Research Article Air-Aided Shear on a Thin Film Subjected to ...e e ect of air shear on the...

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; rizwan.sadiq.i@gmail.com

Received 28 June 2013; Accepted 22 August 2013

Academic Editors: L. E. Oxman and W.-H. Steeb

Copyright © 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:

ℎ𝑡+A (ℎ) ℎ

𝑥+

𝜕

𝜕𝑥{B (ℎ) ℎ

𝑥+C (ℎ) ℎ

𝑥𝑥𝑥} = 0, (1)

whereA,B, andC are arbitrary continuous functions of theinterfacial thickness ℎ(𝑥, 𝑡), 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 [1–18] 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, 32–43].

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 andD’Alessio[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, 𝑔, is considered. The flow is oriented towardsthe 𝑥-axis, and the plane makes an angle 𝛽 with the horizon.Properties of the fluid like density (𝜌), viscosity (𝜇), andsurface-tension (𝜎) are constants. The magnetic flux densityis defined by the vectorB = 𝐵

0⃗𝑗, where𝐵

0is themagnitude of

ISRNMathematical Physics 3

Y

𝛽

h(x, t)B0

𝜆

g

𝜏a

X

h0

Figure 1: Sketch of the flow configuration. When 𝜏𝑎> 0, the air

shears the surface along the downhill direction (in the direction ofthe arrow). However, if 𝜏

𝑎< 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 𝜏

𝑎= 0.

the magnetic field imposed along the 𝑦-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 𝜏

𝑎on the interface andmoves tangentially

along the surface. The 𝑦-axis is perpendicular to the planarsubstrate such that, at any instant of time 𝑡, ℎ(𝑥, 𝑡) measuresthe film thickness.

The magnetohydrodynamic phenomena can be modeledby the following equations, which express the momentumand mass balance:

𝜌(𝜕U𝜕𝑡

+ U ⋅ ∇U) = 𝜌G − ∇𝑝 + 𝜇∇2U + F, (2)

∇ ⋅ U = 0. (3)

The last term in (2) arises due to the contribution of Lorenzbody force, based on Maxwell’s generalized electromagneticfield equations [28, 54, 55].

In a realistic situation corresponding to a three-dimensional flow, the total current flow can be defined usingOhm’s law as follows:

J = Σ (E + U × B) , (4)

where J, E, U, and Σ represent the current density, electricfield, velocity vector, and electrical conductivity, respectively.The Lorenz force acting on the liquid is defined asF = J×B. 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 𝑥-direction; therefore,

F = (𝐹𝑥, 𝐹𝑦) = (−Σ𝐵

2

0𝑢, 0) , (5)

where G = (𝑔 cos𝛽, −𝑔 sin𝛽) and 𝑝 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 = (𝑢, 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

𝑝𝑎− 𝑝 + (𝜇 (∇U + ∇U𝑇) ⋅ n) ⋅ n

= 2𝜎Π (ℎ) on 𝑦 = ℎ (𝑥, 𝑡) ,(7)

where Π(ℎ) = −(1/2)∇ ⋅ n is the mean film curvature and𝑝𝑎is the pressure afforded by the surrounding air. The unit

outward normal vector at any point on the free surface is n =∇(𝑦 − ℎ)/|∇(𝑦 − ℎ)|, and t represents the unit vector alongthe tangential direction at that point such that n ⋅ t = 0. Thetangential component of the surface-traction is influenced bythe air stress 𝜏

𝑎and reads as

(𝜇 (∇U + ∇U𝑇) ⋅ t) ⋅ n = 𝜏𝑎

on 𝑦 = ℎ (𝑥, 𝑡) . (8)

The location of the interface can be tracked through thefollowing kinematic condition:

𝜕ℎ

𝜕𝑡= V − 𝑢

𝜕ℎ

𝜕𝑥on 𝑦 = ℎ (𝑥, 𝑡) . (9)

In order to remove the units associated with the model(2)–(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 [7–9, 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 𝑙, vertical distance by ℎ

0, streamwise

velocity by ]/ℎ0, transverse velocity by ]/𝑙, pressure by

𝜌]2/ℎ20, time by 𝑙ℎ

0/], and, finally, the shear stress offered by

the wind by 𝜌]2/ℎ20. In addition, the slenderness parameter

(𝜀 = ℎ0/𝑙) is considered small, and a gradient expansion of

the dependent variables is done [7–9]. The horizontal lengthscale, 𝑙, is chosen such that 𝑙 = 𝜆, where 𝜆 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 sin𝛽 (theReynolds number) [9], Ga = 𝑔ℎ3

0/]2 (the Galileo number),

Ha = Σ𝐵20ℎ2

0/𝜇 (the Hartmann number, which measures the

relative importance of the drag force resulting frommagneticinduction to the viscous force arising in the flow), S =𝜎ℎ0/𝜌]2 (the surface-tension parameter), and 𝜏 = 𝜏

𝑎ℎ2

0/𝜌]2

(the shear stress parameter). The surface-tension parameteris usually large; therefore, it is rescaled as 𝜀2S = 𝑆 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 𝜀 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(𝜀) of the form (1) as

ℎ𝑡+ 𝐴 (ℎ) ℎ

𝑥+ 𝜀(𝐵 (ℎ) ℎ

𝑥+ 𝐶 (ℎ) ℎ

𝑥𝑥𝑥)𝑥+ O (𝜀

2) = 0. (10)

The standard procedure for the derivation [7, 9, 14, 15, 28] isskipped here. The expressions for 𝐴(ℎ), 𝐵(ℎ), and 𝐶(ℎ) are,the following:

𝐴 (ℎ) =ReHa

tanh2 (√Ha ℎ)

+𝜏

√Hatanh (√Ha ℎ) sech (√Ha ℎ) ,

(11)

𝐵 (ℎ) = {Re cot𝛽Ha3/2

tanh (√Ha ℎ) −Re cot𝛽ℎ

Ha}

+ {Re2ℎ2Ha5/2

tanh (√Ha ℎ) sech4 (√Ha ℎ)

+Re2ℎHa5/2

tanh (√Ha ℎ) sech2 (√Ha ℎ)

−3Re2

2Ha3tanh2 (√Ha ℎ) sech2 (√Haℎ)}

+ 𝜏 {3Re

2Ha5/2tanh (√Ha ℎ) sech (√Ha ℎ)

−3ReHa5/2

tanh (√Ha ℎ) sech3 (√Ha ℎ)

+Re ℎHa2

(sech5 (√Ha ℎ)

+3

2sech3 (√Ha ℎ)

−sech (√Ha ℎ)) }

+ 𝜏2{ℎ tanh (√Ha ℎ)

Ha3/2( − sech2 (√Ha ℎ)

−1

2sech4 (√Ha ℎ))

+3

2Ha2tanh2 (√Ha) sech2 (√Ha ℎ)} ,

(12)

𝐶 (ℎ) =𝑆ℎ

Ha−

𝑆

Ha3/2tanh (√Ha ℎ) . (13)

It should be remarked that, when 𝜏 = 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 𝐵(ℎ) in Tsai et al. [28] iscorrected here, which as per the convention followed in theirpaper should read as (4𝛼Re ℎ/𝑚5)sech2𝑚ℎ tanh𝑚ℎ. Also,the case corresponding toHa = 0 can be recovered from (11)–(13) when Ha → 0. In this case, the functions in (11)–(13)read as

𝐴 (ℎ) = Re ℎ2 + 𝜏ℎ,

𝐵 (ℎ) =2

15Re ℎ5 (Re ℎ + 𝜏) − 1

3Re cot𝛽ℎ3,

𝐶 (ℎ) = 𝑆ℎ3

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 𝐴(ℎ) 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 𝐵(ℎ) in (12). Therest of the terms in 𝐵(ℎ) affect the mean flow due to inertial,air shear andmagnetic field contributions.The function𝐶(ℎ)corresponds to themean surface-tension effect. Although theeffect of 𝜏 cannot be properly judged based on its appearancein (11)–(13), it is obvious that in the absence of the magneticfield both 𝐴(ℎ) and 𝐵(ℎ) increase (decrease) when 𝜏 > 0 (<0). The stability of the long-wave model (10) subject to (11)–(13) is investigated next.

