research article influence of chemical reaction on heat and mass transfer flow...

Research ArticleInfluence of Chemical Reaction on Heat and Mass TransferFlow of a Micropolar Fluid over a Permeable Channel withRadiation and Heat Generation

Khilap Singh and Manoj Kumar

Department of Mathematics Statistics and Computer Science G B Pant University of Agriculture and TechnologyPantnagar Uttarakhand India

Correspondence should be addressed to Khilap Singh singhkhilap8gmailcom

Received 30 June 2016 Accepted 9 November 2016

Academic Editor Felix Sharipov

Copyright copy 2016 K Singh and M Kumar This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The effects of chemical reaction on heat andmass transfer flowof amicropolar fluid in a permeable channel with heat generation andthermal radiation is studiedThe Rosseland approximations are used to describe the radiative heat flux in the energy equationThemodel contains nonlinear coupled partial differential equations which have been transformed into ordinary differential equationby using the similarity variables The relevant nonlinear equations have been solved by Runge-Kutta-Fehlberg fourth fifth-ordermethod with shooting technique The physical significance of interesting parameters on the flow and heat transfer characteristicsas well as the local skin friction coefficient wall couple stress and the heat transfer rate are thoroughly examined

1 Introduction

Themicropolar fluid theory is the one of the most importantnon-Newtonian fluid models described by Eringen [1] Thistheory showsmicrorotation effects aswell asmicroinertia andhas many applications such as polymer fluids liquid crystalsanimal bloods unusual lubricants colloidal and suspensionsolutions colloidal fluids liquid crystals and polymericsuspension The extensive reviews of the micropolar fluidtheory and its applications can be found in Eringen [2] andLukaszewicz [3] The unsteady mixed convection flow of amicropolar fluid from a vertical surface in the presence ofviscous dissipation and the buoyancy force has been studiedby El-Aziz [4] Ashraf et al [5] investigated the micropolarfluids flow through a porous channel Sheikholeslami etal [6] proposed the flow and heat transfer of micropolarfluid in a permeable channel Prakash and Muthtamilselvan[7] investigated radiation effect on MHD micropolar fluidflow in porous vertical channel Darvishi et al [8] analyzednumerically themicropolar fluid flow from a porous channelSherief et al [9] studied the motion along axis of a circularcylindrical pore in a micropolar fluid of a slip spherical par-ticle Mosayebidorcheh [10] analyzed the flow of micropolar

fluid over a porous channel with changing walls Recentlymany authors [11ndash14] have studied the micropolar fluidflow for different fluid properties over different geometriesHowever the effects of chemical reaction onmicropolar fluidflow over a permeable channel in the presence of radiationand heat generation have not been considered in the aboveinvestigations

Combined heat andmass transfer flows in the presence ofchemical reaction have numerous applications in engineeringand geophysics such as drying geothermal reservoirs dehy-dration at the surface of a water body drying of porous solidsgeothermal pool thermal insulation enhanced oil recoverycooling of nuclear reactors fibrous insulation evaporationat the surface of a water body pollution studies cooling thepolymer production and manufacturing of ceramics energytransfer in a wet cooling tower the flow in a desert andoxidation and synthesismaterialsMohamed andAbo-Dahab[15] studied the effects of chemical reaction and thermalradiation on hydromagnetic free convection heat and masstransfer for a micropolar fluid bounded by a semi-infinitevertical porous plate in the presence of heat generationIn recent years many researchers have studied and reportedthe effect of first-order chemical reaction [16ndash25]

Hindawi Publishing CorporationJournal of ermodynamicsVolume 2016 Article ID 8307980 10 pageshttpdxdoiorg10115520168307980

2 Journal of Thermodynamics

Influence of thermal radiation on flow and heat transferstudy has become more important industrially The heattransfer and temperature profile of a micropolar fluid overdifferent geometries can be affected significantly at hightemperature Bhattacharyya et al [26] considered thermalradiation effect on micropolar fluid flow and heat transferover a porous shrinking sheet Hussain et al [27] analyzedradiation effects on the thermal boundary layer flow ofa micropolar fluid towards a permeable stretching sheetOahimire and Olajuwon [28] investigated the influence ofHall current and thermal radiation on heat and mass transferof a chemically reacting MHD flow of a micropolar fluidthrough a porous medium Mabood et al [29] studiedeffects of nonuniform heat sourcesink and Soret on MHDnon-Darcian convective flow past a stretching sheet in amicropolar fluid with radiation

The effect of heat generation on heat transfer is an impor-tant issue in view of various physical problems Ziabakhshet al [30] analyzed the micropolar fluid flow with heatgeneration Singh and Kumar [31] considered the meltingeffect in stagnation-point flow of micropolar fluid towardsa stretchingshrinking surface Bakr [32] investigated theeffects of chemical reaction and heat source magnetoconvec-tion andmass transfer flow of a micropolar fluid in a rotatingframe of reference The heat generationabsorption effectson MHD flow and heat transfer of micropolar fluid througha stretching surface have been proposed by Mahmoud andWaheed [33] Abbasi et al [34] examined the flow ofMaxwellnanofluid in the presence of heat generationabsorptionMliki et al [35] investigated the influence of nanoparti-cle Brownian motion and heat generationabsorption overlinearsinusoidally heated cavity in the presence of mag-netohydrodynamic natural convection Sheikholeslami andGanji [36] studied three-dimensional heat and mass transferflow of nanofluid over a rotating system Thermal radiationeffects on mixed convection flow and heat transfer of amicropolar fluid through an unsteady stretching surfacewith heat generationabsorption are presented by Singh andKumar [37]

Motivated by the above studies and applications thepresent work explores effects of chemical reaction on heatandmass transfer flow of a micropolar fluid over a permeablechannel in the presence of radiation and heat generationThe equations of continuity momentum angular momen-tum energy and concentration have been reduced to asystem of nonlinear ordinary differential equations by sim-ilarity transforms which are solved by Runge-Kutta-Fehlbergmethod with shooting technique It is expected that theresults obtained from present paper will provide importantinformation to the audience To the best of our knowledgesuch type of study is not investigated before in the scientificliterature

2 Mathematical Formulation

The heat and mass transfer flow of a micropolar fluid in apermeable channel with chemical reaction is considered inthe present work The thermal radiation and heat source areincorporated in the energy equation The graphical model of

Micropolar fluid flow betweenpermeable walls

xy

z

V0

V0

C1 T1

C2 T2

2ℎ

W

(a)

x

y

V0

V0

C1 T1

C2 T2

2ℎ

(b)

Figure 1 (a) Geometry of problem (b) 119909-119910 view at 119911 = 1198822

the problem has been given along with flow configurationand coordinate system in Figure 1 The assumptions ofthe problem in detail can be found in [6] The governingequations of boundary layer are given in the following form

120597119906120597119909 +

120597V120597119910 = 0 (1)

119906120597119906120597119909 + V120597119906120597119910 = minus1120588

120597119875120597119909 +

(120583 + 120581)120588 (12059721199061205971199092 +

12059721199061205971199102)

+ 120581120588120597119873120597119910

(2)

119906 120597V120597119909 + V120597V120597119910 = minus

1120588120597119875120597119910 +

(120583 + 120581)120588 ( 1205972V1205971199092 +

1205972V1205971199102)

minus 120581120588120597119873120597119910

(3)

119906120597119873120597119909 + V120597119873120597119910 = 120583119904120588119895 (

12059721198731205971199092 +

12059721198731205971199102 )

minus 120581120588119895 (2119873 + 120597119906120597119910 minus

120597V120597119909)

(4)

119906120597119879120597119909 + V120597119879120597119910 = 119896

12058811986211990112059721198791205971199102 minus

1120588119862119901

120597119902119903120597119910 + 120583120588119862119901 (

120597119906120597119910)2

+ 1198760120588119862119901 (119879 minus 1198792) (5)

