1205902 = 1radic712059011205902 = 1radic171205901

1205902 = 1radic371205901

Figure 3 Critical Rayleigh number as a function of 1205822for different

frequencies ratio 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and 120590

1= 5

1205902= 0 In the absence of gravitational modulation (120590

1= 1205902=

0) we find the same result (119877119888asymp 27) as in [8 14] It can be

observed that the frequencies ratio of the QP gravitationalmodulation has a significant influence on the convectiveinstability of the reaction front As indicated increasing thefrequencies ratio the stability domain becomes larger Inparticular this can be seen in the range of the frequency1205901belonging to the interval (0 10) For large values of the

frequency 1205901 the frequencies ratio 120590

21205901has no effect on

the convective instability It is also worthy to notice thatthe QP gravitational modulation has a stabilizing effect forsufficiently large frequencies ratio (120590

2= radic37120590

1) comparing

to the periodic modulation case (1205902= 0)

The critical Rayleigh number as a function of the ampli-tude 120582

2is shown in Figure 2 for different values of 120582

1 We can

conclude that for a fixed value of 1205822and increased values of

1205821 the polymerization front loses its stability monotonically

revealing that the stability of the reaction front can becontrolled by acting on the amplitudes of the QPmodulation

Figure 3 illustrates the critical Rayleigh number as afunction of the amplitude 120582

2for different frequencies ratio

It can be observed that increasing the frequencies ratio thestability domain becomes large for certain values of 120582

2 For

large values of 1205822(1205822ge 100) the effect of the frequencies

ratio on the convective instability boundary is insignificant

10 Mathematical Problems in Engineering

0 25 50 75 100 125 150

k = 15

k = 2

k = 1

0

50

100

150

200

250

300

350

Rc

1205822

Figure 4 Critical Rayleigh number as a function of 1205822for different

values of number wave 1205902= radic7120590

1 1205901= 10 119875 = 05 119885 = 8 and

1205821= 5

0 25 50 75 100 125 1500

255075

100125150175200

Rc

1205822

1205902 = radic21205901

1205902 = radic71205901

Figure 5 Critical Rayleigh number as a function of 1205822for two

different frequencies ratios 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and

1205901= 10

Figure 4 shows for a fixed frequencies ratio 12059021205901= radic7

the influence of the wave number on the critical Rayleighnumber versus 120582

2 The plots indicate that for large values of

the wave number the reaction front gains stabilityThe influence of 120582

2on the stability of the reaction front is

given in Figure 5 for two different frequencies ratio From thisfigure one can conclude that an increase of the frequenciesratio has a stabilizing effect

7 Conclusion

In this work we have studied the influence of the QP vibra-tions on the convective instability of liquid-liquid polymer-ization frontWe have used themodel which includes the heatequation the concentration equation and the Navier-Stokesequations under Boussinesq approximation The ZeldovichFrank-Kamenetskii method has been used assuming that thereaction occurs in a narrow zone To obtain the convec-tive instability threshold using the linear stability analysisthe reduced system of equations has been discretized andresolved using the finite-difference method with implicitscheme The results have shown that for fixed values ofamplitudes an increase of the frequencies ratio stabilizes thereaction front and for large values of the frequency 120590

1the

critical Rayleigh number tends to the unmodulated criticalvalue In addition it is observed that for a fixed frequencies

ratio and for a given amplitude 1205822an increase of the ampli-

tude1205821destabilizes the reaction front Also the reaction front

becomes more stable by increasing the wave number

Appendix

The algorithm of the considered problem uses the finite-difference approximation with implicit scheme The velocityis computed from the previous time values of the problemThe discretization in space is given by 119911

119894= 119894ℎ where ℎ is the

space step and 119894 is a positive integerThe discretization in timeis given by 119905

119894= 119894120591 where 120591 is sufficiently small time step The

variables of discretization are 119908119895119894= 119908(119911

119894 119905119895) V119895119894= V(119911

119894 119905119895)

and 120579119895119894= 120579(119911119894 119905119895) One obtains the following

In the liquid monomer 119911 lt 120577

(minus1

ℎ2) 120579119895+1

119894minus1+ (

1

120591+

2

ℎ2+ 1198962+119906

ℎ) 120579119895+1

119894

+ (minus1

ℎ2minus119906

ℎ) 120579119895+1

119894+1=1

120591120579119895

119894

(minus119875

ℎ2)119908119895+1

119894minus1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

119894

+ (minus119875

ℎ2minus119906

ℎ)119908119895+1

119894+1=1

120591119908119895

119894

minus 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895)) 120579119895

119894

(1

ℎ2) V119895+1119894minus1

+ (minus2

ℎ2minus 1198962) V119895+1119894

+ (1

ℎ2) V119895+1119894+1

= 119908119895

119894

(A1)

In the liquid polymer 119911 gt 120577

(minus1

ℎ2)119879119895+1

119894minus1+ (

1

120591+

2

ℎ2+ 1198962+119906

ℎ)119879119895+1

119894

+ (minus1

ℎ2minus119906

ℎ)119879119895+1

119894+1=1

120591119879119895

119894+ 119906 exp (minus119906119911

119894) 119881119895

119894

(minus119875

ℎ2)119882119895+1

119894minus1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119882119895+1

119894

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

119894+1=1

120591119882119895

119894

minus 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895)) 119879119895

119894

(1

ℎ2)119881119895+1

119894minus1+ (minus

2

ℎ2minus 1198962)119881119895+1

119894+ (

1

ℎ2)119881119895+1

119894+1= 119882119895

119894

(A2)

From the jump conditions we have also the followingsystems

(minus1

ℎ2) 120579119895+1

1

+ (1

120591+

2

ℎ2+ 1198962+119906

ℎ+ (

1198881

1198882

)(1

ℎ2+119906

ℎ)) 120579119895+1

0

Mathematical Problems in Engineering 11

= (minus1

120591) 120579119895

0minus (

1

1199061205911198882

)(1

ℎ2+119906

ℎ) (119879119895

0minus 120579119895

0)

minus (V0

1198882

)(1

ℎ2+119906

ℎ)

(minus119875

ℎ2)119908119895+1

1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

0

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

0= (

minus1

120591)119908119895

1

+ 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895))

sdot ((minus1198882

1198881

)1198790minus

1

1199061205911198881

(119879119895

0minus 120579119895

0) minus

V0

1198881

)

(A3)

(1

120591+

2

ℎ2+ 1198962+119906

ℎ+

1198882

ℎ21198881

)119879119895+1

0+ (

minus1

ℎ2minus119906

ℎ)119879119895+1

1

= (minus1

120591)119879119895

0minus 119906 exp (minus119906ℎ) minus ( 1

119906120591ℎ21198881

) (119879119895

0minus 120579119895

0)

minusV0

ℎ21198881

(A4)

