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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 © 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
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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
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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.
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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.
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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.
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