119906120597119862120597119909 + V120597119862120597119910 = 119863lowast 12059721198621205971199102 minus 1205740 (119862 minus 1198622) (6)

Journal of Thermodynamics 3

where 119906 and V indicate the velocity components in the 119909and 119910 directions respectively 120588 is the fluid density 120583 isthe dynamic viscosity 120581 is the material parameter 119873 is theangular or microrotation velocity 119875 is the fluid pressure 119895 isthe microinertia density 120583119904 = (120583 + 1205812)119895 is the microrotationviscosity 119879 is the fluid temperature 119862119901 is the specific heatat constant pressure 119862 is the fluid concentration 119896 is thethermal conductivity 119902119903 is the radiative heat flux 1198760 is theheat generation coefficient 119863lowast is the molecular diffusivityand 1205740 is the chemical reaction rate coefficient

Using Rosselandrsquos approximation for radiation we obtain

119902119903 = minus( 412059031198960)1205971198794120597119910 (7)

where 120590 is the StefanndashBoltzmann constant and 1198960 is theabsorption coefficient We consider that the temperaturevariationwithin the flow is such that1198794may be expanded in aTaylorrsquos series Expanding1198794 about119879infin and neglecting higherorder terms we get 1198794 = 41198793infin119879 minus 31198794infin Now (5) reduces to

119906120597119879120597119909 + V120597119879120597119910 = 119896

12058811986211990112059721198791205971199102 +

161205901198793infin3119896112058811986211990112059721198791205971199102

+ 120583120588119862119901 (

120597119906120597119910)2 + 1198760120588119862119901 (119879 minus 1198792)

(8)

The appropriate boundary conditions for the flow are

119906 = V = 0119873 = minus119899120597119906120597119910 119879 = 1198791119862 = 1198621

at 119910 = minusℎ119906 = V0119909ℎ V = 0119873 = V0119909ℎ2 119879 = 1198792119862 = 1198622

at 119910 = minusℎ

(9)

where boundary parameters 119899 (0 le 119899 le 1) indicate thedegree to which the microelements are free to rotate nearthe channel walls The case when 119899 = 0 is called strongconcentration of microelements which implies 119873 = 0 nearthe wall surface This represents concentrated particle flowwhere the microelements close to the wall surface are unableto rotate In the case when 119899 = 12 this indicates vanishingof the antisymmetric part of the stress tensor and denoted

weak concentration of microelement and 119899 = 1 is used forthe modeling of turbulent boundary layer flow In this paperthe authors considered 119899 = 12 for which the governingequations can be reduced to the classical problem of steadyboundary layer flow of a viscous incompressible fluid near thechannel wall

Equations (2) (3) (4) (6) and (8) can be transformedinto a set of nonlinear ordinary differential equations by usingthe following similarity transformations

120578 = 119910ℎ

120595 = minus1205920119909119891 (120578) 119873 = 1205920119909ℎ2 119892 (120578)

120579 (120578) = 119879 minus 11987921198791 minus 1198792

120601 (120578) = 119862 minus 11986221198621 minus 1198622

(10)

where 1198792 = 1198791 minus 119860119909 and 1198622 = 1198621 minus 119861119909 with 119860 and 119861 asconstants The stream function 120595 is defined as

119906 = 120597120595120597119910

V = minus120597120595120597119909 (11)

The coupled system of transformed nonlinear ordinary dif-ferential equations is

(1 + 1198731) 119891IV minus 1198731119892 minus Re (119891119891101584010158401015840 minus 119891101584011989110158401015840) = 0119873211989210158401015840 + 1198731 (11989110158401015840 minus 2119892) minus 1198733Re (1198911198921015840 minus 1198911015840119892) = 0(1 + 119877) 12057910158401015840 + Peℎ [Ec119891101584010158402 + 1198911015840120579 minus 1198911205791015840 + 119867120579] = 0

12060110158401015840 + Pe119898 (1198911015840120601 minus 1198911206011015840 minus 120574120601) = 0

(12)

Boundary conditions in nondimensional form are

119891 (minus1) = 1198911015840 (minus1) = 119892 (minus1) = 0120579 (minus1) = 120601 (minus1) = 1119891 (1) = 120579 (1) = 120601 (1) = 01198911015840 (1) = minus1119892 (1) = 1

(13)

where 1198731 = 120581120583 is the coupling number 1198732 = V119904120583ℎ2 is thespin-gradient viscosity parameter 1198733 = 119895ℎ2 is the microp-olar material constant Ec = V20119909120592ℎ3119888119901119860 is the local Eckertnumber 119867 = 1205790ℎ120588119862119901V0 is the heat generation parameterPeℎ = Pr Re and Pe119898 = Sc Re are the Peclet numbers forthe diffusion of heat and the diffusion of mass Re = (V0120592)ℎ


is the Reynolds number 119877 = 31198960119896161205901198793infin is the thermalradiation parameter Gr = 119892120573119879119860ℎ41205922 is theGrashof numberPr = 120592120588119888119901119896 is the Prandtl number Sc = 120592119863lowast is the Schmidtnumber and 120574 = 1205740ℎV0 is the chemical reaction parameter

The other parameters of physical interest are the localNusselt Nu119909 and Sherwood Sh119909 numbers which are definedas follows

Nu119909 = 119909119902119908119896 (1198791 minus 1198792) Sh119909 = 119909119898119908119863lowast (1198621 minus 1198622)

(14)

where 119902119908 and 119898119908 are the local heat flux and mass fluxrespectively which are defined as

119902119908 = minus119896(120597119879120597119910 )119910=minusℎ

119898119908 = 119863lowast (120597119862120597119910 )119910=minusℎ (15)

Now using (10) and (15) in (14) we get

Nu119909 = minus1205791015840 (minus1) Sh119909 = minus1206011015840 (minus1)

(16)

3 Method of Solution

In this present paper Runge-Kutta-Fehlberg fourth fifth-order method has been employed to solve the system of non-linear ordinary differential equations (12) with the boundaryconditions given by (13) for different values of governingparametersTheRKF45method has a procedure to determineif the appropriate step size ℎ is being used The formula offifth-order Runge-Kutta-Fehlberg method can be defined asfollows

119911119899+1 = 119911119899 + ( 161351198960 +6656128251198962 +

28561564301198963 minus

9501198964

+ 2551198965) ℎ

(17)

where the coefficients 1198960 to 1198965 are given by

1198960 = 119891 (119909119899 119910119899) 1198961 = 119891(119909119899 + 14ℎ 119910119899 +

14ℎ1198960)

1198962 = 119891(119909119899 + 38ℎ 119910119899 + (3321198960 +

9321198961) ℎ)

1198963 = 119891(119909119899 + 1213ℎ 119910119899+ (193221971198960 minus

720021971198961 +

729621971198962) ℎ)

1198964 = 119891(119909119899 + ℎ 119910119899+ (4392161198960 minus 81198961 +

3680513 1198962 minus

84541041198963) ℎ)

1198965 = 119891(119909119899 + 12ℎ 119910119899+ (minus 8271198960 + 21198961 minus

354425651198962 +

185941041198963 minus

11401198964) ℎ)

(18)

The computation of the error can be achieved by subtractingthe fifth-order from the fourth-order method

119910119899+1 = 119910119899 + ( 252161198960 +140825651198962 +

219741011198963 minus

151198964) ℎ (19)

If the error goes beyond a specified antechamber the resultscan be recalculated using a smaller step size The approach tocomputing the new step size is shown as follows

ℎnew = ℎold ( 120576ℎold2 1003816100381610038161003816119911119899+1 minus 119910119899+11003816100381610038161003816)14

(20)