(minus119875

ℎ2)119908119895+1

1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

0

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

0= (

minus1

120591)119908119895

0

+ 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895))

sdot ((minus1198881

1198882

)1205790minus

1

1199061205911198882

)(119879119895

0minus 120579119895

0minusV0

1198882

)

(A5)

with

1198881= (

1

ℎminus119885

119906minus

1

119906120591)

1198882= (minus

1

ℎ+

1

119906120591)

(A6)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] B B Khanukaev M A Kozhushner N S Enikolopyan and NM Chechilo ldquoTheory of the propagation of a polymerizationfrontrdquo Combustion Explosion and Shock Waves vol 10 no 1pp 17ndash21 1974

[2] J A Pojman and T-C-M Qui Nonlinear Dynamics with Pol-ymers Fundamentals Methods and Applications John Wiley ampSons 2011

[3] I D Robertson H L Hernandez S R White and J S MooreldquoRapid stiffening of a microfluidic endoskeleton via frontalpolymerizationrdquo ACS Applied Materials and Interfaces vol 6no 21 pp 18469ndash18474 2014

[4] Vi Volpert and Vl Volpert ldquoPropagation of frontal polym-erizationmdashcrystallization wavesrdquo European Journal of AppliedMathematics vol 5 no 2 pp 201ndash215 1994

[5] G Bowden M Garbey V M Ilyashenko et al ldquoEffect of con-vection on a propagating front with a solid product comparisonof theory and experimentsrdquoThe Journal of Physical Chemistry Bvol 101 no 4 pp 678ndash686 1997

[6] M Garbey A Taik and V Volpert ldquoLinear stability analysisof reaction fronts in liquidsrdquo Quarterly of Applied Mathematicsvol 54 no 2 pp 225ndash247 1996

[7] K Allali V Volpert and J A Pojman ldquoInfluence of vibrationson convective instability of polymerization frontsrdquo Journal ofEngineering Mathematics vol 41 no 1 pp 13ndash31 2001

[8] K Allali F Bikany A Taik and V Volpert ldquoInfluence ofvibrations on convective instability of reaction fronts in liquidsrdquoMathematical Modelling of Natural Phenomena vol 5 no 7 pp35ndash41 2010

[9] K Allali M Belhaq and K El Karouni ldquoInfluence of quasi-periodic gravitational modulation on convective instability ofreaction fronts in porous mediardquo Communications in NonlinearScience and Numerical Simulation vol 17 no 4 pp 1588ndash15962012

[10] KAllali S Assiyad andM Belhaq ldquoConvection of polymeriza-tion front with solid product under quasi-periodic gravitationalmodulationrdquo Nonlinear Dynamics and Systems Theory vol 14no 4 pp 323ndash334 2014

[11] M Menzinger and R Wolfgang ldquoThe meaning and use of theArrhenius activation energyrdquoAngewandte Chemie InternationalEdition in English vol 8 no 6 pp 438ndash444 1969

[12] B V Novozhilov ldquoThe rate of propagation of the front of anexothermic reaction in a condensed phaserdquo Doklady PhysicalChemistry vol 141 pp 836ndash838 1961

[13] Y B Zeldovich and D A Frank-Kamenetskii ldquoA theory ofthermal propagation of amerdquo Acta Physicochimica USSR vol 9pp 341ndash350 1938

[14] M Garbey A Taık and V Volpert ldquoInfluence of natural con-vection on stability of reaction fronts in liquidsrdquo Quarterly ofApplied Mathematics vol 56 no 1 pp 1ndash35 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Subtituting (73) into (64) (72) one obtains the first-orderterms as follows

In the liquid monomer 1199112gt 120585

120597120579

120597119905= Δ120579 + 119906

120597120579

1205971199112

minus V1199111205791015840

119904 (74)

120597V120597119905

= minusnabla119901 + 119875ΔV + 119906120597120579

1205971199112

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579120574

(75)

div (V) = 0 (76)

In the liquid polymer 1199112lt 120585

120597120579

120597119905= Δ120579 + 119906

120597120579

1205971199112

(77)

120597V120597119905

= minusnabla119901 + 119875ΔV + 119906120597V1205971199112

+ 119876 (1 + 1205821sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579120574

(78)

div (V) = 0 (79)

where 120585 is defined by 120585 = 120577 minus 119906119905We note

(120579 V119911) =

(1205791 V1199111) for 119911

2lt 120585

(1205792 V1199112) for 119911

2lt 120585

(80)

and consider the perturbation as follows

120579119894= 120579119894(1199112 119905) exp (119895 (119896

1119909 + 1198962119910))

V119911119894= V119911119894(1199112 119905) exp (119895 (119896

1119909 + 1198962119910))

120585 = 1205981(119905) exp (119895 (119896

1119909 + 1198962119910))

(81)

where 119896119894 119894 = 1 2 and 120598

1denote respectively the wave

numbers (in 119909 and 119910 directions) and the amplitude of theperturbation and 119895

2= minus1 Then jump conditions (69) are

linearized by taking into account the fact that

120579|120585=plusmn0 = 120579119904(plusmn0) + 120585120579

1015840

119904(plusmn0) + 120579 (plusmn0)

120597120579

1205971199112

10038161003816100381610038161003816100381610038161003816120585=plusmn0

= 1205791015840

119904(plusmn0) + 120585120579

10158401015840

119904(plusmn0) +

120597120579

1205971199112

100381610038161003816100381610038161003816100381610038161003816120585=plusmn0

(82)

Then the higher-order terms are given as

1205792

100381610038161003816100381610038161199112=0minus 1205791

100381610038161003816100381610038161199112=0= 119906120585 (83)

1205971205792

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

minus1205971205791

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

= minus1199062120585 minus

120597120585

120597119905+ V119911

10038161003816100381610038161199112=0 (84)

1199062120585 +

1205971205792

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

= minus119885

1199061205791

100381610038161003816100381610038161199112=0 (85)

V2119911

10038161003816100381610038161199112=0= V1119911

10038161003816100381610038161199112=0 (86)

120597V1199112

1205971199112

1003816100381610038161003816100381610038161003816100381610038161199112=0

=120597V1119911

1205971199112

100381610038161003816100381610038161003816100381610038161199112=0

(87)

1205972V2119911

12059711991122

1003816100381610038161003816100381610038161003816100381610038161199112=0

=1205972V1119911

12059711991122

1003816100381610038161003816100381610038161003816100381610038161199112=0

(88)

1205973V2119911

12059711991132

1003816100381610038161003816100381610038161003816100381610038161199112=0

=1205973V1119911

12059711991132

1003816100381610038161003816100381610038161003816100381610038161199112=0

(89)

