Hindawi Publishing CorporationISRNMathematical PhysicsVolume 2013, Article ID 748613, 24 pageshttp://dx.doi.org/10.1155/2013/748613
Research ArticleAir-Aided Shear on a Thin Film Subjected to a TransverseMagnetic Field of Constant Strength: Stability and Dynamics
Mohammed Rizwan Sadiq Iqbal
Department of Mathematics and Statistics, University of Constance, 78457 Constance, Germany
Correspondence should be addressed to Mohammed Rizwan Sadiq Iqbal; [email protected]
Received 28 June 2013; Accepted 22 August 2013
Academic Editors: L. E. Oxman and W.-H. Steeb
Copyright ยฉ 2013 Mohammed Rizwan Sadiq Iqbal. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.
The effect of air shear on the hydromagnetic instability is studied through (i) linear stability, (ii) weakly nonlinear theory, (iii)sideband stability of the filtered wave, and (iv) numerical integration of the nonlinear equation. Additionally, a discussion on theequilibria of a truncated bimodal dynamical system is performed. While the linear and weakly nonlinear analyses demonstrate thestabilizing (destabilizing) tendency of the uphill (downhill) shear, the numerics confirm the stability predictions. They show that(a) the downhill shear destabilizes the flow, (b) the time taken for the amplitudes corresponding to the uphill shear to be dominatedby the one corresponding to the zero shear increases with magnetic fields strength, and (c) among the uphill shear-induced flows, ittakes a long time for the wave amplitude corresponding to small shear values to become smaller than the one corresponding to largeshear values when the magnetic field intensity increases. Simulations show that the streamwise and transverse velocities increasewhen the downhill shear acts in favor of inertial force to destabilize the flowmechanism. However, the uphill shear acts oppositely.It supports the hydrostatic pressure and magnetic field in enhancing films stability. Consequently, reduced constant flow rates anduniform velocities are observed.
1. Introduction
A nonlinear fourth-order degenerate parabolic differentialequation of the form:
โ๐ก+A (โ) โ
๐ฅ+
๐
๐๐ฅ{B (โ) โ
๐ฅ+C (โ) โ
๐ฅ๐ฅ๐ฅ} = 0, (1)
whereA,B, andC are arbitrary continuous functions of theinterfacial thickness โ(๐ฅ, ๐ก), represents a scalar conservationlaw associated to the flow of a thin viscous layer on an inclineunder different conditions. The stability and dynamics ofequations of the type of (1) is a subject of major interest [1โ18] because of their robustness in regimes where viscositydominates inertia [19]. Such studies have focused attentionprimarily on the isothermal and nonisothermal instabilityanalysis, mainly for nonconducting fluids.
Since the investigation of Chandrasekar [20] on thestability of a flow between coaxial rotating cylinders in thepresence of a magnetic field held in the axial direction, thelaminar flow of an electrically conducting fluid under the
presence of a magnetic field has been studied extensively. Forinstance, Stuart [21] has reported on the stability of a pressureflow between parallel plates under the application of a parallelmagnetic field. Among other earlier investigations, Lock [22]examined the stability when the magnetic field is appliedperpendicular to the flow direction and to the boundaryplanes. Hsieh [23] found that the magnetic field stabilizesthe flow through Hartmann number when the electricallyconducting fluid is exposed to a transverse magnetic field,provided that the surface-tension effects are negligible ina horizontal film. Ladikov [24] studied a problem in thepresence of longitudinal and transverse magnetic fields.The author observed that the longitudinal magnetic fieldplays a stabilizing role and that the effect of instability atsmall wave numbers could be removed if the longitudinalmagnetic field satisfies certain conditions. Lu and Sarma [25]investigated the transverse effects of the magnetic field inmagnetohydrodynamic gravity-capillary waves. The flow ofan electrically conducting fluid over a horizontal plane inthe presence of tangential electric and magnetic fields was
2 ISRNMathematical Physics
reported byGordeev andMurzenko [26].They found that theflow suffers from instability not due to the Reynolds numberbut due to the strength of the external electric field.
In applications such as magnetic-field-controlled mate-rial-processing systems, aeronautics, plasma engineering,MEMS technology, andmagnetorheological lubrication tech-nologies, the hydromagnetic effects are important. Also,liquid metal film flows are used to protect the solid structuresfrom thermonuclear plasma in magnetic confinement fusionreactors, and this application requires a better understandingof the instability mechanism arising in a thin magnetohydro-dynamic flow over planar substrates [27]. Furthermore, thepresence of an external magnetic field regulates the thicknessof a coating film. It prevents any form of direct electrical ormechanical contact with the fluid thereby reducing the riskof contamination [28]. Renardy and Sun [29] pointed outthat the magnetic fluid is effective in controlling the flowof ordinary fluids and in reducing hydraulic resistance. Thereason is that the magnetic fluids can be easily controlledwith external magnetic fields and that coating streamlinedbodies with a layer of less-viscousmagnetic fluid significantlyreduces the shear stress in flow boundaries. Magnetohydro-dynamic flow can be a viable option for transporting weaklyconducting fluids in microscale systems such as flows insidea micro-channel network of a lab-on-a-chip device [30, 31].In all of the above applications, considering the associatedstability problem is important because it gives guidance inchoosing the flow parameters for practical purposes. In thisregard, in the past two decades, the emerging studies onthe hydromagnetic effects have focused their attention onstability problems and on analyzing the flow characteristics[28, 29, 32โ43].
The shearing effect of the surrounding air on the fluidin realistic situations induces stress tangentially on the inter-facial surface. The hydrodynamic instability in thin films inthe presence of an external air stream attracted attention inthe mid 1960s, which led Craik [44] to conduct laboratoryresearch and study theory. He found that the instabilityoccurs regardless of the magnitude of the air stream whenthe film is sufficiently thin. Tuck and Vanden-Broeck [45]reported on the effect of air stream in industrial applica-tions of thin films of infinite extent in coating technology.Sheintuch and Dukler [46] did phase plane and bifurcationanalyses of thin wavy films subjected to shear from counter-current gas flows and found satisfactory agreement of theirresults connected to the wave velocity along the floodingcurvewith the experimental results of Zabaras [47]. Althoughthe experimental results related to the substrate thickness didnot match exactly with the theoretical predictions, still theresults gave qualitative information about the model. Thinliquid layer supported by steady air-flow surface-tractionwas reported by King and Tuck [48]. Their study modelsthe surface-traction-supported fluid drops observed on thewindscreen of a moving car on a rainy day. Incorporating thesuperficial shear offered by the air, Pascal [49] studied a prob-lemwhichmodels themechanismofwind-aided spreading ofoil on the sea and found quantitative information regardingthe maximum upwind spread of the gravity current. Wilsonand Duffy [50] studied the steady unidirectional flow of a
thin rivulet on a vertical substrate subjected to a prescribeduniform longitudinal shear stress on the free surface. Theycategorized the possible flow patterns and found that thedirection of the prescribed shear stress affects the velocityin the entire rivulet. The generation of roll waves on thefree boundary of a non-Newtonian liquid was numericallyassessed using a finite volumemethod by Pascal andDโAlessio[51], revealing the significant effect of air shear on theevolution of the flow. Their study showed that the instabilitycriteria was conditional and depends on the directionallyinduced shear. Kalpathy et al. [52] investigated an idealizedmodel suitable for lithographic printing by examining theshear-induced suppression of two stratified thin liquid filmsconfined between parallel plates taking into account the vander Waals force. A film thickness equation for the liquid-liquid interface was derived in their study using lubricationapproximation.They found that the effect of shear affects theimaginary part of the growth rate, indicating the existenceof traveling waves. Furthermore, they also observed a criticalshear rate value beyond which the rupture mechanism couldbe suppressed. This study motivated Davis et al. [53] toconsider the effect of unidirectional air shear on a singlefluid layer. For a two-dimensional ultrathin liquid, Daviset al. [53] showed that the rupture mechanism induced bythe London van der Waals force could be suppressed whenthe magnitude of the wind shear exceeds a critical value, asobserved byKalpathy et al. [52]. Recently, Uma [18]measurednumerically the profound effect of unidirectional wind stresson the stability of a condensate/evaporating power-law liquidflowing down an incline.
In the present investigation, the effects of downhill anduphill air shear on a thin falling film in the presence of a trans-verse magnetic field are studied. Such an investigation willillustrate the realistic influence of the natural environmentacting upon the flows. Or, it may illustrate the need to controlthe flow mechanism through artificial techniques, whichblow air when the hydromagnetic effects are considered.The outline of this paper is organized as follows. Section 1presents the introduction. In Section 2, mathematical equa-tions governing the physical problem are presented. Section 3discusses the long-wave Benney-type equation. In Section 4,linear stability analysis, weakly nonlinear stability analysis,and the instability arising due to sideband disturbances areanalyzed. The equilibria of a truncated bimodal dynamicalsystem is mathematically presented in Section 5. While theresults in Section 6 discuss nonlinear simulations of the filmthickness evolution, Section 7 highlights the main conclu-sions of the study and includes future perspectives.
2. Problem Description
A thin Newtonian liquid layer of an infinite extent fallingfreely over a plane under the influence of gravitationalacceleration, ๐, is considered. The flow is oriented towardsthe ๐ฅ-axis, and the plane makes an angle ๐ฝ with the horizon.Properties of the fluid like density (๐), viscosity (๐), andsurface-tension (๐) are constants. The magnetic flux densityis defined by the vectorB = ๐ต
0โ๐, where๐ต
0is themagnitude of
ISRNMathematical Physics 3
Y
๐ฝ
h(x, t)B0
๐
g
๐a
X
h0
Figure 1: Sketch of the flow configuration. When ๐๐> 0, the air
shears the surface along the downhill direction (in the direction ofthe arrow). However, if ๐
๐< 0, the air shears the surface along the
uphill direction (in this case, the arrow would point to the oppositedirection). The air shear effect is zero when ๐
๐= 0.
the magnetic field imposed along the ๐ฆ-direction (Figure 1).It is assumed that there is no exchange of heat between theliquid and the surrounding air, but an air flow (either inthe uphill or in the downhill direction) induces a constantstress ofmagnitude ๐
๐on the interface andmoves tangentially
along the surface. The ๐ฆ-axis is perpendicular to the planarsubstrate such that, at any instant of time ๐ก, โ(๐ฅ, ๐ก) measuresthe film thickness.
The magnetohydrodynamic phenomena can be modeledby the following equations, which express the momentumand mass balance:
๐(๐U๐๐ก
+ U โ โU) = ๐G โ โ๐ + ๐โ2U + F, (2)
โ โ U = 0. (3)
The last term in (2) arises due to the contribution of Lorenzbody force, based on Maxwellโs generalized electromagneticfield equations [28, 54, 55].
In a realistic situation corresponding to a three-dimensional flow, the total current flow can be defined usingOhmโs law as follows:
J = ฮฃ (E + U ร B) , (4)
where J, E, U, and ฮฃ represent the current density, electricfield, velocity vector, and electrical conductivity, respectively.The Lorenz force acting on the liquid is defined asF = JรB. Inthe rest of the analysis, a short circuited systemcorrespondingto a two-dimensional problem is considered by assuming thatE = 0.This assumption simplifies the last term correspondingto the electromagnetic contribution in (2). In this case, thepondermotive force acting on the flow (the last term in (2))has only one nonvanishing term in the ๐ฅ-direction; therefore,
F = (๐น๐ฅ, ๐น๐ฆ) = (โฮฃ๐ต
2
0๐ข, 0) , (5)
where G = (๐ cos๐ฝ, โ๐ sin๐ฝ) and ๐ is the pressure.