Thenumerical computationswere carried outwithΔ120578 = 001The variation of the dimensionless velocity microrotationtemperature and concentration is ensured to be less than 10minus6between any two successive iterations for the convergencecriterion

4 Results and Discussion

In order to study the effects of various governing physicalparameters on the flow heat and mass transfer numericalcomputations are carried out for 01 le 1198731 = 1198732 = 1198733 le 10 le 119867 le 10 minus2 le Re le 5 0 le 119877 le 4 0 le Peℎ le 2 0 lePe119898 le 2 and 0 le 120574 le 10 while the Eckert number Ec = 001is fixed A critical analysis with previously published work isdone in Table 1 and these results are found to be in very goodagreement

The effects of the Reynolds number Re on velocity 1198911015840(120578)andmicrorotation119892(120578) are shown in Figures 2 and 3 It is seenfrom Figure 2 that velocity profile 1198911015840(120578) decreases near thelower channel wall while it increases near the upper channelwall when the Reynolds number Re increases It is clear fromFigure 3 that an increase in the magnitude of Re leads todecrease in microrotation profile 119892(120578)


Table 1 Comparison of values of 120601(120578) for various values of 120578 Pe119898 and Re when Peℎ = 02 1198731 = 1198732 = 1198733 = 01 1198731 = 1198732 = 1198733 = 01and Ec = 119877 = 120574 = 0120578 Sheikholeslami et al [6] Present results

Pe119898 = 02 Re = 05 Pe119898 = 05 Re = 1 Pe119898 = 02 Re = 05 Pe119898 = 05 Re = 1minus10 1 1 1 1minus06 0814341 0835986 0814346 0835992minus02 0621160 0653497 0621165 065350100 0521109 0553582 0521114 055359002 0419033 0448495 0419039 044849906 0210678 0227352 0210684 022735810 0 0 0 0

Re = minus2

Re = 1

Re = 2

Re = 5

00 05 10minus05minus10

120578

minus10

minus08

minus06

minus04

minus02

00

02

04

f998400 (120578)

N1 = N2 = N3 = 01 Peℎ = Pem = 02 H = R = 120574 = 2

Figure 2 Velocity profile 1198911015840(120578) for various values of Re

g(120578

)

minus05 00 05 10minus10

120578

00

02

04

06

08

10

Re = minus2

Re = 1

Re = 2

Re = 5

N1 = N2 = N3 = 01 Peℎ = Pem = 02 H = R = 120574 = 2

Figure 3 Microrotation profile 119892(120578) for various values of Re

g(120578

)

N1 = 01

N1 = 03

N1 = 05

N1 = 07


120578

minus04

minus02

00

02

04

06

08

10N2 = N3 = 01 Re = 05 Peℎ = Pem = 02 H = R = 120574 = 2

Figure 4 Microrotation profile 119892(120578) for various values of1198731

The microrotation profile 119892(120578) for the coupling num-ber 1198731 the spin-gradient viscosity parameter 1198732 and themicropolar material constant 1198733 are displayed in Figures 4ndash6 It is noticed from Figure 4 that the microrotation profileincreases with 1198731 at lower channel wall for 0 le 120578 lt minus065but reverse trend is found for the case 120578 gt minus065 It isdepicted from Figure 5 that the value of microrotation profile119892(120578) increases as the value of1198732 increases It is evident fromFigure 6 that the values of microrotation are lower for highervalues of1198733

Figures 7ndash9 show the temperature distribution 120579(120578) withcollective variation in heat source parameter 119867 thermalradiation parameter 119877 and Peclet number Peℎ Figure 7exhibits that temperature 120579(120578) considerably increases withan increase in 119867 Figure 8 indicates that an increase in119877 induces a decrease in temperature 120579(120578) It is noticedfrom Figure 9 that a rise in Peℎ causes rapid increase intemperature

The effects of the Peclet number Pe119898 and chemicalreaction parameter 120574 on the concentration profile 120601(120578) areshown in Figures 10 and 11 Figures 10 and 11 indicate that


N2 = 01

N2 = 03

N2 = 05

N2 = 10


120578

g(120578

)

00

02

04

06

08

10N1 = N3 = 01 Re = 05 Peℎ = Pem = 02

H = R = 120574 = 2


g(120578

)

N3 = 01

N3 = 03

N3 = 05

N3 = 10


120578

00

02

04

06

08

10N1 = N2 = 01 Re = 05 Peℎ = Pem = 02 H = R = 120574 = 2


concentration decreases with increasing values of Pe119898 and120574The variation of the Nusselt number (dimensionless

heat transfer rate at the surface) is displayed for differentparameters in Figures 12ndash14 Figure 12 depicts the behaviorof the Nusseltrsquos number against Reynolds number Re withvarious values of heat source parameter 119867 It is clear thatwith the increasing of 119867 the heat transfer rate decreasesFigure 13 enlightens the variation of the heat transfer rateswith Reynolds number Re for various values of thermalradiation parameter 119877 It is evident from Figure 13 thatNusselt number monotonically increases with a rise in 119877values Figure 14 depicts the variation of Nusselt number asfunction of Reynolds number Re and Peclet number Peℎ It

H = 0

H = 2

H = 5

H = 10

120579(120578

)

00

02

04

06

08

10

minus08 minus06 minus04 minus02 00 02 04 06 08 10minus10

120578

N1 = N2 = N3 = 01

Re = 1 Peℎ = Pem = 02 R = 120574 = 2

Figure 7 Temperature profile 120579(120578) for various values of119867

120579(120578

)

R = 0

R = 1

R = 2

R = 4


120578

00

02

04

06

08

10

N1 = N2 = N3 = 01

Re = 05 Peℎ = Pem = 02 H = 120574 = 2

Figure 8 Temperature profile 120579(120578) for various values of 119877

is noted that heat transfer rate decreases with a rise in Peℎvalues Also a negligible change in heat transfer rate withincreasing values of Reynolds number Re is seen fromFigures12ndash14

Figures 15 and 16 displayed the variation of the Sherwoodnumber ormass transfer rate for different parameters Figures15 and 16 depict the behavior of the mass transfer rate againstReynolds number Re with different values of Peclet numberPe119898 and chemical reaction parameter 120574 The mass transferrate rises with increasing values of the Peclet number Pe119898and chemical reaction parameter 120574 It is also clear from thesefigures that there is no change in mass transfer rate with risein Reynolds number Re values

Journal of Thermodynamics 7120579(120578

)

Peℎ = 0

Peℎ = 02

Peℎ = 05

Peℎ = 1

Peℎ = 2


120578

00

02

04

06

08

10

12

14

16

N1 = N2 = N3 = 01

Re = 05 Pem = 02 H = R = 120574 = 2

Figure 9 Temperature profile 120579(120578) for various values of Peℎ

120601(120578

)


120578

00

02

04

06

08

10

Pem = 0

Pem = 02

Pem = 05

Pem = 1

Pem = 2

N1 = N2 = N3 = 01

Re = 1 Peℎ = 02 H = R = 120574 = 2

Figure 10 Concentration profile 120601(120578) for various values of Pe119898

5 Conclusions

The present paper deals with numerical analysis of chem-ical reaction effects on heat and mass transfer flow of amicropolar fluid over a permeable channel in the presenceof radiation and heat generation The system of nonlinearpartial differential equations was converted to a system ofordinary differential equations and then is solved numericallyusing the Runge-Kutta-Fehlbergmethod along with shootingmethod From the above discussion the important results aresummarized as follows

(i) Velocity and Reynolds number are inversely propor-tional to each other at lower channel wall while

120601(120578

)

