International Journal of Non-Linear Mechanics 48 (2013) 37–43

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International Journal of Non-Linear Mechanics

0020-74

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Viscous flowing film instability down an inclined plane in the presenceof constant electromagnetic field

Kadry Zakaria a, Yasser Gamiel b,n

a Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egyptb Department of Engineering Mathematics and Physics, Faculty of Engineering, Tanta University, Tanta, Egypt

a r t i c l e i n f o

Article history:

Received 10 June 2012

Accepted 23 July 2012Available online 4 August 2012

Keywords:

Falling film

Electromagnetic field

Hopf bifurcation

Solitons

62/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.ijnonlinmec.2012.07.005

esponding author. Tel.: þ20 01006247152.

ail address: y_egcglass@hotmail.com (Y. Gam

a b s t r a c t

The present work deals with temporal stability properties of a falling liquid film down an inclined plane

in the presence of constant electromagnetic field. Using the Karman approximation, the problem is

reduced to the study of the evolution equation for the free surface of the liquid film derived through a

long-wave approximation. A linear stability analysis of the base flow is performed. Also, the solutions of

stationary waves and Shkadov waves are introduced and discussed analytically by analyzing the

linearized instability of the fixed points and Hopf bifurcation.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The influence of electric field on a thin liquid film produces asignificant class of problems that has attracted much attention ofseveral researchers due to its practical applications such as, innuclear energy equipments and different cooling systems. Also, inlaser cutting process, where the surface waves are undesirable atthe molten interface hence, applying a magnetic field counteractsthe inertia force, the instability could be prevented to maintainthe smooth flow. The presence of electric field introduces addi-tional physical effects on the flow dynamics such as body forcedue to a current in conducting fluids and the Maxwell stress at thefree interfaces. Gonzalez and Castellanos [1] studied the non-linear stability of a perfectly conducting film flowing down aninclined plane in presence of normal electric field. They derived anon-linear evolution equation within the limit of small Reynoldsnumber and conclude the destabilizing effect of the electric fieldin the finite amplitude. Recently, Mukhopadhyay and Dandapat[2] extended the study of Gonzalez and Castellanos [1] within theregime of large Reynolds number and confirmed the existence ofsubcritical unstable and supercritical stable zones.

Due to the relevant impact of the instability of flow down anincline, it is important to theoretically determine the criticalconditions for the onset of instability and to predict the evolutionof the unstable flow. By performing a linear stability analysis on thegoverning equations, one determines the range of the flow para-meters for which infinitesimal disturbances grow exponentially in

ll rights reserved.

iel).

time. However, as the disturbances grow the non-linear interac-tions become significant and a non-linear analysis must be con-ducted in order to predict the subsequent evolution.

It turns out that, for our model we are interested in, theelevation of the waves remains small when compared to theirwavelength. This motivates a long standing practice [3] of study-ing them by means of asymptotic expansions in powers of a smallparameter E usually called the film parameter.

The current paper is organized as follows. In Section 2, theproblem formulation and the evolution equations are derivedaccurate to order OðEÞ, without the assumption of hydrostaticpressure. Karman’s momentum integral method is used. Linearstability analysis of the interfacial waves is discussed in Section 3.In Section 4, the solutions of stationary waves and Shkadov wavesare introduced and discussed by analyzing the linearized instabil-ity of the fixed points and Hopf bifurcation. The finial section isconcerned with the concluding remarks.

2. The problem statement

2.1. Mathematical formulation

In this work we considered a flow of thin viscous conductingliquid film of thickness h0 down an inclined non-conducting flatplane of inclination y with the horizon under the action of gravityin the presence of an electromagnetic field. The full range ofinclination is allowed so that 0oyrp=2. A Cartesian co-ordinatesystem is introduced such that the x-axis coinciding withthe plane bottom and the y-axis pointing vertically upwards fromthe inclined plane, the geometry of the flow is depicted in Fig. 1.

y

x

z

θ

0

0

x

y

h0

hu

E0

B0

Fig. 1. Schematic representation of thin film flow down an inclined plate.