Substituting (81) into (83)ndash(89) leads to

1205792(0 119905) minus 120579

1(0 119905) = 119906120598

1(119905)

1205791015840

2(0 119905) minus 120579

1015840

1(0 119905) = minus120598

1 (119905) 1199062minus 1205981015840

1(119905) + V

1 (0 119905)

1205981 (119905) 119906

2+ 1205791015840

2(0 119905) = minus

119885

1199061205791 (0 119905)

V(119894)2(0 119905) = V(119894)

1(0 119905) 119894 = 0 1 2 3

(90)

where

V(119894) =120597119894V120597V119894

1205791015840=120597120579119894

120597119911

1205981015840

1(119905) =

1198891205981(119905)

119889119905

(91)

After simplification (75) and (78) become

120597

120597119905ΔV119911minus 119906

120597

1205971199112

ΔV119911= 119875ΔΔV

119911+ 119876(1 + 120582

1sin (120590

1119905)

+ 1205822sin (120590

2119905) (

1205972

1205971199092+

1205972

1205971199102))120579

(92)

Substituting (71) (87) (88) into (74) (77) (92) andintroducing the variable 119908 = V10158401015840 minus 1198962V where 119896 = radic1198962

1+ 11989622

one obtains two systems of equations as follows

1199112lt 120585

120597119908

120597119905minus 1199061199081015840minus 119875 (119908

10158401015840minus 1198962119908)

= minus1198761198962(1 + 120582

1sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579

119908 = V10158401015840 minus 1198962V

120597120579

120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 0

(93)

Mathematical Problems in Engineering 9

0 5 10 15 20 25 300

50

100

150

200

250

300

350

Rc

1205901

1205902 = radic212059011205902 = radic71205901

1205902 = 0

1205902 = radic371205901

Figure 1 Critical Rayleigh number as a function of 1205901for different

frequencies ratio and for 1205902= 0 120582

1= 1205822= 5 119875 = 05 119885 = 8 and

119896 = 07

1199112gt 120585

120597119908

120597119905minus 1199061199081015840minus 119875 (119908

10158401015840minus 1198962119908)

= minus1198761198962(1 + 120582

1sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579

119908 = V10158401015840 minus 1198962V

120597120579

120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 119906 exp (minus119906119911

2) V

(94)

with the following boundary conditions

1205792(0 119905) minus 120579

1(0 119905) = 119906120598

1(119905) (95)

1205791015840

2(0 119905) minus 120579

1015840

1(0 119905) = minus119906

21205761(119905) minus 120576

1015840

1(119905) + V

1(0 119905) (96)

1205791015840

2(0 119905) +

119885

1199061205791 (0 119905) = minus119906

21205761 (119905) (97)

V(119894)1(0 119905) = V(119894)

2(0 119905) 119894 = 0 1 2 3 (98)

Combining the three boundary conditions (95)ndash(97)yields

1205791015840

1(0 119905) +

119885

1199061205791(0 119905) =

1

119906(1205792(0 119905) minus 120579

1(0 119905))

1015840

119905

minus V1(0 119905)

(99)

6 Numerical Results

The numerical simulations of the problem are performedusing the finite-difference approximation with implicitscheme The algorithm is given in Appendix

Evolution of maximum temperature versus time providesthe onset of stability of the polymerization front The con-vective instability occurs when a jump from bounded tounbounded values of maximum temperature is achieved

Figure 1 shows the critical Rayleigh number as a functionof the frequency 120590

1for different frequencies ratio and for

0 25 50 75 100 125 1500

5

10

15

20

25

Rc

1205822

1205821 = 20

1205821 = 50

1205821 = 80

1205821 = 150

Figure 2 Critical Rayleigh number as a function of 1205822for different

values of 1205821 1205902= radic2120590

1 1205901= 5 119875 = 05 119885 = 8 and 119896 = 07

0 25 50 75 100 125 150 175 2000

50

100

150

200

250

Rc

1205822

1205902 = radic371205901

1205902 = radic1712059011205902 = radic71205901

1205902 = 1radic712059011205902 = 1radic171205901

1205902 = 1radic371205901

Figure 3 Critical Rayleigh number as a function of 1205822for different

frequencies ratio 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and 120590

1= 5

1205902= 0 In the absence of gravitational modulation (120590

1= 1205902=

0) we find the same result (119877119888asymp 27) as in [8 14] It can be

observed that the frequencies ratio of the QP gravitationalmodulation has a significant influence on the convectiveinstability of the reaction front As indicated increasing thefrequencies ratio the stability domain becomes larger Inparticular this can be seen in the range of the frequency1205901belonging to the interval (0 10) For large values of the

frequency 1205901 the frequencies ratio 120590

21205901has no effect on

the convective instability It is also worthy to notice thatthe QP gravitational modulation has a stabilizing effect forsufficiently large frequencies ratio (120590

2= radic37120590

1) comparing

to the periodic modulation case (1205902= 0)

The critical Rayleigh number as a function of the ampli-tude 120582

2is shown in Figure 2 for different values of 120582

1 We can

conclude that for a fixed value of 1205822and increased values of

1205821 the polymerization front loses its stability monotonically

revealing that the stability of the reaction front can becontrolled by acting on the amplitudes of the QPmodulation

Figure 3 illustrates the critical Rayleigh number as afunction of the amplitude 120582

2for different frequencies ratio

It can be observed that increasing the frequencies ratio thestability domain becomes large for certain values of 120582

2 For

large values of 1205822(1205822ge 100) the effect of the frequencies

ratio on the convective instability boundary is insignificant

10 Mathematical Problems in Engineering

0 25 50 75 100 125 150

k = 15

k = 2

k = 1

0

50

100

150

200

250

300

350

Rc

1205822

Figure 4 Critical Rayleigh number as a function of 1205822for different

values of number wave 1205902= radic7120590

1 1205901= 10 119875 = 05 119885 = 8 and

1205821= 5

0 25 50 75 100 125 1500

255075

100125150175200

Rc

1205822

1205902 = radic21205901

1205902 = radic71205901

Figure 5 Critical Rayleigh number as a function of 1205822for two

different frequencies ratios 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and

1205901= 10

Figure 4 shows for a fixed frequencies ratio 12059021205901= radic7

the influence of the wave number on the critical Rayleighnumber versus 120582

2 The plots indicate that for large values of

the wave number the reaction front gains stabilityThe influence of 120582

2on the stability of the reaction front is

given in Figure 5 for two different frequencies ratio From thisfigure one can conclude that an increase of the frequenciesratio has a stabilizing effect