120574 = 2

120574 = 0 120574 = 5

120574 = 10

00

02

04

06

08

10


120578

N1 = N2 = N3 = 01

Re = 05 Peℎ = Pem = 02 H = R = 2

Figure 11 Concentration profile 120601(120578) for various values of 120574

H = 0H = 5H = 10

06040200 08 10minus04minus06minus08 minus02minus10

Re

minus02

00

02

04

06N

u x

Figure 12 Variation of Nusseltrsquos number with119867 and Re

velocity and Reynolds number are proportional toeach other at upper channel wall

(ii) Microrotation decreases with increase in the value ofcoupling number micropolar material constant andReynolds number but it increases with increase in thevalue of spin-gradient viscosity parameter

(iii) Temperature increases with heat generation param-eter and Peℎ and Pe119898 are the Peclet numbers forthe diffusion of heat and the diffusion of massand temperature decreases with thermal radiationparameter

(iv) Concentration is inversely proportional to the Pecletnumber for the diffusion of mass and chemical reac-tion parameter


R = 0

R = 2

R = 4


Re

015

020

025

030

035

040

045

050

Nu x

Figure 13 Variation of Nusseltrsquos number with 119877 and Re

Peh = 1

Peh = 2

Peh = 0


Re

minus15

minus10

minus05

00

05

10

Nu x

Figure 14 Variation of Nusseltrsquos number with Peℎ and Re

(v) The rate of heat transfer increases with thermal radia-tion parameter and rate of heat transfer decreaseswithPeclet number Peℎ and heat generation parameter

(vi) The rate of mass transfer rises with increase Pecletnumber Pe119898 and chemical reaction parameter

Nomenclature

119862 Species concentration119863lowast Molecular diffusivity119891 Dimensionless stream function119892 Dimensionless microrotationℎ Width of the channel119895 Microinertia density

Pem = 0

Pem = 1

Pem = 2

minus05minus10 05 1000Re

00

05

10

15

20

25

Shx

0 2 4 6 8 10

Figure 15 Variation of Sherwood number with Pe119898 and Re

120574 = 0

120574 = 2

120574 = 5

080602minus06 minus04 00 04minus08 10minus02minus10

Re

04

05

06

07

08

09

10

11

Shx

Figure 16 Variation of Sherwood number with 120574 and Re

119873 Microrotationangular velocity119873123 Dimensionless parameterNu Nusselt numberSh Sherwood numberSc Schmidt number119901 PressurePt Prandtl numberPe Peclet number119902 Mass transfer parameter119902119903 Radiative heat fluxRe Reynold number119879 Fluid temperature119904 Microrotation boundary condition


(119906 V) Cartesian velocity components(119909 119910) Cartesian coordinate components parallel

and normal to cannel axis respectively

Greek Symbols

120578 Similarity variable120579 Dimensionless temperature120583 Dynamic viscosity120581 Material parameter120588 Fluid density120583119904 Microrotationspin-gradient viscosity120595 Stream function

Competing Interests

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

Acknowledgments

Thefirst author gratefully acknowledges the financial supportof UGC India under F 17-972008 (SA-I) for pursuing thiswork

References

[1] A C Eringen ldquoTheory ofmicropolar fluidrdquo Jour ofMathematicsand Mechanics vol 16 pp 1ndash18 1966

[2] A C Eringen ldquoTheory of thermomicrofluidsrdquo Journal ofMathematical Analysis and Applications vol 38 no 2 pp 480ndash496 1972

[3] G Lukaszewicz Micropolar Fluids Theory and ApplicationBirkhauser Basel Switzerland 1999

[4] M A El-Aziz ldquoMixed convection flow of a micropolar fluidfrom an unsteady stretching surface with viscous dissipationrdquoJournal of the Egyptian Mathematical Society vol 21 no 3 pp385ndash394 2013

[5] M Ashraf M A Kamal and K S Syed ldquoNumerical studyof asymmetric laminar flow of micropolar fluids in a porouschannelrdquo Computers amp Fluids vol 38 no 10 pp 1895ndash19022009

[6] M Sheikholeslami M Hatami and D D Ganji ldquoMicropolarfluid flow and heat transfer in a permeable channel usinganalytical methodrdquo Journal of Molecular Liquids vol 194 pp30ndash36 2014

[7] D Prakash and M Muthtamilselvan ldquoEffect of radiation ontransient MHD flow of micropolar fluid between porous ver-tical channel with boundary conditions of the third kindrdquo AinShams Engineering Journal vol 5 no 4 pp 1277ndash1286 2014

[8] M T Darvishi F Khani F G Awad A A Khidir and PSibanda ldquoNumerical investigation of the flow of a micropolarfluid through a porous channel with expanding or contractingwallsrdquo Propulsion and Power Research vol 3 no 3 pp 133ndash1422014

[9] H H Sherief M S Faltas E A Ashmawy andM G NashwanldquoSlow motion of a slip spherical particle along the axis ofa circular cylindrical pore in a micropolar fluidrdquo Journal ofMolecular Liquids vol 200 pp 273ndash282 2014

[10] SMosayebidorcheh ldquoAnalytical investigation of themicropolarflow through a porous channel with changing wallsrdquo Journal ofMolecular Liquids vol 196 pp 113ndash119 2014

[11] M Fakour A Vahabzadeh D D Ganji and M HatamildquoAnalytical study of micropolar fluid flow and heat transfer ina channel with permeable wallsrdquo Journal of Molecular Liquidsvol 204 pp 198ndash204 2015

[12] A Tetbirt M N Bouaziz and M T Abbes ldquoNumericalstudy of magnetic effect on the velocity distribution field in amacromicro-scale of a micropolar and viscous fluid in verticalchannelrdquo Journal of Molecular Liquids vol 216 pp 103ndash1102016

[13] M Ramzan M Farooq T Hayat and J D Chung ldquoRadiativeand Joule heating effects in the MHD flow of a micropolar fluidwith partial slip and convective boundary conditionrdquo Journal ofMolecular Liquids vol 221 pp 394ndash400 2016

[14] N S Gibanov M A Sheremet and I Pop ldquoNatural convectionof micropolar fluid in a wavy differentially heated cavityrdquoJournal of Molecular Liquids vol 221 pp 518ndash525 2016

[15] R A Mohamed and S M Abo-Dahab ldquoInfluence of chemicalreaction and thermal radiation on the heat and mass transfer inMHDmicropolar flow over a vertical moving porous plate in aporous medium with heat generationrdquo International Journal ofThermal Sciences vol 48 no 9 pp 1800ndash1813 2009

[16] E Magyari and A J Chamkha ldquoCombined effect of heatgeneration or absorption and first-order chemical reaction onmicropolar fluid flows over a uniformly stretched permeablesurface the full analytical solutionrdquo International Journal ofThermal Sciences vol 49 no 9 pp 1821ndash1828 2010

[17] K Das ldquoEffect of chemical reaction and thermal radiation onheat and mass transfer flow of MHD micropolar fluid in arotating frame of referencerdquo International Journal of Heat andMass Transfer vol 54 no 15-16 pp 3505ndash3513 2011

[18] A A Bakr ldquoEffects of chemical reaction on MHD free convec-tion andmass transfer flowof amicropolar fluidwith oscillatoryplate velocity and constant heat source in a rotating frame ofreferencerdquoCommunications inNonlinear Science andNumericalSimulation vol 16 no 2 pp 698ndash710 2011

[19] K Das ldquoInfluence of thermophoresis and chemical reactionon MHD micropolar fluid flow with variable fluid propertiesrdquoInternational Journal of Heat and Mass Transfer vol 55 no 23-24 pp 7166ndash7174 2012

[20] A M Rashad S Abbasbandy and A J Chamkha ldquoMixed con-vection flow of a micropolar fluid over a continuously movingvertical surface immersed in a thermally and solutally stratifiedmediumwith chemical reactionrdquo Journal of the Taiwan Instituteof Chemical Engineers vol 45 no 5 pp 2163ndash2169 2014

