Electrohydrodynamic Instability of Two Thin Viscous Leaky...

International Scholarly Research NetworkISRN Applied MathematicsVolume 2011, Article ID 498718, 35 pagesdoi:10.5402/2011/498718

Review ArticleElectrohydrodynamic Instability ofTwo Thin Viscous Leaky Dielectric Fluid Films ina Porous Medium

M. F. El-Sayed,1, 2 M. H. M. Moussa,1 A. A. A. Hassan,1and N. M. Hafez1

1 Department of Mathematics, Faculty of Education, Ain Shams University, Heliopolis (Roxy),Cairo 11757, Egypt

2 Department of Mathematics, College of Science, Qassim University, P.O. Box 6644,Buraidah 51452, Saudi Arabia

Correspondence should be addressed to M. F. El-Sayed, [email protected]

Received 23 March 2011; Accepted 4 May 2011

Academic Editor: R. Cardoso

Copyright q 2011 M. F. El-Sayed et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

The effect of an applied electric field on the stability of the interface between two thin viscous leakydielectric fluid films in porousmedium is analyzed in the long-wave limit. A systematic asymptoticexpansion is employed to derive coupled nonlinear evolution equations of the interface andinterfacial free charge distribution. The linearized stability of these equations is determined andthe effects of various parameters are examined in detail. For perfect-perfect dielectrics, the variousparameters affect only for small wavenumber values. For dielectrics, the various parameters affectonly for small wavenumber values. For effect for small wavenumbers, and a stabilizing effectafterwards, and for high wavenumber values for the other physical parameters, new regions ofstability or instability appear. For leaky-leaky dielectrics, the conductivity of upper fluid has adestabilizing effect for small or high wavenumbers, while it has a dual role on the stability ofthe system in a wavenumber range between them. The effects of all other physical parametersbehave in the same manner as in the case of perfect-leaky dielectrics, except that in the later case,the stability or instability regions occur more faster than the corresponding case of leaky-leakydielectrics.

1. Introduction

The effect of electric fields on the stability and dynamics of fluid-fluid interfaces has been anarea of extensive research, beginning from the classic works of Taylor and McEwan [1] andMelcher and Smith [2]. These works and subsequent studies, see, for example, the reviewsof Saville [3] and Griffiths [4], have amply demonstrated the role of electrical stresses on

2 ISRN Applied Mathematics

fluid interfaces and the associated electrohydrodynamic instabilities in such systems. Oneof the basic problems here is to understand the stability of the interface between two fluidlayers bounded on the top and bottom by rigid plates, and this has been the subject of manyprevious studies. Mohamed et al. [5] concentrated on two superposed viscous fluids in achannel subjected to a normal electric field, where the upper fluid is highly conducting,while the lower fluid is dielectric, and they performed the long-wave linear stability analysis,[6, 7], and showed that the electric field always has a destabilizing effect on the flow.Abdella and Rasmussen [8] studied Couette flow of two viscous fluids with different viscos-ities, densities, conductivities and permittivities, in an unbounded domain subjected to anormal electric field. They studied, following Melcher [9], two special cases in detail: theelectrohydrodynamic free-charge configuration (EH-If) and the electrohydrodynamic polari-zation charge configuration (EH-If).

These studies have largely considered systems in which gravitational effects are im-portant, and therefore, a critical applied voltage is required to cause the instability, verylong waves are stabilized by interfacial tension, and waves of intermediate lengths becomeunstable. These earlier studies have focused on how the critical voltage required for insta-bility is affected by the nature of the fluids, namely. whether they are perfect dielectrics, orwhether they are leaky dielectrics in which there is the possibility of free charge conductionin the fluids, and also the possibility of accumulation and redistribution of charges on theinterface between two fluids. Recently, there has been a renewed interest in this area, in partdue to the relevance of such phenomena in the formation of well-controlled patterns usingthe application of electric fields to thin liquid films [10–12], which has demonstrated thatthe application of an external electric field to polymer-air or polymer-polymer interfacesenhances the spontaneous fluctuations at the interface leading to an instability. However, forleaky dielectrics with surface charges at the interface and two fluids that are not perfectlyconducting, we must bear in mind that there is an electrical tangential shear stress at theinterface, induced by the electric field, and hence, it changes the stability of the two-fluidlayer system as previously investigated by Ozen et al. [13]. Wu and Chou [14] performedlinear stability analysis of a leaky dielectric viscoelastic fluid whose constitutive behaviorwas described using the Jeffrey’s model [15]. The surface instability of a Newtonian fluid(modeled as both leaky and perfect dielectrics) under the effect of electric field is nowwell understood [16]. For recent developments of this topic, see the investigations ofPapageorgiou and Petropoulos [17], Shankar and Sharma [18], Craster and Matar [19], Liet al. [20], Tomar et al. [21], and Supeene et al. [22].

Theoretical studies that have considered the linear stability characteristics of thinfluid films subjected to electric fields were restricted to the following configurations: (i) theinterface between a perfect dielectric liquid and air [10], (ii) the interface between two perfectdielectric liquids [11], (iii) the interface between a dielectric fluid of finite thickness and a con-ducting fluid of much larger thickness [23], and (iv) the interface between a leaky dielectricliquid and air [24]. Except the study of Pease III and Russel [24], all these studies have focusedonly on perfect dielectric systems. The presence of conductivity in one or both of the liquidscould have a significant impact on the length scales and growth rates. It should be noted thatin all the above-mentioned studies, the medium has been considered to be nonporous.

Porous media theories, on one hand, play an important role in many branches ofengineering, including material science, petroleum industry, chemical engineering, and soilmechanics as well as biomechanics. The flow through porous media has gained considerableinterest in recent years, particularly among geophysical fluid dynamicists. It is well knownthat in Darcy’s law, which relates the pressure gradient, the bulk viscous resistance, and

ISRN Applied Mathematics 3

the gravitational force in a porous medium, the usual viscous term in the equation of motionis replaced by the resistive force −(μ/k1)v, where μ is the fluid viscosity, k1 is the mediumpermeability, and v is the Darcian velocity of the fluid. Much of the recent studies on thistopic are given by Ingham and Pop [25], Vafai [26], Del Rio and Whitaker [27], Pop andIngham [28], and Nield and Bejan [29]. On the other hand, electrohydrodynamic instabilitystudies for flows in porous media has attracted little attention in the scientific literature [30–36] despsite their applications in various diverse fields with great interest. Thus, there is agrowing need for original research in the updated electrohydrodynamic phenomena whichhave some physical and engineering applications.

In this study, we consider the most general case of the effect of an external appliedelectric field on the stability of the interface between two thin leaky dielectric fluids with arbi-trary viscosities and conductivities. We first use a systematic long-wave asymptotic analysisto derive the nonlinear evolution equations for the interface position and interfacial chargedistribution, and we subsequently study the linearized stability of the nonlinear differentialequations. This paper is organized in the following manner: Section 2 discusses the relevantgoverning equations and boundary conditions, and in Section 3, we nondimensionalize theseequations and conditions. In Sections 4 and 5, we outline the long-wave asymptotic analysisused to derive the nonlinear evolution equations for the interface position and charge, and inSection 6, we develop the linear stability analysis of the nonlinear equations, and discuss theimportant representative studies results obtained from the general case of two leaky dielectricinterface and the limiting cases of both perfect-leaky dielectric, and perfect-perfect dielectricinterfaces. Finally, the salient conclusions of the present study are discussed in Section 7.