7 Conclusion

In this work we have studied the influence of the QP vibra-tions on the convective instability of liquid-liquid polymer-ization frontWe have used themodel which includes the heatequation the concentration equation and the Navier-Stokesequations under Boussinesq approximation The ZeldovichFrank-Kamenetskii method has been used assuming that thereaction occurs in a narrow zone To obtain the convec-tive instability threshold using the linear stability analysisthe reduced system of equations has been discretized andresolved using the finite-difference method with implicitscheme The results have shown that for fixed values ofamplitudes an increase of the frequencies ratio stabilizes thereaction front and for large values of the frequency 120590

1the

critical Rayleigh number tends to the unmodulated criticalvalue In addition it is observed that for a fixed frequencies

ratio and for a given amplitude 1205822an increase of the ampli-

tude1205821destabilizes the reaction front Also the reaction front

becomes more stable by increasing the wave number

Appendix

The algorithm of the considered problem uses the finite-difference approximation with implicit scheme The velocityis computed from the previous time values of the problemThe discretization in space is given by 119911

119894= 119894ℎ where ℎ is the

space step and 119894 is a positive integerThe discretization in timeis given by 119905

119894= 119894120591 where 120591 is sufficiently small time step The

variables of discretization are 119908119895119894= 119908(119911

119894 119905119895) V119895119894= V(119911

119894 119905119895)

and 120579119895119894= 120579(119911119894 119905119895) One obtains the following

In the liquid monomer 119911 lt 120577

(minus1

ℎ2) 120579119895+1

119894minus1+ (

1

120591+

2

ℎ2+ 1198962+119906

ℎ) 120579119895+1

119894

+ (minus1

ℎ2minus119906

ℎ) 120579119895+1

119894+1=1

120591120579119895

119894

(minus119875

ℎ2)119908119895+1

119894minus1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

119894

+ (minus119875

ℎ2minus119906

ℎ)119908119895+1

119894+1=1

120591119908119895

119894

minus 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895)) 120579119895

119894

(1

ℎ2) V119895+1119894minus1

+ (minus2

ℎ2minus 1198962) V119895+1119894

+ (1

ℎ2) V119895+1119894+1

= 119908119895

119894

(A1)

In the liquid polymer 119911 gt 120577

(minus1

ℎ2)119879119895+1

119894minus1+ (

1

120591+

2

ℎ2+ 1198962+119906

ℎ)119879119895+1

119894

+ (minus1

ℎ2minus119906

ℎ)119879119895+1

119894+1=1

120591119879119895

119894+ 119906 exp (minus119906119911

119894) 119881119895

119894

(minus119875

ℎ2)119882119895+1

119894minus1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119882119895+1

119894

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

119894+1=1

120591119882119895

119894

minus 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895)) 119879119895

119894

(1

ℎ2)119881119895+1

119894minus1+ (minus

2

ℎ2minus 1198962)119881119895+1

119894+ (

1

ℎ2)119881119895+1

119894+1= 119882119895

119894

(A2)

From the jump conditions we have also the followingsystems

(minus1

ℎ2) 120579119895+1

1

+ (1

120591+

2

ℎ2+ 1198962+119906

ℎ+ (

1198881

1198882

)(1

ℎ2+119906

ℎ)) 120579119895+1

0

Mathematical Problems in Engineering 11

= (minus1

120591) 120579119895

0minus (

1

1199061205911198882

)(1

ℎ2+119906

ℎ) (119879119895

0minus 120579119895

0)

minus (V0

1198882

)(1

ℎ2+119906

ℎ)

(minus119875

ℎ2)119908119895+1

1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

0

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

0= (

minus1

120591)119908119895

1

+ 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895))

sdot ((minus1198882

1198881

)1198790minus

1

1199061205911198881

(119879119895

0minus 120579119895

0) minus

V0

1198881

)

(A3)

(1

120591+

2

ℎ2+ 1198962+119906

ℎ+

1198882

ℎ21198881

)119879119895+1

0+ (

minus1

ℎ2minus119906

ℎ)119879119895+1

1

= (minus1

120591)119879119895

0minus 119906 exp (minus119906ℎ) minus ( 1

119906120591ℎ21198881

) (119879119895

0minus 120579119895

0)

minusV0

ℎ21198881

(A4)

(minus119875

ℎ2)119908119895+1

1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

0

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

0= (

minus1

120591)119908119895

0

+ 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895))

sdot ((minus1198881

1198882

)1205790minus

1

1199061205911198882

)(119879119895

0minus 120579119895

0minusV0

1198882

)

(A5)

with

1198881= (

1

ℎminus119885

119906minus

1

119906120591)

1198882= (minus

1

ℎ+

1

119906120591)

(A6)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] B B Khanukaev M A Kozhushner N S Enikolopyan and NM Chechilo ldquoTheory of the propagation of a polymerizationfrontrdquo Combustion Explosion and Shock Waves vol 10 no 1pp 17ndash21 1974

[2] J A Pojman and T-C-M Qui Nonlinear Dynamics with Pol-ymers Fundamentals Methods and Applications John Wiley ampSons 2011

[3] I D Robertson H L Hernandez S R White and J S MooreldquoRapid stiffening of a microfluidic endoskeleton via frontalpolymerizationrdquo ACS Applied Materials and Interfaces vol 6no 21 pp 18469ndash18474 2014

[4] Vi Volpert and Vl Volpert ldquoPropagation of frontal polym-erizationmdashcrystallization wavesrdquo European Journal of AppliedMathematics vol 5 no 2 pp 201ndash215 1994

[5] G Bowden M Garbey V M Ilyashenko et al ldquoEffect of con-vection on a propagating front with a solid product comparisonof theory and experimentsrdquoThe Journal of Physical Chemistry Bvol 101 no 4 pp 678ndash686 1997

[6] M Garbey A Taik and V Volpert ldquoLinear stability analysisof reaction fronts in liquidsrdquo Quarterly of Applied Mathematicsvol 54 no 2 pp 225ndash247 1996

[7] K Allali V Volpert and J A Pojman ldquoInfluence of vibrationson convective instability of polymerization frontsrdquo Journal ofEngineering Mathematics vol 41 no 1 pp 13ndash31 2001

[8] K Allali F Bikany A Taik and V Volpert ldquoInfluence ofvibrations on convective instability of reaction fronts in liquidsrdquoMathematical Modelling of Natural Phenomena vol 5 no 7 pp35ndash41 2010

[9] K Allali M Belhaq and K El Karouni ldquoInfluence of quasi-periodic gravitational modulation on convective instability ofreaction fronts in porous mediardquo Communications in NonlinearScience and Numerical Simulation vol 17 no 4 pp 1588ndash15962012

[10] KAllali S Assiyad andM Belhaq ldquoConvection of polymeriza-tion front with solid product under quasi-periodic gravitationalmodulationrdquo Nonlinear Dynamics and Systems Theory vol 14no 4 pp 323ndash334 2014

[11] M Menzinger and R Wolfgang ldquoThe meaning and use of theArrhenius activation energyrdquoAngewandte Chemie InternationalEdition in English vol 8 no 6 pp 438ndash444 1969