[21] F Mabood W A Khan and A I M Ismail ldquoMHD stagnationpoint flow and heat transfer impinging on stretching sheet withchemical reaction and transpirationrdquo Chemical EngineeringJournal vol 273 pp 430ndash437 2015

[22] K Singh and M Kumar ldquoThe effect of chemical reaction anddouble stratification on MHD free convection in a micropolarfluid with heat generation and Ohmic heatingrdquo Jordan Journalof Mechanical and Industrial Engineering vol 9 no 4 pp 279ndash288 2015

[23] S Srinivas A Gupta S Gulati and A S Reddy ldquoFlow andmass transfer effects on viscous fluid in a porous channelwith movingstationary walls in presence of chemical reactionrdquoInternational Communications in Heat and Mass Transfer vol48 pp 34ndash39 2013


[24] J C Umavathi M A Sheremet and S Mohiuddin ldquoCombinedeffect of variable viscosity and thermal conductivity on mixedconvection flow of a viscous fluid in a vertical channel in thepresence of first order chemical reactionrdquo European Journal ofMechanicsmdashBFluids vol 58 pp 98ndash108 2016

[25] J C Misra and S D Adhikary ldquoMHD oscillatory channel flowheat and mass transfer in a physiological fluid in presence ofchemical reactionrdquo Alexandria Engineering Journal vol 55 no1 pp 287ndash297 2016

[26] K Bhattacharyya S Mukhopadhyay G C Layek and I PopldquoEffects of thermal radiation on micropolar fluid flow and heattransfer over a porous shrinking sheetrdquo International Journal ofHeat and Mass Transfer vol 55 no 11-12 pp 2945ndash2952 2012

[27] M Hussain M Ashraf S Nadeem and M Khan ldquoRadiationeffects on the thermal boundary layer flow of a micropolar fluidtowards a permeable stretching sheetrdquo Journal of the FranklinInstitute vol 350 no 1 pp 194ndash210 2013

[28] J I Oahimire and B I Olajuwon ldquoEffect of Hall current andthermal radiation on heat and mass transfer of a chemicallyreacting MHD flow of a micropolar fluid through a porousmediumrdquo Journal of King Saud UniversitymdashEngineering Sci-ences vol 26 no 2 pp 112ndash121 2014

[29] FMabood S M IbrahimMM Rashidi M S Shadloo and GLorenzini ldquoNon-uniform heat sourcesink and Soret effects onMHD non-Darcian convective flow past a stretching sheet in amicropolar fluid with radiationrdquo International Journal of Heatand Mass Transfer vol 93 pp 674ndash682 2016

[30] Z Ziabakhsh G Domairry and H Bararnia ldquoAnalytical solu-tion of non-Newtonian micropolar fluid flow with uniformsuctionblowing and heat generationrdquo Journal of the TaiwanInstitute of Chemical Engineers vol 40 no 4 pp 443ndash451 2009

[31] K Singh and M Kumar ldquoMelting heat transfer in boundarylayer stagnation point flow ofMHDmicro-polar fluid towards astretchingshrinking surfacerdquo Jordan Journal of Mechanical andIndustrial Engineering vol 8 no 6 pp 403ndash408 2014


[33] M A A Mahmoud and S E Waheed ldquoMHD flow and heattransfer of a micropolar fluid over a stretching surface withheat generation (absorption) and slip velocityrdquo Journal of theEgyptian Mathematical Society vol 20 no 1 pp 20ndash27 2012

[34] F M Abbasi S A Shehzad T Hayat and B Ahmad ldquoDoublystratifiedmixed convection flowofMaxwell nanofluidwith heatgenerationabsorptionrdquo Journal of Magnetism and MagneticMaterials vol 404 pp 159ndash165 2016

[35] B Mliki M A Abbassi A Omri and B Zeghmati ldquoEffectsof nanoparticles Brownian motion in a linearlysinusoidallyheated cavity with MHD natural convection in the presence ofuniform heat generationabsorptionrdquo Powder Technology vol29 pp 69ndash83 2016

[36] M Sheikholeslami and D D Ganji ldquoThree dimensional heatand mass transfer in a rotating system using nanofluidrdquo PowderTechnology vol 253 pp 789ndash796 2014

[37] K Singh and M Kumar ldquoEffects of thermal radiation onmixed convection flow of a micro-polar fluid from an unsteadystretching surface with viscous dissipation and heat genera-tionabsorptionrdquo International Journal of Chemical Engineeringvol 2016 Article ID 8190234 10 pages 2016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


Influence of thermal radiation on flow and heat transferstudy has become more important industrially The heattransfer and temperature profile of a micropolar fluid overdifferent geometries can be affected significantly at hightemperature Bhattacharyya et al [26] considered thermalradiation effect on micropolar fluid flow and heat transferover a porous shrinking sheet Hussain et al [27] analyzedradiation effects on the thermal boundary layer flow ofa micropolar fluid towards a permeable stretching sheetOahimire and Olajuwon [28] investigated the influence ofHall current and thermal radiation on heat and mass transferof a chemically reacting MHD flow of a micropolar fluidthrough a porous medium Mabood et al [29] studiedeffects of nonuniform heat sourcesink and Soret on MHDnon-Darcian convective flow past a stretching sheet in amicropolar fluid with radiation

The effect of heat generation on heat transfer is an impor-tant issue in view of various physical problems Ziabakhshet al [30] analyzed the micropolar fluid flow with heatgeneration Singh and Kumar [31] considered the meltingeffect in stagnation-point flow of micropolar fluid towardsa stretchingshrinking surface Bakr [32] investigated theeffects of chemical reaction and heat source magnetoconvec-tion andmass transfer flow of a micropolar fluid in a rotatingframe of reference The heat generationabsorption effectson MHD flow and heat transfer of micropolar fluid througha stretching surface have been proposed by Mahmoud andWaheed [33] Abbasi et al [34] examined the flow ofMaxwellnanofluid in the presence of heat generationabsorptionMliki et al [35] investigated the influence of nanoparti-cle Brownian motion and heat generationabsorption overlinearsinusoidally heated cavity in the presence of mag-netohydrodynamic natural convection Sheikholeslami andGanji [36] studied three-dimensional heat and mass transferflow of nanofluid over a rotating system Thermal radiationeffects on mixed convection flow and heat transfer of amicropolar fluid through an unsteady stretching surfacewith heat generationabsorption are presented by Singh andKumar [37]

Motivated by the above studies and applications thepresent work explores effects of chemical reaction on heatandmass transfer flow of a micropolar fluid over a permeablechannel in the presence of radiation and heat generationThe equations of continuity momentum angular momen-tum energy and concentration have been reduced to asystem of nonlinear ordinary differential equations by sim-ilarity transforms which are solved by Runge-Kutta-Fehlbergmethod with shooting technique It is expected that theresults obtained from present paper will provide importantinformation to the audience To the best of our knowledgesuch type of study is not investigated before in the scientificliterature

2 Mathematical Formulation

The heat and mass transfer flow of a micropolar fluid in apermeable channel with chemical reaction is considered inthe present work The thermal radiation and heat source areincorporated in the energy equation The graphical model of

Micropolar fluid flow betweenpermeable walls

xy

z

V0

V0

C1 T1

C2 T2

2ℎ

W

(a)

x

y

V0

V0

C1 T1

C2 T2

2ℎ

(b)

Figure 1 (a) Geometry of problem (b) 119909-119910 view at 119911 = 1198822

the problem has been given along with flow configurationand coordinate system in Figure 1 The assumptions ofthe problem in detail can be found in [6] The governingequations of boundary layer are given in the following form

120597119906120597119909 +

120597V120597119910 = 0 (1)

119906120597119906120597119909 + V120597119906120597119910 = minus1120588

120597119875120597119909 +

(120583 + 120581)120588 (12059721199061205971199092 +

12059721199061205971199102)