Boundary conditions on the planar surface and theinterface are added to complete the problem definition of (2)-(3). On the solid substrate, the no-slip and the no-penetrationconditions are imposed, which read as
U = (๐ข, V) = 0. (6)
The jump in the normal component of the surface-tractionacross the interface is balanced by the capillary pressure(product of themean surface-tension coefficient and the localcurvature of the interface), which is expressed as
๐๐โ ๐ + (๐ (โU + โU๐) โ n) โ n
= 2๐ฮ (โ) on ๐ฆ = โ (๐ฅ, ๐ก) ,(7)
where ฮ (โ) = โ(1/2)โ โ n is the mean film curvature and๐๐is the pressure afforded by the surrounding air. The unit
outward normal vector at any point on the free surface is n =โ(๐ฆ โ โ)/|โ(๐ฆ โ โ)|, and t represents the unit vector alongthe tangential direction at that point such that n โ t = 0. Thetangential component of the surface-traction is influenced bythe air stress ๐
๐and reads as
(๐ (โU + โU๐) โ t) โ n = ๐๐
on ๐ฆ = โ (๐ฅ, ๐ก) . (8)
The location of the interface can be tracked through thefollowing kinematic condition:
๐โ
๐๐ก= V โ ๐ข
๐โ
๐๐ฅon ๐ฆ = โ (๐ฅ, ๐ก) . (9)
In order to remove the units associated with the model(2)โ(9) involving the physical variables, reference scales mustbe prescribed. In principle, one can nondimensionalize thesystem based on the nature of the problem by choosingone of the following scales [9, 16]: (a) kinematic viscositybased scales, (b) gravitational acceleration as the flow agentbased scales, (c) mean surface-tension based scales, and (d)Marangoni effect as the flow agent based scales. However,for very thin falling films, the main characteristic time is theviscous one [7โ9, 16]. An advantage of choosing the viscousscale is that either both the large and the small inclinationangles could be considered by maintaining the sine of theangle and the Galileo number as separate entities [7, 16] orthe Reynolds number as a product of Galileo number andsine of the inclination angle could be defined as a single entity[9]. Also, such a scale plays a neutral role while comparingthe action of gravity and Marangoni effect in nonisothermalproblems [16]. Choosing the viscous scale, the horizontaldistance is scaled by ๐, vertical distance by โ
0, streamwise
velocity by ]/โ0, transverse velocity by ]/๐, pressure by
๐]2/โ20, time by ๐โ
0/], and, finally, the shear stress offered by
the wind by ๐]2/โ20. In addition, the slenderness parameter
(๐ = โ0/๐) is considered small, and a gradient expansion of
the dependent variables is done [7โ9]. The horizontal lengthscale, ๐, is chosen such that ๐ = ๐, where ๐ is a typicalwavelength larger than the film thickness.
4 ISRNMathematical Physics
The dimensionless system is presented in Appendix A(the same symbols have been used to avoid new nota-tions). The set of nondimensional parameters arising duringnondimensionalization procedure are Re = Ga sin๐ฝ (theReynolds number) [9], Ga = ๐โ3
0/]2 (the Galileo number),
Ha = ฮฃ๐ต20โ2
0/๐ (the Hartmann number, which measures the
relative importance of the drag force resulting frommagneticinduction to the viscous force arising in the flow), S =๐โ0/๐]2 (the surface-tension parameter), and ๐ = ๐
๐โ2
0/๐]2
(the shear stress parameter). The surface-tension parameteris usually large; therefore, it is rescaled as ๐2S = ๐ and setas O(1) in accordance to the waves observed in laboratoryexperiments. All of the other quantities are considered O(1).The long-wave equation is derived in the next step.
3. Long-Wave Equation
The dependent variables are asymptotically expanded interms of the slenderness parameter ๐ to derive the long-wave equation. Using the symbolic math toolbox availablein MATLAB, the zeroth and the first-order systems aresolved. These solutions are then substituted in the kinematiccondition (A.5) to derive a Benney-typemodel accurate up toO(๐) of the form (1) as
โ๐ก+ ๐ด (โ) โ
๐ฅ+ ๐(๐ต (โ) โ
๐ฅ+ ๐ถ (โ) โ
๐ฅ๐ฅ๐ฅ)๐ฅ+ O (๐
2) = 0. (10)
The standard procedure for the derivation [7, 9, 14, 15, 28] isskipped here. The expressions for ๐ด(โ), ๐ต(โ), and ๐ถ(โ) are,the following:
๐ด (โ) =ReHa
tanh2 (โHa โ)
+๐
โHatanh (โHa โ) sech (โHa โ) ,
(11)
๐ต (โ) = {Re cot๐ฝHa3/2
tanh (โHa โ) โRe cot๐ฝโ
Ha}
+ {Re2โ2Ha5/2
tanh (โHa โ) sech4 (โHa โ)
+Re2โHa5/2
tanh (โHa โ) sech2 (โHa โ)
โ3Re2
2Ha3tanh2 (โHa โ) sech2 (โHaโ)}
+ ๐ {3Re
2Ha5/2tanh (โHa โ) sech (โHa โ)
โ3ReHa5/2
tanh (โHa โ) sech3 (โHa โ)
+Re โHa2
(sech5 (โHa โ)
+3
2sech3 (โHa โ)
โsech (โHa โ)) }
+ ๐2{โ tanh (โHa โ)
Ha3/2( โ sech2 (โHa โ)
โ1
2sech4 (โHa โ))
+3
2Ha2tanh2 (โHa) sech2 (โHa โ)} ,
(12)
๐ถ (โ) =๐โ
Haโ
๐
Ha3/2tanh (โHa โ) . (13)
It should be remarked that, when ๐ = 0, the above termsagree with the long-wave equation derived by Tsai et al.[28] when the phase-change effects on the interface areneglected. The dimensionless parameters differ from thenondimensional set presented in Tsai et al. [28] becauseviscous scales are employed here. Although one can considerdifferent scales, in principle, the structure of the evolutionequation remains unchanged regardless of the dimensionlessparameters appearing in the problem. Furthermore, the errorin the second term associated with ๐ต(โ) in Tsai et al. [28] iscorrected here, which as per the convention followed in theirpaper should read as (4๐ผRe โ/๐5)sech2๐โ tanh๐โ. Also,the case corresponding toHa = 0 can be recovered from (11)โ(13) when Ha โ 0. In this case, the functions in (11)โ(13)read as
๐ด (โ) = Re โ2 + ๐โ,
๐ต (โ) =2
15Re โ5 (Re โ + ๐) โ 1
3Re cot๐ฝโ3,
๐ถ (โ) = ๐โ3
3,
(14)
which match with the evolution equation derived by Miladi-nova et al. [9] in the absence of Marangoni and air shearingeffect. The effect of magnetic field and air shear affectsthe leading order solution through ๐ด(โ) term in (11). Thisterm contributes towards wave propagation and steepeningmechanism.The effect of hydrostatic pressure is measured bythe terms within the first flower bracket in ๐ต(โ) in (12). Therest of the terms in ๐ต(โ) affect the mean flow due to inertial,air shear andmagnetic field contributions.The function๐ถ(โ)corresponds to themean surface-tension effect. Although theeffect of ๐ cannot be properly judged based on its appearancein (11)โ(13), it is obvious that in the absence of the magneticfield both ๐ด(โ) and ๐ต(โ) increase (decrease) when ๐ > 0 (<0). The stability of the long-wave model (10) subject to (11)โ(13) is investigated next.
ISRNMathematical Physics 5
4. Stability Analysis
The Nusselt solution corresponding to the problem is
โ0= 1, (15a)
๐ข0=
sinh (โHa๐ฆ)โHa
(ReโHa
tanhโHa + ๐ sechโHa)
+ReHa
(1 โ cosh (โHa๐ฆ)) ,(15b)
V0= 0, (15c)
๐0= Re cot๐ฝ (1 โ ๐ฆ) . (15d)
For a parallel shear flow, (10) subject to (11)โ(13) admitsnormal mode solutions of the form โ = 1 + ๐ป(๐ฅ, ๐ก), where๐ป(๐ฅ, ๐ก) is the unsteady part of the film thickness representingthe disturbance component such that ๐ป โช 1 [56, 57].Inserting โ = 1 + ๐ป(๐ฅ, ๐ก) in (10) and invoking a Taylorseries expansion about โ = 1, the unsteady nonlinearequation representing a slight perturbation to the free surfaceis obtained as
๐ป๐ก+ ๐ด1๐ป๐ฅ+ ๐๐ต1๐ป๐ฅ๐ฅ
+ ๐๐ถ1๐ป๐ฅ๐ฅ๐ฅ๐ฅ
= โ[๐ด
1๐ป +
๐ด
1
2๐ป2+๐ด
1
6๐ป3]๐ป๐ฅ
โ ๐ [๐ต
1๐ป +
๐ต
1
2๐ป2]๐ป๐ฅ๐ฅ
โ ๐(๐ถ
1๐ป +
๐ถ
1
2๐ป2)๐ป๐ฅ๐ฅ๐ฅ๐ฅ
โ ๐ [๐ต
1+ ๐ต
1๐ป] (๐ป
๐ฅ)2
โ ๐ (๐ถ
1+ ๐ถ
1๐ป)๐ป๐ฅ๐ป๐ฅ๐ฅ๐ฅ
+ O (๐๐ป4, ๐ป5, ๐2๐ป) ,
(16)
where (a prime denotes the order of the derivative withrespect to โ)
๐ด1= ๐ด (โ = 1) , ๐ด
1= ๐ด(โ = 1) ,
๐ด
1= ๐ด(โ = 1) ,
๐ต1= ๐ต (โ = 1) , ๐ต
1= ๐ต(โ = 1) ,
๐ต
1= ๐ต(โ = 1) ,
๐ถ1= ๐ถ (โ = 1) , ๐ถ
1= ๐ถ(โ = 1) ,
๐ถ
1= ๐ถ(โ = 1) .
(17)
It should be remarked that, while expanding the Taylor series,there are two small parameters, namely, ๐ โช 1 and ๐ป โช 1whose orders of magnitude should be considered such that๐ โช ๐ป โช 1. When O(๐๐ป3) terms are retained and since
๐๐ป3โช ๐ป4, (๐ด1/6)๐ป3๐ป๐ฅappears as a unique contribution of
order๐ป4 in (16). This term, although present in the unsteadyequation (16), does not contributewhen amultiple-scale anal-ysis is done (refer to Section 4.2.1 and Appendix B), whereequations only up to O(๐ผ3) are considered while deriving acomplex Ginzburg-Landau-type equation [12, 14, 17, 56, 57].In addition, such a term did not appear in earlier studies[12, 14, 17, 56, 57] because ๐ด(โ) was a mathematical functionof second degree in โ, whose higher-order derivatives arezero.
Equation (16) forms the starting point for the linear sta-bility analysis and describes the behavior of finite-amplitudedisturbances of the film. Such an equation predicts theevolution of timewise behavior of an initially sinusoidaldisturbance given to the film. It is important to note thatthe constant film thickness approximation with long-waveperturbations is a reasonable approximation only for certainsegments of flow and implies that (16) is only locally valid.
4.1. Linear Stability. To assess the linear stability, the linearterms in (16) are considered. The unsteady part of the filmthickness is decomposed as (a tilde denotes the complexconjugate)
๐ป(๐ฅ, ๐ก) = ๐๐๐(๐๐ฅโ๐ถ๐ฟ๐ก)+๐ถ๐ ๐ก + ๐๐
โ๐(๐๐ฅโ๐ถ๐ฟ๐ก)โ๐ถ๐ ๐ก, (18)
where ๐ (โช 1) is a complex disturbance amplitude indepen-dent of ๐ฅ and ๐ก. The complex eigenvalue is given by ๐ =๐ถ๐ฟ+ ๐๐ถ๐ such that ๐ โ [0, 1] represents the streamwise
wavenumber. The linear wave velocity and the linear growthrate (amplification rate) of the disturbance are ๐ถ
๐ฟ/๐, ๐ถ๐
โ
(โโ,โ), respectively. Explicitly, they are found as
๐ถ๐ฟ๐
=๐ถ๐ฟ
๐= ๐ด1,
๐ถ๐ = ๐๐2(๐ต1โ ๐2๐ถ1) .