2. Problem Formulation and Governing Equations

The system of interest consists of two leaky dielectric fluids in porous medium of arbitraryviscosities occupying the regions −H < y < 0 (fluid 2) and 0 < y < βH (fluid 1) in the initialunperturbed state, see Figure 1, where β is the ratio of the thicknesses of top and bottomfluids. The perturbed interface between the two fluids is denoted by y = h(x). The two fluidsare stationary in the initial state, with viscosities μi, dielectric constants εi, conductivitiesσi, porosity of porous medium ε and medium permeability k1, where i = 1, 2. Fluid 2 isbounded at the bottom y = −H by a rigid plate which is maintained at an electric potentialψ = ψb, while fluid 1 is bounded at the top y = βH by a rigid plate maintained at an electricpotential ψ = 0. In the ensuing analysis, we assume that the material properties of the fluidsuch as viscosities μi, dielectric constants εi, conductivities σi, porosity of porous medium ε,and medium permeability k1 are constants and independent of spatial position. Followingthe leaky dielectric model formulation of Saville [3], we assume that electroneutrality is validin the bulk, while free charge is assumed to accumulate at the fluids interface. We also neglectthe diffusion of free charge within the interface.

For leaky dielectric fluids of constant conductivities and with zero net charge in thebulk, the following governing equations are appropriate for the electric field E in the twofluids 1 and 2 [3]

∇ · Ei = 0, i = 1, 2. (2.1)


ψ = ψb

n

tx

y

y = −H

Fluid 1

Fluid 2

ψ = 0

ɛ1 μ1 σ1 ε k1

ɛ2 μ2 σ2 ε k1

y = h(X)

y = βH

Figure 1: Schematic diagram showing the configuration and coordinate system.

Since the electric fields are irrotational, Ei = −∇ψi, where ψi is the electric potential: in the fluidi. Substituting this into (2.1) gives the following Laplace’s equation for the electric potential:ψi

∇2ψi = 0. (2.2)

These governing equations are supplemented by the following boundary conditions.

(1) The normal component of the electric field, at the interface y = h(x), satisfies

ε2ε0(∇ψ2 · n

) − ε1ε0(∇ψ1 · n

)= q(x, t) at y = h(x), (2.3)

where n is the unit normal to the interface at y = h(x) (see Figure 1), εi (i = 1, 2) isthe dielectric constant in fluid i, ε0 is the permittivity of free space, and q(x, t) is thesurface charge density of free charges at the interface.

(2) The continuity of the tangential component of the electric field at the interface y =h(x) translates to the continuity of the electric potentials; that is,

ψ1 = ψ2 at y = h(x). (2.4)

(3) The electric potentials satisfy the following conditions at the rigid boundaries:

ψ2 = ψb at y = −H,

ψ1 = 0 at y = βH.(2.5)


We next turn to the equations governing the motion of the two fluids. Owing to the relativelysmall thicknesses of the fluids, we ignore inertial effects in both fluids, and hence, thegoverning equations are the Stokes equations for continuity and momentum balance

∇ ·Vi = 0, ∇ · Ti = 0, (2.6)

where Vi and Ti are the velocity field and the total stress tensor, respectively, in fluid i. Boththe fluids are assumed to be irrotational, then the fluid velocity Vi can be derived from ascalar velocity potential ϕi such that Vi = −∇ϕi. We have neglected the effects of gravity onthe length scales of interest here. In addition, the effects of van der Waals dispersion forcesare negligible for the films considered in the experimental studies [11]. The total stress tensorT is given by a sum of isotropic pressure, deviatoric viscous stresses for the Newtonian fluid,the electrical Maxwell stress tensor, and a Darcy’s law term describing the isotropic porousmedium

Ti = −piI + μi(∇Vi +∇VT

i

)+mi +

μik1ϕiI, i = 1, 2, (2.7)

where pi is the pressure in fluid i, I is the identity tensor, and the Maxwell stress tensor mi isgiven by Saville [3]

mi = εiε0[EiEi − 1

2(Ei · Ei)I

]. (2.8)

The divergence of the Maxwell stress tensor ∇ ·mi = 0, because the bulk of the fluid is free ofnet charge, and the dielectric constants are independent of spatial position in the two fluids;that is,

∇ ·mi = ∇ ·{εiε0

[EiEi − 1

2(Ei · Ei)I

]}= −ε0Ei

2· Ei∇εi + ρfEi = 0, (2.9)

with ρf is the bulk free charge. Thus, the Maxwell stress tensor will not appear in the momen-tum balance but will affect the flow only through the conditions at the interface. The govern-ing momentum equations in the two fluids, therefore, become

ρ

ε

DVi

Dt= −∇pi +∇ ·mi +

μiε∇2Vi −

μik1

Vi, i = 1, 2. (2.10)

Since there is no time scale, then Dvi/Dt = 0, and we get from the above two equations that

∇pi −μiε∇2Vi +

μik1

Vi = 0, i = 1, 2. (2.11)

The fluid velocities satisfy no-slip and no-penetration conditions at the top and bottom plates

V1(y = βH

)= 0, V2

(y = −H) = 0. (2.12)


At the interface y = h(x) between the two fluids, continuity of velocities and stresses apply

(V · n)1 = (V · n)2, (2.13)

(V · t)1 = (V · t)2, (2.14)

(n · T · n)2 = (n · T · n)1 + γκ, (2.15)

(t · T · n)2 = (t · T · n)1, (2.16)

where γ is the interfacial tension between the two fluids, t is the unit tangent to the interface(see Figure 1), and κ = ∂2h(x)/∂x2 is the mean curvature of the interface. We restrict ourattention to two-dimensional systems which are invariant in the z direction and denote thevelocity components in the x and y directions by u and υ, respectively; that is, Vi = (ui, υi).Upon substituting the expression for the Maxwell stress tensor (2.8) in (2.15) and (2.16),respectively, we get

(n · T · n)i = −pi + 2μi∂υi∂y

+ εiε0[EN

2

i − 12E2i

]+μik1ϕi,

(t · T · n)i = μi(∂ui∂y

+∂υi∂x

)+ εiε0ETi E

Ni ,

(2.17)

where ETi and ENi are the tangential and normal components of the electric field Ei in thefluid i, respectively. Hence, we obtain the following conditions to be applied at the interfacey = h(x):

[p1 − 2μ1

∂υ1∂y

]−[p2 − 2μ2

∂υ2∂y

]+

1k1

[μ2ϕ2 − μ1ϕ1

]

= γκ +ε1ε02

[(∇ψ1 · n)2 − (∇ψ1 · t

)2] − ε2ε02

[(∇ψ2 · n)2 − (∇ψ2 · t

)2],

(2.18)

for the normal stress continuity, and

μ2

[∂u2∂y

+∂υ2∂x

]− μ1

[∂u1∂y

+∂υ1∂x

]= −(∇ψ1 · t

)q(x, t), (2.19)

for the tangential stress continuity. In the last equation, we have used the normal electric fieldcontinuity condition, (2.3), to simplify the right-hand side. The kinematic condition at theinterface prescribes the evolution of the interface position h(x, t),

υ1(y = h(x)

)= υ2(y = h(x)

)= ε

∂h

∂t+V · ∇sh, (2.20)


where ∇s is the gradient operator along the interface y = h(x). Finally, the interfacial chargeis governed by a conservation equation

ε∂q

∂t+V · ∇sq − qn · (n · ∇)V = ε([σ2E2 · n] − [σ1E1 · n]), (2.21)

where the terms on the left-hand side represent, respectively, the accumulation, convection,and variation of the charge due to dilation of the interface, while the right side represents themigration of charge to or from the interface due to ion conduction in the bulk [3].