K. Zakaria, Y. Gamiel / International Journal of Non-Linear Mechanics 48 (2013) 37–4338

The magnetic field acts parallel to the y-axis and the electric fieldacts normal to the x–y plane. Therefore, the electric and magneticfields are taken as E0 ¼ ð0,0,E0Þ and B0 ¼ ð0,B0,0Þ, respectively.Dynamic influence of air flow above the film is ignored. Thegoverning equations in cartesian coordinates are

r � U¼ 0, ð1Þ

r @U

@tþU � rU

� �¼�rPþmr2Uþ J� Bþrg, ð2Þ

where g is the gravitational acceleration, r is the density, mdenotes the dynamic viscosity and U¼ ðu,vÞ is the liquid filmvelocity vector. The electromagnetic body force J� B where J andB are the current density and the magnetic field, respectively. Thecurrent density is defined from Ohm’s law as

J¼ sðEþU� BÞ, ð3Þ

where s is the electrical conductivity, the fields E and B aredefined by Maxwell’s equation.

2.2. Boundary conditions

(i) The no-slip condition: At the plate surface we have

U¼ ðu,v,0Þ ¼ 0 at y¼ 0: ð4Þ

(ii) The kinematical condition: We first specify a kinematic bound-ary condition: fluid particles can only move tangentially to thefluid interface i.e. the normal velocity at the interface vanished.The function that defines the perturbed interface is F¼y�h, hencethe kinematical condition is

DF

Dt¼ 0 at y¼ h: ð5Þ

(iii) Tangent condition: The shear stress on the free surface mustbe vanished [4], then the continuity of shear stress condition onthe surface is

n � S � t¼ 0 at y¼ h, ð6Þ

where S is the stress

S¼�PIþs, ð7Þ

where P is the liquid pressure, I is the identity tensor and s isthe viscous stress tensor, whose individual components are

sij ¼12 ð@Vi=@xjþ@Vj=@xiÞ. The outward normal unit vector to the

interface is given from the relation, n¼rF=9rF9, and has the

following explicit form:

n¼ð�E@h=@x,1,0Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þE2ð@h=@xÞ2

q : ð8Þ

Then, a tangent unit vector could be written as:

t¼ð1,E@h=@x,0Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þE2ð@h=@xÞ2

q : ð9Þ

(V) Normal condition: The problem that we focus on takes theeffect of surface tension of the perturbed surface into considera-tion. So, we can write the dynamical condition as follows:

n � S � n¼�Tr � n at y¼ h, ð10Þ

where T is the surface tension coefficient.Before solving the problem we want to rewrite the problem

precisely in dimensionless form. We define the dimensionlessquantities as

x¼ l0xn, ðh,yÞ ¼ h0ðhn,ynÞ, t¼ ðl0=u0Þt

n,

P¼ ru20Pn,u¼ u0un, v¼ ðh0=l0Þu0vn:

The variables that are associated with a superscript ‘‘n’’ stand for adimensionless quantities, where we assume l0 as the character-istic longitudinal length scale whose order may be considered tobe same as the wavelength, the undisturbed film thickness h0 asthe length scale in transverse direction i.e. the film parameter,E� h0=l0 is small and u0 as the characteristic velocity.

We reduce the governing equations and the boundary condi-tions correct to zeroth and first order in the small parameter E andafter dropping the n for reader convenience as

@u

@xþ@v

@y¼ 0, ð11Þ

E @u

@tþu

@u

@xþv

@u

@y

� �¼�E @P

@xþ

1

Re

@2u

@y2�uM2

�w� �

, ð12Þ

0¼@P

@yþ

3 cot yRe

, ð13Þ

where the Reynolds number Re ¼ rh0u0=m, the Hartmann number

M¼ B0h0

ffiffiffiffiffiffiffiffiffis=m

p, Ep ¼ E0=B0u0 is the electric parameter and

w¼M2Ep�3.

Let us point out here that our interest in the present problemis to study the hydrodynamic stability of the conducting thin filmflow in the presence of electromagnetic field within the range ofsmall M, (M2

51). At the plate surface y¼0

u¼ 0, v¼ 0: ð14Þ

At the interface surface y¼h

@h

@tþu

@h

@x�v¼ 0, ð15Þ

@u

@y¼ 0, ð16Þ

PþE2

We

@2h

@x2¼ 0, ð17Þ

where, the Weber number We ¼ rh0u20=T .