[12] B V Novozhilov ldquoThe rate of propagation of the front of anexothermic reaction in a condensed phaserdquo Doklady PhysicalChemistry vol 141 pp 836ndash838 1961

[13] Y B Zeldovich and D A Frank-Kamenetskii ldquoA theory ofthermal propagation of amerdquo Acta Physicochimica USSR vol 9pp 341ndash350 1938

[14] M Garbey A Taık and V Volpert ldquoInfluence of natural con-vection on stability of reaction fronts in liquidsrdquo Quarterly ofApplied Mathematics vol 56 no 1 pp 1ndash35 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0 5 10 15 20 25 300

50

100

150

200

250

300

350

Rc

1205901

1205902 = radic212059011205902 = radic71205901

1205902 = 0

1205902 = radic371205901

Figure 1 Critical Rayleigh number as a function of 1205901for different

frequencies ratio and for 1205902= 0 120582

1= 1205822= 5 119875 = 05 119885 = 8 and

119896 = 07

1199112gt 120585

120597119908

120597119905minus 1199061199081015840minus 119875 (119908

10158401015840minus 1198962119908)

= minus1198761198962(1 + 120582

1sin (120590

1119905) + 120582

2sin (120590

2119905)) 120579

119908 = V10158401015840 minus 1198962V

120597120579

120597119905minus 12057910158401015840minus 1199061205791015840+ 1198962120579 = 119906 exp (minus119906119911

2) V

(94)

with the following boundary conditions

1205792(0 119905) minus 120579

1(0 119905) = 119906120598

1(119905) (95)

1205791015840

2(0 119905) minus 120579

1015840

1(0 119905) = minus119906

21205761(119905) minus 120576

1015840

1(119905) + V

1(0 119905) (96)

1205791015840

2(0 119905) +

119885

1199061205791 (0 119905) = minus119906

21205761 (119905) (97)

V(119894)1(0 119905) = V(119894)

2(0 119905) 119894 = 0 1 2 3 (98)

Combining the three boundary conditions (95)ndash(97)yields

1205791015840

1(0 119905) +

119885

1199061205791(0 119905) =

1

119906(1205792(0 119905) minus 120579

1(0 119905))

1015840

119905

minus V1(0 119905)

(99)

6 Numerical Results

The numerical simulations of the problem are performedusing the finite-difference approximation with implicitscheme The algorithm is given in Appendix

Evolution of maximum temperature versus time providesthe onset of stability of the polymerization front The con-vective instability occurs when a jump from bounded tounbounded values of maximum temperature is achieved

Figure 1 shows the critical Rayleigh number as a functionof the frequency 120590

1for different frequencies ratio and for

0 25 50 75 100 125 1500

5

10

15

20

25

Rc

1205822

1205821 = 20

1205821 = 50

1205821 = 80

1205821 = 150

Figure 2 Critical Rayleigh number as a function of 1205822for different

values of 1205821 1205902= radic2120590

1 1205901= 5 119875 = 05 119885 = 8 and 119896 = 07

0 25 50 75 100 125 150 175 2000

50

100

150

200

250

Rc

1205822

1205902 = radic371205901

1205902 = radic1712059011205902 = radic71205901

1205902 = 1radic712059011205902 = 1radic171205901

1205902 = 1radic371205901

Figure 3 Critical Rayleigh number as a function of 1205822for different

frequencies ratio 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and 120590

1= 5

1205902= 0 In the absence of gravitational modulation (120590

1= 1205902=

0) we find the same result (119877119888asymp 27) as in [8 14] It can be

observed that the frequencies ratio of the QP gravitationalmodulation has a significant influence on the convectiveinstability of the reaction front As indicated increasing thefrequencies ratio the stability domain becomes larger Inparticular this can be seen in the range of the frequency1205901belonging to the interval (0 10) For large values of the

frequency 1205901 the frequencies ratio 120590

21205901has no effect on

the convective instability It is also worthy to notice thatthe QP gravitational modulation has a stabilizing effect forsufficiently large frequencies ratio (120590

2= radic37120590

1) comparing

to the periodic modulation case (1205902= 0)

The critical Rayleigh number as a function of the ampli-tude 120582

2is shown in Figure 2 for different values of 120582

1 We can

conclude that for a fixed value of 1205822and increased values of

1205821 the polymerization front loses its stability monotonically

revealing that the stability of the reaction front can becontrolled by acting on the amplitudes of the QPmodulation

Figure 3 illustrates the critical Rayleigh number as afunction of the amplitude 120582

2for different frequencies ratio

It can be observed that increasing the frequencies ratio thestability domain becomes large for certain values of 120582

2 For

large values of 1205822(1205822ge 100) the effect of the frequencies

ratio on the convective instability boundary is insignificant

10 Mathematical Problems in Engineering

0 25 50 75 100 125 150

k = 15

k = 2

k = 1

0

50

100

150

200

250

300

350

Rc

1205822

Figure 4 Critical Rayleigh number as a function of 1205822for different

values of number wave 1205902= radic7120590

1 1205901= 10 119875 = 05 119885 = 8 and

1205821= 5

0 25 50 75 100 125 1500

255075

100125150175200

Rc

1205822

1205902 = radic21205901

1205902 = radic71205901

Figure 5 Critical Rayleigh number as a function of 1205822for two

different frequencies ratios 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and

1205901= 10

Figure 4 shows for a fixed frequencies ratio 12059021205901= radic7

the influence of the wave number on the critical Rayleighnumber versus 120582

2 The plots indicate that for large values of

the wave number the reaction front gains stabilityThe influence of 120582

2on the stability of the reaction front is

given in Figure 5 for two different frequencies ratio From thisfigure one can conclude that an increase of the frequenciesratio has a stabilizing effect

7 Conclusion

In this work we have studied the influence of the QP vibra-tions on the convective instability of liquid-liquid polymer-ization frontWe have used themodel which includes the heatequation the concentration equation and the Navier-Stokesequations under Boussinesq approximation The ZeldovichFrank-Kamenetskii method has been used assuming that thereaction occurs in a narrow zone To obtain the convec-tive instability threshold using the linear stability analysisthe reduced system of equations has been discretized andresolved using the finite-difference method with implicitscheme The results have shown that for fixed values ofamplitudes an increase of the frequencies ratio stabilizes thereaction front and for large values of the frequency 120590

1the

critical Rayleigh number tends to the unmodulated criticalvalue In addition it is observed that for a fixed frequencies

ratio and for a given amplitude 1205822an increase of the ampli-

tude1205821destabilizes the reaction front Also the reaction front

becomes more stable by increasing the wave number

Appendix

The algorithm of the considered problem uses the finite-difference approximation with implicit scheme The velocityis computed from the previous time values of the problemThe discretization in space is given by 119911