3. Nondimensional Forms

It is useful at this point to nondimensionalize the governing equations and boundary condi-tions by setting

ψ ′i =

ψiψb, p′i =

piH2

ε0ψ2b

, T ′i =

TiH2

ε0ψ2b

, x′ =x

H, y′ =

y

H,

E′i =

EiHψb

, V′i =

Viμ2H

ε0ψ2b

, t′ =tε0ψ

2b

μ2H2, ϕ′

i =μ2ϕi

ε0ψ2b

,

q′ =qH

ε0ψb, κ′ = κH, k′1 =

k1H2

, h′ =h

H.

(3.1)

Upon using these scales to nondimensionalize the above governing equations and boundaryconditions, we end up with the following nondimensional set of equations. Without loss ofclarity, and for the sake of brevity, we represent nondimensional variables with the samenotation in the ensuing discussion by dropping dashes.

The nondimensional governing equations for the electric potentials ψi are

∇2ψi = 0, i = 1, 2, (3.2)

with the following boundary conditions at the interface y = h(x),

[ε2∇ψ2 · n

] − [ε1∇ψ1 · n]= q, ψ1 = ψ2, (3.3)

and the following boundary conditions at the top and bottom boundaries

ψ1(y = β

)= 0, ψ2

(y = −1) = 1. (3.4)


Similarly, the nondimensional equations governing the fluid motion are

∇ ·V1 = 0, ∇ ·V2 = 0, (3.5)

∇p1 −μrε∇2V1 +

μrk1

V1 = 0,

∇p2 − 1ε∇2V2 +

1k1

V2 = 0.(3.6)

with μr = μ1/μ2 being the ratio of viscosities of the two fluids. The nondimensional normaland tangential stress continuity conditions at the interface become

[p1 − 2μr

∂υ1∂y

]−[p2 − 2

∂υ2∂y

]+

1k1

[ϕ2 − μrϕ1

]

= γκ +ε12

[(∇ψ1 · n)2 − (∇ψ1 · t

)2] − ε22

[(∇ψ2 · n)2 − (∇ψ2 · t

)2],

(3.7)

[∂u2∂y

+∂υ2∂x

]− μr[∂u1∂y

+∂υ1∂x

]= −(∇ψ1 · t

)q(x, t), (3.8)

where γ = γH/ε0ψ2bis the nondimensional interfacial tension. The boundary conditions for

the velocities at the top and bottom plates become

u1(y = β

)= 0, υ1

(y = β

)= 0,

u2(y = −1) = 0, υ2

(y = −1) = 0.

(3.9)

The nondimensional kinematic condition at the interface is

υ1(y = h(x)

)= υ2(y = h(x)

)= ε

∂h

∂t+V · ∇sh, (3.10)

and the nondimensional charge conservation equation at the interface is

ε∂q

∂t+V · ∇sq − qn · (n · ∇)V = ε

([S1∇ψ1 · n

] − [S2∇ψ2 · n]), (3.11)

where Si = σiμ2H2/ε20ψ

2b, i = 1, 2 are the nondimensional conductivities in the two fluids.

This completes the specification of the governing equations and boundary conditions,which are highly coupled. Due to the negligible effect of gravity at length scales of interesthere, the above system of equations undergoes a long-wave instability. We now carry out along-wave asymptotic analysis to make the above system of equations tractable, and therebyderive coupled nonlinear evolution equations for the interface position h(x, t) and chargeq(x, t). While the main focus of this paper is to analyze the stability of the linearized equa-tions, it is nonetheless useful to first derive the nonlinear evolution equations, since theseequations can be used in future studies to understand (by numerical simulations) the non-linear evolution processes that occur after the linear instability.


4. Long-Wave Asymptotic Analysis

In the long-wave limit, the wavelength L of the fastest growing modes is much larger thanthe transverse length scale H in the system, and it is useful to define a small parameter δ =H/L � 1. The lateral length scale L is determined self-consistently in the following analysis

to be√γH3/ε0ψ

2b, and this is further estimated below to be much larger than H. Similarly,

a slow time scale is necessary to describe the dynamics of the interface motion at such largelength scales, and this is introduced a little later in (4.15). In the limitH � L, the derivativesin the x direction should be scaled with L. To this end, we define the slowly varying scale χin the following manner:

∂

∂x= δ

∂

∂χ, (4.1)

and ∂/∂χ ∼ O(1). When we apply the above scalings, (4.1), to the Laplace equation, (3.2), forthe electric potential ψi, it simplifies in the limit δ � 1 to

∂2ψi

∂y2= 0, i = 1, 2. (4.2)

The continuity condition for the normal component of the electric field at the interfacey = h(χ), (3.3), is similarly simplified in the long-wave limit as

ε2∂ψ2

∂y− ε1

∂ψ1

∂y= q, (4.3)

while the other interface condition (second equation in (3.3)) and the boundary conditions,(3.4), remain unchanged in the long-wave limit.

We now turn to the simplification of the momentum equations (3.6) for the fluidmotion in the long-wave limit. It is useful to define the variable μr,i such that μr,i = μr fori = 1 and μr,i = 1 for i = 2. The x-momentum equation can be simplified in the long-wavelimit as

δ∂pi∂χ

− μr,iε

∂2ui∂y2

+μr,ik1

ui = 0. (4.4)

This suggests that ui ∼ O(δ)pi. In order to make this explicit and to make ordering of vari-ous quantities simpler, we represent the pressure pi and the x-component velocity ui in thefollowing manner:

pi = p(0)i , ui = δu

(0)i . (4.5)

The above variables are the leading order quantities in an asymptotic expansion in δ, and wewill be concerned only with the leading order variables in this paper. The continuity equation


in both fluids (3.5) becomes, upon using ∂/∂x ∼ δ∂/∂χ and using the above expansion forui,

δ2∂u

(0)i

∂χ+∂υi∂y

= 0, (4.6)

which suggests the following expansion for υi:

υi = δ2υ(0)i. (4.7)

Upon using this expansion, the nondimensional y component of the momentum equationyields to leading order in δ, ∂p(0)i /∂y = 0, implying that the pressure is constant in both filmsacross the y direction, and so pi = pi(χ, t). The simplified x-momentum equation, therefore,is given by

dp(0)i

dχ− μr,i

ε

∂2u(0)i

∂y2+μr,ik1

u(0)i = 0. (4.8)