The order of magnitude assignment E2=We �Oð1Þ, correspondsto the so-called ‘‘strong surface tension limit’’ frequently invokedin falling film studies, implying that the surface tension is ofleading-order importance in (17). It is evident that for a wide

K. Zakaria, Y. Gamiel / International Journal of Non-Linear Mechanics 48 (2013) 37–43 39

range of situations, the Reynolds number is high [5]. Therefore inthis work we shall assume

Re ¼OðE�1Þ, E2=We ¼Oð1Þ:

The pressure distribution in the flow layer can be deduced byintegrating the y-momentum Eq. (13) with respect to y from 0 toh, and applying the normal stress boundary condition (17) yields:

Pðx,y,tÞ ¼3 cot y

Reðh�yÞ�

E2

We

@2h

@x2: ð18Þ

System (11)–(17) admits a trivial solution corresponding to asteady constant-thickness film, often called the Nusselt solution.Assuming @=@t� 0 and @=@x� 0, a film of thickness h0 yields

uðyÞ ¼w

M2ðcoshðM�MyÞsech M�1Þ,

PðyÞ ¼3 cot y

Reðh�yÞ:

9>>=>>; ð19Þ

Laboratory experiments show that the Nusselt solution may notbe relevant, being possibly unstable against waves at the surfaceof the film. However, for low flow rates (or, equivalently, lowvalues of the Reynolds number), the interface remains smooth atthe scale of the film thickness as measured locally by h(x,t).

Defining the local instantaneous flow rate as

qðx,tÞ ¼

Z h

0uðx,y,tÞ dy: ð20Þ

Now, the idea is to correct the parabolic velocity profile bylinearly independent combinations of polynomials appropriatelychosen for the ease of algebraic calculations. This way a fullagreement with the long wave theory can be expected byconstruction. Let us then expand u as

uðx,y,tÞ ¼ A1ðx,tÞþA2ðx,tÞeMyþA3ðx,tÞeMy, ð21Þ

where the unknown fields A1, A2 and A3 are supposed to be slowlyvarying functions of x and t, which are utterly determined usingEqs. (14), (16) and (20) as, respectively,

A1ðx,tÞ ¼ð�M4h4

þ210M2h2þ525Þq

175M2h3,

A2ðx,tÞ ¼ðM4h4

þ35M3h3�210M2h2

þ525Mh�525Þq

350M2h3,

A3ðx,tÞ ¼ðM4h4

�35M3h3�210M2h2

�525Mh�525Þq

350M2h3,

9>>>>>>>>>=>>>>>>>>>;

ð22Þ

which combined with (21) gives

uðx,y,tÞ ¼yð2h�yÞð5M2y2�10M2hyþ4h2M2

þ60Þq

40h3: ð23Þ

Noteworthy is that for M-0, we obtain semiparabolic velocityprofile corresponding to the viscous film flow.

The velocity component in the y-direction is obtained fromEqs. (11), (14) and (23) as

vðx,y,tÞ ¼y2

40h4ðð2h�yÞð3M2y2�4M2hyþ60Þqhxþ½M

2hy3�5M2h2y2

þ4hð2M2h2þ5Þy�4h2

ðM2h2þ15Þ�qxÞ: ð24Þ

In this work we use the Karman approximation to obtain theevolution equations of the flow. Integrating (11) over the interval(0,h) then using the no slip condition Eq. (14), the kinematicalcondition Eq. (15) and the local instantaneous flow rate Eq. (20),we arrive at the integral condition

qxþht ¼ 0: ð25Þ

Similarly, from the integrated x-momentum Eq. (12) one gets

qtþ3ðM2h2

þ420Þ

1400hqqxþ

3ð2M2h2þ5Þ

5h2ERe

q

þðM2h2

�60Þ

40hqht þ

3h cot yRe

�2ðh2M2

þ105Þq2

175h2

!hx

�E2

Wehhxxxþ

hðM2Ep�3Þ

ERe¼ 0, ð26Þ

where the subscripts x, t and xxx are used to represent variouspartial derivatives of the associated underlying variable. Eqs. (25)and (26) form a set of two differential equations in (x,t) for theperturbed film thickness h and the local instantaneous flow rate q.

The numerical simulations of (25) and (26) are investigated byDemekhin and Shkadov, Chang et al. [6,7], it is found that thelong-wave approximation is always obeyed, the blow-up behavioris never observed and E remains small.