ISRNMathematical Physics 5

4. Stability Analysis

The Nusselt solution corresponding to the problem is

ℎ0= 1, (15a)

𝑢0=

sinh (√Ha𝑦)√Ha

(Re√Ha

tanh√Ha + 𝜏 sech√Ha)

+ReHa

(1 − cosh (√Ha𝑦)) ,(15b)

V0= 0, (15c)

𝑝0= Re cot𝛽 (1 − 𝑦) . (15d)

For a parallel shear flow, (10) subject to (11)–(13) admitsnormal mode solutions of the form ℎ = 1 + 𝐻(𝑥, 𝑡), where𝐻(𝑥, 𝑡) is the unsteady part of the film thickness representingthe disturbance component such that 𝐻 ≪ 1 [56, 57].Inserting ℎ = 1 + 𝐻(𝑥, 𝑡) in (10) and invoking a Taylorseries expansion about ℎ = 1, the unsteady nonlinearequation representing a slight perturbation to the free surfaceis obtained as

𝐻𝑡+ 𝐴1𝐻𝑥+ 𝜀𝐵1𝐻𝑥𝑥

+ 𝜀𝐶1𝐻𝑥𝑥𝑥𝑥

= −[𝐴

1𝐻 +

𝐴

1

2𝐻2+𝐴

1

6𝐻3]𝐻𝑥

− 𝜀 [𝐵

1𝐻 +

𝐵

1

2𝐻2]𝐻𝑥𝑥

− 𝜀(𝐶

1𝐻 +

𝐶

1

2𝐻2)𝐻𝑥𝑥𝑥𝑥

− 𝜀 [𝐵

1+ 𝐵

1𝐻] (𝐻

𝑥)2

− 𝜀 (𝐶

1+ 𝐶

1𝐻)𝐻𝑥𝐻𝑥𝑥𝑥

+ O (𝜀𝐻4, 𝐻5, 𝜀2𝐻) ,

(16)

where (a prime denotes the order of the derivative withrespect to ℎ)

𝐴1= 𝐴 (ℎ = 1) , 𝐴

1= 𝐴(ℎ = 1) ,

𝐴

1= 𝐴(ℎ = 1) ,

𝐵1= 𝐵 (ℎ = 1) , 𝐵

1= 𝐵(ℎ = 1) ,

𝐵

1= 𝐵(ℎ = 1) ,

𝐶1= 𝐶 (ℎ = 1) , 𝐶

1= 𝐶(ℎ = 1) ,

𝐶

1= 𝐶(ℎ = 1) .

(17)

It should be remarked that, while expanding the Taylor series,there are two small parameters, namely, 𝜀 ≪ 1 and 𝐻 ≪ 1whose orders of magnitude should be considered such that𝜀 ≪ 𝐻 ≪ 1. When O(𝜀𝐻3) terms are retained and since

𝜀𝐻3≪ 𝐻4, (𝐴1/6)𝐻3𝐻𝑥appears as a unique contribution of

order𝐻4 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(𝛼3) 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 𝐴(ℎ) was a mathematical functionof second degree in ℎ, 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)

𝐻(𝑥, 𝑡) = 𝜂𝑒𝑖(𝑘𝑥−𝐶𝐿𝑡)+𝐶𝑅𝑡 + 𝜂𝑒

−𝑖(𝑘𝑥−𝐶𝐿𝑡)−𝐶𝑅𝑡, (18)

where 𝜂 (≪ 1) is a complex disturbance amplitude indepen-dent of 𝑥 and 𝑡. The complex eigenvalue is given by 𝜆 =𝐶𝐿+ 𝑖𝐶𝑅such that 𝑘 ∈ [0, 1] represents the streamwise

wavenumber. The linear wave velocity and the linear growthrate (amplification rate) of the disturbance are 𝐶

𝐿/𝑘, 𝐶𝑅

∈

(−∞,∞), respectively. Explicitly, they are found as

𝐶𝐿𝑉

=𝐶𝐿

𝑘= 𝐴1,

𝐶𝑅= 𝜀𝑘2(𝐵1− 𝑘2𝐶1) .

(19)

The disturbances grow (decay) when 𝐶𝑅> 0 (𝐶

𝑅< 0).

However, when 𝐶𝑅

= 0, the curve 𝑘 = 0 and the positivebranch of 𝑘2 = 𝐵

1/𝐶1represent the neutral stability curves.

Identifying the positive branch of 𝐵1− 𝑘2𝐶1

= 0 as 𝑘𝑐=

√(𝐵1/𝐶1) (𝑘𝑐is the critical wavenumber), the wavenumber

𝑘𝑚corresponding to the maximal growth rate is obtained

from (𝜕/𝜕𝑘)𝐶𝑅

= 0. This gives 𝐵1− 2𝑘2

𝑚𝐶1= 0 such that

𝑘𝑚

= 𝑘𝑐/√2. The linear amplification of the most unstable

mode is calculated from 𝐶𝑅|𝑘=𝑘𝑚

.A parametric study considering the elements of the set

S = {Re,Ha, 𝑆, 𝛽, 𝜏} 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

𝑐) varying as a function of the shear

parameter is presented in Figure 2. For each Ha, there is aRe𝑐below 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 𝜏 > 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

−5

𝜏

Ha = 0.5

Ha = 0.1

Ha = 0

Rec

(a)

0 50

1

2

3

4

5

6

7

8

9

10

Stable

Unstable

−5

𝜏

Ha = 0.5

Ha = 0.1Ha = 0

Rec

(b)

Figure 2: Critical Reynolds number as a function of 𝜏 when 𝑆 = 0: (a) 𝛽 = 45∘; (b) 𝛽 = 80∘.

Figures 3 and 4 display the neutral stability curves, whichdivide the 𝑘 − Re and 𝑘 − 𝜏 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 (𝜏 > 0) and increases the instability thresholdcompared to the case corresponding to 𝜏 = 0 (Figures 3and 4). However, the flow mechanism is better stabilizedwhen the applied shear stress offered by the air is in theuphill direction (𝜏 < 0) than when 𝜏 = 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 𝐶𝐿and the decreasing

plots of 𝐶𝐿𝑉

with respect to 𝑘 and Ha, respectively, arepresented in Figure 6. The effect of inertia doubles the linearwave speed, 𝐶

𝐿𝑉. 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 𝜏 = 0 and 𝜏 < 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]):

𝑡1= 𝛼𝑡, 𝑡

2= 𝛼2𝑡, 𝑥

1= 𝛼𝑥, (20)

such that

𝜕

𝜕𝑡→

𝜕

𝜕𝑡+ 𝛼

𝜕

𝜕𝑡1

+ 𝛼2 𝜕

𝜕𝑡2

,

𝜕

𝜕𝑥→

𝜕

𝜕𝑥+ 𝛼

𝜕

𝜕𝑥1

.

(21)

Here, 𝛼 is a small parameter independent of 𝜀 and measuresthe distance from criticality such that 𝐶

𝑅∼ O(𝛼2) (or, 𝐵

1−

𝑘2𝐶1

∼ O(𝛼2)). 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

𝜏 = 0 𝜏 = 1

0.2

0.4

0.6

0.8

0 2 4 6 8 100

1

1 2 3 ST

US

kc

kc

kc

𝜏 = −1

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

𝜏 = 0 𝜏 = 1 𝜏 = −1

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 𝑆 = 1; (2)Ha = 0.2 and 𝑆 = 1; (3)Ha = 0.2 and 𝑆 = 10. (a) 𝛽 = 30∘ and (b) 𝛽 = 60∘.

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, 𝛼Γ

0, and the

nonlinear wave speed,𝑁𝑐.