The normal stress condition at the interface, (3.7), simplifies to give the following equation inthe long-wave limit:

p(0)1 − p(0)2 +

1k1

(ϕ2 − μrϕ1

)= γδ2

∂2h

∂χ2+ε12

(∂ψ1

∂y

)2

− ε22

(∂ψ2

∂y

)2

. (4.9)

In order for the interfacial tension to be of the same order as the other terms in the aboveequation, we require γδ2 ∼ O(1), where δ = H/L. We set γ(H/L)2 = 1, and from this relation,

we determine the lateral length scale L to be L =√γH2 =

√γH3/ε0ψ

2b . Upon using the rela-

tion γδ2 = 1, (4.9) becomes

p(0)1 − p(0)2 +

1k1

(ϕ2 − μrϕ1

)=∂2h

∂χ2+ε12

(∂ψ1

∂y

)2

− ε22

(∂ψ2

∂y

)2

. (4.10)

The tangential stress continuity, (3.8), simplifies to the following condition in the long-wavelimit:

∂u(0)2

∂y− μr

∂u(0)1

∂y= −q∂ψ1

∂χ. (4.11)

The nondimensional kinematic condition at the interface, (3.10), after using the asymptoticexpansion for υi (4.7), yields

δ2υ(0)1

(y = h

(χ))

= ε∂h

∂t+ δ2u(0)1

∂h

∂χ. (4.12)


In order for the time derivative term in the above equation to be of the same order as theother two terms, it is necessary to stipulate a slow time scale in the long-wave limit such that

∂

∂t= δ2

∂

∂τ, (4.13)

where ∂/∂τ is O(1). The kinematic condition thus gives

υ(0)1

(y = h

(χ))

= ε∂h

∂τ+ u(0)1

∂h

∂χ. (4.14)

The dimensional slow time scale is obtained as follows:

∂

∂t=μ2H

2

ε0ψ2b

∂

∂tdim= δ2

∂

∂τ, (4.15)

where tdim is the dimensional time. Thus, in order to nondimensionalize the dimensional timetdim in long-wave limit, the appropriate time scale is μ2H

2/(ε0ψ2bδ2). After using γδ2 = 1 to

eliminate δ2, the time scale becomes μ2H3γ/(ε0ψ2

b)2.

Finally, the nondimensional interfacial charge balance, (3.11) is simplified in the long-wave limit as

εδ2∂q

∂τ+ δ2u(0)1

∂q

∂χ− δ2q∂υ

(0)1

∂y= ε[S1∂ψ1

∂y− S2

∂ψ2

∂y

]. (4.16)

The above equation suggests that the nondimensional conductivities S1 and S2 both shouldscale as δ2 in order to balance the left side of the equation. So, we let Si = δ2S

(0)i , i = 1, 2, where

S(0)i = σiμ2γH

3/(ε30ψ4b). Upon using these rescaled conductivities, the charge conservation

equation becomes, after using the continuity equation, (4.6),

ε∂q

∂τ+∂(u(0)1 q)

∂χ= ε[S(0)1

∂ψ1

∂y− S(0)

2∂ψ2

∂y

]. (4.17)

This completes the derivation of the simplified governing equations in the long-wave limit.

5. Nonlinear Evolution Equations

We now outline the derivation of the nonlinear evolution equations for the interfacial positionh(χ, τ) and surface charge density q(χ, τ). The simplified Laplacian for the potential ψi, (4.2),


is easily solved along with the boundary conditions, (4.3), to give the following expressionsfor the potentials ψi (i = 1, 2):

ψ1(χ, y)=

(β − y)[ε2 +

(1 + h

(χ))q(χ)]

ε1 + βε2 + (ε1 − ε2)h(χ) ,

ψ2(χ, y)=βε2 − ε1y + β

(1 + y

)q(χ)+ h(χ)[ε1 − ε2 −

(1 + y

)q(χ)]

ε1 + βε2 + (ε1 − ε2)h(χ) ,

(5.1)

where the interfacial charge density q(χ, τ) is determined below by the interface chargeconservation equation (4.17). The simplified x-momentum equation (4.8) can be integratedwith respect to y, since dp(0)i /dχ is independent of y and the two constants of integration thatarise are determined by the boundary conditions

u(0)1

(β)= 0, u

(0)1

[y = h

(χ)]

= u(0)int

(χ),

u(0)2 (−1) = 0, u

(0)2

[y = h

(χ)]

= u(0)int

(χ).

(5.2)

Here, u(0)int (χ) is the x component of the velocity at the interface y = h(χ), and this quantitywill eventually be determined by using the tangential stress continuity condition (4.11).For the purposes of keeping the algebra tractable, it is found convenient to keep u

(0)int (χ)

undetermined at present.The solutions for the x-component velocities u(0)i (i = 1, 2) thus obtained are

u(0)1

(χ, y)=k1μr

dp(0)1

dχ

⎡

⎢⎣sinh[√

ε/k1(y − β)

]− sinh

[√ε/k1

(y − h(χ))

]

sinh[√

ε/k1[h(χ) − β]

] − 1

⎤

⎥⎦

+ u(0)int

(χ) sinh

[√ε/k1

(y − β)

]

sinh[√

ε/k1[h(χ) − β]

] ,

(5.3)

u(0)2

(χ, y)= k1

dp(0)2

dχ

⎡

⎢⎣sinh[√

ε/k1(y + 1

)] − sinh[√

ε/k1(y − h(χ))

]

sinh[√

ε/k1(h(χ)+ 1)] − 1

⎤

⎥⎦

+ u(0)int

(χ) sinh

[√ε/k1

(y + 1

)]

sinh[√

ε/k1(h(χ)+ 1)] ,

(5.4)

where the pressure gradients dp(0)i /dχ (i = 1, 2) are determined below.


The continuity equation (4.6), after substituting the asymptotic expansion for υi, (4.7),simplifies to

∂u(0)i

∂χ+∂υ

(0)i

∂y= 0. (5.5)

The above equation is integrated with respect to y from y = β to y = h(χ) for fluid 1 andy = −1 to y = h(χ) for fluid 2, to yield the following expressions for the normal velocities atthe interface υ(0)

i [y = h(χ)] for i = 1, 2, where the integration constant is set to zero in orderto satisfy the no-penetration condition at y = β and y = −1:

υ(0)1

[y = h

(χ)]

= −∫h(χ)

β

∂u(0)1

∂χdy

υ(0)2

[y = h

(χ)]

= −∫h(χ)

−1

∂u(0)2

∂χdy.