3. Linear analysis

For checking with experiments and for later reference, wesketch the standard analysis on the instability of the uniform flow(h¼1, q¼q0) when disturbed by an infinitesimal wavy distur-bance. Let Z and ~q denote infinitesimal disturbances from theuniform flow, i.e.

h¼ 1þZ, q¼ q0þ ~q, Z, ~q51: ð27Þ

Consider a wave-like disturbance

fZðx,tÞ,qðx,tÞg ¼ fc1,c2geiðkx�otÞ, ð28Þ

where c1 and c2 are real quantities and o is the complexfrequency. We are interested on the temporal stability, hencethe wave number k is considered real. Using Eqs. (25)–(28) andelimination of, q0 ¼ 1�M2 2

5 þEp=3� �

, yields the following coupledequations for o:

�o k

B1 B2

!c1

c2

!¼ 0, ð29Þ

where

B1 ¼3ð5M2Epþ4M2

�15Þ

5EReþ i

E2k3

Weþ

166kM2

175þ

3k cot yRe

þ4

5kM2Ep�

6k

5�

1

8oð4M2Epþ5M2

�12Þ

�,

B2 ¼3ð2M2

þ5Þ

5ERe�i oþ3kð140M2Epþ167M2

�420Þ

1400

!:

9>>>>>>>>>>=>>>>>>>>>>;

ð30Þ

Equating the determinant with zero yields the following disper-sion relation:

o2þðB3þ iB4ÞoþðB5þ iB6Þ ¼ 0, ð31Þ

where

B3 ¼4

175kðM2

ð35Epþ43Þ�105Þ,

B4 ¼3ð2M2

þ5Þ

5ERe,

B5 ¼1

175k2 210�M2

ð140Epþ166Þ�175E2k2

We�

525 cot yRe

!,

B6 ¼3kð5M2Epþ4M2

�15Þ

5ERe:

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;ð32Þ

0.0 0.2 0.4 0.6 0.8 1.0

1

2

5

10

20

M→

Re/c

ot�

Unstable

Stable

Ep = 0

Ep = 0.6

Ep = 1.2

Fig. 3. Neutral curves in the Re=cot y versus M plane for various electric

parameter Ep.

K. Zakaria, Y. Gamiel / International Journal of Non-Linear Mechanics 48 (2013) 37–4340

Solving Eq. (31) we obtain the roots of the frequency o as follows:

o¼ 12½�B3�iB47

ffiffiffiffiffiffiffiffiffiffiffiffiaþ ib

p�, ð33Þ

where

a¼ B23�B2

4�4B5, b¼ 2B3B4�4B6:

From this we obtain,

or ¼12½�B37

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12ð

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2þb2

qþaÞ

r�, ð34Þ

and

oi ¼12½�B47

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12ð

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2þb2

q�aÞ

r�, ð35Þ

where or and oi represent the real and imaginary parts of Eq.(33) respectively [8]. According to Eq. (28) the flow is stable if thegrowth rate oio0, otherwise it turns to unstable state.

We deduce first the marginal stability conditions whichcorrespond to the wave number k with null growth rate, i.e. oi¼0

Re

cot y¼�

3ð4M2þ5Þ

k2ð4M2

þ5ÞE2

WeþM2

ð10Epþ14Þ�15

: ð36Þ

Substitution of M¼0 and k¼1 the above condition then gives

Re

cot y¼ 1þ

E2

3We: ð37Þ

Therefore, the condition (37) fully agrees with the Lee and Meimodel (6) [5], as all neglected terms are of order higher than Eexcept for the term containing We, i.e., by construction.

The variation of the Reynolds number Re with inclination angley, for the following dimensionless parameters; E¼ 0:3,k¼ 0:2,We ¼ 0:01 and Ep¼0.3 for various values of the Hartmann numberM is depicted in Fig. 2. One observes from this figure that thestable area increases in a monotonic fashion with the increase inthe Hartmann number M, especially for higher values of inclina-tion angle. Moreover, in the case of M¼0, the fluid flow is a simplefilm flow with no applied magnetic field [5]. Recall that theelectromagnetic force, like that of gravity, is a body force indepen-dent of the velocity of flow. Unlike gravity, it is also independent ofthe inclination of the film flow. Thus the electromagnetic force

M = 0

M = 0.4

M = 0.6M

0.0 0.2 0.4 0.6 0.8 1.0

1

2

5

10

20

50

100

200

Re

�/�→

Stable

Unstable

Fig. 2. Neutral curves in the Re versus y plane for various Hartmann number

values.

stabilizes the flow when gravity is ineffective, as noted when theinclination angle is increased.