+ 120581120588120597119873120597119910

(2)

119906 120597V120597119909 + V120597V120597119910 = minus

1120588120597119875120597119910 +

(120583 + 120581)120588 ( 1205972V1205971199092 +

1205972V1205971199102)

minus 120581120588120597119873120597119910

(3)

119906120597119873120597119909 + V120597119873120597119910 = 120583119904120588119895 (

12059721198731205971199092 +

12059721198731205971199102 )

minus 120581120588119895 (2119873 + 120597119906120597119910 minus

120597V120597119909)

(4)

119906120597119879120597119909 + V120597119879120597119910 = 119896

12058811986211990112059721198791205971199102 minus

1120588119862119901

120597119902119903120597119910 + 120583120588119862119901 (

120597119906120597119910)2

+ 1198760120588119862119901 (119879 minus 1198792) (5)

119906120597119862120597119909 + V120597119862120597119910 = 119863lowast 12059721198621205971199102 minus 1205740 (119862 minus 1198622) (6)




119902119903 = minus( 412059031198960)1205971198794120597119910 (7)


119906120597119879120597119909 + V120597119879120597119910 = 119896

12058811986211990112059721198791205971199102 +

161205901198793infin3119896112058811986211990112059721198791205971199102

+ 120583120588119862119901 (

120597119906120597119910)2 + 1198760120588119862119901 (119879 minus 1198792)

(8)


119906 = V = 0119873 = minus119899120597119906120597119910 119879 = 1198791119862 = 1198621

at 119910 = minusℎ119906 = V0119909ℎ V = 0119873 = V0119909ℎ2 119879 = 1198792119862 = 1198622


(9)




120578 = 119910ℎ

120595 = minus1205920119909119891 (120578) 119873 = 1205920119909ℎ2 119892 (120578)

120579 (120578) = 119879 minus 11987921198791 minus 1198792

120601 (120578) = 119862 minus 11986221198621 minus 1198622

(10)


119906 = 120597120595120597119910

V = minus120597120595120597119909 (11)



12060110158401015840 + Pe119898 (1198911015840120601 minus 1198911206011015840 minus 120574120601) = 0

(12)



(13)






(14)


119902119908 = minus119896(120597119879120597119910 )119910=minusℎ

119898119908 = 119863lowast (120597119862120597119910 )119910=minusℎ (15)



(16)



119911119899+1 = 119911119899 + ( 161351198960 +6656128251198962 +

28561564301198963 minus

9501198964

+ 2551198965) ℎ

(17)


1198960 = 119891 (119909119899 119910119899) 1198961 = 119891(119909119899 + 14ℎ 119910119899 +

14ℎ1198960)

1198962 = 119891(119909119899 + 38ℎ 119910119899 + (3321198960 +

9321198961) ℎ)

1198963 = 119891(119909119899 + 1213ℎ 119910119899+ (193221971198960 minus

720021971198961 +

729621971198962) ℎ)

1198964 = 119891(119909119899 + ℎ 119910119899+ (4392161198960 minus 81198961 +

3680513 1198962 minus

84541041198963) ℎ)

1198965 = 119891(119909119899 + 12ℎ 119910119899+ (minus 8271198960 + 21198961 minus

354425651198962 +

185941041198963 minus

11401198964) ℎ)

(18)


119910119899+1 = 119910119899 + ( 252161198960 +140825651198962 +

219741011198963 minus

151198964) ℎ (19)



(20)








Re = minus2

Re = 1

Re = 2

Re = 5


120578

minus10

minus08

minus06

minus04

minus02

00

02

04

f998400 (120578)

N1 = N2 = N3 = 01 Peℎ = Pem = 02 H = R = 120574 = 2


g(120578

)


120578

00

02

04

06

08

10

Re = minus2

Re = 1

Re = 2

Re = 5

N1 = N2 = N3 = 01 Peℎ = Pem = 02 H = R = 120574 = 2


g(120578

)

N1 = 01

N1 = 03

N1 = 05

N1 = 07


120578

minus04

minus02

00

02

04

06

08

10N2 = N3 = 01 Re = 05 Peℎ = Pem = 02 H = R = 120574 = 2






N2 = 01

N2 = 03

N2 = 05

N2 = 10


120578

g(120578

)

00

02

04

06

08

10N1 = N3 = 01 Re = 05 Peℎ = Pem = 02

H = R = 120574 = 2


g(120578

)

N3 = 01

N3 = 03

N3 = 05

N3 = 10


120578

00

02

04

06

08

10N1 = N2 = 01 Re = 05 Peℎ = Pem = 02 H = R = 120574 = 2




H = 0

H = 2

H = 5

H = 10

120579(120578

)

00

02

04

06

08

10


120578

N1 = N2 = N3 = 01

Re = 1 Peℎ = Pem = 02 R = 120574 = 2


120579(120578

)

R = 0

R = 1

R = 2

R = 4


120578

00

02

04

06

08

10

N1 = N2 = N3 = 01

Re = 05 Peℎ = Pem = 02 H = 120574 = 2





)

Peℎ = 0

Peℎ = 02

Peℎ = 05

Peℎ = 1

Peℎ = 2


120578

00

02

04

06

08

10

12

14

16

N1 = N2 = N3 = 01

Re = 05 Pem = 02 H = R = 120574 = 2


120601(120578

)


120578

00

02

04

06

08

10

Pem = 0

Pem = 02

Pem = 05

Pem = 1

Pem = 2

N1 = N2 = N3 = 01

Re = 1 Peℎ = 02 H = R = 120574 = 2


5 Conclusions



120601(120578

)

120574 = 2

120574 = 0 120574 = 5

120574 = 10

00

02

04

06

08

10


120578

N1 = N2 = N3 = 01

Re = 05 Peℎ = Pem = 02 H = R = 2


H = 0H = 5H = 10


Re

minus02

00

02

04

06N

u x







R = 0

R = 2

R = 4


Re

015

020

025

030

035

040

045

050

Nu x


Peh = 1

Peh = 2

Peh = 0


Re

minus15

minus10

minus05

00

05

10

Nu x




Nomenclature


Pem = 0

Pem = 1

Pem = 2


00

05

10

15

20

25

Shx

0 2 4 6 8 10


120574 = 0

120574 = 2

120574 = 5


Re

04

05

06

07

08

09

10

11

Shx






Greek Symbols


Competing Interests


Acknowledgments


References












































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119902119903 = minus( 412059031198960)1205971198794120597119910 (7)


119906120597119879120597119909 + V120597119879120597119910 = 119896

12058811986211990112059721198791205971199102 +

161205901198793infin3119896112058811986211990112059721198791205971199102

+ 120583120588119862119901 (

120597119906120597119910)2 + 1198760120588119862119901 (119879 minus 1198792)

(8)


119906 = V = 0119873 = minus119899120597119906120597119910 119879 = 1198791119862 = 1198621

at 119910 = minusℎ119906 = V0119909ℎ V = 0119873 = V0119909ℎ2 119879 = 1198792119862 = 1198622


(9)




120578 = 119910ℎ

120595 = minus1205920119909119891 (120578) 119873 = 1205920119909ℎ2 119892 (120578)

120579 (120578) = 119879 minus 11987921198791 minus 1198792

120601 (120578) = 119862 minus 11986221198621 minus 1198622

(10)


119906 = 120597120595120597119910

V = minus120597120595120597119909 (11)



12060110158401015840 + Pe119898 (1198911015840120601 minus 1198911206011015840 minus 120574120601) = 0

(12)



(13)






(14)


119902119908 = minus119896(120597119879120597119910 )119910=minusℎ

119898119908 = 119863lowast (120597119862120597119910 )119910=minusℎ (15)



(16)