(5.6)

Note that the normal velocities of the two fluids are equal at the interface (normal velocitycontinuity condition), and so,

υ(0)1

[y = h

(χ)]

= υ(0)2

[y = h

(χ)]

= υ(0)int . (5.7)

Therefore, we equate the two integrals in the above equation and apply Leibnitz rule

∂

∂χ

∫h(χ)

β

u(0)1 dy − ∂h

∂χu(0)1 h(χ)=

∂

∂χ

∫h(χ)

−1u(0)2 dy − ∂h

∂χu(0)2 h(χ). (5.8)

Noting that the x-component velocities are equal at the interface, u(0)1 [h(χ)] = u(0)2 [h(χ)], the

above equation is simplified to

∂

∂χ

[∫h(χ)

β

u(0)1 dy −

∫h(χ)

−1u(0)2 dy

]

= 0. (5.9)

We next integrate the above equation with respect to χ, and set the integration constant(which is at most a function of time) to zero. The constant of integration is zero, becausethe pressure gradients dp(0)i /dχ in the two fluids should be zero in the absence of electricfield, and when the interfaces are flat. We then substitute the expressions for u(0)i (y), (5.3)and (5.4), in the above equation, carry out the integrations with respect to y, and substitutethe simplified normal stress continuity condition (4.10) to eliminate dp(0)2 /dχ in terms ofdp

(0)1 /dχ. Prior to determining dp(0)1 /dχ, it is useful to determine the x-component of the fluid

velocity at the interface u(0)int from the simplified tangential stress continuity condition, (4.11).Once u(0)int is determined, the pressure gradient dp(0)1 /dχ is determined from the integrated


version of (5.9), and thus the velocity profile ((5.3) and (5.4)), is known completely. Therefore,we get

dp(0)1

dχ=

⎧⎨

⎩k1(1 + h(x))

[h′′′(χ) −M(χ)] −GF

√k1ε

−k1[h′′′(χ) −M(χ)]

⎡

⎣G +

√4k1ε

⎤

⎦ tanh[√

ε

4k1

[1 + h

(χ)]]⎫⎬

⎭

{

k1

⟨

1 + h(χ)+

1μr

[β − h(χ)]

−G(tanh

[√ε

4k1

[1 + h

(χ)]]− tanh

[√ε

4k1

[h(χ) − β]

])

−√

4k1ε

(tanh

[√ε

4k1

[1 + h

(χ)]]− 1μr

tanh[√

ε

4k1

[h(χ) − β]

])⟩⎫⎬

⎭

−1

,

(5.10)

dp(0)2

dχ=dp

(0)1

(χ)

dχ− ∂3h

(χ)

∂χ3+u(0)int

(χ)

k1

(μr − 1

)+M

(χ), (5.11)

u(0)int

(χ)=

⎧⎨

⎩

√k1εF − k1

dp(0)1

dχ

(tanh

[√ε

4k1

[1 + h

(χ)]]− tanh

[√ε

4k1

[h(χ) − β]

])

+k1 tanh[√

ε

4k1

[1 + h

(χ)]](

∂3h(χ)

∂χ3−M

)}

×{coth[√

ε

k1

[1 + h

(χ)]]− μr coth

[√ε

k1

[h(χ) − β]

]

+(μr − 1

)tanh

[√ε

4k1

[1 + h

(χ)]]}−1

,

(5.12)

where G,M, and F are functions of χ, and they are defined as

G(χ)={coth[√

ε

k1

[1 + h

(χ)]]− μr coth

[√ε

k1

[h(χ) − β]

]

+(μr − 1

)tanh

[√ε

4k1

[1 + h

(χ)]]}−1


×⎧⎨

⎩(μr − 1

)[1 + h

(χ)]

+

√k1ε

((1 − 2μr

)tanh

[√ε

4k1

[1 + h

(χ)]]

+ tanh[√

ε

4k1

[h(χ) − β]

])}

M(χ)=[ε1 + βε2 + (ε1 − ε2)h

(χ)]−3{

ε1ε2h′(χ)[q(χ)(1 + β

)+ ε2 − ε1

]}

× {ε2 + ε1 + q(χ)[1 + 2h

(χ) − β]} + [ε1 + βε2 + (ε1 − ε2)h

(χ)]−2

×(q(χ)q′(χ){ε1[1 + h

(χ)]2 + ε2

[β − h(χ)]2

}+ ε1ε2q′

(χ)[1 + 2h

(χ) − β]

),

F(χ)= −q

(χ)h′(χ)ε2(β − h(χ))[q(χ)(β + 1

) − ε1 + ε2]

[ε1 + βε2 + (ε1 − ε2)h

(χ)]2

− q(χ)q′(χ)(β − h(χ))(1 + h(χ))

[ε1 + βε2 + (ε1 − ε2)h

(χ)] .

(5.13)

Finally, the kinematic condition at the interface, (4.14), is used to derive the evolution equa-tion for h(χ, τ), as follows. We first substitute the expressions for the normal fluid velocitiesat the interface, (5.6), after using Leibnitz rule on the integral

ε∂h

∂τ+ u(0)1

[h(χ)]∂h∂χ

= − ∂

∂χ

∫h(χ)

β

u(0)1 dy + u(0)1

[h(χ)]∂h∂χ

,

(5.14)

which is simplified to give

ε∂h

∂τ+

∂

∂χ

∫h(χ)

β

u(0)1 dy = 0, (5.15)

where u(0)1 is substituted from (5.3), after using the expressions for dp

(0)1 /dχ and u

(0)int

determined using the procedure outlined above. The evolution equation for the interfacialcharge density q(χ, τ) is obtained from (4.17) after substituting the expressions for u(0)int and


the gradients of the potential (from (5.1)) in that equation. The nonlinear evolution equationsfor h(χ, τ) and q(χ, τ), respectively, take the forms

ε∂h

∂τ+

1μr

{(

k1dp

(0)1

dχ+μr2u(0)int

(χ))∂h

∂χsech2

[12

√ε

k1

(β − h)

]

+ k1

[(β − h)d

2p(0)1

dχ2− dp

(0)1

dχ

∂h

∂χ

]

−√k1ε

(

2k1d2p

(0)1

dχ2+ μru

′(0)int

(χ))

× tanh[12

√ε

k1

(β − h)

] }

= 0,

ε∂q

∂τ+ u(0)int

(χ) ∂q∂χ

− q∂h∂χ

{√εk1μr

dp(0)1

dχtanh

[12

√ε

k1

(h − β)

]+√

ε

k1u(0)int

(χ)coth[√

ε

k1

(h − β)

]}

+ qu′(0)int

(χ)

+εS

(0)1

[(1 + h)q + ε2

]

ε1 + βε2 + (ε1 − ε2)h +εS

(0)2

[(β − h)q − ε1

]

ε1 + βε2 + (ε1 − ε2)h = 0.

(5.16)

These coupled nonlinear equations can be solved numerically with appropriate initialconditions to determine the evolution of the interface in the presence of electric fields andporous medium. In the present work, however, we restrict ourselves to studying the linearstability properties of these equations, an issue we turn to next.