(19)
The disturbances grow (decay) when ๐ถ๐ > 0 (๐ถ
๐ < 0).
However, when ๐ถ๐
= 0, the curve ๐ = 0 and the positivebranch of ๐2 = ๐ต
1/๐ถ1represent the neutral stability curves.
Identifying the positive branch of ๐ต1โ ๐2๐ถ1
= 0 as ๐๐=
โ(๐ต1/๐ถ1) (๐๐is the critical wavenumber), the wavenumber
๐๐corresponding to the maximal growth rate is obtained
from (๐/๐๐)๐ถ๐
= 0. This gives ๐ต1โ 2๐2
๐๐ถ1= 0 such that
๐๐
= ๐๐/โ2. The linear amplification of the most unstable
mode is calculated from ๐ถ๐ |๐=๐๐
.A parametric study considering the elements of the set
S = {Re,Ha, ๐, ๐ฝ, ๐} is done in order to trace the neutralstability and linear amplification curves by assuming theslenderness parameter to be 0.1. Only those curves which arerelevant in drawing an opinion are presented.
The influence of the magnetic field on the criticalReynolds number (Re
๐) varying as a function of the shear
parameter is presented in Figure 2. For each Ha, there is aRe๐below which the flow is stable. The critical Reynolds
number decreases when the angle of inclination increases,and, therefore, the flow destabilizes. The stabilizing effect ofthemagnetic field is also seenwhenHa increases.There existsa certain ๐ > 0 such that the flow remains unstable beyond it.
6 ISRNMathematical Physics
0 50
1
2
3
4
5
6
7
8
9
10
Unstable
Stable
โ5
๐
Ha = 0.5
Ha = 0.1
Ha = 0
Rec
(a)
0 50
1
2
3
4
5
6
7
8
9
10
Stable
Unstable
โ5
๐
Ha = 0.5
Ha = 0.1Ha = 0
Rec
(b)
Figure 2: Critical Reynolds number as a function of ๐ when ๐ = 0: (a) ๐ฝ = 45โ; (b) ๐ฝ = 80โ.
Figures 3 and 4 display the neutral stability curves, whichdivide the ๐ โ Re and ๐ โ ๐ planes into regions of stableand unstable domains. On the other hand, Figure 5 showsthe linear amplification curves. The shear stress offered bythe air destabilizes the flow when it flows along the downhilldirection (๐ > 0) and increases the instability thresholdcompared to the case corresponding to ๐ = 0 (Figures 3and 4). However, the flow mechanism is better stabilizedwhen the applied shear stress offered by the air is in theuphill direction (๐ < 0) than when ๐ = 0 (Figures 3and 4). As seen from Figure 3, the portion of the axiscorresponding to the unstable Reynolds numbers increasesand extends towards the left when the angle of inclinationincreases, thereby reducing the stabilizing effect offered by thehydrostatic pressure at small inclination angles. From curves2 and 3 corresponding to Figure 3, it is observed that the forceof surface-tension stabilizes the flow mechanism. Figure 4supports the information available from Figure 3. When themagnitude of the Hartmann number increases, the instabilityregion decreases because the value of the critical wavenumberdecreases. Comparing Figure 4(a) with Figure 4(c), it isalso observed that the inertial force destabilizes the flowmechanism. The growth rate curves (Figure 5) agree withthe results offered by the neutral stability curves (Figures 3and 4).
The linearly increasing graphs of ๐ถ๐ฟand the decreasing
plots of ๐ถ๐ฟ๐
with respect to ๐ and Ha, respectively, arepresented in Figure 6. The effect of inertia doubles the linearwave speed, ๐ถ
๐ฟ๐. But when the magnitude of Ha, increases,
the linear wave speed decreases. The downhill effect of theshear stress on the interface makes the linear wave speedlarger than the cases corresponding to ๐ = 0 and ๐ < 0.
The linear stability results give only a firsthand infor-mation about the stability mechanism. The influence ofair-induced shear on the stability of the flow under theapplication of a transverse magnetic field will be betterunderstood only when the nonlinear effects are additionallyconsidered. To analyze and illustrate the nonlinear effects onthe stability threshold, a weakly nonlinear study is performedin the next step.
4.2. Multiple-Scale Analysis
4.2.1. Weakly NonlinearTheory. In order to domultiple-scaleanalysis, the following slow scales are introduced followingSadiq and Usha [14] (the justification for stretching the scalesis provided in Lin [56] and in Krishna and Lin [57]):
๐ก1= ๐ผ๐ก, ๐ก
2= ๐ผ2๐ก, ๐ฅ
1= ๐ผ๐ฅ, (20)
such that
๐
๐๐กโ
๐
๐๐ก+ ๐ผ
๐
๐๐ก1
+ ๐ผ2 ๐
๐๐ก2
,
๐
๐๐ฅโ
๐
๐๐ฅ+ ๐ผ
๐
๐๐ฅ1
.
(21)
Here, ๐ผ is a small parameter independent of ๐ and measuresthe distance from criticality such that ๐ถ
๐ โผ O(๐ผ2) (or, ๐ต
1โ
๐2๐ถ1
โผ O(๐ผ2)). The motivation behind such a study liesin deriving the complex Ginzburg-Landau equation (CGLE),which describes the evolution of amplitudes of unstablemodes for any process exhibiting a Hopf bifurcation. Usingsuch an analysis, it is possible to examine whether the
ISRNMathematical Physics 7
0 2 4 6 8 100
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
1
0 2 4 6 8 100
1
1 2
3
ST
US
1 2 3
ST
US
๐ = 0 ๐ = 1
0.2
0.4
0.6
0.8
0 2 4 6 8 100
1
1 2 3 ST
US
kc
kc
kc
๐ = โ1
Re Re Re
(a)
0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10
1 1 1 2 2 2 3 3
3
ST
ST
ST
US US US
0
0.2
0.4
0.6
0.8
1
kc
0
0.2
0.4
0.6
0.8
1
kc
0
0.2
0.4
0.6
0.8
1
kc
๐ = 0 ๐ = 1 ๐ = โ1
Re Re Re
(b)
Figure 3: Neutral stability curves. The symbols ST and US indicate the stable and unstable regions. Here, (1) Ha = 0.001 and ๐ = 1; (2)Ha = 0.2 and ๐ = 1; (3)Ha = 0.2 and ๐ = 10. (a) ๐ฝ = 30โ and (b) ๐ฝ = 60โ.
nonlinear waves in the vicinity of criticality attain a finiteheight and remain stable or continue to grow in time andeventually become unstable. Refer to Appendix B for thedetailed derivations of the threshold amplitude, ๐ผฮ
0, and the
nonlinear wave speed,๐๐.
The threshold amplitude subdivides the flow domainaccording to the signs of ๐ถ
๐ and ๐ฝ
2[14]. If ๐ถ
๐ < 0 and
๐ฝ2
< 0, the flow exhibits subcritical instability. However,when ๐ถ
๐ > 0 and ๐ฝ
2> 0, the Landau state is supercritically
stable. If, on the other hand, ๐ถ๐ < 0 and ๐ฝ
2> 0, the flow is
subcritically stable. A blow-up supercritical explosive state isobserved when ๐ถ
๐ > 0 and ๐ฝ
2< 0.
Different regions of the instability threshold obtainedthroughmultiple-scale analysis are illustrated in Figure 7.Thesubcritical unstable region is affected due to the variation of ๐.For ๐ < 0, such a region is larger than the ones correspondingto ๐ โฅ 0. In addition, the subcritical unstable and thestable regions increase whenHa increases.The explosive stateregion (also called the nonsaturation zone) decreases when๐ < 0 than when ๐ โฅ 0. The strip enclosing asterisks is the
supercritical stable region where the flow, although linearlyunstable, exhibits a finite-amplitude behavior and saturatesas time progresses [8, 9, 14]. The bottom line of the strip isthe curve ๐
๐ = ๐๐/2 which separates the supercritical stable
region from the explosive region [56, 58, 59].Within the strip๐๐ < ๐ < ๐
๐, different possible shapes of the waves exist
[7, 9, 14].Tables 1 and 2 show the explosive and equilibration
state values for different flow parameters. The wavenumbervalue corresponding to the explosive state increases when theinertial effects increase. This increases the unstable region0 < ๐ < ๐
๐ (Table 1). Such a wavenumber decreases either
when the hydromagnetic effect is increased or when the airshear is in the uphill direction. It is evident from Table 1 thatthe explosive state occurs at small values of Re and ๐ฝ, which isalso true for large values ofHawhen ๐ < 0. Also, it is observedthat either when the Hartmann number or the value of theuphill shear is increased, the critical wavenumber becomeszero (Table 2). Therefore, ๐
๐ = 0 (Table 1).
8 ISRNMathematical Physics
00
5
ST
US
1
2
3
4
1
0.8
0.6
0.4
0.2
โ5
kc
๐
(a)
00
5
ST
US
1 2
3
4
1
0.8
0.6
0.4
0.2
โ5
kc
๐
(b)
00
5
ST
US
1 4
2
3
1
0.8
0.6
0.4
0.2
โ5
kc
๐
(c)
Figure 4: Neutral stability curves in the ๐ โ ๐ plane for ๐ = 1: (1) Ha = 0; (2) Ha = 0.05; (3) Ha = 0.1; and (4) Ha = 0.15. (a) ๐ฝ = 60โ andRe = 1; (b) ๐ฝ = 75โ and Re = 1; and (c) ๐ฝ = 60โ and Re = 2.5.
The threshold amplitude (๐ผฮ0) profiles display an asym-
metric structure (Figure 8), increasing up to a certainwavenumber (>๐
๐) and then decreasing beyond it in the
supercritical stable region. The ๐ > 0 induced amplitudesshow larger peak amplitude than the cases corresponding to๐ = 0 and ๐ < 0. The peak amplitude value decreases whenHa increases.Thenonlinearwave speed (๐
๐) curves represent
a 90โ counterclockwise rotation of the mirror image of thealphabet ๐ฟ when Ha is small. The nonlinear speed decreasesto a particular value in the vertical direction and thereaftertraces a constant value beyond it as thewavenumber increasesin the supercritical stable region. However, when the effectof the magnetic field is increased, the nonlinear wave speeddecreases in magnitude and sketches an almost linear con-stant profile. The magnitude of ๐
๐in the supercritical stable
region corresponding to ๐ = 0 remains inbetween the valuescorresponding to ๐ < 0 and ๐ > 0.
4.2.2. Sideband Instability. Let us consider the quasi-mono-chromatic wave of (B.4) exhibiting no spatial modulation ofthe form
๐ = ฮ๐ (๐ก2) = ฮ0๐โ๐๐๐ก2 , (22)
where ฮ0and ๐ are defined by (B.10) and (B.11) such that
๐๐ + ๐ is the wave frequency and ๐(> 0) is the modulation
wavenumber.If one considers a band of frequencies centered around๐
๐ ,
the interaction of one side-mode with the second harmonicwould be resonant with the other side-mode causing thefrequency to amplify. This leads to an instability known assideband instability [60, 61].
To investigate such an instability, ฮ๐ (๐ก2) is subjugated
to sideband disturbances of bandwidth ๐๐ผ [12, 56, 57].