Fig. 3 shows that an increase in the electric parameter, Ep,induces an expansion of the area of the unstable region. It is clearthat as Ep increases for any fixed value of M, the critical Reynoldsnumber decreases implying the destabilizing role of Ep, whileit increases with the increase in M exhibiting the stabilizingrole of M.

From a simple geometric point of view a physical explanationcan be presented as follows: For the basic flow state there is onlydownstream component. While, in the perturbed state conditionthe magnitude of the downstream velocity component is muchlarger than the transverse velocity component. The connectionbetween part of Lorentz force and the electric field, accelerate theflow in the downstream direction. While the other part of theLorentz force, in response to the interaction of the velocity andthe magnetic field, is directed upstream to face the downstreamflow. The magnetic lines of force act like elastic strings which tendto resist any deviation from the mean flow due to the perturba-tion motion. As M increases, the field strength increases toprovide more restoring force in suppressing disturbances. Toconclude, the electrical field destabilizes the film flow whereasmagnetic field stabilizes it.

4. Traveling wave solutions

Traveling waves are computed as stationary solutions in areference frame moving at the speed of the wave, denoted c. Toobtain the equations governing the traveling wave solutions weintroduce the moving coordinate transformation, x¼ x�ct, andwe set @t ¼�c@x for the waves to be stationary in the movingframe.

In this section, our main interest focuses on transformation ofthe approximate system (25) and (26) to the moving coordinatesystem. To this end, we consider a particular type of wave formthat travels at a constant celerity c without changing its shape.This type of solutions which are characterized by their celerityexists under suitable conditions, when the influence of dispersionis balanced exactly by the effect due to the nonlinearity. Changingthe parameter c in some range allows to explore the entirespectrum of asymptotic behavior of these traveling waves. Thekinematic condition Eq. (25) is transformed to qðxÞ�chðxÞ ¼ C,where C is the integration constant. Since q¼ q0 for h¼1 (becauseof our scaling), one obtain:

qðxÞ ¼ q0þcðhðxÞ�1Þ: ð38Þ

Lee and Mei [5]Present Model

0 1 2 3 4 50

1

2

3

4

5

c

H1

H(1)

H(2)

Fig. 4. Fixed points of the present model compared with that obtained by Lee and

Mei model [5].

K. Zakaria, Y. Gamiel / International Journal of Non-Linear Mechanics 48 (2013) 37–43 41

Therefore, eliminating q in favor of h from (26), yielding a third-order ordinary differential equation for the flow depth h

E2h3

Wehxxxþ

1

1756M2c2h4

� 8M2cðc�1Þþ525 cot y

Re

� �h3

�

þð2M2ðc�1Þ2�35c2Þh2

þ14ðc�1Þð10EpM2

þ3ð4M2þ5c�5ÞÞ

ihx þ

1

5ERe½ð15�ð6cþ5EpÞM

2Þh3

þ6ðc�1ÞM2h2�15chþð15ðc�1Þþð6þ5EpÞM

2Þ� ¼ 0: ð39Þ

Introducing new state variables

H1 ¼ h, H2 ¼ hx, H3 ¼ hxx: ð40Þ

The ordinary differential Eq. (39) can be recast into the dynamicalsystem

dH1

dx¼H2,

dH2

dx¼H3,

dH3

dx¼

G1ðH1,H2; E,Re,cot y=Re,Ep,M,cÞ

G2ðH1; E,WeÞ, ð41Þ

with

G1 ¼ �6c2M2H41þ 8cðc�1ÞM2

þ525cot y

Re

� �H3

1

�

þð35c2�2ðc�1Þ2M2ÞH2

1�14ðc�1Þð2ð5Epþ6ÞM2þ15ðc�1ÞÞ

iH2

þ35

ERe½ðM2ð6cþ5EpÞ�15ÞH3

1�6ðc�1ÞM2H21

þ15cH1�ðð5Epþ6ÞM2þ15ðc�1ÞÞ, ð42Þ

G2 ¼ 175E2

WeH3

1: ð43Þ

Physically H1 corresponds to the flow depth, H2 to the surfaceslope, and H3 to the surface curvature.