6. Stability Analysis and Discussion

Before linearizing the coupled nonlinear equations, it is necessary to first determine the basestate about which we perturb. The steady base state we consider is that of stationary fluidswith a flat interface h(χ, τ) = 0, and with a constant interfacial charge density q0 which isindependent of χ and τ . This base state interfacial charge q0 is determined from (4.17) withthe left side set to zero, since ∂/∂τ = 0 and u(0)i = 0 in the base state

S(0)1

[∂ψ1

∂y

]

y=0= S(0)

2

[∂ψ2

∂y

]

y=0. (6.1)

The derivatives of the potentials in the above equation are calculated from (5.1), with h(χ)set to zero. This yields the following expression for the steady interfacial charge density:

q0 =ε1S

(0)2 − ε2S(0)

1

S(0)1 + βS(0)

2

. (6.2)


The variables h and q are now perturbed about their base state values

h(χ, τ)= h1 exp

[ikχ +ωτ

],

q(χ, τ)= q0 + q1 exp

[ikχ +ωτ

],

(6.3)

where h1 and q1 are the amplitudes of the perturbations which are independent of χ and τ ,

k is the nondimensional wavenumber based on the lateral length scale L =√γH3/ε0ψ

2band

ω is the nondimensional growth rate based on the time scale μ2H3γ/(ε0ψ2

b)2. We substitute

the expressions for dp(0)1 /dχ and u(0)int from (5.10), (5.12), and (6.3) in the nonlinear evolutionequations (5.16), and apply Taylor expansion on the hyperbolic functions about h(χ) = 0. Wethen linearize the resulting coupled nonlinear evolution equations with respect to h1 and q1,to obtain a set linear homogeneous equations for h1 and q1 of the form

H̃1h1 + Q̃1q1 = 0,

H̃2h1 + Q̃2q1 = 0,(6.4)

where

H̃1 = −(R14 + k4R13R1

)β4ε42 + ε

41

(R14 + k4R13R1 − k2R13R1ε2

)

− ε21ε22[6(R14 + k4R13R1

)β2 + k2R13R1ε2

]

+ ε31ε2[R15 − 4β

(R14 + k4R13R1

)+ k2R13R1βε2

]

− ε1ε32[R15 + 4β3

(R14 + k4R13R1

)+ k2R13R1βε2

]

+ q20(1 + β

)ε2

⎧⎨

⎩− R6R10R12 + k2R13

(ε1 + βε2

)

×⎡

⎣

⎛

⎝R1(β − 1

) − C1

B1β

√k1ε

⎞

⎠ε1 − C1

B1β2

√k1εε2

⎤

⎦

⎫⎬

⎭

− q0ε2

⎧⎨

⎩R8R10R12 + k2R13

(ε1 + βε2

)

×⎡

⎣β

⎛

⎝2R1 − C1

B1

√k1ε

⎞

⎠ε21 +

⎛

⎝2R1 − C1

B1β(β − 1

)√k1ε

⎞

⎠ε1ε2 +C1

B1β2

√k1εε22

⎤

⎦

⎫⎬

⎭

+ B1R5ωε3/2μr,


Q̃1 =(β − 1

)ε1ε2(ε1 + βε2

)[R15 + k2R1R13

(ε1 + βε2

)]

− q0

⎧⎨

⎩R7R10R12 + k2R13

(ε1 + βε2

)2

×⎡

⎣

⎛

⎝R1 + βC1

B1

√k1ε

⎞

⎠ε1 + β2⎛

⎝R1 +C1

B1

√k1ε

⎞

⎠ε2

⎤

⎦

⎫⎬

⎭,

H̃2 =(ε1 + βε2

)⎛

⎝ε

⎛

⎝εR2

[R18 − B1R4S

(0)2

]ε41μr + ε31(R16 + R3R4 + B1R2R4R34)

+ ε21ε2[R17 + 3βR3R4 + 3B1R2R4R35

]

+ βε2

⎛

⎝R30R22

√k1εq0μr

+ βε22(R19 + βR3R4 + εβR2

×(B1R4q0

[S(0)1 + βS(0)

2

]+[R18 + B1R4S

(0)1

]ε2)μr)⎞

⎠

+ ε1

⎛

⎝R30R22

√k1εq0μr

+ βε22(R20 + 3βR3R4 + εβR2

(3B1R4q0

[S(0)1 + βS(0)

2

]

+[4βR18 + 3B1R4S

(0)1 − βB1R4S

(0)2

]ε2)μr)⎞

⎠

⎞

⎠

−R2q0μr(R9[R21 + R23 + R25 + R28 − R29] + R32R36)

⎞

⎠,

Q̃2 = R2(ε1 + βε2

)μr{B1R2ε

2(ε1 + βε2)3[

S(0)1 + βS(0)

2 +ω(ε1 + βε2

)]

−q0[R9(R22 + R24 + R26 + R27) − R31R2

√εk1(ε1 + βε2

)+ R33R36

]},

(6.5)

in which B1, B2, C1, C2, and R1 − R49 are given in the appendix.The set of linear homogeneous equations (6.4) for h1 and q1 is written in the matrix

form M · CT = 0, where the vector C = [h1, q1] and the determinant of this matrix M isset to zero for nontrivial solutions in order to obtain the characteristic equation for ω. This


characteristic equation, which gives the growth rate ω as a function of k, k1, β, μr , S(0)1 , S(0)

2 , ε,ε1, and ε2, is a quadratic equation for ω. The roots of the characteristic equation for ω can bewritten as

ω1 = −R48

R49−

√R2

48 + 2H̃2Q̃1R49 − 2R41R47R49

R49, (6.6)

ω2 = −R48

R49+

√R2

48 + 2H̃2Q̃1R49 − 2R41R47R49

R49. (6.7)

The roots ω1 and ω2 are always real, with one of them ω1 is always negative, and the otheroneω2 can be positive or negative depending on the choice of the system parameters. We cantake ω = ω2 as a function of k only in (6.7) if the values of the other parameters are known.

Now, to see the effects of various parameters on the stability of the considered system,we calculate the growth rate ω given by (6.7) as a function of the wavenumber k for differentvalues of all physical parameters included in the analysis. These calculations are presentedin Figures 2–7 for the general case of leaky-leaky dielectric fluids, Figures 8 and 9 for thelimiting case of perfect-leaky dielectric fluids in which S(0)

1 = 0, and Figures 10–13 for anotherlimiting case of perfect-perfect dielectric fluids in which S(0)

1 = S(0)2 = 0, where we have given

the growth rate ω against the wavenumber k for the porosity of porous medium ε, mediumpermeability k1, nondimensional conductivities S(0)

1 , S(0)2 , dielectric constants ε1, ε2, the ratio

of the thicknesses of top and bottom fluids β, and the ratio of viscosities of the two fluidsμr = μ1/μ2, respectively.

6.1. Leaky Dielectric-Leaky Dielectric Interface

This is the general case in which S(0)1 /= 0 and S(0)

2 /= 0, which is discussed in Figures 2–7, whenthe conductivities of the upper and lower fluids are present in the analysis. Figures 2(a) and2(b) shows the variation of the growth rate ω versus the wavenumber k for various valuesof the porosity of porous medium ε. It is clear from Figure 2(b) that for small values of theporosity (e.g., ε = 0.1), the growth rate ω increases by increasing the wavenumber k till amaximum value of ε after which ω decreases by increasing k; that is, there exists a maximummode of instability only for small values of the porosity ε, while for any other value of ε, wefound that ω decreases by increasing k, that is. the system is always stable in this case. It isclear also from Figure 2(a), by increasing the porosity values and at any wavenumber value,that the porosity of porous medium has a stabilizing effect for the wavenumber range 0 ≤ k ≤15, and it has a destabilizing effect for higher wavenumber values k > 15; that is, the porosityof porous medium has a dual role on the stability of the considered system (stabilizing andthen destabilizing) depends on the wavenumber range (less than or hifher than a criticalwavenumber value k = 15, resp.). Figure 3 shows the variation of growth rate ω with thewavenumber k for different values of the medium permeability k1. We conclude from thisfigure that for any wavenumber value k ≤ 20, the growth rate ω increases by increasingthe medium permeability k1 values, while for wavenumber values k > 20, all the curvescorrespond to different values of medium permeability k1 coincide. This means that themedium permeability k1 has a destabilizing effect for the wavenumber range 0 < k ≤ 20 and ithas no effect on the stability of the considered system for higher wavenumber values k > 20.