The threshold amplitude subdivides the flow domainaccording to the signs of 𝐶

𝑅and 𝐽

2[14]. If 𝐶

𝑅< 0 and

𝐽2

< 0, the flow exhibits subcritical instability. However,when 𝐶

𝑅> 0 and 𝐽

2> 0, the Landau state is supercritically

stable. If, on the other hand, 𝐶𝑅< 0 and 𝐽

2> 0, the flow is

subcritically stable. A blow-up supercritical explosive state isobserved when 𝐶

𝑅> 0 and 𝐽

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 𝜏.For 𝜏 < 0, such a region is larger than the ones correspondingto 𝜏 ≥ 0. In addition, the subcritical unstable and thestable regions increase whenHa increases.The explosive stateregion (also called the nonsaturation zone) decreases when𝜏 < 0 than when 𝜏 ≥ 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 𝑘

𝑠= 𝑘𝑐/2 which separates the supercritical stable

region from the explosive region [56, 58, 59].Within the strip𝑘𝑠< 𝑘 < 𝑘

𝑐, 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 < 𝑘 < 𝑘

𝑠(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 𝛽, which isalso true for large values ofHawhen 𝜏 < 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, 𝑘

𝑠= 0 (Table 1).

8 ISRNMathematical Physics

00

5

ST

US

1

2

3

4

1

0.8

0.6

0.4

0.2

−5

kc

𝜏

(a)

00

5

ST

US

1 2

3

4

1

0.8

0.6

0.4

0.2

−5

kc

𝜏

(b)

00

5

ST

US

1 4

2

3

1

0.8

0.6

0.4

0.2

−5

kc

𝜏

(c)

Figure 4: Neutral stability curves in the 𝑘 − 𝜏 plane for 𝑆 = 1: (1) Ha = 0; (2) Ha = 0.05; (3) Ha = 0.1; and (4) Ha = 0.15. (a) 𝛽 = 60∘ andRe = 1; (b) 𝛽 = 75∘ and Re = 1; and (c) 𝛽 = 60∘ and Re = 2.5.

The threshold amplitude (𝛼Γ0) profiles display an asym-

metric structure (Figure 8), increasing up to a certainwavenumber (>𝑘

𝑚) and then decreasing beyond it in the

supercritical stable region. The 𝜏 > 0 induced amplitudesshow larger peak amplitude than the cases corresponding to𝜏 = 0 and 𝜏 < 0. The peak amplitude value decreases whenHa increases.Thenonlinearwave speed (𝑁

𝑐) curves represent

a 90∘ counterclockwise rotation of the mirror image of thealphabet 𝐿 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 𝑁

𝑐in the supercritical stable

region corresponding to 𝜏 = 0 remains inbetween the valuescorresponding to 𝜏 < 0 and 𝜏 > 0.

4.2.2. Sideband Instability. Let us consider the quasi-mono-chromatic wave of (B.4) exhibiting no spatial modulation ofthe form

𝜂 = Γ𝑠(𝑡2) = Γ0𝑒−𝑖𝑏𝑡2 , (22)

where Γ0and 𝑏 are defined by (B.10) and (B.11) such that

𝜔𝑠+ 𝑏 is the wave frequency and 𝑘(> 0) is the modulation

wavenumber.If one considers a band of frequencies centered around𝜔

𝑠,

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, Γ𝑠(𝑡2) is subjugated

to sideband disturbances of bandwidth 𝑘𝛼 [12, 56, 57].

ISRNMathematical Physics 9

0 0.2 0.4 0.6 0.8 1

0

0.01

0.02

0.03

−0.01

−0.02

−0.03

𝜏 = 1

𝜏 = 0

𝜏 = −1

CR

k

0 1

0

0.0025

0.005

−0.0025

−0.005

𝜏 = 1

𝜏 = 0

𝜏 = −1

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

−0.01

−0.005

−0.01

−0.015

𝜏 = 1 𝜏 = 1

𝜏 = 0

𝜏 = 0

𝜏 = −1

𝜏 = −1

CR

kk

Ha = 0.001

Re = 2

Ha = 0.3

Re = 2

(b)

Figure 5: Growth rate curves when We = 1: (a) 𝛽 = 75∘ and (b) 𝛽 = 90∘.

The explicit expression for the eigenvalues is found as (referto Appendix C)

𝜗 =1

2(tr (M) ± √tr2 (M) − 4 det (M))

= −𝑐𝑖+ 𝑘2𝐽1𝑟±1

2√4 (𝑐2𝑖+ 𝑘2V2) −

8𝑖𝐽4𝑐𝑖V𝑘

𝐽2

= 𝜛1± 𝜛2,

(23)

where

tr (M) = 2𝑐𝑖+ 2𝑘2𝐽1𝑟− 4𝐽2

Γ𝑠2

= −2𝑐𝑖+ 2𝑘2𝐽1𝑟,

det (M) = 𝐽21𝑘4− (V2 + 2𝑐

𝑖𝐽1) 𝑘2−2𝑖𝑐𝑖V𝐽4

𝐽2

𝑘.

(24)

It should be remarked that the above expression for 𝜗 istrue only if V ∼ O(𝛼). However, when V = 0, it is easily

seen from (23) that 𝜗1= 𝜛1+ 𝜛2= 𝑘2𝐽1𝑟

< 0 and 𝜗2=

𝜛1− 𝜛2= −2𝑐

𝑖+ 𝑘2𝐽1𝑟

< 0 (when 𝑐𝑖> 0), implying that

the system is stable to the sideband disturbances as 𝑡2→ ∞.

For nonzero V, the eigenvalues depend on the dimensionlessflow parameters. If 𝜛

2> 0 and is less than the absolute value

of 𝜛1, the sideband modes stabilize the system as 𝑡

2→ ∞.

However, if 𝜛2> 0 and is greater than the absolute value of

𝜛1, only one of the modes is sideband stable. On the other

hand, if (𝑐2𝑖+ 𝑘2V2) − (8𝑖𝐽

4𝑐𝑖V𝑘 \ 𝐽

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

𝜏 = 0

𝜏 = 1

𝜏 = −1

𝜏 = 0

𝜏 = 1

𝜏 = −1

𝜏 = 0

𝜏 = 1

𝜏 = −1

𝜏 = 0

𝜏 = 1

𝜏 = −1

𝜏 = 0

𝜏 = 1

𝜏 = −1

𝜏 = 0

𝜏 = 1

𝜏 = −1

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, 𝐶𝐿𝑉, with respect to Ha, and 𝐶

𝐿versus 𝑘.

ISRNMathematical Physics 11

∗ ∗

∗ ∗

∗

∗

∗

∗

∗ ∗

∗

∗

∗

∗

∗

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

Re

k

1

2

3

Ha = 0.05𝜏 = 0

∗

∗

∗ ∗

∗

∗

∗ ∗ ∗

∗ ∗

∗ ∗

∗ ∗

∗

∗ ∗

∗

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

Re

k

1

2 3

Ha = 0.2𝜏 = 0

∗ ∗

∗

∗ ∗ ∗

∗ ∗

∗ ∗ ∗

∗

∗ ∗

∗

∗

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

Re

k

1

2 3

Ha = 0.4𝜏 = 0

∗ ∗ ∗ ∗

∗ ∗

∗ ∗ ∗

∗

∗ ∗ ∗

∗

∗ ∗

∗

∗

∗

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

Re

k

1

2

3

Ha = 0.05𝜏 = 1

∗ ∗ ∗

∗ ∗

∗ ∗ ∗

∗ ∗

∗ ∗

∗ ∗ ∗

∗

∗

∗

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

Re

k

1

2

3

Ha = 0.2𝜏 = 1

∗ ∗

∗ ∗

∗ ∗ ∗

∗

∗

∗

∗ ∗

∗ ∗ ∗

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

Re

k

1

23

Ha = 0.4𝜏 = 1

∗ ∗ ∗

∗ ∗

∗ ∗

∗ ∗

∗ ∗

∗ ∗

∗

∗

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

Re

k

Ha = 0.05𝜏 = −1

∗ ∗ ∗

∗

∗ ∗

∗

∗

∗ ∗

∗ ∗ ∗

∗ ∗

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

Re

k

1

1

2

2

3

3

Ha = 0.2𝜏 = −1

∗ ∗ ∗ ∗

∗ ∗ ∗

∗ ∗ ∗

∗ ∗ ∗

∗

∗ ∗

∗ ∗ ∗

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

Re

k

1

2 3

Ha = 0.4𝜏 = −1

Figure 7: Variation of 𝑘 with respect to Re when 𝛽 = 50∘ and 𝑆 = 5: (1) 𝐶𝑅< 0 and 𝐽

2< 0; (2) 𝐶

𝑅< 0 and 𝐽

2> 0; (3) 𝐶

𝑅> 0 and 𝐽

2< 0. The

strip enclosing asterisks indicates that 𝐶𝑅> 0 and 𝐽

2> 0. The broken line within the strip represents 𝑘

𝑚= 𝑘𝑐/√2.