119911119899+1 = 119911119899 + ( 161351198960 +6656128251198962 +

28561564301198963 minus

9501198964

+ 2551198965) ℎ

(17)


1198960 = 119891 (119909119899 119910119899) 1198961 = 119891(119909119899 + 14ℎ 119910119899 +

14ℎ1198960)

1198962 = 119891(119909119899 + 38ℎ 119910119899 + (3321198960 +

9321198961) ℎ)

1198963 = 119891(119909119899 + 1213ℎ 119910119899+ (193221971198960 minus

720021971198961 +

729621971198962) ℎ)

1198964 = 119891(119909119899 + ℎ 119910119899+ (4392161198960 minus 81198961 +

3680513 1198962 minus

84541041198963) ℎ)

1198965 = 119891(119909119899 + 12ℎ 119910119899+ (minus 8271198960 + 21198961 minus

354425651198962 +

185941041198963 minus

11401198964) ℎ)

(18)


119910119899+1 = 119910119899 + ( 252161198960 +140825651198962 +

219741011198963 minus

151198964) ℎ (19)



(20)








Re = minus2

Re = 1

Re = 2

Re = 5


120578

minus10

minus08

minus06

minus04

minus02

00

02

04

f998400 (120578)

N1 = N2 = N3 = 01 Peℎ = Pem = 02 H = R = 120574 = 2


g(120578

)


120578

00

02

04

06

08

10

Re = minus2

Re = 1

Re = 2

Re = 5

N1 = N2 = N3 = 01 Peℎ = Pem = 02 H = R = 120574 = 2


g(120578

)

N1 = 01

N1 = 03

N1 = 05

N1 = 07


120578

minus04

minus02

00

02

04

06

08

10N2 = N3 = 01 Re = 05 Peℎ = Pem = 02 H = R = 120574 = 2






N2 = 01

N2 = 03

N2 = 05

N2 = 10


120578

g(120578

)

00

02

04

06

08

10N1 = N3 = 01 Re = 05 Peℎ = Pem = 02

H = R = 120574 = 2


g(120578

)

N3 = 01

N3 = 03

N3 = 05

N3 = 10


120578

00

02

04

06

08

10N1 = N2 = 01 Re = 05 Peℎ = Pem = 02 H = R = 120574 = 2




H = 0

H = 2

H = 5

H = 10

120579(120578

)

00

02

04

06

08

10


120578

N1 = N2 = N3 = 01

Re = 1 Peℎ = Pem = 02 R = 120574 = 2


120579(120578

)

R = 0

R = 1

R = 2

R = 4


120578

00

02

04

06

08

10

N1 = N2 = N3 = 01

Re = 05 Peℎ = Pem = 02 H = 120574 = 2





)

Peℎ = 0

Peℎ = 02

Peℎ = 05

Peℎ = 1

Peℎ = 2


120578

00

02

04

06

08

10

12

14

16

N1 = N2 = N3 = 01

Re = 05 Pem = 02 H = R = 120574 = 2


120601(120578

)


120578

00

02

04

06

08

10

Pem = 0

Pem = 02

Pem = 05

Pem = 1

Pem = 2

N1 = N2 = N3 = 01

Re = 1 Peℎ = 02 H = R = 120574 = 2


5 Conclusions



120601(120578

)

120574 = 2

120574 = 0 120574 = 5

120574 = 10

00

02

04

06

08

10


120578

N1 = N2 = N3 = 01

Re = 05 Peℎ = Pem = 02 H = R = 2


H = 0H = 5H = 10


Re

minus02

00

02

04

06N

u x







R = 0

R = 2

R = 4


Re

015

020

025

030

035

040

045

050

Nu x


Peh = 1

Peh = 2

Peh = 0


Re

minus15

minus10

minus05

00

05

10

Nu x




Nomenclature


Pem = 0

Pem = 1

Pem = 2


00

05

10

15

20

25

Shx

0 2 4 6 8 10


120574 = 0

120574 = 2

120574 = 5


Re

04

05

06

07

08

09

10

11

Shx






Greek Symbols


Competing Interests


Acknowledgments


References












































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







(14)


119902119908 = minus119896(120597119879120597119910 )119910=minusℎ

119898119908 = 119863lowast (120597119862120597119910 )119910=minusℎ (15)



(16)



119911119899+1 = 119911119899 + ( 161351198960 +6656128251198962 +

28561564301198963 minus

9501198964

+ 2551198965) ℎ

(17)


1198960 = 119891 (119909119899 119910119899) 1198961 = 119891(119909119899 + 14ℎ 119910119899 +

14ℎ1198960)

1198962 = 119891(119909119899 + 38ℎ 119910119899 + (3321198960 +

9321198961) ℎ)

1198963 = 119891(119909119899 + 1213ℎ 119910119899+ (193221971198960 minus

720021971198961 +

729621971198962) ℎ)

1198964 = 119891(119909119899 + ℎ 119910119899+ (4392161198960 minus 81198961 +

3680513 1198962 minus

84541041198963) ℎ)

1198965 = 119891(119909119899 + 12ℎ 119910119899+ (minus 8271198960 + 21198961 minus

354425651198962 +

185941041198963 minus

11401198964) ℎ)

(18)


119910119899+1 = 119910119899 + ( 252161198960 +140825651198962 +

219741011198963 minus

151198964) ℎ (19)



(20)








Re = minus2

Re = 1

Re = 2

Re = 5


120578

minus10

minus08

minus06

minus04

minus02

00

02

04

f998400 (120578)

N1 = N2 = N3 = 01 Peℎ = Pem = 02 H = R = 120574 = 2


g(120578

)


120578

00

02

04

06

08

10

Re = minus2

Re = 1

Re = 2

Re = 5

N1 = N2 = N3 = 01 Peℎ = Pem = 02 H = R = 120574 = 2


g(120578

)

N1 = 01

N1 = 03

N1 = 05

N1 = 07


120578

minus04

minus02

00

02

04

06

08

10N2 = N3 = 01 Re = 05 Peℎ = Pem = 02 H = R = 120574 = 2






N2 = 01

N2 = 03

N2 = 05

N2 = 10


120578

g(120578

)

00

02

04

06

08

10N1 = N3 = 01 Re = 05 Peℎ = Pem = 02

H = R = 120574 = 2


g(120578

)

N3 = 01

N3 = 03

N3 = 05

N3 = 10


120578

00

02

04

06

08

10N1 = N2 = 01 Re = 05 Peℎ = Pem = 02 H = R = 120574 = 2




H = 0

H = 2

H = 5

H = 10

120579(120578

)

00

02

04

06

08

10


120578

N1 = N2 = N3 = 01

Re = 1 Peℎ = Pem = 02 R = 120574 = 2


120579(120578

)

R = 0

R = 1

R = 2

R = 4


120578

00

02

04

06

08

10

N1 = N2 = N3 = 01

Re = 05 Peℎ = Pem = 02 H = 120574 = 2





)

Peℎ = 0

Peℎ = 02

Peℎ = 05

Peℎ = 1

Peℎ = 2


120578

00

02

04

06

08

10

12

14

16

N1 = N2 = N3 = 01

Re = 05 Pem = 02 H = R = 120574 = 2


120601(120578

)


120578

00

02

04

06

08

10

Pem = 0

Pem = 02

Pem = 05

Pem = 1

Pem = 2

N1 = N2 = N3 = 01

Re = 1 Peℎ = 02 H = R = 120574 = 2


5 Conclusions



120601(120578

)