0

−250

−500

−750

−1000

−1250

−1500

−1750

ε = 0.1ε = 0.3ε = 0.5

2.5 5 7.5 10

k

12.5 15 17.5 20

ω

(a)

6

4

2

ω

0

−2

−4

2

k

3 4 51

ε = 0.1ε = 0.3ε = 0.5

(b)

Figure 2: Variation of growth rate ω with wavenumber k for various values of ε when k1 = 3, S(0)1 = 103,

S(0)2 = 104, ε1 = 3, ε2 = 4, β = 0.5, μr = 1, and ε = 0.1, 0.3, 0.5, respectively, for the wavenumber ranges (a)

0 ≤ k ≤ 20 and (b) 0 ≤ k ≤ 5.

Figures 4(a) and 4(b) shows the variation of the growth rateω versus the wavenumberk for various values of the conductivity of upper fluid S

(0)1 . It is clear from this figure that

the conductivity S(0)1 has a stabilizing effect for small wavenumber values and a destabilizing

effect for high wavenumber values, as the growth ratesω decrease and increase by increasingthe conductivity S(0)

1 values, respectively. It is also seen that between these two wavenumberranges, the conductivity of upper fluid S(0)

1 has a dual role on the stability of the consideredsystem; that is, it has a destabilizing effect for S(0)

1 values greater than 102, while it has astabilizing effect for S(0)

1 values greater than 104. Therefore, we conclude that the conductivity


500

0

−500

ω −1000

−1500

−2000

−2500

k1 = 5k1 = 10k1 = 12

5 10 15 20

k

25 30 35 40

Figure 3: Variation of growth rate ω with wavenumber k for various values of k1 when ε = 0.2, S(0)1 = 103,

S(0)2 = 104, ε1 = 3, ε2 = 4, β = 0.5, μr = 1, and k1 = 5, 10, 12, respectively.

of upper fluid S(0)1 has different effects on the stability of the system depending on the chosen

wavenumber range. Figures 5(a) and 5(b) shows the variation of growth rate ω with thewavenumber k for different values of the conductivity of lower fluid S(0)

2 . It indicates that theconductivity S(0)

1 has a destabilizing effect for the wavenumber range 0 < k < 10, while it hasa stabilizing effect for higher wavenumber values k ≥ 10, since the growth rate ω increases inthe first wavenumber range and decreases in the secondwavenumber range by increasing theconductivity of lower fluid S

(0)2 . We conclude also from Figure 5(b) that there exists a mode

of maximum instability for small values of the conductivity S(0)2 which disappears for high

values of the conductivity of lower fluid S(0)2 ≥ 104.

Figure 6 shows the variation of growth rate ω with the wavenumber k for differentvalues of the dielectric constant of upper fluid ε1. It is seen from this figure that there existsa critical wavenumber value k = 4.6, before which the growth rates decrease by increasingthe dielectric constant ε1, and after which they increase by increasing ε1 values. Thus, thedielectric constant of upper fluid ε1 has a stabilizing as well as a destabilizing effects, forwavenumber ranges before and after this critical wavenumber value, respectively. Similarly,the effects of both the dielectric constant of lower fluid ε2, and the ratio of thicknesses of upperand lower fluids β on the stability of the considered system are found to have opposite effectsto the effect of the dielectric constant ε1 given by Figure 6, but the corresponding figures arenot given here. In other words, we conclude that the dielectric constant of lower fluid ε2 has adestabilizing as well as a stabilizing effects for wavenumber ranges before and after the samecritical wavenumber value k = 4.6, respectively, while the ratio of thicknesses of upper andlower fluids β has also a destabilizing as well as a stabilizing effects for wavenumber rangesbefore and after a less critical wavenumber value k = 2.5, respectively. Figures 7(a) and 7(b)shows the variation of the growth rate ω versus the wavenumber k for various values ofthe ratio of viscosity of upper and lower fluids μr . It indicates that the viscosity ratio μr hasa stabilizing effect for small wavenumber values, and it has also a destabilizing effect forhigher wavenumber values, since the growth rate ω decreases and increases by increasingthe increase of μr , respectively. It is clear also from Figure 7(b) that for μr > 1, that is, when


0

−2000

ω −4000

−6000

k

60 80 100

S(0)1 = 102

S(0)1 = 103

S(0)1 = 104

4020

(a)

ω

220

S(0)1 = 102

S(0)1 = 103

S(0)1 = 104

1 2 3

k

5 6 7

300

280

260

240

200

4

(b)

Figure 4: Variation of growth rate ω with wavenumber k for various values of S(0)1 when ε = 0.4, k1 = 10,

S(0)2 = 104, ε1 = 3, ε2 = 4, β = 0.5, μr = 1, and S(0)

1 = 102, S(0)1 = 103, 104, respectively, for the wavenumber

ranges (a) 0 ≤ k ≤ 100 and (b) 0 ≤ k ≤ 10.

the viscosity of upper fluid is larger than the viscosity of lower fluid, there exists a mode ofmaximum instability which disappear for μr ≤ 1.

6.2. Perfect Dielectric-Leaky Dielectric Interface

This is the limiting case in which S(0)1 = 0 and S

(0)2 /= 0, which is discussed in Figures 8 and

9, when the conductivity of the upper fluid is not included in the analysis. Figures 8(a) and8(b) show the variation of the growth rate ω against the wavenumber k for various values of


ω

0

−2000

−4000

−6000

−8000

−10000

20 40

k

60

S(0)2 = 102

S(0)2 = 104

S(0)2 = 105

80 100

(a)

ω

305.5

305.4

305.3

305.2

305.1

305

304.9

1.5

k

2 2.5 310.5

S(0)2 = 102

S(0)2 = 104

S(0)2 = 105

(b)

Figure 5: Variation of growth rate ω with wavenumber k for various values of S(0)2 when ε = 0.4, k1 = 10,

S(0)1 = 103, ε1 = 3, ε2 = 4, β = 0.5, μr = 1, and S(0)

2 = 102, 104, 105, respectively, for the wavenumber ranges(a) 0 ≤ k ≤ 100 and (b) 0 ≤ k ≤ 3.

the porosity of porous medium ε. In comparison with Figures 2(a) and 2(b), we conclude thatthe porosity of porous medium ε behaves in the samemanner as in the previous case of leaky-leaky dielectric fluids, except that the values of growth rates ω are higher than their valuesin the previous case and the corresponding curves intersect the k-axis at larger wavenumbervalues than in the previous case. We conclude also that for small wavenumber values, thereexists a mode of maximum instability for porosity values ε ≤ 0.3. Similarly, the effects ofmedium permeability k1, the conductivity of lower fluid S