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 𝑘 − Re plane theequations 𝑏

111= 0 and 𝜕[𝑘b

111]/𝜕𝑘 = 0 give the neutral

stability curve and the curve corresponding to the maximumrate of amplification for linear disturbances. The expression

V𝑐=

𝑑

𝑑𝑡𝜃1(𝑡) = 𝑐

111+ [𝑐121

cos𝜙 (𝑡) − 𝑏121

sin𝜙 (𝑡)]

× 𝑎2(𝑡) + 𝑐

131𝑎2

1(𝑡) + 𝑐

132𝑎2

2(𝑡)

(25)

measures the velocity along the direction of themeanflowof asteady finite-amplitude wave. The surface deflection, ℎ(𝑥, 𝑡),moving with velocity V

𝑐along the mean flow direction in a

coordinate system 𝑥 = 𝑥 + 𝜃1is calculated from (D.1) and

reads as

ℎ (𝑥, 𝑡) = 1 + 2 [𝐵1cos𝑥 + 𝐵

2cos (2𝑥 − 𝜙)] . (26)

The steady solutions of the system (D.2a) and (D.2b)correspond to the fixed points of (D.4a)–(D.4c). In additionto the trivial solution 𝑎

1(𝑡) = 0, 𝑎

2(𝑡) = 0, and 𝜙(𝑡) =

0 (𝑧1(𝑡) = 0; 𝑧

2(𝑡) = 0), system (D.4a)–(D.4c) offers

nontrivial fixed points [62–64]. 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 𝑓

𝑖(𝑎1, 𝑎2, 𝜙) = 0, 𝑖 = 1, 2, 3, in (D.4a)–(D.4c). This

gives 𝑎1(𝑡) = 0 and 𝑎2

2(𝑡) = −𝑏

211/𝑏232

. The solution existsif sgn(𝑏

211𝑏232

) = −1. For convenience, 𝑎22(𝑡) = 𝑎

2 is set. From

12 ISRNMathematical Physics

Table 1: Parameter values for which the nonlinear wave explodes.

𝛽 = 55∘

𝛽 = 75∘

Re = 3 Re = 6 Re = 3 Re = 6

𝜏 = 1

Ha = 0 𝑘𝑠= 0.3671 𝑘

𝑠= 0.7933 𝑘

𝑠= 0.4467 𝑘

𝑠= 0.8707

Ha = 0.2 𝑘𝑠= 0.2309 𝑘

𝑠= 0.6056 𝑘

𝑠= 0.3438 𝑘

𝑠= 0.7045

Ha = 0.4 𝑘𝑠= 0.0268 𝑘

𝑠= 0.4399 𝑘

𝑠= 0.2561 𝑘

𝑠= 0.5686

𝜏 = 0

Ha = 0 𝑘𝑠= 0.2733 𝑘

𝑠= 0.7132 𝑘

𝑠= 0.3734 𝑘

𝑠= 0.7992

Ha = 0.2 𝑘𝑠= 0.1536 𝑘

𝑠= 0.5517 𝑘

𝑠= 0.2974 𝑘

𝑠= 0.6588

Ha = 0.4 𝑘𝑠= 0 𝑘

𝑠= 0.4073 𝑘

𝑠= 0.2319 𝑘

𝑠= 0.5436

𝜏 = −1

Ha = 0 𝑘𝑠= 0.1211 𝑘

𝑠= 0.6237 𝑘

𝑠= 0.2823 𝑘

𝑠= 0.7205

Ha = 0.2 𝑘𝑠= 0 𝑘

𝑠= 0.4861 𝑘

𝑠= 0.2301 𝑘

𝑠= 0.6049

Ha = 0.4 𝑘𝑠= 0 𝑘

𝑠= 0.3602 𝑘

𝑠= 0.1835 𝑘

𝑠= 0.5094

Table 2: Parameter values for which the nonlinear wave attains a finite amplitude.

𝛽 = 55∘

𝛽 = 75∘

Re = 3 Re = 6 Re = 3 Re = 6

𝜏 = 1

Ha = 0 𝑘𝑐= 0.7342 𝑘

𝑐= 1.5866 𝑘

𝑐= 0.8933 𝑘

𝑐= 1.7415

Ha = 0.2 𝑘𝑐= 0.4618 𝑘

𝑐= 1.2112 𝑘

𝑐= 0.6875 𝑘

𝑐= 1.4092

Ha = 0.4 𝑘𝑐= 0.0536 𝑘

𝑐= 0.8799 𝑘

𝑐= 0.5121 𝑘

𝑐= 1.1371

𝜏 = 0

Ha = 0 𝑘𝑐= 0.5467 𝑘

𝑐= 1.4263 𝑘

𝑐= 0.7468 𝑘

𝑐= 1.5984

Ha = 0.2 𝑘𝑐= 0.3071 𝑘

𝑐= 1.1034 𝑘

𝑐= 0.5947 𝑘

𝑐= 1.3177

Ha = 0.4 𝑘𝑐= 0 𝑘

𝑐= 0.8145 𝑘

𝑐= 0.4638 𝑘

𝑐= 1.0872

𝜏 = −1

Ha = 0 𝑘𝑐= 0.2421 𝑘

𝑐= 1.2475 𝑘

𝑐= 0.5646 𝑘

𝑐= 1.4411

Ha = 0.2 𝑘𝑐= 0 𝑘

𝑐= 0.9722 𝑘

𝑐= 0.4602 𝑘

𝑐= 1.2099

Ha = 0.4 𝑘𝑐= 0 𝑘

𝑐= 0.7205 𝑘

𝑐= 0.1834 𝑘

𝑐= 1.0187

(D.4c), 𝑐121

cos𝜙(𝑡) − 𝑏121

sin𝜙(𝑡) = (𝑘/2)𝐴1𝑎 is obtained.

From this condition, the fixed point for 𝜙(𝑡) is derived as

𝜙 (𝑡) = 𝜙 (𝑡) = tan−1 [(4𝑏121

𝑐121

± 𝑘𝐴

1𝑎

× {4𝑏2

121+ 4𝑐2

121− 𝑘2𝐴2

1𝑎2}1/2

)

× (4𝑏2

121− 𝑘2𝐴2

1𝑎2)−1

] ,

(27)

provided that the quantity within the square root is realand positive. The stability of the nonlinear dynamical system(D.4a)–(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)–(D.4c) can be represented in matrix notationas

((

(

𝑑𝑎1(𝑡)

𝑑𝑡

𝑑𝑎2(𝑡)

𝑑𝑡

𝑑𝜙 (𝑡)

𝑑𝑡

))

)

= (

𝑝1

0 0

0 𝑝2

0

0 𝑝3

𝑝4

)(

𝑎1(𝑡)

𝑎2(𝑡)

𝜙 (𝑡)

) + (

𝑞1

𝑞2

𝑞3

) ,

(28)

such that

𝑝1= 𝑏111

+ (𝑏121

cos𝜙 + 𝑐121

sin𝜙) 𝑎 + 𝑏132

𝑎2;

𝑝2= 𝑏211

+ 3𝑏232

𝑎2,

𝑝3= −2𝑏

121sin𝜙 + 2𝑐

121cos𝜙 − 2𝑘𝐴

1𝑎;

𝑝4= − (2𝑏

121cos𝜙 + 2𝑐

121sin𝜙) 𝑎,

𝑞1= 0; 𝑞

2= −2𝑏

232𝑎3;

𝑞3= 𝑘𝐴

1𝑎2+ (2𝑏121

cos𝜙 + 2𝑐121

sin𝜙) 𝜙𝑎.