ISRNMathematical Physics 9
0 0.2 0.4 0.6 0.8 1
0
0.01
0.02
0.03
โ0.01
โ0.02
โ0.03
๐ = 1
๐ = 0
๐ = โ1
CR
k
0 1
0
0.0025
0.005
โ0.0025
โ0.005
๐ = 1
๐ = 0
๐ = โ1
CR
0.2 0.4 0.6 0.8k
Ha = 0.001
Re = 2
Ha = 0.3
Re = 2
(a)
0 0.2 0.4 0.6 0.8 1
0
0.01
0.02
0.03
0.04
0.05
0 0.2 0.4 0.6 0.8 1
0
0.005
0.01
0.015
CR
โ0.01
โ0.005
โ0.01
โ0.015
๐ = 1 ๐ = 1
๐ = 0
๐ = 0
๐ = โ1
๐ = โ1
CR
kk
Ha = 0.001
Re = 2
Ha = 0.3
Re = 2
(b)
Figure 5: Growth rate curves when We = 1: (a) ๐ฝ = 75โ and (b) ๐ฝ = 90โ.
The explicit expression for the eigenvalues is found as (referto Appendix C)
๐ =1
2(tr (M) ยฑ โtr2 (M) โ 4 det (M))
= โ๐๐+ ๐2๐ฝ1๐ยฑ1
2โ4 (๐2๐+ ๐2V2) โ
8๐๐ฝ4๐๐V๐
๐ฝ2
= ๐1ยฑ ๐2,
(23)
where
tr (M) = 2๐๐+ 2๐2๐ฝ1๐โ 4๐ฝ2
ฮ๐ 2
= โ2๐๐+ 2๐2๐ฝ1๐,
det (M) = ๐ฝ21๐4โ (V2 + 2๐
๐๐ฝ1) ๐2โ2๐๐๐V๐ฝ4
๐ฝ2
๐.
(24)
It should be remarked that the above expression for ๐ istrue only if V โผ O(๐ผ). However, when V = 0, it is easily
seen from (23) that ๐1= ๐1+ ๐2= ๐2๐ฝ1๐
< 0 and ๐2=
๐1โ ๐2= โ2๐
๐+ ๐2๐ฝ1๐
< 0 (when ๐๐> 0), implying that
the system is stable to the sideband disturbances as ๐ก2โ โ.
For nonzero V, the eigenvalues depend on the dimensionlessflow parameters. If ๐
2> 0 and is less than the absolute value
of ๐1, the sideband modes stabilize the system as ๐ก
2โ โ.
However, if ๐2> 0 and is greater than the absolute value of
๐1, only one of the modes is sideband stable. On the other
hand, if (๐2๐+ ๐2V2) โ (8๐๐ฝ
4๐๐V๐ \ ๐ฝ
2) < 0, again, one of the
modes is sideband stable.
5. Equilibria of a Bimodal Dynamical System
Considering the initial thickness of the amplitude to beone, Gjevik [58] analyzed the amplitude equations by rep-resenting the amplitude using a truncated Fourier seriesand by imposing restrictions on its coefficients. The velocityalong the mean flow direction and the corresponding surfacedeflection moving with this velocity were deduced by posing
10 ISRNMathematical Physics
0
1
2
3
4
5
6
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
3
k
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
k
0 0.2 0.4 0.6 0.8 1
k
0 0.2 0.4 0.6 0.8 1
k
๐ = 0
๐ = 1
๐ = โ1
๐ = 0
๐ = 1
๐ = โ1
๐ = 0
๐ = 1
๐ = โ1
๐ = 0
๐ = 1
๐ = โ1
๐ = 0
๐ = 1
๐ = โ1
๐ = 0
๐ = 1
๐ = โ1
CL
0
0.5
1
1.5
2
2.5
CL
CL
0
1
2
3
4
5
CL
0
1
2
3
4
5
6
CLV
0
1
2
3
4
5
6
CLV
Ha = 0.001
Re = 2
Ha = 0.5
Re = 2
Re = 2
Ha = 0.5
Re = 5
Re = 5
Ha = 0.001
Re = 5
Ha Ha
Figure 6: Variation of the linear wave velocity, ๐ถ๐ฟ๐, with respect to Ha, and ๐ถ
๐ฟversus ๐.
ISRNMathematical Physics 11
โ โ
โ โ
โ
โ
โ
โ
โ โ
โ
โ
โ
โ
โ
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
Re
k
1
2
3
Ha = 0.05๐ = 0
โ
โ
โ โ
โ
โ
โ โ โ
โ โ
โ โ
โ โ
โ
โ โ
โ
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
Re
k
1
2 3
Ha = 0.2๐ = 0
โ โ
โ
โ โ โ
โ โ
โ โ โ
โ
โ โ
โ
โ
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
Re
k
1
2 3
Ha = 0.4๐ = 0
โ โ โ โ
โ โ
โ โ โ
โ
โ โ โ
โ
โ โ
โ
โ
โ
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
Re
k
1
2
3
Ha = 0.05๐ = 1
โ โ โ
โ โ
โ โ โ
โ โ
โ โ
โ โ โ
โ
โ
โ
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
Re
k
1
2
3
Ha = 0.2๐ = 1
โ โ
โ โ
โ โ โ
โ
โ
โ
โ โ
โ โ โ
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
Re
k
1
23
Ha = 0.4๐ = 1
โ โ โ
โ โ
โ โ
โ โ
โ โ
โ โ
โ
โ
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
Re
k
Ha = 0.05๐ = โ1
โ โ โ
โ
โ โ
โ
โ
โ โ
โ โ โ
โ โ
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
Re
k
1
1
2
2
3
3
Ha = 0.2๐ = โ1
โ โ โ โ
โ โ โ
โ โ โ
โ โ โ
โ
โ โ
โ โ โ
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
Re
k
1
2 3
Ha = 0.4๐ = โ1
Figure 7: Variation of ๐ with respect to Re when ๐ฝ = 50โ and ๐ = 5: (1) ๐ถ๐ < 0 and ๐ฝ
2< 0; (2) ๐ถ
๐ < 0 and ๐ฝ
2> 0; (3) ๐ถ
๐ > 0 and ๐ฝ
2< 0. The
strip enclosing asterisks indicates that ๐ถ๐ > 0 and ๐ฝ
2> 0. The broken line within the strip represents ๐
๐= ๐๐/โ2.
the problem into a dynamical system. Gottlieb and Oron[62] and Dandapat and Samanta [12] also expanded theevolution equation using a truncated Fourier series to derivea modal dynamical system. The results in Gottlieb and Oron[62] showed that a two-mode model was found to coincidewith the numerical solution along the Hopf bifurcationcurve. Based on this confidence, the stability of the bimodaldynamical system was assessed in Dandapat and Samanta[12].
In this section, the stability of a truncated bimodaldynamical system is analyzed using the approach followed bythe above authors, but using the assumptions considered inGjevik [58].The coupled dynamical system and its entries arelisted in Appendix D.
It should be remarked that in the ๐ โ Re plane theequations ๐
111= 0 and ๐[๐b
111]/๐๐ = 0 give the neutral
stability curve and the curve corresponding to the maximumrate of amplification for linear disturbances. The expression
V๐=
๐
๐๐ก๐1(๐ก) = ๐
111+ [๐121
cos๐ (๐ก) โ ๐121
sin๐ (๐ก)]
ร ๐2(๐ก) + ๐
131๐2
1(๐ก) + ๐
132๐2
2(๐ก)
(25)
measures the velocity along the direction of themeanflowof asteady finite-amplitude wave. The surface deflection, โ(๐ฅ, ๐ก),moving with velocity V
๐along the mean flow direction in a
coordinate system ๐ฅ = ๐ฅ + ๐1is calculated from (D.1) and
reads as
โ (๐ฅ, ๐ก) = 1 + 2 [๐ต1cos๐ฅ + ๐ต
2cos (2๐ฅ โ ๐)] . (26)
The steady solutions of the system (D.2a) and (D.2b)correspond to the fixed points of (D.4a)โ(D.4c). In additionto the trivial solution ๐
1(๐ก) = 0, ๐
2(๐ก) = 0, and ๐(๐ก) =
0 (๐ง1(๐ก) = 0; ๐ง
2(๐ก) = 0), system (D.4a)โ(D.4c) offers
nontrivial fixed points [62โ64]. These fixed points can beclassified as pure-mode fixed points (where one of the fixedpoints is zero and the others are nonzero), mixed-mode fixedpoints (nonzero fixed points with a zero or nonzero phasedifference), and traveling waves with nonzero constant phasedifference.
5.1. Pure-Mode. The fixed points in this case are obtained bysetting ๐
๐(๐1, ๐2, ๐) = 0, ๐ = 1, 2, 3, in (D.4a)โ(D.4c). This
gives ๐1(๐ก) = 0 and ๐2
2(๐ก) = โ๐
211/๐232
. The solution existsif sgn(๐
211๐232
) = โ1. For convenience, ๐22(๐ก) = ๐
2 is set. From
12 ISRNMathematical Physics
Table 1: Parameter values for which the nonlinear wave explodes.
๐ฝ = 55โ
๐ฝ = 75โ
Re = 3 Re = 6 Re = 3 Re = 6
๐ = 1
Ha = 0 ๐๐ = 0.3671 ๐
๐ = 0.7933 ๐
๐ = 0.4467 ๐
๐ = 0.8707
Ha = 0.2 ๐๐ = 0.2309 ๐
๐ = 0.6056 ๐
๐ = 0.3438 ๐
๐ = 0.7045
Ha = 0.4 ๐๐ = 0.0268 ๐
๐ = 0.4399 ๐
๐ = 0.2561 ๐
๐ = 0.5686
๐ = 0
Ha = 0 ๐๐ = 0.2733 ๐
๐ = 0.7132 ๐
๐ = 0.3734 ๐
๐ = 0.7992
Ha = 0.2 ๐๐ = 0.1536 ๐
๐ = 0.5517 ๐
๐ = 0.2974 ๐
๐ = 0.6588
Ha = 0.4 ๐๐ = 0 ๐
๐ = 0.4073 ๐
๐ = 0.2319 ๐
๐ = 0.5436
๐ = โ1
Ha = 0 ๐๐ = 0.1211 ๐
๐ = 0.6237 ๐
๐ = 0.2823 ๐
๐ = 0.7205
Ha = 0.2 ๐๐ = 0 ๐
๐ = 0.4861 ๐
๐ = 0.2301 ๐
๐ = 0.6049
Ha = 0.4 ๐๐ = 0 ๐
๐ = 0.3602 ๐
๐ = 0.1835 ๐
๐ = 0.5094
Table 2: Parameter values for which the nonlinear wave attains a finite amplitude.
๐ฝ = 55โ
๐ฝ = 75โ
Re = 3 Re = 6 Re = 3 Re = 6
๐ = 1
Ha = 0 ๐๐= 0.7342 ๐
๐= 1.5866 ๐
๐= 0.8933 ๐
๐= 1.7415
Ha = 0.2 ๐๐= 0.4618 ๐
๐= 1.2112 ๐
๐= 0.6875 ๐
๐= 1.4092
Ha = 0.4 ๐๐= 0.0536 ๐
๐= 0.8799 ๐
๐= 0.5121 ๐
๐= 1.1371
๐ = 0
Ha = 0 ๐๐= 0.5467 ๐
๐= 1.4263 ๐
๐= 0.7468 ๐
๐= 1.5984
Ha = 0.2 ๐๐= 0.3071 ๐
๐= 1.1034 ๐
๐= 0.5947 ๐
๐= 1.3177
Ha = 0.4 ๐๐= 0 ๐
๐= 0.8145 ๐
๐= 0.4638 ๐
๐= 1.0872
๐ = โ1
Ha = 0 ๐๐= 0.2421 ๐
๐= 1.2475 ๐
๐= 0.5646 ๐
๐= 1.4411
Ha = 0.2 ๐๐= 0 ๐
๐= 0.9722 ๐
๐= 0.4602 ๐
๐= 1.2099
Ha = 0.4 ๐๐= 0 ๐
๐= 0.7205 ๐
๐= 0.1834 ๐
๐= 1.0187
(D.4c), ๐121
cos๐(๐ก) โ ๐121
sin๐(๐ก) = (๐/2)๐ด1๐ is obtained.