The fixed points of Eq. (41) are obtained as solutions to

½15�M2ð6cþ5EpÞ�H

31þ6ðc�1ÞM2H2

1�15cH1þ15ðc�1ÞþM2ð5Epþ6Þ ¼ 0,

ð44Þ

c4max5

2M2�

5Ep

6,1�

2M2

5�

M2Ep

3,15þð6�5EpÞM

2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi225�30ð5Ep

q12 M2

8<:

co15þð6�5EpÞM

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi225�30ð5Epþ3ÞM2

q12M2

, 8Epo3

M2�

6

5:

8>>>>>>><>>>>>>>:

which gives three solutions

Hð1Þ : ðH1,H2,H3Þ ¼ ð1,0,0Þ,

Hð2Þ : ðH1,H2,H3Þ ¼�a1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

1�4a0a2

q2a0

,0,0

0@

1A,

Hð3Þ : ðH1,H2,H3Þ ¼�a1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

1�4a0a2

q2a0

,0,0

0@

1A,

9>>>>>>>>>>=>>>>>>>>>>;

ð45Þ

where

a0 ¼ ð6cþ5EpÞM2�15,

a1 ¼ ð6þ5EpÞM2�15,

a2 ¼ ð6þ5EpÞM2þ15ðc�1Þ:

It is worth noting that:

(i)

ffiffiffiffiffiffiffiffiffiffiþ3Þ

in addition to the trivial fixed point H(1)¼(1, 0, 0), corre-

sponds to the uniform primary flow, at M¼0.2 and Ep¼0.4the stationary wave solution exhibits an additional positivereal solution H(2) which does really exist only for c4 3

4.Hence, the dynamical system (41) admits two fixed points,H(1) and H(2).

(ii)

Eqs. (45) show that the fixed points are independent of theparameters y, E, Re and We and that the fixed points cor-respond to the uniform primary flow depending on the electricparameter Ep and the Hartmann number M.

(iii)

as M tends to zero, the fixed points H(1) and H(2) reduced tothat obtained by Lee and Mei (4.7) and (4.8) [5].

Fig. 4 shows the fixed points obtained from the present modelat M ¼0.2, Ep¼0.4 and that presented by Lee and Mei [5]. Theyhave shown that the fixed point H(2) is a function of c and is realand positive only for c41, at c¼3, Hð1Þ ¼Hð2Þ and so the fixedpoints cross each other. This suggests that a c¼3 is a transcriticalbifurcation at which the two fixed points should exchange theirstability properties. For the model derived in this work with aconstant prescribed electromagnetic field, the fixed points crosseach other twice at c¼0.75 and 2.908.

4.1. Existence of fixed points

In this section we examine the solutions of Eq. (44) to obtainthe conditions required for these solutions to be real and positive,i.e. we discuss the existence of fixed points. According to Eqs. (45)the solutions will be positive real only for:

(i)

a040, a1o0, a240 and a142ffiffiffiffiffiffiffiffiffiffia0a2p

, then we have threefixed points

HðjÞ ¼ ð1,0,0Þ,

ffiffiffiffiffia2

1

q8

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

1�4a0a2

q2a0

,0,0

0@

1A

8<:

9=;, j¼ 1,2,3,

satisfying the following inequalities:

ffiffiffiffiffiffiffiM29=;,

ð46Þ

K. Zakaria, Y. Gamiel / International Journal of Non-Linear Mechanics 48 (2013) 37–4342

a0o0, a1o0 and a240, then we have two fixed points

(ii)

HðjÞ ¼ ð1,0,0Þ,

ffiffiffiffiffia2

1

q�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

1þ4ffiffiffiffiffia2

0

qa2

r

�2ffiffiffiffiffia2

0

q ,0,0

0BB@

1CCA

8>><>>:

9>>=>>;, j¼ 1,3,

such that

1�2 M2

5�

M2Ep

3oco

5

2M2�

5Ep

68Epo

3

M2�

6

5: ð47Þ

Ep = 0.8

Ep = 1.2

0.0 0.1 0.2 0.3 0.4 0.5

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3.0

M

c Hop

f

Ep = 0.4

Fig. 6. Plot showing the effect of the Hartmann number M and the electric

parameter Ep on stationary wave Hopf speed cHopf.