(29)

The stability of the linear system (28) depends on theeigenvalues of the coefficient matrix with entries 𝑝

𝑖. The

eigenvalues are found as

𝜆1= 𝑏211

+ 3𝑎2𝑏232

= −2𝑏211

,

𝜆2= −2𝑎 (𝑏

121cos𝜙 + 𝑐

121sin𝜙) ,

𝜆3= 𝑏111

+ 𝑎 (𝑏121

cos𝜙 + 𝑐121

sin𝜙) + 𝑏132

𝑎2.

(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

𝛼Γ0

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

𝛼Γ0

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 𝛽 = 50∘ and 𝑆 = 5: (1) 𝜏 = −1, (2) 𝜏 = 0, and(3) 𝜏 = 1.

TheHopf bifurcation at the critical threshold is defined by𝐵1= 𝑘2𝐶1, and this yields 𝜆

1= 24𝜀𝑘

4𝐶1> 0. This eigenvalue

being independent of 𝜏 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 (𝐶

𝑅< 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 eigenvalue𝜆

1. Although themagnitude of𝜆

1remains

larger for 𝜏 < 0 than for 𝜏 ≥ 0, being positive, it plays adestabilizing role.

5.2. Mixed-Mode. When 𝑎1(𝑡), 𝑎2(𝑡) ̸= 0, the fixed points in

this case correspond to mixed-mode. Considering 𝜙(𝑡) ̸= 0and 𝑎∗1, 𝑎∗2, and 𝜙∗ to be the fixed points of (D.4a)–(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 entries𝑡𝑖𝑗(𝑖, 𝑗 = 1, 2, 3) of the 3 × 3 Jacobian matrix arising due to

linearization are the following:

𝑡11

= 𝑏111

+ (𝑏121

cos𝜙∗ + 𝑐121

sin𝜙∗) 𝑎∗2

+ 3𝑏131

𝑎∗2

1+ 𝑏132

𝑎∗2

2,

𝑡12

= (𝑏121

cos𝜙∗ + 𝑐121

sin𝜙∗) 𝑎∗1+ 2𝑏132

𝑎∗

1𝑎∗

2,

𝑡13

= (−𝑏121

sin𝜙∗ + 𝑐121

cos𝜙∗) 𝑎∗1𝑎∗

2,

𝑡21

= 2 (𝑏221

cos𝜙∗ − 𝑐221

sin𝜙∗) 𝑎∗1+ 2𝑏231

𝑎∗

1𝑎∗

2,

𝑡22

= 𝑏211

+ 𝑏231

𝑎∗2

1+ 3𝑏232

𝑎∗2

2,

𝑡23

= − (𝑏221

sin𝜙∗ + 𝑐221

cos𝜙∗) 𝑎∗21,

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

𝜆1=−2b

211

k

𝜏 = 1

𝜏 = 0

𝜏 = −1

𝜏 = −2

𝜏 = 2

Ha = 0.3

Re = 2

Figure 9: Eigenvalue𝜆1as a function of 𝑘whenWe = 1 and𝛽 = 75∘.

𝑡31

= 2𝑘𝐴

1𝑎∗

1−2𝑎∗

1

𝑎∗2

(𝑏221

sin𝜙∗ + 𝑐221

cos𝜙∗) ,

𝑡32

= − 2𝑘𝐴

1𝑎∗

2+ 2 (𝑐121

cos𝜙∗ − 𝑏121

sin𝜙∗)

+𝑎∗2

1

𝑎∗22

(𝑏221

sin𝜙∗ + 𝑐221

cos𝜙∗) ,

𝑡33

= − 2 (𝑐121

sin𝜙∗ + 𝑏121

cos𝜙∗) 𝑎∗2

− (𝑏221

cos𝜙∗ − 𝑐221

sin𝜙∗)𝑎∗2

1

𝑎∗2

.

(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 𝑎

1(𝑡), 𝑎2(𝑡) ̸= 0, but by evaluating the

Jacobian first by imposing 𝑎1(𝑡), 𝑎2(𝑡), 𝜙(𝑡) ̸= 0 and then by

substituting 𝜙 = 0 in the computed Jacobian. If one considers𝑎1(𝑡), 𝑎2(𝑡), 𝜙(𝑡) = 0, an overdetermined system is obtained.

5.3. Traveling Waves. Fixed points of (D.4a)–(D.4c) with 𝜙being a nonzero constant correspond to traveling waves. Themodal amplitudes are considered small in the neighborhoodof the neutral stability limit. After rescaling 𝑎

1(𝑡) → 𝛼𝑚

1(𝑡)

and 𝑎2(𝑡) → 𝛼

2𝑚2(𝑡), the following system is obtained:

𝑑𝑚1(𝑡)

𝑑𝑡= 𝑏111

𝑚1(𝑡) + 𝛼

2

× [𝑏121

cos𝜙 (𝑡) + 𝑐121

sin𝜙 (𝑡)]𝑚1(𝑡)𝑚2(𝑡)

+ 𝛼2𝑏131

𝑚3

1(𝑡) + O (𝛼

4) ,

(32a)

𝑑𝑚2(𝑡)

𝑑𝑡= 𝑏211

𝑚2(𝑡)

+ [𝑏221

cos𝜙 (𝑡) − 𝑐221

sin𝜙 (𝑡)]𝑚21(𝑡)

+ 𝛼2𝑏231

𝑚2

1(𝑡)𝑚2(𝑡) + O (𝛼

4) ,

(32b)

𝑑𝜙 (𝑡)

𝑑𝑡= − [𝑏

221sin𝜙 (𝑡) + 𝑐

221cos𝜙 (𝑡)]

𝑚2

1(𝑡)

𝑚2(𝑡)

+ 𝛼2𝑘𝐴

1𝑚2

1(𝑡) + 2𝛼

2

× [𝑐121

cos𝜙 (𝑡) − 𝑏121

sin𝜙 (𝑡)]𝑚2(𝑡) + O (𝛼

4) .

(32c)

To the O(𝛼2), 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(𝛼2):

tan𝜙 (𝑡) =(−𝑙1𝑙2± 𝑙3(𝑙2

1+ 𝑙2

2− 𝑙2

3)1/2

)

(𝑙21− 𝑙23) ,

,(33)

where 𝑙1= (1 + 2𝛼

2𝑏121

𝑚2

2(𝑡)/𝑏221

𝑚2

1(𝑡)), 𝑙2= (𝑐221

/𝑏221

) −

(2𝛼2𝑐121

𝑚2

2(𝑡)/𝑏221

𝑚2

1(𝑡)), and 𝑙

3= 𝛼2𝑘𝐴

1𝑚2(𝑡)/𝑏221

. In theabove equation, if only the leading order effect is considered,the fixed point is found as

𝜙∗(𝑡) = tan−1 (

−𝑐221

𝑏221

) + O (𝛼2) . (34)

Equating (32a) to zero and using (34), it is found that

𝛼2𝑏131

𝑚2

1(𝑡) = −𝑏

111− 𝛼2𝜑1𝑚2(𝑡) . (35)

Such a solution exists if [𝑏111

+ 𝛼2𝜑1𝑚2(𝑡)]𝑏131

< 0, where𝜑1= (𝑏121

cos𝜙∗ + 𝑐121

sin𝜙∗). A quadratic equation in𝑚2(𝑡)

is obtained by considering (32b) as

𝛼4𝑏231

𝜑1𝑚2

2(𝑡) + 𝛼

2(𝑏111

𝑏231

− 𝑏211

𝑏131

+ 𝜑1𝜑2)

× 𝑚2(𝑡) + 𝜑

2𝑏111

= 0,

(36)

where 𝜑2

= 𝑏221

cos𝜙∗ − 𝑐221

sin𝜙∗. For this equation,there are two solutions. The existence of such solutions tobe a real number demands (𝑏

111𝑏231

− 𝑏211

𝑏131

+ 𝜑1𝜑2)2−

4𝛼4𝜑1𝜑2𝑏111

𝑏231

> 0. Since 𝑏111

= 0 in the neutral stabilitylimit, (36) gives the amplitude of the nonzero traveling waveas

𝑚2(𝑡) =

𝑏211

𝑏131

− 𝜑1𝜑2

𝛼2𝑏231

𝜑1

=3𝑘2𝐶1𝐽2

𝛼2 (𝐵1− 4𝑘2𝐶

1) 𝜑1

. (37)

When 𝐽2= 0, 𝑚

2(𝑡) 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

−𝜋/k x 𝜋/k

t = 15 t = 14.5

h(x,t)

(a)

0.8

1.0

1.2

−𝜋/k x 𝜋/k

t = 29.95

t = 29.85

h(x,t)

(b)

0.8

1.0

1.2

−𝜋/k x 𝜋/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.