From this condition, the fixed point for ๐(๐ก) is derived as
๐ (๐ก) = ๐ (๐ก) = tanโ1 [(4๐121
๐121
ยฑ ๐๐ด
1๐
ร {4๐2
121+ 4๐2
121โ ๐2๐ด2
1๐2}1/2
)
ร (4๐2
121โ ๐2๐ด2
1๐2)โ1
] ,
(27)
provided that the quantity within the square root is realand positive. The stability of the nonlinear dynamical system(D.4a)โ(D.4c) can be locally evaluated using the eigenvaluesof the matrix obtained after linearizing the system aroundthe fixed points. The linear approximation of the dynamicalsystem (D.4a)โ(D.4c) can be represented in matrix notationas
((
(
๐๐1(๐ก)
๐๐ก
๐๐2(๐ก)
๐๐ก
๐๐ (๐ก)
๐๐ก
))
)
= (
๐1
0 0
0 ๐2
0
0 ๐3
๐4
)(
๐1(๐ก)
๐2(๐ก)
๐ (๐ก)
) + (
๐1
๐2
๐3
) ,
(28)
such that
๐1= ๐111
+ (๐121
cos๐ + ๐121
sin๐) ๐ + ๐132
๐2;
๐2= ๐211
+ 3๐232
๐2,
๐3= โ2๐
121sin๐ + 2๐
121cos๐ โ 2๐๐ด
1๐;
๐4= โ (2๐
121cos๐ + 2๐
121sin๐) ๐,
๐1= 0; ๐
2= โ2๐
232๐3;
๐3= ๐๐ด
1๐2+ (2๐121
cos๐ + 2๐121
sin๐) ๐๐.
(29)
The stability of the linear system (28) depends on theeigenvalues of the coefficient matrix with entries ๐
๐. The
eigenvalues are found as
๐1= ๐211
+ 3๐2๐232
= โ2๐211
,
๐2= โ2๐ (๐
121cos๐ + ๐
121sin๐) ,
๐3= ๐111
+ ๐ (๐121
cos๐ + ๐121
sin๐) + ๐132
๐2.
(30)
ISRNMathematical Physics 13
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
2
1
3
0.2 0.4 0.80.6 1 1.2 1.4
k
๐ผฮ0
3.5
4
4.5
5
5.5
6
6.5
7
2
1
3
0.2 0.4 0.80.6 1 1.2 1.4
k
0
0.005
0.01
0.015
0.02
2
1
3
0.2 0.4 0.80.6 1 1.2 1.4
k
๐ผฮ0
3.5
4
4.5
5
5.5
6
6.5
7
2
1
3
0.2 0.4 0.80.6 1 1.2 1.4
k
Nc
Nc
Ha = 0.05
Re = 5
Ha = 0.05
Re = 5
Ha = 0.2
Re = 5
Ha = 0.2
Re = 5
Figure 8: Threshold amplitude and nonlinear wave speed in the supercritical stable region for ๐ฝ = 50โ and ๐ = 5: (1) ๐ = โ1, (2) ๐ = 0, and(3) ๐ = 1.
TheHopf bifurcation at the critical threshold is defined by๐ต1= ๐2๐ถ1, and this yields ๐
1= 24๐๐
4๐ถ1> 0. This eigenvalue
being independent of ๐ increases when the effects of surface-tension and Hartmann number increase. Therefore, at thecritical threshold, the fixed points corresponding to thepure-mode are unstable. Beyond the neutral stability limitwhere the flow is linearly stable (๐ถ
๐ < 0) and where
the stable wavenumber region decreases when the directionof the shear offered by the wind changes from uphill todownhill direction, the eigenvalue may remain positive andstill destabilize the system as shown in Figure 9. The stablewavenumbers corresponding to the linear stability threshold(refer to Figure 5 withHa = 0.3 and Re = 2) are considered toplot the eigenvalue๐
1. Although themagnitude of๐
1remains
larger for ๐ < 0 than for ๐ โฅ 0, being positive, it plays adestabilizing role.
5.2. Mixed-Mode. When ๐1(๐ก), ๐2(๐ก) ฬธ= 0, the fixed points in
this case correspond to mixed-mode. Considering ๐(๐ก) ฬธ= 0and ๐โ1, ๐โ2, and ๐โ to be the fixed points of (D.4a)โ(D.4c), the
nonlinear system can be linearized around the fixed points.Then, the stability (instability) of the fixed points demandsall of the eigenvalues of the linearized Jacobian matrix to benegative (at least one of them to be positive).The nine entries๐ก๐๐(๐, ๐ = 1, 2, 3) of the 3 ร 3 Jacobian matrix arising due to
linearization are the following:
๐ก11
= ๐111
+ (๐121
cos๐โ + ๐121
sin๐โ) ๐โ2
+ 3๐131
๐โ2
1+ ๐132
๐โ2
2,
๐ก12
= (๐121
cos๐โ + ๐121
sin๐โ) ๐โ1+ 2๐132
๐โ
1๐โ
2,
๐ก13
= (โ๐121
sin๐โ + ๐121
cos๐โ) ๐โ1๐โ
2,
๐ก21
= 2 (๐221
cos๐โ โ ๐221
sin๐โ) ๐โ1+ 2๐231
๐โ
1๐โ
2,
๐ก22
= ๐211
+ ๐231
๐โ2
1+ 3๐232
๐โ2
2,
๐ก23
= โ (๐221
sin๐โ + ๐221
cos๐โ) ๐โ21,
14 ISRNMathematical Physics
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
๐1=โ2b
211
k
๐ = 1
๐ = 0
๐ = โ1
๐ = โ2
๐ = 2
Ha = 0.3
Re = 2
Figure 9: Eigenvalue๐1as a function of ๐whenWe = 1 and๐ฝ = 75โ.
๐ก31
= 2๐๐ด
1๐โ
1โ2๐โ
1
๐โ2
(๐221
sin๐โ + ๐221
cos๐โ) ,
๐ก32
= โ 2๐๐ด
1๐โ
2+ 2 (๐121
cos๐โ โ ๐121
sin๐โ)
+๐โ2
1
๐โ22
(๐221
sin๐โ + ๐221
cos๐โ) ,
๐ก33
= โ 2 (๐121
sin๐โ + ๐121
cos๐โ) ๐โ2
โ (๐221
cos๐โ โ ๐221
sin๐โ)๐โ2
1
๐โ2
.
(31)
It should be remarked that Samanta [65] also discussedthe mixed-mode for the flow of a thin film on a nonuni-formly heated vertical wall. However, it should be notedthat the Jacobian was computed not by considering a zerophase difference with ๐
1(๐ก), ๐2(๐ก) ฬธ= 0, but by evaluating the
Jacobian first by imposing ๐1(๐ก), ๐2(๐ก), ๐(๐ก) ฬธ= 0 and then by
substituting ๐ = 0 in the computed Jacobian. If one considers๐1(๐ก), ๐2(๐ก), ๐(๐ก) = 0, an overdetermined system is obtained.
5.3. Traveling Waves. Fixed points of (D.4a)โ(D.4c) with ๐being a nonzero constant correspond to traveling waves. Themodal amplitudes are considered small in the neighborhoodof the neutral stability limit. After rescaling ๐
1(๐ก) โ ๐ผ๐
1(๐ก)
and ๐2(๐ก) โ ๐ผ
2๐2(๐ก), the following system is obtained:
๐๐1(๐ก)
๐๐ก= ๐111
๐1(๐ก) + ๐ผ
2
ร [๐121
cos๐ (๐ก) + ๐121
sin๐ (๐ก)]๐1(๐ก)๐2(๐ก)
+ ๐ผ2๐131
๐3
1(๐ก) + O (๐ผ
4) ,
(32a)
๐๐2(๐ก)
๐๐ก= ๐211
๐2(๐ก)
+ [๐221
cos๐ (๐ก) โ ๐221
sin๐ (๐ก)]๐21(๐ก)
+ ๐ผ2๐231
๐2
1(๐ก)๐2(๐ก) + O (๐ผ
4) ,
(32b)
๐๐ (๐ก)
๐๐ก= โ [๐
221sin๐ (๐ก) + ๐
221cos๐ (๐ก)]
๐2
1(๐ก)
๐2(๐ก)
+ ๐ผ2๐๐ด
1๐2
1(๐ก) + 2๐ผ
2
ร [๐121
cos๐ (๐ก) โ ๐121
sin๐ (๐ก)]๐2(๐ก) + O (๐ผ
4) .
(32c)
To the O(๐ผ2), there are no nontrivial fixed points in theabove system. The phase evolution is governed by equatingthe right-hand side of (32c) to zero by considering terms upto O(๐ผ2):
tan๐ (๐ก) =(โ๐1๐2ยฑ ๐3(๐2
1+ ๐2
2โ ๐2
3)1/2
)
(๐21โ ๐23) ,
,(33)
where ๐1= (1 + 2๐ผ
2๐121
๐2
2(๐ก)/๐221
๐2
1(๐ก)), ๐2= (๐221
/๐221
) โ
(2๐ผ2๐121
๐2
2(๐ก)/๐221
๐2
1(๐ก)), and ๐
3= ๐ผ2๐๐ด
1๐2(๐ก)/๐221
. In theabove equation, if only the leading order effect is considered,the fixed point is found as
๐โ(๐ก) = tanโ1 (
โ๐221
๐221
) + O (๐ผ2) . (34)
Equating (32a) to zero and using (34), it is found that
๐ผ2๐131
๐2
1(๐ก) = โ๐
111โ ๐ผ2๐1๐2(๐ก) . (35)
Such a solution exists if [๐111
+ ๐ผ2๐1๐2(๐ก)]๐131
< 0, where๐1= (๐121
cos๐โ + ๐121
sin๐โ). A quadratic equation in๐2(๐ก)
is obtained by considering (32b) as
๐ผ4๐231
๐1๐2
2(๐ก) + ๐ผ
2(๐111
๐231
โ ๐211
๐131
+ ๐1๐2)
ร ๐2(๐ก) + ๐
2๐111
= 0,
(36)
where ๐2
= ๐221
cos๐โ โ ๐221
sin๐โ. For this equation,there are two solutions. The existence of such solutions tobe a real number demands (๐
111๐231
โ ๐211
๐131
+ ๐1๐2)2โ
4๐ผ4๐1๐2๐111
๐231
> 0. Since ๐111
= 0 in the neutral stabilitylimit, (36) gives the amplitude of the nonzero traveling waveas
๐2(๐ก) =
๐211
๐131
โ ๐1๐2
๐ผ2๐231
๐1
=3๐2๐ถ1๐ฝ2
๐ผ2 (๐ต1โ 4๐2๐ถ
1) ๐1
. (37)
When ๐ฝ2= 0, ๐
2(๐ก) is zero, and this corresponds to finding
the critical Reynolds number in the neutral stability limit [12].