120574 = 2

120574 = 0 120574 = 5

120574 = 10

00

02

04

06

08

10


120578

N1 = N2 = N3 = 01

Re = 05 Peℎ = Pem = 02 H = R = 2


H = 0H = 5H = 10


Re

minus02

00

02

04

06N

u x







R = 0

R = 2

R = 4


Re

015

020

025

030

035

040

045

050

Nu x


Peh = 1

Peh = 2

Peh = 0


Re

minus15

minus10

minus05

00

05

10

Nu x




Nomenclature


Pem = 0

Pem = 1

Pem = 2


00

05

10

15

20

25

Shx

0 2 4 6 8 10


120574 = 0

120574 = 2

120574 = 5


Re

04

05

06

07

08

09

10

11

Shx






Greek Symbols


Competing Interests


Acknowledgments


References












































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






Re = minus2

Re = 1

Re = 2

Re = 5


120578

minus10

minus08

minus06

minus04

minus02

00

02

04

f998400 (120578)

N1 = N2 = N3 = 01 Peℎ = Pem = 02 H = R = 120574 = 2


g(120578

)


120578

00

02

04

06

08

10

Re = minus2

Re = 1

Re = 2

Re = 5

N1 = N2 = N3 = 01 Peℎ = Pem = 02 H = R = 120574 = 2


g(120578

)

N1 = 01

N1 = 03

N1 = 05

N1 = 07


120578

minus04

minus02

00

02

04

06

08

10N2 = N3 = 01 Re = 05 Peℎ = Pem = 02 H = R = 120574 = 2






N2 = 01

N2 = 03

N2 = 05

N2 = 10


120578

g(120578

)

00

02

04

06

08

10N1 = N3 = 01 Re = 05 Peℎ = Pem = 02

H = R = 120574 = 2


g(120578

)

N3 = 01

N3 = 03

N3 = 05

N3 = 10


120578

00

02

04

06

08

10N1 = N2 = 01 Re = 05 Peℎ = Pem = 02 H = R = 120574 = 2




H = 0

H = 2

H = 5

H = 10

120579(120578

)

00

02

04

06

08

10


120578

N1 = N2 = N3 = 01

Re = 1 Peℎ = Pem = 02 R = 120574 = 2


120579(120578

)

R = 0

R = 1

R = 2

R = 4


120578

00

02

04

06

08

10

N1 = N2 = N3 = 01

Re = 05 Peℎ = Pem = 02 H = 120574 = 2





)

Peℎ = 0

Peℎ = 02

Peℎ = 05

Peℎ = 1

Peℎ = 2


120578

00

02

04

06

08

10

12

14

16

N1 = N2 = N3 = 01

Re = 05 Pem = 02 H = R = 120574 = 2


120601(120578

)


120578

00

02

04

06

08

10

Pem = 0

Pem = 02

Pem = 05

Pem = 1

Pem = 2

N1 = N2 = N3 = 01

Re = 1 Peℎ = 02 H = R = 120574 = 2


5 Conclusions



120601(120578

)

120574 = 2

120574 = 0 120574 = 5

120574 = 10

00

02

04

06

08

10


120578

N1 = N2 = N3 = 01

Re = 05 Peℎ = Pem = 02 H = R = 2


H = 0H = 5H = 10


Re

minus02

00

02

04

06N

u x







R = 0

R = 2

R = 4


Re

015

020

025

030

035

040

045

050

Nu x


Peh = 1

Peh = 2

Peh = 0


Re

minus15

minus10

minus05

00

05

10

Nu x




Nomenclature


Pem = 0

Pem = 1

Pem = 2


00

05

10

15

20

25

Shx

0 2 4 6 8 10


120574 = 0

120574 = 2

120574 = 5


Re

04

05

06

07

08

09

10

11

Shx






Greek Symbols


Competing Interests


Acknowledgments


References












































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




N2 = 01

N2 = 03

N2 = 05

N2 = 10


120578

g(120578

)

00

02

04

06

08

10N1 = N3 = 01 Re = 05 Peℎ = Pem = 02

H = R = 120574 = 2


g(120578

)

N3 = 01

N3 = 03

N3 = 05

N3 = 10


120578

00

02

04

06

08

10N1 = N2 = 01 Re = 05 Peℎ = Pem = 02 H = R = 120574 = 2




H = 0

H = 2

H = 5

H = 10

120579(120578

)

00

02

04

06

08

10


120578

N1 = N2 = N3 = 01

Re = 1 Peℎ = Pem = 02 R = 120574 = 2


120579(120578

)

R = 0

R = 1

R = 2

R = 4


120578

00

02

04

06

08

10

N1 = N2 = N3 = 01

Re = 05 Peℎ = Pem = 02 H = 120574 = 2





)

Peℎ = 0

Peℎ = 02

Peℎ = 05

Peℎ = 1

Peℎ = 2


120578

00

02

04

06

08

10

12

14

16

N1 = N2 = N3 = 01

Re = 05 Pem = 02 H = R = 120574 = 2


120601(120578

)


120578

00

02

04

06

08

10

Pem = 0

Pem = 02

Pem = 05

Pem = 1

Pem = 2

N1 = N2 = N3 = 01

Re = 1 Peℎ = 02 H = R = 120574 = 2


5 Conclusions



120601(120578

)

120574 = 2

120574 = 0 120574 = 5

120574 = 10

00

02

04

06

08

10


120578

N1 = N2 = N3 = 01

Re = 05 Peℎ = Pem = 02 H = R = 2


H = 0H = 5H = 10


Re

minus02

00

02

04

06N

u x







R = 0

R = 2

R = 4


Re

015

020

025

030

035

040

045

050

Nu x


Peh = 1

Peh = 2

Peh = 0


Re

minus15

minus10

minus05

00

05

10

Nu x




Nomenclature


Pem = 0

Pem = 1

Pem = 2


00

05

10

15

20

25

Shx

0 2 4 6 8 10


120574 = 0

120574 = 2

120574 = 5


Re

04

05

06

07

08

09

10

11

Shx






Greek Symbols


Competing Interests


Acknowledgments


References












































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




)

Peℎ = 0

Peℎ = 02

Peℎ = 05

Peℎ = 1

Peℎ = 2


120578

00

02

04

06

08

10

12

14

16

N1 = N2 = N3 = 01

Re = 05 Pem = 02 H = R = 120574 = 2


120601(120578

)


120578

00

02

04

06

08

10

Pem = 0

Pem = 02

Pem = 05

Pem = 1

Pem = 2

N1 = N2 = N3 = 01

Re = 1 Peℎ = 02 H = R = 120574 = 2


5 Conclusions



120601(120578

)

120574 = 2

120574 = 0 120574 = 5

120574 = 10

00

02

04

06

08

10


120578

N1 = N2 = N3 = 01

Re = 05 Peℎ = Pem = 02 H = R = 2


H = 0H = 5H = 10


Re

minus02

00

02

04

06N

u x







R = 0

R = 2

R = 4


Re

015

020

025

030

035

040

045

050

Nu x


Peh = 1

Peh = 2

Peh = 0


Re

minus15

minus10

minus05

00

05

10

Nu x




Nomenclature


Pem = 0

Pem = 1

Pem = 2


00

05

10

15

20

25

Shx

0 2 4 6 8 10


120574 = 0

120574 = 2

120574 = 5


Re

04

05

06

07

08

09

10

11

Shx






Greek Symbols


Competing Interests


Acknowledgments


References












































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




R = 0

R = 2

R = 4


Re

015

020

025

030

035

040

045

050

Nu x


Peh = 1

Peh = 2

Peh = 0


Re

minus15

minus10

minus05

00

05

10

Nu x




Nomenclature


Pem = 0

Pem = 1

Pem = 2


00

05

10

15

20

25

Shx

0 2 4 6 8 10


120574 = 0

120574 = 2

120574 = 5


Re

04

05

06

07

08

09

10

11

Shx






Greek Symbols


Competing Interests


Acknowledgments


References












































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






Greek Symbols


Competing Interests


Acknowledgments


References












































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



research article influence of chemical reaction on heat and mass transfer flow...

Documents