If𝑚∗1and𝑚∗

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:

𝐽 = (

2𝛼2𝑏131

𝑚∗2

1𝛼2𝜑1𝑚∗

1−𝛼2 sin𝜙∗ (𝑏

121+𝑐121

𝑏221

𝑐221

)𝑚∗

1𝑚∗

2

2 (𝜑2+ 𝛼2𝑏231

𝑚∗

2)𝑚∗

1𝛼2𝑏211

𝑚∗2

10

2𝛼2𝑘𝐴

1𝑚∗

1−2𝛼2(𝑏121

+𝑐121

𝑏221

𝑐221

) sin𝜙∗ −(2𝛼2𝜑1𝑚∗

2+ 𝜑2

𝑚∗2

1

𝑚∗2

)

). (38)

6. Nonlinear Development ofthe Interfacial Surface

The nonlinear interactions are studied numerically beyondthe linear stability threshold in a periodic domain, D =

{(𝑥, 𝑡) : 𝑥 ∈ (−𝜋/𝑘𝑚

, 𝜋/𝑘𝑚

), 𝑡 ∈ [0,∞)}, by solving (10)subject to the initial condition ℎ(𝑥, 0) = 1 − 0.1 cos(𝑘

𝑚𝑥).

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

0−𝜋/k

x

𝜋/k

1.06

1.04

1.02

1

0.98

0.96

0.94

h(x,t)

0−𝜋/k

x

𝜋/k

1.06

1.04

1.02

1

0.98

0.96

0.94

h(x,t)

0−𝜋/k

x

𝜋/k

1.06

1.04

1.02

1

0.98

0.96

0.94

h(x,t)

0−𝜋/k

x

𝜋/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, 𝛽 = 45∘, and 𝑆 = 5 when 𝜏 = 0 at time 𝑡 = 250.

0 50 100 150 200 2501

1.05

1.1

1.15

𝜏 =1

𝜏 = 0

𝜏 = −1

𝜏 = −1.5

Hm

ax

t

(a)

0 100 200 300 400 500 600 7001

1.05

1.1

1.15

𝜏 = 0

𝜏 = 1

𝜏 = −1 𝜏 = −1.5

Hm

ax

t

(b)

Figure 12: Maximum film thickness profiles at Re = 5, 𝛽 = 45∘, and 𝑆 = 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

𝜏 = 0

𝜏 = 0

𝜏 = −1𝜏 = −1

Hm

in

t

1

1.05

1.1

1.15

𝜏 = 0𝜏 = −1

𝜏 = 0 𝜏 = −1

Hm

ax

0 200 400 600 800 1000 1200 1400

t

Figure 13:Maximumandminimumamplitude profiles correspond-ing to Re = 5, 𝛽 = 45∘, 𝑆 = 5, and Ha = 0.3.

The local truncation error for this numerical scheme is T =O(Δ𝑡, Δ𝑥2). The computations are performed with a smalltime increment, Δ𝑡 = 10−3. 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 𝑁 = 800 such that the spatial step length isΔ𝑥 = 2𝜋/𝑁𝑘

𝑚. An error tolerance of 10−10 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 𝜏 > 0 remains smaller than the ones corresponding to𝜏 ≤ 0. Also, for different Ha, the amplitudes correspondingto 𝜏 < 0 remain larger than those corresponding to 𝜏 = 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 𝜏 < 0 to decrease below the amplitudecorresponding to 𝜏 = 0 increases (Figures 12(a), 12(b), and13). In addition, it is also observed that the time needed for

the amplitude profiles corresponding to 𝜏 = −1 to dominatethe amplitude profiles corresponding to 𝜏 = −1.5 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 𝜏 = 0, a wave whichhas a stretched front is observed after a long time. When𝜏 = −1, 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, 𝑞(𝑥, 𝑡) = ∫ℎ

0𝑢(𝑥, 𝑡)𝑑𝑦, for different

values of 𝜏 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 𝜏 > 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 𝜏 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 [37–39, 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

−3𝜋/k −𝜋/k 3𝜋/k𝜋/k

x

h(x,t)

(a)

0.998

0.999

1

1.001

1.002

1.003

1.004

1.005Time = 1299

Time = 1300

−3𝜋/k −𝜋/k 3𝜋/k𝜋/k

x

h(x,t)

(b)

Figure 14: Evolution of the surface-wave instability at Re = 5, 𝛽 = 45∘, 𝑆 = 5, and Ha = 0.3: (a) 𝜏 = 0 and (b) 𝜏 = −1. 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 (𝐽

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, 37–40, 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

−𝜋/k 𝜋/k

x

−𝜋/k 𝜋/k

x

0−0.5

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)

𝜏 = 1

0−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

u(x,t),(x,t),q(x,t)

𝜏 = 0

0−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

u(x,t),(x,t),q(x,t)

𝜏 = −1

0−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

u(x,t),(x,t),q(x,t)

𝜏 = 0

1

2

3

0−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

u(x,t),(x,t),q(x,t)

𝜏 = 1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

u(x,t),(x,t),q(x,t)

𝜏 = −1

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, 𝛽 = 45∘, and 𝑆 = 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

𝑢𝑥+ V𝑦= 0,

𝜀 (𝑢𝑡+ 𝑢𝑢𝑥+ V𝑢𝑦) = −𝜀𝑝

𝑥+ 𝜀2𝑢𝑥𝑥

+ 𝑢𝑦𝑦

+ Re−Ha𝑢,

𝜀2(V𝑡+ 𝑢V𝑥+ VV𝑦) = −𝑝

𝑦+ 𝜀3V𝑥𝑥

+ 𝜀V𝑦𝑦

− Re cot𝛽,

(A.1)

and the respective boundary conditions are

𝑢 = 0, V = 0 on 𝑦 = 0, (A.2)

𝑝 + 2𝜀(1 − 𝜀

2ℎ2

𝑥)

(1 + 𝜀2ℎ2𝑥)𝑢𝑥+

2𝜀ℎ𝑥

1 + 𝜀2ℎ2𝑥

(𝑢𝑦+ 𝜀2V𝑥)

= −𝜀2Sℎ𝑥𝑥

(1 + 𝜀2ℎ2𝑥)3/2

on 𝑦 = ℎ,

(A.3)

(1 − 𝜀2ℎ2

𝑥)

(1 + 𝜀2ℎ2𝑥)(𝑢𝑦+ 𝜀2V𝑥) −

4𝜀2ℎ𝑥

(1 + 𝜀2ℎ2𝑥)𝑢𝑥= 𝜏

on 𝑦 = ℎ,

(A.4)

V = ℎ𝑡+ 𝑢ℎ𝑥

on 𝑦 = ℎ. (A.5)

B. Multiple Scale Analysis

The film thickness,𝐻, is expanded as

𝐻(𝜀, 𝑥, 𝑥1, 𝑡, 𝑡1, 𝑡2) =

∞

∑

𝑖=1

𝛼𝑖𝐻𝑖(𝜀, 𝑥, 𝑥

1, 𝑡, 𝑡1, 𝑡2) . (B.1)