ISRNMathematical Physics 15
0.8
1.0
1.2
โ๐/k x ๐/k
t = 15 t = 14.5
h(x,t)
(a)
0.8
1.0
1.2
โ๐/k x ๐/k
t = 29.95
t = 29.85
h(x,t)
(b)
0.8
1.0
1.2
โ๐/k x ๐/k
t = 2t = 0
h(x,t)
(c)
Figure 10: (a), (b), and (c) represent the surface-wave instability curves reproduced from Joo et al. [7] corresponding to Figures 5(b), 7(a),and 9(a), respectively, in their study.
If๐โ1and๐โ
2are the fixed points obtained from (35) and (36),
the stability or instability of the fixed points corresponding totraveling waves depends on the sign of the eigenvalues of thefollowing matrix:
๐ฝ = (
2๐ผ2๐131
๐โ2
1๐ผ2๐1๐โ
1โ๐ผ2 sin๐โ (๐
121+๐121
๐221
๐221
)๐โ
1๐โ
2
2 (๐2+ ๐ผ2๐231
๐โ
2)๐โ
1๐ผ2๐211
๐โ2
10
2๐ผ2๐๐ด
1๐โ
1โ2๐ผ2(๐121
+๐121
๐221
๐221
) sin๐โ โ(2๐ผ2๐1๐โ
2+ ๐2
๐โ2
1
๐โ2
)
). (38)
6. Nonlinear Development ofthe Interfacial Surface
The nonlinear interactions are studied numerically beyondthe linear stability threshold in a periodic domain, D =
{(๐ฅ, ๐ก) : ๐ฅ โ (โ๐/๐๐
, ๐/๐๐
), ๐ก โ [0,โ)}, by solving (10)subject to the initial condition โ(๐ฅ, 0) = 1 โ 0.1 cos(๐
๐๐ฅ).
A central difference scheme, which is second-order accuratein space, and a backward Euler method, which is implicit inforward time, are adopted to solve the problem in MATLAB.
16 ISRNMathematical Physics
0โ๐/k
x
๐/k
1.06
1.04
1.02
1
0.98
0.96
0.94
h(x,t)
0โ๐/k
x
๐/k
1.06
1.04
1.02
1
0.98
0.96
0.94
h(x,t)
0โ๐/k
x
๐/k
1.06
1.04
1.02
1
0.98
0.96
0.94
h(x,t)
0โ๐/k
x
๐/k
1.06
1.04
1.02
1
0.98
0.96
0.94
h(x,t)
Ha = 0.05Ha = 0
Ha = 0.2 Ha = 0.3
Figure 11: Free surface profiles at Re = 5, ๐ฝ = 45โ, and ๐ = 5 when ๐ = 0 at time ๐ก = 250.
0 50 100 150 200 2501
1.05
1.1
1.15
๐ =1
๐ = 0
๐ = โ1
๐ = โ1.5
Hm
ax
t
(a)
0 100 200 300 400 500 600 7001
1.05
1.1
1.15
๐ = 0
๐ = 1
๐ = โ1 ๐ = โ1.5
Hm
ax
t
(b)
Figure 12: Maximum film thickness profiles at Re = 5, ๐ฝ = 45โ, and ๐ = 5: (a) Ha = 0.05 and (b) Ha = 0.2.
ISRNMathematical Physics 17
0 200 400 600 800 1000 1200 1400
1
0.98
0.96
0.94
0.92
0.9
0.88
๐ = 0
๐ = 0
๐ = โ1๐ = โ1
Hm
in
t
1
1.05
1.1
1.15
๐ = 0๐ = โ1
๐ = 0 ๐ = โ1
Hm
ax
0 200 400 600 800 1000 1200 1400
t
Figure 13:Maximumandminimumamplitude profiles correspond-ing to Re = 5, ๐ฝ = 45โ, ๐ = 5, and Ha = 0.3.
The local truncation error for this numerical scheme is T =O(ฮ๐ก, ฮ๐ฅ2). The computations are performed with a smalltime increment, ฮ๐ก = 10โ3. The numerical scheme is alwaysstable (since a backward Euler method is used) and doesnot build up errors. The number of nodes along the spatialdirection is ๐ = 800 such that the spatial step length isฮ๐ฅ = 2๐/๐๐
๐. An error tolerance of 10โ10 is set, and
the simulations are stopped once the absolute value of theerror becomes smaller than this value. Also, the numericalsimulations show no particular deviation from the resultsobtained neither when the spatial grid points are doublednor when the time step is further reduced. Figures 5(b), 7(a),and 9(a) available in Joo et al. [7] are reproduced in Figure 10to show the correctness of the numerical scheme. This givesconfidence in applying it to the evolution equation consideredhere.
The effect of the transverse magnetic field is illustratedin Figure 11. The wave structure when the magnetic fieldstrength is zero displays a steep curvier wave than whenHa >0. The surface-wave instability decreases when the strengthof the applied magnetic field increases. This reveals thestabilizing mechanism of the transversely applied magneticfield.
Figure 12 displays maximum amplitude profiles of thesheared flow for two different values of Ha on a semi-inclinedplane. The maximum amplitude initially increases and thendecreases. Due to saturation of the nonlinear interactions, theinitial perturbation of the free surface is damped after a longtime. For a short time, the amplitude profile correspondingto ๐ > 0 remains smaller than the ones corresponding to๐ โค 0. Also, for different Ha, the amplitudes correspondingto ๐ < 0 remain larger than those corresponding to ๐ = 0 upto a certain time (see also Figure 13). However, these trendschange over time. When the magnitude of the Hartmannnumber increases, the time required for the amplitudescorresponding to ๐ < 0 to decrease below the amplitudecorresponding to ๐ = 0 increases (Figures 12(a), 12(b), and13). In addition, it is also observed that the time needed for
the amplitude profiles corresponding to ๐ = โ1 to dominatethe amplitude profiles corresponding to ๐ = โ1.5 increaseswhen the value of Ha increases (Figures 12(a) and 12(b)).
The surface waves emerging at the free surface arecaptured and presented in Figure 14. For ๐ = 0, a wave whichhas a stretched front is observed after a long time. When๐ = โ1, a one-humped solitary-like wave is formed whena large amplitude wave and a small capillary ripple coalescetogether (Figure 14(b)) at the time of saturation (a certaintime after which all of the waves have the same structure andshape). Figure 14 demonstrates that the instability measuredby the wave height decreases when the shear is induced alongthe uphill direction.
Tsai et al. [28] pointed out that, when the intensity of themagnetic field increases, the flow retards and stabilizes thesystem. The interfacial surface subjected to air shear affectsthe velocity and flow rate, ๐(๐ฅ, ๐ก) = โซโ
0๐ข(๐ฅ, ๐ก)๐๐ฆ, for different
values of ๐ and Ha (Figure 15). The numerical interactionsreveal that the velocity and the flow rate can either increaseor decrease depending on the direction the air shears thedeformable free interface. In conjunction to the nonlinearwave speed traced in Figure 8, the velocity profiles traced inFigure 15 show a similar response. For Ha = 0 and ๐ > 0,the streamwise velocity and the flow rate increase. Also, thetransverse velocity is larger for Ha = 0 than when Ha > 0.Furthermore, the velocity and the flow rate decrease whenthe strength of the applied magnetic field increases. Whenthe magnitude of the shear induced in the uphill directionis increased (by considering a small negative ๐ value) forlarge values of Ha, constant velocity and flow rate profiles areobserved.
It is inferred from the nonlinear simulations that theeffect of magnetic field and wind shear greatly affects thethickness of the interfacial free surface. When the strength ofthe magnetic field increases, the flow has a uniform velocityand constant discharge. For Ha > 0, the transverse velocityshows a small growth in the positive direction.The growth ofthe unwanted development of the free surface is retarded bythemagnetic field, when it acts opposite to the flow direction.Such a growth-reducing mechanism of the amplitude, inparticular, can be further enhanced by applying uphill shearon the free surface. This inclusion causes further reductionin the values of streamwise and transverse velocities, therebyleading to constant flux situation and improved stability.
7. Conclusions and Perspectives
Linear and nonlinear analyses on the stability of a thin filmsubjected to air shear on a free surface in the presence ofa transversely applied external magnetic field of constantstrength have been studied. Instead of obtaining the solutionsat different orders of approximation using a power seriesapproach [37โ39, 43], the solutions of equations arising atvarious orders have been straightforwardly solved [28].
The linear stability results gave firsthand informationabout the stability mechanism. The modal interaction
18 ISRNMathematical Physics
0.995
1.005
1.01
1
0.99
Time = 1299
Time = 1300
โ3๐/k โ๐/k 3๐/k๐/k
x
h(x,t)
(a)
0.998
0.999
1
1.001
1.002
1.003
1.004
1.005Time = 1299
Time = 1300
โ3๐/k โ๐/k 3๐/k๐/k
x
h(x,t)
(b)
Figure 14: Evolution of the surface-wave instability at Re = 5, ๐ฝ = 45โ, ๐ = 5, and Ha = 0.3: (a) ๐ = 0 and (b) ๐ = โ1. The waves are plottedat a gap of 0.1 time units in a periodic domain.
phenomenon was studied by deriving a complex Ginzburg-Landau-type equation using the method of multiple scales.The stability regimes identified by the linear theory werefurther categorized using the weakly nonlinear theory byconsidering a filtered wave to be the solution of the complexGinzburg-Landau equation. When the nonlinear amplifica-tion rate (๐ฝ
2) is positive, an infinitesimal disturbance in the
linearly unstable region attains a finite-amplitude equilibriumstate. The threshold amplitude and the nonlinear wave speedexist in the supercritical stable region. Stability of a fil-tered wave subject to sideband disturbances was considered,and the conditions under which the eigenvalues decay intime were mathematically identified. Considering the initialthickness to be one and imposing restrictions upon theamplitude coefficients, a truncated Fourier series was used toderive a bimodal dynamical system.The stability of the pure-mode, mixed-mode, and the traveling wave solution has beenmathematically discussed.
Although there are several investigations based onweaklynonlinear theory [28, 37โ40, 43], the complete nonlinearevolution equation was not studied numerically beyond thelinear stability threshold in the presence of an externalmagnetic field, especially in a short-circuited system. In thisregard, the nonlinear interactions have been numericallyassessed using finite-difference technique with an implicittime-stepping procedure. Using such a scheme, the evolutionof disturbances to a small monochromatic perturbation was
analyzed.The destabilizing mechanism of the downhill shearwas identified with the help of computer simulation. Also, ittakes a long time for the amplitude profiles corresponding tothe zero-shear-induced effect to dominate those correspond-ing to the uphill shear, when the strength of the magneticfield increases. Furthermore, among the uphill shear-inducedflows, the smaller the uphill shear-induced effect is, the longerit takes for the respective amplitude to be taken over bythe amplitude corresponding to large values of uphill shear,provided that the magnetic fields intensity increases.
The velocity and the flow rate profiles gave a clearunderstanding of the physical mechanism involved in theprocess. When the magnetic field effect and the air shearare not considered, the gravitational acceleration, the inertialforce, and the hydrostatic pressure determine the mechanismof long-wave instability, by competing against each other.These forces trigger the flow, amplify the film thickness,and prevent the wave formation, respectively [66]. Whenthe magnetic field effect is included, it competes with otherforces to decide the stability threshold. The applied magneticfield retards the flow considerably by reducing the transversevelocity. This phenomenon reduces the wave thickness, butit favors hydrostatic pressure and surface-tension to promotestability on a semi-inclined plane. However, when the airshear is considered, it can either stabilize or destabilizethe system depending on the direction it shears the freesurface. For downhill shear, the transverse velocity increases
ISRNMathematical Physics 19
โ๐/k ๐/k
x
โ๐/k ๐/k
x
0โ0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
1
1
1
1
1
2
2
2
2
2
3
3
33
3
u(x,t),(x,t),q(x,t)
๐ = 1
0โ0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
u(x,t),(x,t),q(x,t)
๐ = 0
0โ0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
u(x,t),(x,t),q(x,t)
๐ = โ1
0โ0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
u(x,t),(x,t),q(x,t)
๐ = 0
1
2
3
0โ0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
u(x,t),(x,t),q(x,t)
๐ = 1
โ0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
u(x,t),(x,t),q(x,t)
๐ = โ1
0
Ha = 0
Ha = 0
Ha = 0
Ha = 0.2
Ha = 0.2
Ha = 0.2
Figure 15: (1) Streamwise velocity, (2) flow rate, and (3) transverse velocity profiles at Re = 5, ๐ฝ = 45โ, and ๐ = 5 in a periodic domain.
resulting in increased accumulation of the fluid under atypical cusp, thereby favoring inertial force. Overall, thismechanism causes the film thickness to increase. For uphillshear, the results are opposite.