Fig. 5 clarifies the existence of fixed points derived fromEq. (45), for M¼0.4 and Ep¼0.8. In Fig. 5(a) it is clearly shown thatthe system exhibits three fixed points according to the first case,satisfying the inequalities (46), i.e. 15:1764c414:958. Whereaspart(b) of Fig. 5 illustrates the second case exhibiting two fixedpoints, satisfying the inequalities (47) for 14:9584c40:893.

To show the dynamical behavior of the previous fixed pointslet us consider the asymptotic solutions near these points. Nearthe fixed points, the solution can be represented in the formhðxÞ ¼HðjÞ1 þ

~H , j¼ 1,2,3, where ~H51. Substituting it into Eq. (39)and neglecting the terms of higher order of ~H , we obtain

~HxxxþG3~HxþG4

~H ¼ 0, ð48Þ

with the characteristic equation

O3þG3OþG4 ¼ 0, ð49Þ

where

G3 ¼We

175E2ReðHðjÞ1 Þ

3ð6c2 M2ReðH

ðjÞ1 Þ

4�½8cðc�1ÞReM2

þ525 cot y�ðHðjÞ1 Þ3þRe½ð2M2

�35Þc2�4M2cþ2M2�ðHðjÞ1 Þ

2

þ42Re½5c2þ2ð2M2�5Þc�4M2

þ5�þ140Reðc�1ÞM2EpÞ,

G4 ¼�3We

5E3ReðHðjÞ1 Þ

3ð½ð6cþ5EpÞM

2�15�ðHðjÞ1 Þ

2�4M2

ðc�1ÞHðjÞ1 þ5cÞ:

Eq. (49) has the roots

O1 ¼ Z1þZ2, O2,3 ¼�1

2ðZ1þZ2Þ7 i

ffiffiffi3p

2ðZ1�Z2Þ,

where

Zk ¼ ð�1Þk�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiG3

3

� �3

þG4

2

� �2s

�G4

2

0@

1A

1=3

, k¼ 1,2:

From the Cardano formulas it follows that Z1þZ240 if G4o0and vice versa. It means that the stability of the fixed pointscan be classified according to the sign of G4. The characteristicEq. (49) describing the behavior of the small solutions near the

xx x

x

xx

T

15.00 15.05 15.10 15.15 15.20 15.25 15.301

510

50100

c

H1

H(1)

H(3)

H(2)

Fig. 5. Plot showing the existence of fixed points derived from Eq. (45

fixed points. On this basis we can conclude that if G4o0, there isa one-dimensional unstable manifold and two-dimensional stablemanifold. Otherwise, if G440, there is a one-dimensional stablemanifold and two-dimensional unstable manifold.

The eigenvalue properties of the fixed points are displayed inFig. 5. The positions of the eigenvalues in the complex planeðRðOÞ,IðOÞÞ are indicated by crosses and transcritical (T) bifurca-tion is observed at c¼15.175 and 2.55 for parts (a) and (b) of thesame figure respectively.

Needless to say, there is a close relation between the linearstability analysis of the flat film and the stability analysis of thefixed points. In fact, the changes @x-ik and @t-io associatedwith the linear stability of the flat film are formally equivalent tothe sequence @x-@x and @t-�c@x, respectively.

4.2. Hopf bifurcations

The term Hopf bifurcation refers to the local birth or deathof a periodic solution from an equilibrium as a parameter crossesa critical value. The Andronov–Hopf bifurcation of an equilibriumis characterized by one bifurcation condition, namely, the pre-sence of a purely imaginary pair of eigenvalues at this equili-brium.At Z1þZ2 ¼ 0, we have two imaginary conjugate roots, O1,2 ¼

7 iðffiffiffi3p

=2ÞðZ1�Z2Þ. In this case, the Hopf bifurcation is a dominatestate. To this end, G4 ¼ 0 reveals the following condition:

c¼5ð3�M2EpÞðH

ðjÞ1 Þ

2�4M2HðjÞ1

6M2ðHðjÞ1 Þ

2�4M2HðjÞ1 þ5

: ð50Þ

xx x

xx

x

T

0 2 4 6 8 10 12 14

1.0

0.5

2.0

0.2

5.010.0

20.0

c

H1

H(3)

H(1)

), (a) and (b) represent the first and the second case respectively.