Exploiting (16) using (20) and (B.1), the following equation isobtained:

(𝐿0+ 𝛼𝐿1+ 𝛼2𝐿2) (𝛼𝐻

1+ 𝛼2𝐻2+ 𝛼3𝐻3) = −𝛼

2𝑁2− 𝛼3𝑁3,

(B.2)

where the operators 𝐿0, 𝐿1, and 𝐿

2and the quantities𝑁

2and

𝑁3are given as follows

𝐿0=

𝜕

𝜕𝑡+ 𝐴1

𝜕

𝜕𝑥+ 𝜀𝐵1

𝜕2

𝜕𝑥2+ 𝜀𝐶1

𝜕4

𝜕𝑥4, (B.3a)

𝐿1=

𝜕

𝜕𝑡1

+ 𝐴1

𝜕

𝜕𝑥1

+ 2𝜀𝐵1

𝜕2

𝜕𝑥𝜕𝑥1

+ 4𝜀𝐶1

𝜕4

𝜕𝑥3𝜕𝑥1

, (B.3b)

𝐿2=

𝜕

𝜕𝑡2

+ 𝜀𝐵1

𝜕2

𝜕𝑥21

+ 6𝜀𝐶1

𝜕4

𝜕𝑥2𝜕𝑥21

, (B.3c)

𝑁2= 𝐴

1𝐻1𝐻1𝑥

+ 𝜀𝐵

1𝐻1𝐻1𝑥𝑥

+ 𝜀𝐶

1𝐻1𝐻1𝑥𝑥𝑥𝑥

+ 𝜀𝐵

1𝐻2

1𝑥+ 𝜀𝐶

1𝐻1𝑥𝐻1𝑥𝑥𝑥

,

(B.3d)

𝑁3= 𝐴

1(𝐻1𝐻2𝑥

+ 𝐻1𝐻1𝑥1

+ 𝐻2𝐻1𝑥)

+ 𝜀𝐵

1(𝐻1𝐻2𝑥𝑥

+ 2𝐻1𝐻1𝑥𝑥1

+ 𝐻2𝐻1𝑥𝑥

)

+ 𝜀𝐶

1(𝐻1𝐻2𝑥𝑥𝑥𝑥

+ 4𝐻1𝐻1𝑥𝑥𝑥𝑥1

+ 𝐻2𝐻1𝑥𝑥𝑥𝑥

)

+ 𝜀𝐵

1(2𝐻1𝑥𝐻2𝑥

+ 2𝐻1𝑥𝐻1𝑥1

)

+ 𝜀𝐶

1(𝐻2𝑥𝑥𝑥

𝐻1𝑥

+ 3𝐻1𝑥𝑥𝑥1

𝐻1𝑥

+𝐻1𝑥𝑥𝑥

𝐻2𝑥

+ 𝐻1𝑥𝑥𝑥

𝐻1𝑥1

)

+𝐴

1

2𝐻2

1𝐻1𝑥

+ 𝜀𝐵

1

2𝐻2

1𝐻1𝑥𝑥

+ 𝜀𝐶

1

2𝐻2

1𝐻1𝑥𝑥𝑥𝑥

+ 𝜀𝐵

1𝐻1𝐻2

1𝑥+ 𝜀𝐶

1𝐻1𝐻1𝑥𝐻1𝑥𝑥𝑥

.

(B.3e)

The equations are solved order-by-order up to 𝑂(𝛼3) [12,14, 56, 57] to arrive at an equation related to the com-plex Ginzburg-Landau type for the perturbation amplitude𝜂(𝑥1, 𝑡1, 𝑡2):

𝜕𝜂

𝜕𝑡2

− 𝛼−2𝐶𝑅𝜂 + 𝜀 (𝐵

1− 6𝑘2𝐶1)𝜕2𝜂

𝜕𝑥21

+ (𝐽2+ 𝑖𝐽4)𝜂2

𝜂 = 0,

(B.4)

where

𝑒𝑟=

2 (𝐵

1− 𝑘2𝐶

1)

(−4𝐵1+ 16𝑘2𝐶

1), 𝑒

𝑖=

−𝐴

1

𝜀 (−4𝑘𝐵1+ 16𝑘3𝐶

1),

𝐽2= 𝜀(7𝑘

4𝑒𝑟𝐶

1− 𝑘2𝑒𝑟𝐵

1−𝑘2

2𝐵

1+𝑘4

2𝐶

1) − 𝑘𝑒

𝑖𝐴

1,

𝐽4= 𝜀 (7𝑘

4𝑒𝑖𝐶

1− 𝑘2𝑒𝑖𝐵

1) + 𝑘𝑒

𝑟𝐴

1+ 𝑘

𝐴

1

2.

(B.5)

It should be remarked that in (B.4) an imaginary, apparentlyconvective term of the form

𝑖V𝜕𝜂

𝜕𝑥1

= 2𝑖𝑘𝜀 (𝐵1− 2𝑘2𝐶1) 𝛼−1 𝜕𝜂

𝜕𝑥1

(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(𝛼) and V < 0. Such aterm arises in the secular condition at𝑂(𝛼3) as a contributionarising from one of the terms from 𝐿

1𝐻1at 𝑂(𝛼2). 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 𝜂 = Γ

0(𝑡2)𝑒[−𝑖𝑏(𝑡2)𝑡2].

ISRNMathematical Physics 21

This leads to a nonlinear ordinary differential equation for Γ0,

namely,

𝑑Γ0

𝑑𝑡2

− 𝑖Γ0

𝑑

𝑑𝑡2

[𝑏 (𝑡2) 𝑡2] − 𝛼−2𝐶𝑅Γ0+ (𝐽2+ 𝑖𝐽4) Γ3

0= 0,

(B.7)

where the real and imaginary parts, when separated from(37), give

𝑑 [Γ0(𝑡2)]

𝑑𝑡2

= (𝛼−2𝐶𝑅− 𝐽2Γ2

0) Γ0, (B.8)

𝑑 [𝑏 (𝑡2) 𝑡2]

𝑑𝑡2

= 𝐽4Γ2

0. (B.9)

If 𝐽2

= 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 𝐶

𝑅and

𝐽2. The perturbed wave speed caused by the infinitesimal

disturbances appearing in the nonlinear system can bemodified using (B.8).The threshold amplitude 𝛼Γ

0from (B.8)

is obtained as (when Γ0is nonzero and independent of 𝑡

2)

𝛼Γ0= √

𝐶𝑅

𝐽2

. (B.10)

Equation (B.9) with the use of (B.10) gives

𝑏 =𝐶𝑅𝐽4

𝛼2𝐽2

. (B.11)

The speed𝑁𝑐of the nonlinearwave is nowobtained from (26)

by substituting 𝜂 = Γ0𝑒[−𝑖𝑏𝑡2] and by using 𝑡

2= 𝛼2𝑡. This gives

the nonlinear wave speed as

𝑁𝑐= 𝐶𝐿𝑉

+ 𝛼2𝑏 = 𝐶

𝐿𝑉+ 𝐶𝑅

𝐽4

𝐽2

. (B.12)

C. Sideband Stability

The sideband instability is analyzed by subjecting Γ𝑠(𝑡2) to

sideband disturbances in the form of (Δ ≪ 1) [12, 56, 57]

𝜂 = Γ𝑠(𝑡2) + (ΔΓ

+(𝑡2) 𝑒𝑖𝑘𝑥

+ ΔΓ−(𝑡2) 𝑒−𝑖𝑘𝑥

) 𝑒−𝑖𝑏𝑡2 , (C.1)

in (B.4). Neglecting the nonlinear terms and separating thecoefficients of Δ𝑒𝑖(𝑘𝑥−𝑏𝑡2) and Δ𝑒−𝑖(𝑘𝑥−𝑏𝑡2), the following set ofequations are derived (V = (2𝜀𝑘/𝛼)(𝐵