The present investigation suggests the inclusion of super-ficial shear stress on the interfacial surface to enhance thestability of the film, or on the other hand, shows the natural
effect of air shear on the films stability under natural condi-tions, in particular to those films subjected to a transverselyapplied uniform external magnetic field in a short-circuitedsystem. Future research activities may include (i) assessingnonisothermal effects, (ii) studying non-Newtonian films,and (iii) considering the contribution of electric field. Thelast point would demand considering Poisson equation to
20 ISRNMathematical Physics
describe electric potential together with appropriate bound-ary conditions.
Appendices
A. Dimensionless Equations
The dimensionless equations are given by
๐ข๐ฅ+ V๐ฆ= 0,
๐ (๐ข๐ก+ ๐ข๐ข๐ฅ+ V๐ข๐ฆ) = โ๐๐
๐ฅ+ ๐2๐ข๐ฅ๐ฅ
+ ๐ข๐ฆ๐ฆ
+ ReโHa๐ข,
๐2(V๐ก+ ๐ขV๐ฅ+ VV๐ฆ) = โ๐
๐ฆ+ ๐3V๐ฅ๐ฅ
+ ๐V๐ฆ๐ฆ
โ Re cot๐ฝ,
(A.1)
and the respective boundary conditions are
๐ข = 0, V = 0 on ๐ฆ = 0, (A.2)
๐ + 2๐(1 โ ๐
2โ2
๐ฅ)
(1 + ๐2โ2๐ฅ)๐ข๐ฅ+
2๐โ๐ฅ
1 + ๐2โ2๐ฅ
(๐ข๐ฆ+ ๐2V๐ฅ)
= โ๐2Sโ๐ฅ๐ฅ
(1 + ๐2โ2๐ฅ)3/2
on ๐ฆ = โ,
(A.3)
(1 โ ๐2โ2
๐ฅ)
(1 + ๐2โ2๐ฅ)(๐ข๐ฆ+ ๐2V๐ฅ) โ
4๐2โ๐ฅ
(1 + ๐2โ2๐ฅ)๐ข๐ฅ= ๐
on ๐ฆ = โ,
(A.4)
V = โ๐ก+ ๐ขโ๐ฅ
on ๐ฆ = โ. (A.5)
B. Multiple Scale Analysis
The film thickness,๐ป, is expanded as
๐ป(๐, ๐ฅ, ๐ฅ1, ๐ก, ๐ก1, ๐ก2) =
โ
โ
๐=1
๐ผ๐๐ป๐(๐, ๐ฅ, ๐ฅ
1, ๐ก, ๐ก1, ๐ก2) . (B.1)
Exploiting (16) using (20) and (B.1), the following equation isobtained:
(๐ฟ0+ ๐ผ๐ฟ1+ ๐ผ2๐ฟ2) (๐ผ๐ป
1+ ๐ผ2๐ป2+ ๐ผ3๐ป3) = โ๐ผ
2๐2โ ๐ผ3๐3,
(B.2)
where the operators ๐ฟ0, ๐ฟ1, and ๐ฟ
2and the quantities๐
2and
๐3are given as follows
๐ฟ0=
๐
๐๐ก+ ๐ด1
๐
๐๐ฅ+ ๐๐ต1
๐2
๐๐ฅ2+ ๐๐ถ1
๐4
๐๐ฅ4, (B.3a)
๐ฟ1=
๐
๐๐ก1
+ ๐ด1
๐
๐๐ฅ1
+ 2๐๐ต1
๐2
๐๐ฅ๐๐ฅ1
+ 4๐๐ถ1
๐4
๐๐ฅ3๐๐ฅ1
, (B.3b)
๐ฟ2=
๐
๐๐ก2
+ ๐๐ต1
๐2
๐๐ฅ21
+ 6๐๐ถ1
๐4
๐๐ฅ2๐๐ฅ21
, (B.3c)
๐2= ๐ด
1๐ป1๐ป1๐ฅ
+ ๐๐ต
1๐ป1๐ป1๐ฅ๐ฅ
+ ๐๐ถ
1๐ป1๐ป1๐ฅ๐ฅ๐ฅ๐ฅ
+ ๐๐ต
1๐ป2
1๐ฅ+ ๐๐ถ
1๐ป1๐ฅ๐ป1๐ฅ๐ฅ๐ฅ
,
(B.3d)
๐3= ๐ด
1(๐ป1๐ป2๐ฅ
+ ๐ป1๐ป1๐ฅ1
+ ๐ป2๐ป1๐ฅ)
+ ๐๐ต
1(๐ป1๐ป2๐ฅ๐ฅ
+ 2๐ป1๐ป1๐ฅ๐ฅ1
+ ๐ป2๐ป1๐ฅ๐ฅ
)
+ ๐๐ถ
1(๐ป1๐ป2๐ฅ๐ฅ๐ฅ๐ฅ
+ 4๐ป1๐ป1๐ฅ๐ฅ๐ฅ๐ฅ1
+ ๐ป2๐ป1๐ฅ๐ฅ๐ฅ๐ฅ
)
+ ๐๐ต
1(2๐ป1๐ฅ๐ป2๐ฅ
+ 2๐ป1๐ฅ๐ป1๐ฅ1
)
+ ๐๐ถ
1(๐ป2๐ฅ๐ฅ๐ฅ
๐ป1๐ฅ
+ 3๐ป1๐ฅ๐ฅ๐ฅ1
๐ป1๐ฅ
+๐ป1๐ฅ๐ฅ๐ฅ
๐ป2๐ฅ
+ ๐ป1๐ฅ๐ฅ๐ฅ
๐ป1๐ฅ1
)
+๐ด
1
2๐ป2
1๐ป1๐ฅ
+ ๐๐ต
1
2๐ป2
1๐ป1๐ฅ๐ฅ
+ ๐๐ถ
1
2๐ป2
1๐ป1๐ฅ๐ฅ๐ฅ๐ฅ
+ ๐๐ต
1๐ป1๐ป2
1๐ฅ+ ๐๐ถ
1๐ป1๐ป1๐ฅ๐ป1๐ฅ๐ฅ๐ฅ
.
(B.3e)
The equations are solved order-by-order up to ๐(๐ผ3) [12,14, 56, 57] to arrive at an equation related to the com-plex Ginzburg-Landau type for the perturbation amplitude๐(๐ฅ1, ๐ก1, ๐ก2):
๐๐
๐๐ก2
โ ๐ผโ2๐ถ๐ ๐ + ๐ (๐ต
1โ 6๐2๐ถ1)๐2๐
๐๐ฅ21
+ (๐ฝ2+ ๐๐ฝ4)๐2
๐ = 0,
(B.4)
where
๐๐=
2 (๐ต
1โ ๐2๐ถ
1)
(โ4๐ต1+ 16๐2๐ถ
1), ๐
๐=
โ๐ด
1
๐ (โ4๐๐ต1+ 16๐3๐ถ
1),
๐ฝ2= ๐(7๐
4๐๐๐ถ
1โ ๐2๐๐๐ต
1โ๐2
2๐ต
1+๐4
2๐ถ
1) โ ๐๐
๐๐ด
1,
๐ฝ4= ๐ (7๐
4๐๐๐ถ
1โ ๐2๐๐๐ต
1) + ๐๐
๐๐ด
1+ ๐
๐ด
1
2.
(B.5)
It should be remarked that in (B.4) an imaginary, apparentlyconvective term of the form
๐V๐๐
๐๐ฅ1
= 2๐๐๐ (๐ต1โ 2๐2๐ถ1) ๐ผโ1 ๐๐
๐๐ฅ1
(B.6)
could be additionally considered on the left-hand side of(B.4) as done in Dandapat and Samanta [12] and Samanta[65], provided that such a term is of O(๐ผ) and V < 0. Such aterm arises in the secular condition at๐(๐ผ3) as a contributionarising from one of the terms from ๐ฟ
1๐ป1at ๐(๐ผ2). The
solution of (B.4) for a filtered wave in which the spatialmodulation does not exist and the diffusion term in (B.4)becomes zero is obtained by considering ๐ = ฮ
0(๐ก2)๐[โ๐๐(๐ก2)๐ก2].
ISRNMathematical Physics 21
This leads to a nonlinear ordinary differential equation for ฮ0,
namely,
๐ฮ0
๐๐ก2
โ ๐ฮ0
๐
๐๐ก2
[๐ (๐ก2) ๐ก2] โ ๐ผโ2๐ถ๐ ฮ0+ (๐ฝ2+ ๐๐ฝ4) ฮ3
0= 0,
(B.7)
where the real and imaginary parts, when separated from(37), give
๐ [ฮ0(๐ก2)]
๐๐ก2
= (๐ผโ2๐ถ๐ โ ๐ฝ2ฮ2
0) ฮ0, (B.8)
๐ [๐ (๐ก2) ๐ก2]
๐๐ก2
= ๐ฝ4ฮ2
0. (B.9)
If ๐ฝ2
= 0 in (B.8), the Ginzburg-Landau equation isreduced to a linear ordinary differential equation for theamplitude of a filtered wave. The second term on the right-hand side in (B.8) induced by the effect of nonlinearitycan either accelerate or decelerate the exponential growthof the linear disturbance depending on the signs of ๐ถ
๐ and
๐ฝ2. The perturbed wave speed caused by the infinitesimal
disturbances appearing in the nonlinear system can bemodified using (B.8).The threshold amplitude ๐ผฮ
0from (B.8)
is obtained as (when ฮ0is nonzero and independent of ๐ก
2)
๐ผฮ0= โ
๐ถ๐
๐ฝ2
. (B.10)
Equation (B.9) with the use of (B.10) gives
๐ =๐ถ๐ ๐ฝ4
๐ผ2๐ฝ2
. (B.11)
The speed๐๐of the nonlinearwave is nowobtained from (26)
by substituting ๐ = ฮ0๐[โ๐๐๐ก2] and by using ๐ก
2= ๐ผ2๐ก. This gives
the nonlinear wave speed as
๐๐= ๐ถ๐ฟ๐
+ ๐ผ2๐ = ๐ถ
๐ฟ๐+ ๐ถ๐
๐ฝ4
๐ฝ2
. (B.12)
C. Sideband Stability
The sideband instability is analyzed by subjecting ฮ๐ (๐ก2) to
sideband disturbances in the form of (ฮ โช 1) [12, 56, 57]
๐ = ฮ๐ (๐ก2) + (ฮฮ
+(๐ก2) ๐๐๐๐ฅ
+ ฮฮโ(๐ก2) ๐โ๐๐๐ฅ
) ๐โ๐๐๐ก2 , (C.1)
in (B.4). Neglecting the nonlinear terms and separating thecoefficients of ฮ๐๐(๐๐ฅโ๐๐ก2) and ฮ๐โ๐(๐๐ฅโ๐๐ก2), the following set ofequations are derived (V = (2๐๐/๐ผ)(๐ต
1โ 2๐2๐ถ1) < 0, ๐
๐=
๐ผโ2๐ถ๐ , and ๐ฝ
1๐= ๐(๐ต
1โ 6๐2๐ถ1) < 0 since ๐ต
1โ ๐2๐ถ1
=
O(๐ผ2)):
๐ฮ+(๐ก2)
๐๐ก2
= (๐P + ๐V + ๐๐+ ๐2๐ฝ1๐โ 2 (๐ฝ2+ ๐๐ฝ4)ฮ๐
2
)
ร ฮ+(๐ก2) โ (๐ฝ2+ ๐๐ฝ4)ฮ๐
2
ฮโ(๐ก2) ,
(C.2a)
๐ฮโ(๐ก2)
๐๐ก2
= (๐P โ ๐V + ๐๐+ ๐2๐ฝ1๐โ 2 (๐ฝ2+ ๐๐ฝ4)ฮ๐
2
)
ร ฮโ(๐ก2) โ (๐ฝ2+ ๐๐ฝ4)ฮ๐
2
ฮ+(๐ก2) ,
(C.2b)
where the barred quantities are the complex conjugatescorresponding to their counterparts. For convenience, theabove system is represented as
(
๐ฮ+(๐ก2)
๐๐ก2
๐ฮโ(๐ก2)
๐๐ก2
) = M(
ฮ+(๐ก2)
ฮโ(๐ก2)
) , (C.3)
whereM is a 2 ร 2 matrix whose entries are
๐11
= ๐P + ๐V + ๐๐+ ๐2๐ฝ1๐โ 2 (๐ฝ2+ ๐๐ฝ4)ฮ๐
2
,
๐22
= โ๐P โ ๐V + ๐๐+ ๐2๐ฝ1๐โ 2 (๐ฝ2โ ๐๐ฝ4)ฮ๐
2
,
๐12
= โ (๐ฝ2+ ๐๐ฝ4)ฮ๐
2
, ๐21
= ๐12.