119894= 119894ℎ where ℎ is the

space step and 119894 is a positive integerThe discretization in timeis given by 119905

119894= 119894120591 where 120591 is sufficiently small time step The

variables of discretization are 119908119895119894= 119908(119911

119894 119905119895) V119895119894= V(119911

119894 119905119895)

and 120579119895119894= 120579(119911119894 119905119895) One obtains the following

In the liquid monomer 119911 lt 120577

(minus1

ℎ2) 120579119895+1

119894minus1+ (

1

120591+

2

ℎ2+ 1198962+119906

ℎ) 120579119895+1

119894

+ (minus1

ℎ2minus119906

ℎ) 120579119895+1

119894+1=1

120591120579119895

119894

(minus119875

ℎ2)119908119895+1

119894minus1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

119894

+ (minus119875

ℎ2minus119906

ℎ)119908119895+1

119894+1=1

120591119908119895

119894

minus 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895)) 120579119895

119894

(1

ℎ2) V119895+1119894minus1

+ (minus2

ℎ2minus 1198962) V119895+1119894

+ (1

ℎ2) V119895+1119894+1

= 119908119895

119894

(A1)

In the liquid polymer 119911 gt 120577

(minus1

ℎ2)119879119895+1

119894minus1+ (

1

120591+

2

ℎ2+ 1198962+119906

ℎ)119879119895+1

119894

+ (minus1

ℎ2minus119906

ℎ)119879119895+1

119894+1=1

120591119879119895

119894+ 119906 exp (minus119906119911

119894) 119881119895

119894

(minus119875

ℎ2)119882119895+1

119894minus1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119882119895+1

119894

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

119894+1=1

120591119882119895

119894

minus 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895)) 119879119895

119894

(1

ℎ2)119881119895+1

119894minus1+ (minus

2

ℎ2minus 1198962)119881119895+1

119894+ (

1

ℎ2)119881119895+1

119894+1= 119882119895

119894

(A2)

From the jump conditions we have also the followingsystems

(minus1

ℎ2) 120579119895+1

1

+ (1

120591+

2

ℎ2+ 1198962+119906

ℎ+ (

1198881

1198882

)(1

ℎ2+119906

ℎ)) 120579119895+1

0

Mathematical Problems in Engineering 11

= (minus1

120591) 120579119895

0minus (

1

1199061205911198882

)(1

ℎ2+119906

ℎ) (119879119895

0minus 120579119895

0)

minus (V0

1198882

)(1

ℎ2+119906

ℎ)

(minus119875

ℎ2)119908119895+1

1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

0

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

0= (

minus1

120591)119908119895

1

+ 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895))

sdot ((minus1198882

1198881

)1198790minus

1

1199061205911198881

(119879119895

0minus 120579119895

0) minus

V0

1198881

)

(A3)

(1

120591+

2

ℎ2+ 1198962+119906

ℎ+

1198882

ℎ21198881

)119879119895+1

0+ (

minus1

ℎ2minus119906

ℎ)119879119895+1

1

= (minus1

120591)119879119895

0minus 119906 exp (minus119906ℎ) minus ( 1

119906120591ℎ21198881

) (119879119895

0minus 120579119895

0)

minusV0

ℎ21198881

(A4)

(minus119875

ℎ2)119908119895+1

1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

0

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

0= (

minus1

120591)119908119895

0

+ 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895))

sdot ((minus1198881

1198882

)1205790minus

1

1199061205911198882

)(119879119895

0minus 120579119895

0minusV0

1198882

)

(A5)

with

1198881= (

1

ℎminus119885

119906minus

1

119906120591)

1198882= (minus

1

ℎ+

1

119906120591)

(A6)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] B B Khanukaev M A Kozhushner N S Enikolopyan and NM Chechilo ldquoTheory of the propagation of a polymerizationfrontrdquo Combustion Explosion and Shock Waves vol 10 no 1pp 17ndash21 1974

[2] J A Pojman and T-C-M Qui Nonlinear Dynamics with Pol-ymers Fundamentals Methods and Applications John Wiley ampSons 2011

[3] I D Robertson H L Hernandez S R White and J S MooreldquoRapid stiffening of a microfluidic endoskeleton via frontalpolymerizationrdquo ACS Applied Materials and Interfaces vol 6no 21 pp 18469ndash18474 2014

[4] Vi Volpert and Vl Volpert ldquoPropagation of frontal polym-erizationmdashcrystallization wavesrdquo European Journal of AppliedMathematics vol 5 no 2 pp 201ndash215 1994

[5] G Bowden M Garbey V M Ilyashenko et al ldquoEffect of con-vection on a propagating front with a solid product comparisonof theory and experimentsrdquoThe Journal of Physical Chemistry Bvol 101 no 4 pp 678ndash686 1997

[6] M Garbey A Taik and V Volpert ldquoLinear stability analysisof reaction fronts in liquidsrdquo Quarterly of Applied Mathematicsvol 54 no 2 pp 225ndash247 1996

[7] K Allali V Volpert and J A Pojman ldquoInfluence of vibrationson convective instability of polymerization frontsrdquo Journal ofEngineering Mathematics vol 41 no 1 pp 13ndash31 2001

[8] K Allali F Bikany A Taik and V Volpert ldquoInfluence ofvibrations on convective instability of reaction fronts in liquidsrdquoMathematical Modelling of Natural Phenomena vol 5 no 7 pp35ndash41 2010

[9] K Allali M Belhaq and K El Karouni ldquoInfluence of quasi-periodic gravitational modulation on convective instability ofreaction fronts in porous mediardquo Communications in NonlinearScience and Numerical Simulation vol 17 no 4 pp 1588ndash15962012

[10] KAllali S Assiyad andM Belhaq ldquoConvection of polymeriza-tion front with solid product under quasi-periodic gravitationalmodulationrdquo Nonlinear Dynamics and Systems Theory vol 14no 4 pp 323ndash334 2014

[11] M Menzinger and R Wolfgang ldquoThe meaning and use of theArrhenius activation energyrdquoAngewandte Chemie InternationalEdition in English vol 8 no 6 pp 438ndash444 1969

[12] B V Novozhilov ldquoThe rate of propagation of the front of anexothermic reaction in a condensed phaserdquo Doklady PhysicalChemistry vol 141 pp 836ndash838 1961

[13] Y B Zeldovich and D A Frank-Kamenetskii ldquoA theory ofthermal propagation of amerdquo Acta Physicochimica USSR vol 9pp 341ndash350 1938

[14] M Garbey A Taık and V Volpert ldquoInfluence of natural con-vection on stability of reaction fronts in liquidsrdquo Quarterly ofApplied Mathematics vol 56 no 1 pp 1ndash35 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