1− 2𝑘2𝐶1) < 0, 𝑐

𝑖=

𝛼−2𝐶𝑅, and 𝐽

1𝑟= 𝜀(𝐵

1− 6𝑘2𝐶1) < 0 since 𝐵

1− 𝑘2𝐶1

=

O(𝛼2)):

𝑑Γ+(𝑡2)

𝑑𝑡2

= (𝑖P + 𝑘V + 𝑐𝑖+ 𝑘2𝐽1𝑟− 2 (𝐽2+ 𝑖𝐽4)Γ𝑠

2

)

× Γ+(𝑡2) − (𝐽2+ 𝑖𝐽4)Γ𝑠

2

Γ−(𝑡2) ,

(C.2a)

𝑑Γ−(𝑡2)

𝑑𝑡2

= (𝑖P − 𝑘V + 𝑐𝑖+ 𝑘2𝐽1𝑟− 2 (𝐽2+ 𝑖𝐽4)Γ𝑠

2

)

× Γ−(𝑡2) − (𝐽2+ 𝑖𝐽4)Γ𝑠

2

Γ+(𝑡2) ,

(C.2b)

where the barred quantities are the complex conjugatescorresponding to their counterparts. For convenience, theabove system is represented as

(

𝑑Γ+(𝑡2)

𝑑𝑡2

𝑑Γ−(𝑡2)

𝑑𝑡2

) = M(

Γ+(𝑡2)

Γ−(𝑡2)

) , (C.3)

whereM is a 2 × 2 matrix whose entries are

𝑀11

= 𝑖P + 𝑘V + 𝑐𝑖+ 𝑘2𝐽1𝑟− 2 (𝐽2+ 𝑖𝐽4)Γ𝑠

2

,

𝑀22

= −𝑖P − 𝑘V + 𝑐𝑖+ 𝑘2𝐽1𝑟− 2 (𝐽2− 𝑖𝐽4)Γ𝑠

2

,

𝑀12

= − (𝐽2+ 𝑖𝐽4)Γ𝑠

2

, 𝑀21

= 𝑀12.

(C.4)

Setting Γ+(𝑡2) ∝ 𝑒

𝜗𝑡2 and Γ−(𝑡2) ∝ 𝑒

𝜗𝑡2 , the eigenvaluesare obtained from |M − 𝜗𝐼| = 0, and the stability of (22)subject to sideband disturbances as 𝑡

2→ ∞ demands that

𝜗 < 0. The eigenvalues are found as follows

𝜗 =1

2(tr (M) ± √tr2 (M) − 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 (𝑧

𝑛(𝑡) is complex, and a bar above it designates its

complex conjugate)

ℎ (𝑥, 𝑡) = 1 +

2

∑

𝑛=1

𝑧𝑛(𝑡) 𝑒𝑛𝑖𝑘𝑥

+ 𝑧𝑛(𝑡) 𝑒−𝑛𝑖𝑘𝑥 (D.1)

and is substituted in the evolution equation (10). A Taylorseries expansion is then sought to be about “1”, and the

22 ISRNMathematical Physics

following coupled dynamical system is obtained from thecoefficients of 𝑒𝑖𝑘𝑥 and 𝑒2𝑖𝑘𝑥 terms:

𝑑

𝑑𝑡𝑧1(𝑡) = 𝑎

111𝑧1(𝑡) + 𝑎

121𝑧1(𝑡) 𝑧2(𝑡)

+ 𝑧1(𝑡) (𝑎131

𝑧12

+ 𝑎132

𝑧22

) ,

(D.2a)

𝑑

𝑑𝑡𝑧2(𝑡) = 𝑎

211𝑧2(𝑡) + 𝑎

221𝑧2

1(𝑡)

+ 𝑧2(𝑡) (𝑎231

𝑧12

+ 𝑎232

𝑧22

) .

(D.2b)

It should be remarked that the higher-order terms arisingwhile deducing (D.2a) and (D.2b) are neglected by assumingthat |𝑧

1| ≪ 1 and |𝑧

2| ≪ |𝑧

1|. The coefficients of the above

system are given by

𝑎111

= 𝜀 (𝑘2𝐵1− 𝑘4𝐶1) − 𝑖𝑘𝐴

1,

𝑎211

= 𝜀 (4𝑘2𝐵1− 16𝑘

4𝐶1) − 2𝑖𝑘𝐴

1,

𝑎121

= 𝜀 (𝑘2𝐵

1− 7𝑘4𝐶

1) − 𝑖𝑘𝐴

1,

𝑎221

= 𝜀 (2𝑘2𝐵

1− 2𝑘4𝐶

1) − 𝑖𝑘𝐴

1,

𝑎131

=𝜀

2(𝑘2𝐵

1− 𝑘4𝐶

1) −

𝑖

2𝑘𝐴

1,

𝑎231

= 𝜀 (4𝑘2𝐵

1− 16𝑘

4𝐶

1) − 2𝑖𝑘𝐴

1,

𝑎132

= 𝜀 (𝑘2𝐵

1− 𝑘4𝐶

1) − 𝑖𝑘𝐴

1,

𝑎232

= 𝜀 (2𝑘2𝐵

1− 8𝑘4𝐶

1) − 𝑖𝑘𝐴

1.

(D.3)

The complex conjugates occurring in system (D.3) canbe avoided by representing 𝑧

𝑛(𝑡) = 𝑎

𝑛(𝑡)𝑒𝑖𝜃𝑛(𝑡) in polar

form, where 𝑎𝑛(𝑡) and 𝜃

𝑛(𝑡) are real functions such that 𝑛 =

1, 2. Using polar notation, the following three equations areobtained:

𝑑𝑎1(𝑡)

𝑑𝑡= 𝑓1(𝑎1(𝑡) , 𝑎2(𝑡) , 𝜙 (𝑡))

= 𝑏111

𝑎1(𝑡) + [𝑏

121cos𝜙 (𝑡) + 𝑐

121sin𝜙 (𝑡)]

× 𝑎1(𝑡) 𝑎2(𝑡) + 𝑎

1(𝑡) [𝑏131

𝑎2

1(𝑡) + 𝑏

132𝑎2

2(𝑡)] ,

(D.4a)

𝑑𝑎2(𝑡)

𝑑𝑡= 𝑓2(𝑎1(𝑡) , 𝑎2(𝑡) , 𝜙 (𝑡))

= 𝑏211

𝑎2(𝑡) + [𝑏

221cos𝜙 (𝑡) − 𝑐

221sin𝜙 (𝑡)]

× 𝑎2

1(𝑡) + 𝑎

2(𝑡) [𝑏231

𝑎2

1(𝑡) + 𝑏

232𝑎2

2(𝑡)] ,

(D.4b)

𝑑𝜙 (𝑡)

𝑑𝑡= 𝑓3(𝑎1(𝑡) , 𝑎2(𝑡) , 𝜙 (𝑡))

= 𝑘𝐴

1(𝑎2

1− 𝑎2

2) + 2 [𝑐

121cos𝜙 (𝑡) − 𝑏

121sin𝜙 (𝑡)]

× 𝑎2(𝑡) − [𝑏

221sin𝜙 (𝑡) + 𝑐

221cos𝜙 (𝑡)]

𝑎2

1(𝑡)

𝑎2(𝑡)

.

(D.4c)

While the coefficient of 𝑒𝑖𝜃𝑛(𝑡) term yields (D.4a) and(D.4b), (D.4c) is obtained from the two equations whichresult as the coefficient of 𝑖𝑒𝑖𝜃𝑛(𝑡) term with the impositionthat the phase relationship is defined as 𝜙(𝑡) = 2𝜃

1(𝑡) − 𝜃

2(𝑡).

The terms 𝑏𝑖𝑗𝑘

and 𝑐𝑖𝑗𝑘

are real numbers and represent the realand imaginary parts of a typical 𝑎

𝑖𝑗𝑘term in (D.3) such that

𝑖, 𝑘 = 1, 2 and 𝑗 = 1, 2, 3.

References

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