(C.4)
Setting ฮ+(๐ก2) โ ๐
๐๐ก2 and ฮโ(๐ก2) โ ๐
๐๐ก2 , the eigenvaluesare obtained from |M โ ๐๐ผ| = 0, and the stability of (22)subject to sideband disturbances as ๐ก
2โ โ demands that
๐ < 0. The eigenvalues are found as follows
๐ =1
2(tr (M) ยฑ โtr2 (M) โ 4 det (M)) . (C.5)
D. Bimodal Dynamical System
In order to analyze the stability of the bimodal dynamicalsystem, the film thickness is expanded as a truncated Fourierseries (๐ง
๐(๐ก) is complex, and a bar above it designates its
complex conjugate)
โ (๐ฅ, ๐ก) = 1 +
2
โ
๐=1
๐ง๐(๐ก) ๐๐๐๐๐ฅ
+ ๐ง๐(๐ก) ๐โ๐๐๐๐ฅ (D.1)
and is substituted in the evolution equation (10). A Taylorseries expansion is then sought to be about โ1โ, and the
22 ISRNMathematical Physics
following coupled dynamical system is obtained from thecoefficients of ๐๐๐๐ฅ and ๐2๐๐๐ฅ terms:
๐
๐๐ก๐ง1(๐ก) = ๐
111๐ง1(๐ก) + ๐
121๐ง1(๐ก) ๐ง2(๐ก)
+ ๐ง1(๐ก) (๐131
๐ง12
+ ๐132
๐ง22
) ,
(D.2a)
๐
๐๐ก๐ง2(๐ก) = ๐
211๐ง2(๐ก) + ๐
221๐ง2
1(๐ก)
+ ๐ง2(๐ก) (๐231
๐ง12
+ ๐232
๐ง22
) .
(D.2b)
It should be remarked that the higher-order terms arisingwhile deducing (D.2a) and (D.2b) are neglected by assumingthat |๐ง
1| โช 1 and |๐ง
2| โช |๐ง
1|. The coefficients of the above
system are given by
๐111
= ๐ (๐2๐ต1โ ๐4๐ถ1) โ ๐๐๐ด
1,
๐211
= ๐ (4๐2๐ต1โ 16๐
4๐ถ1) โ 2๐๐๐ด
1,
๐121
= ๐ (๐2๐ต
1โ 7๐4๐ถ
1) โ ๐๐๐ด
1,
๐221
= ๐ (2๐2๐ต
1โ 2๐4๐ถ
1) โ ๐๐๐ด
1,
๐131
=๐
2(๐2๐ต
1โ ๐4๐ถ
1) โ
๐
2๐๐ด
1,
๐231
= ๐ (4๐2๐ต
1โ 16๐
4๐ถ
1) โ 2๐๐๐ด
1,
๐132
= ๐ (๐2๐ต
1โ ๐4๐ถ
1) โ ๐๐๐ด
1,
๐232
= ๐ (2๐2๐ต
1โ 8๐4๐ถ
1) โ ๐๐๐ด
1.
(D.3)
The complex conjugates occurring in system (D.3) canbe avoided by representing ๐ง
๐(๐ก) = ๐
๐(๐ก)๐๐๐๐(๐ก) in polar
form, where ๐๐(๐ก) and ๐
๐(๐ก) are real functions such that ๐ =
1, 2. Using polar notation, the following three equations areobtained:
๐๐1(๐ก)
๐๐ก= ๐1(๐1(๐ก) , ๐2(๐ก) , ๐ (๐ก))
= ๐111
๐1(๐ก) + [๐
121cos๐ (๐ก) + ๐
121sin๐ (๐ก)]
ร ๐1(๐ก) ๐2(๐ก) + ๐
1(๐ก) [๐131
๐2
1(๐ก) + ๐
132๐2
2(๐ก)] ,
(D.4a)
๐๐2(๐ก)
๐๐ก= ๐2(๐1(๐ก) , ๐2(๐ก) , ๐ (๐ก))
= ๐211
๐2(๐ก) + [๐
221cos๐ (๐ก) โ ๐
221sin๐ (๐ก)]
ร ๐2
1(๐ก) + ๐
2(๐ก) [๐231
๐2
1(๐ก) + ๐
232๐2
2(๐ก)] ,
(D.4b)
๐๐ (๐ก)
๐๐ก= ๐3(๐1(๐ก) , ๐2(๐ก) , ๐ (๐ก))
= ๐๐ด
1(๐2
1โ ๐2
2) + 2 [๐
121cos๐ (๐ก) โ ๐
121sin๐ (๐ก)]
ร ๐2(๐ก) โ [๐
221sin๐ (๐ก) + ๐
221cos๐ (๐ก)]
๐2
1(๐ก)
๐2(๐ก)
.
(D.4c)
While the coefficient of ๐๐๐๐(๐ก) term yields (D.4a) and(D.4b), (D.4c) is obtained from the two equations whichresult as the coefficient of ๐๐๐๐๐(๐ก) term with the impositionthat the phase relationship is defined as ๐(๐ก) = 2๐
1(๐ก) โ ๐
2(๐ก).
The terms ๐๐๐๐
and ๐๐๐๐
are real numbers and represent the realand imaginary parts of a typical ๐
๐๐๐term in (D.3) such that
๐, ๐ = 1, 2 and ๐ = 1, 2, 3.
References
[1] D. J. Benney, โLongwaves on liquid films,โ Journal ofMathemat-ical Physics, vol. 45, pp. 150โ155, 1966.
[2] B. Gjevik, โOccurrence of finite-amplitude surface waves onfalling liquid films,โPhysics of Fluids, vol. 13, no. 8, pp. 1918โ1925,1970.
[3] R. W. Atherton and G. M. Homsy, โOn the derivation ofevolution equations for interfacial waves,โChemical EngineeringCommunications, vol. 2, no. 2, pp. 57โ77, 1976.
[4] S. P. Lin and M. V. G. Krishna, โStability of a liquid filmwith respect to initially finite three-dimensional disturbances,โPhysics of Fluids, vol. 20, no. 12, pp. 2005โ2011, 1977.
[5] A. Pumir, P. Manneville, and T. Pomeau, โOn solitary wavesrunning down an inclined plane,โ Journal of Fluid Mechanics,vol. 135, pp. 27โ50, 1983.
[6] J. P. Burelbach, S. G. Bankoff, and S. H. Davis, โNonlinearstability of evaporating/condensing liquid films,โ Journal ofFluid Mechanics, vol. 195, pp. 463โ494, 1988.
[7] S.W. Joo, S.H.Davis, and S.G. Bankoff, โLong-wave instabilitiesof heated falling films. Two-dimensional theory of uniformlayers,โ Journal of Fluid Mechanics, vol. 230, pp. 117โ146, 1991.
[8] S. W. Joo and S. H. Davis, โInstabilities of three-dimensionalviscous falling films,โ Journal of Fluid Mechanics, vol. 242, pp.529โ547, 1992.
[9] S. Miladinova, S. Slavtchev, G. Lebon, and J.-C. Legros, โLong-wave instabilities of non-uniformly heated falling films,โ Journalof Fluid Mechanics, vol. 453, pp. 153โ175, 2002.
[10] S. Miladinova, D. Staykova, G. Lebon, and B. Scheid, โEffect ofnonuniform wall heating on the three-dimensional secondaryinstability of falling films,โ Acta Mechanica, vol. 156, no. 1-2, pp.79โ91, 2002.
[11] B. Scheid, A. Oron, P. Colinet, U. Thiele, and J. C. Legros,โNonlinear evolution of nonuniformly heated falling liquidfilms,โ Physics of Fluids, vol. 15, no. 2, p. 583, 2003, Erratum in:Physics of Fluids, vol. 14, pp. 4130, 2002.
[12] B. S. Dandapat andA. Samanta, โBifurcation analysis of first andsecond order Benney equations for viscoelastic fluid flowingdown a vertical plane,โ Journal of Physics D, vol. 41, no. 9, ArticleID 095501, 2008.
[13] R. Usha and I. M. R. Sadiq, โWeakly nonlinear stability analysisof a non-uniformly heated non-Newtonian falling film,โ inProceedings of the 5th Joint ASME/JSME Fluids EngineeringSummer Conference (FEDSM โ07), pp. 1661โ1670, August 2007.
ISRNMathematical Physics 23
[14] I. M. R. Sadiq and R. Usha, โThin Newtonian film flow down aporous inclined plane: stability analysis,โ Physics of Fluids, vol.20, no. 2, Article ID 022105, 2008.
[15] I. M. R. Sadiq and R. Usha, โEffect of permeability on theinstability of a non-Newtonian film down a porous inclinedplane,โ Journal of Non-Newtonian Fluid Mechanics, vol. 165, no.19-20, pp. 1171โ1188, 2010.
[16] U. Thiele, B. Goyeau, and M. G. Velarde, โStability analysis ofthin film flow along a heated porous wall,โ Physics of Fluids, vol.21, no. 1, Article ID 014103, 2009.
[17] I. M. R. Sadiq, R. Usha, and S. W. Joo, โInstabilities in a liquidfilm flow over an inclined heated porous substrate,โ ChemicalEngineering Science, vol. 65, no. 15, pp. 4443โ4459, 2010.
[18] B. Uma, โEffect of wind stress on the dynamics and stability ofnon-isothermal power-law film down an inclined plane,โ ISRNMathematical Physics, vol. 2012, Article ID 732675, 31 pages,2012.
[19] B. Scheid, C. Ruyer-Quil, U. Thiele, O. A. Kabov, J. C. Legros,and P. Colinet, โValidity domain of the Benney equationincluding the Marangoni effect for closed and open flows,โJournal of Fluid Mechanics, vol. 527,