0 25 50 75 100 125 150

k = 15

k = 2

k = 1

0

50

100

150

200

250

300

350

Rc

1205822

Figure 4 Critical Rayleigh number as a function of 1205822for different

values of number wave 1205902= radic7120590

1 1205901= 10 119875 = 05 119885 = 8 and

1205821= 5

0 25 50 75 100 125 1500

255075

100125150175200

Rc

1205822

1205902 = radic21205901

1205902 = radic71205901

Figure 5 Critical Rayleigh number as a function of 1205822for two

different frequencies ratios 119896 = 07 1205821= 5 119875 = 05 119885 = 8 and

1205901= 10

Figure 4 shows for a fixed frequencies ratio 12059021205901= radic7

the influence of the wave number on the critical Rayleighnumber versus 120582

2 The plots indicate that for large values of

the wave number the reaction front gains stabilityThe influence of 120582

2on the stability of the reaction front is

given in Figure 5 for two different frequencies ratio From thisfigure one can conclude that an increase of the frequenciesratio has a stabilizing effect

7 Conclusion

In this work we have studied the influence of the QP vibra-tions on the convective instability of liquid-liquid polymer-ization frontWe have used themodel which includes the heatequation the concentration equation and the Navier-Stokesequations under Boussinesq approximation The ZeldovichFrank-Kamenetskii method has been used assuming that thereaction occurs in a narrow zone To obtain the convec-tive instability threshold using the linear stability analysisthe reduced system of equations has been discretized andresolved using the finite-difference method with implicitscheme The results have shown that for fixed values ofamplitudes an increase of the frequencies ratio stabilizes thereaction front and for large values of the frequency 120590

1the

critical Rayleigh number tends to the unmodulated criticalvalue In addition it is observed that for a fixed frequencies

ratio and for a given amplitude 1205822an increase of the ampli-

tude1205821destabilizes the reaction front Also the reaction front

becomes more stable by increasing the wave number

Appendix

The algorithm of the considered problem uses the finite-difference approximation with implicit scheme The velocityis computed from the previous time values of the problemThe discretization in space is given by 119911

119894= 119894ℎ where ℎ is the

space step and 119894 is a positive integerThe discretization in timeis given by 119905

119894= 119894120591 where 120591 is sufficiently small time step The

variables of discretization are 119908119895119894= 119908(119911

119894 119905119895) V119895119894= V(119911

119894 119905119895)

and 120579119895119894= 120579(119911119894 119905119895) One obtains the following

In the liquid monomer 119911 lt 120577

(minus1

ℎ2) 120579119895+1

119894minus1+ (

1

120591+

2

ℎ2+ 1198962+119906

ℎ) 120579119895+1

119894

+ (minus1

ℎ2minus119906

ℎ) 120579119895+1

119894+1=1

120591120579119895

119894

(minus119875

ℎ2)119908119895+1

119894minus1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

119894

+ (minus119875

ℎ2minus119906

ℎ)119908119895+1

119894+1=1

120591119908119895

119894

minus 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895)) 120579119895

119894

(1

ℎ2) V119895+1119894minus1

+ (minus2

ℎ2minus 1198962) V119895+1119894

+ (1

ℎ2) V119895+1119894+1

= 119908119895

119894

(A1)

In the liquid polymer 119911 gt 120577

(minus1

ℎ2)119879119895+1

119894minus1+ (

1

120591+

2

ℎ2+ 1198962+119906

ℎ)119879119895+1

119894

+ (minus1

ℎ2minus119906

ℎ)119879119895+1

119894+1=1

120591119879119895

119894+ 119906 exp (minus119906119911

119894) 119881119895

119894

(minus119875

ℎ2)119882119895+1

119894minus1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119882119895+1

119894

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

119894+1=1

120591119882119895

119894

minus 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895)) 119879119895

119894

(1

ℎ2)119881119895+1

119894minus1+ (minus

2

ℎ2minus 1198962)119881119895+1

119894+ (

1

ℎ2)119881119895+1

119894+1= 119882119895

119894

(A2)

From the jump conditions we have also the followingsystems

(minus1

ℎ2) 120579119895+1

1

+ (1

120591+

2

ℎ2+ 1198962+119906

ℎ+ (

1198881

1198882

)(1

ℎ2+119906

ℎ)) 120579119895+1

0

Mathematical Problems in Engineering 11

= (minus1

120591) 120579119895

0minus (

1

1199061205911198882

)(1

ℎ2+119906

ℎ) (119879119895

0minus 120579119895

0)

minus (V0

1198882

)(1

ℎ2+119906

ℎ)

(minus119875

ℎ2)119908119895+1

1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

0

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

0= (

minus1

120591)119908119895

1

+ 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895))

sdot ((minus1198882

1198881

)1198790minus

1

1199061205911198881

(119879119895

0minus 120579119895

0) minus

V0

1198881

)

(A3)

(1

120591+

2

ℎ2+ 1198962+119906

ℎ+

1198882

ℎ21198881

)119879119895+1

0+ (

minus1

ℎ2minus119906

ℎ)119879119895+1

1

= (minus1

120591)119879119895

0minus 119906 exp (minus119906ℎ) minus ( 1

119906120591ℎ21198881

) (119879119895

0minus 120579119895

0)

minusV0

ℎ21198881

(A4)

(minus119875

ℎ2)119908119895+1

1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

0

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

0= (

minus1

120591)119908119895

0

+ 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895))

sdot ((minus1198881

1198882

)1205790minus

1

1199061205911198882

)(119879119895

0minus 120579119895

0minusV0

1198882

)

(A5)

with

1198881= (

1

ℎminus119885

119906minus

1

119906120591)

1198882= (minus

1

ℎ+

1

119906120591)

(A6)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] B B Khanukaev M A Kozhushner N S Enikolopyan and NM Chechilo ldquoTheory of the propagation of a polymerizationfrontrdquo Combustion Explosion and Shock Waves vol 10 no 1pp 17ndash21 1974

[2] J A Pojman and T-C-M Qui Nonlinear Dynamics with Pol-ymers Fundamentals Methods and Applications John Wiley ampSons 2011

[3] I D Robertson H L Hernandez S R White and J S MooreldquoRapid stiffening of a microfluidic endoskeleton via frontalpolymerizationrdquo ACS Applied Materials and Interfaces vol 6no 21 pp 18469ndash18474 2014

[4] Vi Volpert and Vl Volpert ldquoPropagation of frontal polym-erizationmdashcrystallization wavesrdquo European Journal of AppliedMathematics vol 5 no 2 pp 201ndash215 1994

[5] G Bowden M Garbey V M Ilyashenko et al ldquoEffect of con-vection on a propagating front with a solid product comparisonof theory and experimentsrdquoThe Journal of Physical Chemistry Bvol 101 no 4 pp 678ndash686 1997