K. Zakaria, Y. Gamiel / International Journal of Non-Linear Mechanics 48 (2013) 37–43 43

According to the first case of existence of fixed points, thecondition for Hð2Þ1 and Hð3Þ1 intersection is

M2¼

15ð4c�3Þ

2ð12c2þ10Epc�12c�15Ep�6Þ, ð51Þ

using conditions (51) and (50), after careful mathematical calcu-lation, the condition for the first case to exhibit a Hopf bifurcationis obtained and presented as

c¼20þð6�5EpÞM

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi225�30ð5Epþ3ÞM2

q12M2

: ð52Þ

It is clearly shown that the above condition does not satisfy theinequalities (46), which means that in the first case there is noopportunity for the Hopf bifurcation to occur.

For the second case, substituting Hð1Þ1 ¼Hð3Þ1 ¼ 1 into condition(50) noting that M2

51 yields,

cHopf ¼ 3�ð2þEpÞM2: ð53Þ

Therefore, condition (53) fully agrees with the inequality (47),hence the intersection of the two fixed points in the second caseadmits a transcritical Hopf bifurcation such that cHopf satisfyingcondition (53).

The graph of cHopf as a function of the Hartmann number M forseveral values of the electric parameter Ep is presented in Fig. 6. Itis clear that as Ep increases for any fixed value of M, cHopf increasesimplying the destabilizing role of the electrical field on the filmflow, as it increases the stationary wave speed.

It turns out that as M tends to zero, the above condition isreduced to cHopf ¼3, as we saw earlier in Fig. 4, in agreement withthe model presented by Lee and Mei [5].

4.3. Shkadov solitary-wave solutions

In this section, we present the general solution for Eq. (48) byfollowing Shkadov et al. method [9,10]. To do this, we introducethe following general solution form:

~H ¼ C1eðZ1þZ2ÞxþC2eð�ðZ1þZ2Þ=2Þxcos

ffiffiffi3p

2ðZ1�Z2Þx�c

!, ð54Þ

where C1,C2 and c are the solution constants. It is worth notingthat there are two possible cases:

(1)

Fast wave, c4 ð5ð3�M2EpÞðHðjÞ1 Þ

2�4M2HðjÞ1 Þ=ð6ðH

ðjÞ1 Þ

2M2

�4M2HðjÞ1 þ5Þ, with the following asymptotic solutions forupstream and downstream respectively;

~H ¼ C1eðZ1þZ2Þx as x-�1,

~H ¼ C2eð�ðZ1þZ2Þ=2Þxcos

ffiffiffi3p

2ðZ1�Z2Þx�c

!as x-1:

9>>=>>;ð55Þ

(2)

Slow wave, coð5ð3�M2EpÞðHðjÞ1 Þ

2�4M2HðjÞ1 Þ=ð6ðH

ðjÞ1 Þ

2M2

�4M2HðjÞ1 þ5Þ, with the following asymptotic solutions forupstream and downstream respectively:

~H ¼ C2eð�ðZ1þZ2Þ=2Þx cos

ffiffiffi3p

2ðZ1�Z2Þx�c

!as x-�1,

~H ¼ C1eðZ1þZ2Þx as x-1:

9>>=>>;ð56Þ

The solitary wave is a homoclinic trajectory in the phasespace.

5. Concluding remarks

In this work, we have studied the stability and evolution oflong interfacial waves of a flow of thin viscous conducting liquidfilm down an inclined non-conducting flat plane under the actionof gravity in the presence of an electromagnetic field. UsingKarman approximation, an evolution equation for the film thick-ness profile was derived. In the linear stability stage, a uniformflow of constant depth is stable with respect to the infinitesimalwavy disturbances propagating with the conditional phase velo-city. The linear stability analysis renders the neutral stabilitycurve which separates stable and unstable regions. It is found thatthe electrical field destabilizes the film flow whereas magneticfield stabilizes it. In the non-linear behavior, the attention isfocused on stationary waves of finite amplitude of the third-order.A third-order dynamical system is obtained after changing to theframe of reference moving at the wave propagation speed. Theexistence of fixed points has been studied, as a especial case,when M tends to zero, reduced to that obtained by Lee andMei [5]. The characteristic equation of the dynamical system isobtained to extract the temporal instability cases. The Hopfbifurcation is a dominate state in some cases. We have namedsome cases of a stationary profile as the Shkadov wave. The waveis a homoclinic trajectory in the phase space in some cases.

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