[6] M Garbey A Taik and V Volpert ldquoLinear stability analysisof reaction fronts in liquidsrdquo Quarterly of Applied Mathematicsvol 54 no 2 pp 225ndash247 1996

[7] K Allali V Volpert and J A Pojman ldquoInfluence of vibrationson convective instability of polymerization frontsrdquo Journal ofEngineering Mathematics vol 41 no 1 pp 13ndash31 2001

[8] K Allali F Bikany A Taik and V Volpert ldquoInfluence ofvibrations on convective instability of reaction fronts in liquidsrdquoMathematical Modelling of Natural Phenomena vol 5 no 7 pp35ndash41 2010

[9] K Allali M Belhaq and K El Karouni ldquoInfluence of quasi-periodic gravitational modulation on convective instability ofreaction fronts in porous mediardquo Communications in NonlinearScience and Numerical Simulation vol 17 no 4 pp 1588ndash15962012

[10] KAllali S Assiyad andM Belhaq ldquoConvection of polymeriza-tion front with solid product under quasi-periodic gravitationalmodulationrdquo Nonlinear Dynamics and Systems Theory vol 14no 4 pp 323ndash334 2014

[11] M Menzinger and R Wolfgang ldquoThe meaning and use of theArrhenius activation energyrdquoAngewandte Chemie InternationalEdition in English vol 8 no 6 pp 438ndash444 1969

[12] B V Novozhilov ldquoThe rate of propagation of the front of anexothermic reaction in a condensed phaserdquo Doklady PhysicalChemistry vol 141 pp 836ndash838 1961

[13] Y B Zeldovich and D A Frank-Kamenetskii ldquoA theory ofthermal propagation of amerdquo Acta Physicochimica USSR vol 9pp 341ndash350 1938

[14] M Garbey A Taık and V Volpert ldquoInfluence of natural con-vection on stability of reaction fronts in liquidsrdquo Quarterly ofApplied Mathematics vol 56 no 1 pp 1ndash35 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

= (minus1

120591) 120579119895

0minus (

1

1199061205911198882

)(1

ℎ2+119906

ℎ) (119879119895

0minus 120579119895

0)

minus (V0

1198882

)(1

ℎ2+119906

ℎ)

(minus119875

ℎ2)119908119895+1

1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

0

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

0= (

minus1

120591)119908119895

1

+ 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895))

sdot ((minus1198882

1198881

)1198790minus

1

1199061205911198881

(119879119895

0minus 120579119895

0) minus

V0

1198881

)

(A3)

(1

120591+

2

ℎ2+ 1198962+119906

ℎ+

1198882

ℎ21198881

)119879119895+1

0+ (

minus1

ℎ2minus119906

ℎ)119879119895+1

1

= (minus1

120591)119879119895

0minus 119906 exp (minus119906ℎ) minus ( 1

119906120591ℎ21198881

) (119879119895

0minus 120579119895

0)

minusV0

ℎ21198881

(A4)

(minus119875

ℎ2)119908119895+1

1+ (

1

120591+2119875

ℎ2+ 1198751198962+119906

ℎ)119908119895+1

0

+ (minus119875

ℎ2minus119906

ℎ)119882119895+1

0= (

minus1

120591)119908119895

0

+ 1198761198962(1 + 120582

1sin (120590

1119905119895) + 1205822sin (120590

2119905119895))

sdot ((minus1198881

1198882

)1205790minus

1

1199061205911198882

)(119879119895

0minus 120579119895

0minusV0

1198882

)

(A5)

with

1198881= (

1

ℎminus119885

119906minus

1

119906120591)

1198882= (minus

1

ℎ+

1

119906120591)

(A6)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] B B Khanukaev M A Kozhushner N S Enikolopyan and NM Chechilo ldquoTheory of the propagation of a polymerizationfrontrdquo Combustion Explosion and Shock Waves vol 10 no 1pp 17ndash21 1974

[2] J A Pojman and T-C-M Qui Nonlinear Dynamics with Pol-ymers Fundamentals Methods and Applications John Wiley ampSons 2011

[3] I D Robertson H L Hernandez S R White and J S MooreldquoRapid stiffening of a microfluidic endoskeleton via frontalpolymerizationrdquo ACS Applied Materials and Interfaces vol 6no 21 pp 18469ndash18474 2014

[4] Vi Volpert and Vl Volpert ldquoPropagation of frontal polym-erizationmdashcrystallization wavesrdquo European Journal of AppliedMathematics vol 5 no 2 pp 201ndash215 1994

[5] G Bowden M Garbey V M Ilyashenko et al ldquoEffect of con-vection on a propagating front with a solid product comparisonof theory and experimentsrdquoThe Journal of Physical Chemistry Bvol 101 no 4 pp 678ndash686 1997

[6] M Garbey A Taik and V Volpert ldquoLinear stability analysisof reaction fronts in liquidsrdquo Quarterly of Applied Mathematicsvol 54 no 2 pp 225ndash247 1996

[7] K Allali V Volpert and J A Pojman ldquoInfluence of vibrationson convective instability of polymerization frontsrdquo Journal ofEngineering Mathematics vol 41 no 1 pp 13ndash31 2001

[8] K Allali F Bikany A Taik and V Volpert ldquoInfluence ofvibrations on convective instability of reaction fronts in liquidsrdquoMathematical Modelling of Natural Phenomena vol 5 no 7 pp35ndash41 2010

[9] K Allali M Belhaq and K El Karouni ldquoInfluence of quasi-periodic gravitational modulation on convective instability ofreaction fronts in porous mediardquo Communications in NonlinearScience and Numerical Simulation vol 17 no 4 pp 1588ndash15962012

[10] KAllali S Assiyad andM Belhaq ldquoConvection of polymeriza-tion front with solid product under quasi-periodic gravitationalmodulationrdquo Nonlinear Dynamics and Systems Theory vol 14no 4 pp 323ndash334 2014

[11] M Menzinger and R Wolfgang ldquoThe meaning and use of theArrhenius activation energyrdquoAngewandte Chemie InternationalEdition in English vol 8 no 6 pp 438ndash444 1969

[12] B V Novozhilov ldquoThe rate of propagation of the front of anexothermic reaction in a condensed phaserdquo Doklady PhysicalChemistry vol 141 pp 836ndash838 1961

[13] Y B Zeldovich and D A Frank-Kamenetskii ldquoA theory ofthermal propagation of amerdquo Acta Physicochimica USSR vol 9pp 341ndash350 1938

[14] M Garbey A Taık and V Volpert ldquoInfluence of natural con-vection on stability of reaction fronts in liquidsrdquo Quarterly ofApplied Mathematics vol 56 no 1 pp 1ndash35 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of