(0)2 , the dielectric constants ε1, ε2,

and the ratio of thicknesses of upper and lower fluids β on the stability of the considered


20

10

0

−10ω

−20

−30

−40

−50

ɛ1 = 3ɛ1 = 5ɛ1 = 7

4

k

6 8 102

Figure 6: Variation of growth rate ω with wavenumber k for various values of ε1 when ε = 0.4, k1 = 5,S(0)1 = 103, S(0)

2 = 104, ε2 = 4, β = 0.5, μr = 1, and ε1 = 3, 5, 7, respectively.

system are found to behave in the same manner as their effects in the previous case of leaky-leaky dielectric fluids, but figures are excluded to save space, except that in this case: for theeffect of k1, the growth rate values are higher than their values in the previous case; for theeffect of S(0)

2 , the obtained curves intersect the k-axis at larger wavenumber values than in theprevious case; for the effect of ε1, there exists a mode of maximum instability for all values ofε1; for the effect of ε2, the obtained curves are exactly similar to those obtained in the previouscase; finally, for the effect of β, there is amode ofmaximum instability and the obtained curvesintersect the k-axis at bigger wavenumber values than those obtained in the previous case.Figures 9(a) and 9(b) shows the variation of the growth rate ω versus the wavenumber k forvarious values of the viscosity ratio of upper and lower fluids μr. In comparison with Figures7(a) and 7(b), we conclude that the viscosity ratio μr behaves in the same manner as in theprevious case of leaky-leaky dielectric fluids with the only difference that in this case, for allvalues of μr � 1, there exists a mode of maximum instability and not only for μr > 1 shownin the previous case.

6.3. Perfect Dielectric-Perfect Dielectric Interface

This is the limiting case in which S(0)1 = S(0)

2 = 0, which is discussed in Figures 10–13, when theconductivities of the upper and lower fluids are absent in the analysis. Figure 10 shows thevariation of the growth rate ω versus the wavenumber k for different values of the porosityof porous medium ε. In view of the above discussion, we conclude from this figure that theporosity ε has a slightly stabilizing effect for small wavenumber values k ≤ 2.5 and that ithas no effect on the stability of the considered system afterwards for higher wavenumbervalues. Figure 11 shows the variation of growth rate ω with the wavenumber k for variousvalues of the medium permeability k1. It is clear from this figure that the permeability k1 has adestabilizing effect, since the growth rate ω increases by increasing the medium permeabilityvalues at any fixed wavenumber value. Similarly, the effect of dielectric constant of the


ω

0

−250

−500

−750

−1000

−1250

−1500

μr = 1μr = 2μr = 4

5 10 15k

20 25 30

(a)

2

1

ω 0

−1

1 2

k

3 4 5

μr = 1μr = 2μr = 4

(b)

Figure 7: Variation of growth rate ω with wavenumber k for various values of μr when ε = 0.4, k1 = 3,S(0)1 = 103, S(0)

2 = 104, ε1 = 3, ε2 = 4, β = 0.5, and μr = 1, 2, 4, respectively, for the wavenumber ranges (a)0 ≤ k ≤ 30, and (b) 0 ≤ k ≤ 5.

upper fluid ε1 on the stability of the system is illustrated in Figure 12, and it shows that thedielectric constant ε1 has a stabilizing effect since the growth rate ω decreases by increasingthe dielectric constant ε1 at any wavenumber value. Figure 13 shows the variation of growthrate ω with the wavenumber k for different values of the dielectric constant of lower fluidε2, and from which we conclude thatthe dielectric constant ε1, and after which they increaseby increasing ε2 has a destabilizing effect in the wavenumber range 0 < k ≤ 4.6, and it hasno effect on the stability of the considered system afterwards for higher wavenumber valuesk > 4.6. Similarly, the effects of both the ratio of thicknesses of upper and lower fluids β, andthe ratio of viscosities of upper and lower fluids μr , respectively, on the stability of the system


0

−500

ω −1000

−1500

−2000

ε = 0.1ε = 0.3ε = 0.5

2.5 5 7.5 10

k

12.5 15 17.5 20

(a)

20

10

ω 0

−10

ε = 0.1ε = 0.3ε = 0.5

1 2

k

3 4 5

(b)


S(0)2 = 104, ε1 = 3, ε2 = 4, β = 0.5, μr = 1, and ε = 0.1, 0.3, 0.5, respectively, for the wavenumber ranges (a)

0 ≤ k ≤ 20, and (b) 0 ≤ k ≤ 5.

are found to has the same and the opposite effects as the effect of the dielectric constant ε2shown in Figure 13, but the corresponding figures are not given here to avoid any kind ofrepitation; that is, the parameters β and μr have destabilizing and stabilizing effects for smallwavenumber values, respectively.

7. Concluding Remarks

In conclusion, we have provided a general formulation for analyzing the effect of an exter-nally applied electric field on the stability and dynamics of the interface between two leaky


μr = 1μr = 2μr = 4

ω

0

−250

−500

−750

−1000

−1250

−15005 10 15

k

20 25 30

(a)

3

2

1

0

−1

ω

1 2

k

3 4 5

μr = 1μr = 2μr = 4

(b)

Figure 9: Variation of growth rate ω with wavenumber k for various values of μr when ε = 0.4, k1 = 3,S(0)1 = 0, S(0)

2 = 104, ε1 = 3, ε2 = 4, β = 0.5, and μr = 1, 2, 4, respectively, for the wavenumber ranges (a)0 ≤ k ≤ 30 and (b) 0 ≤ k ≤ 10.

dielectric fluids of arbitrary viscosities and conductivities in porous medium. A systematiclong-wave asymptotic analysis was used to derive coupled nonlinear evolution equations forthe position of the interface and free charge density at the interface. Attention was restrictedto linearized stability of the coupled nonlinear equations and the effect of a variety of systemparameters on the stability of the considered system. Two limiting cases are also studied,that is, the case of perfect-leaky dielectric fluids and the case of two perfect dielectric fluids,and recovered the previous studies in absence of porous medium. The obtained results inthese limiting cases and the general case of two leaky dielectric fluids can be summarized asfollows:


ω

ε = 0.1ε = 0.3ε = 0.5

0.5 1

k

1.5 2 2.5

0.5

1

1.5

2

2.5


S(0)2 = 0, ε1 = 3, ε2 = 4, β = 0.5, μr = 1, and ε = 0.1, 0.3, 0.5, respectively.

ω

600

500

400

300

200

100

k1 = 5k1 = 10k1 = 12

2 4 6

k

8 10

Figure 11: Variation of growth rate ω with wavenumber k for various values of k1 when ε = 0.2, S(0)1 = 0,

S(0)2 = 0, ε1 = 3, ε2 = 4, β = 0.5, μr = 1, and k1 = 5, 10, 12, respectively.

(I) For perfect-perfect dielectric fluids, we conclude for small wavenumbers that

(i) the porosity of porous medium ε, the dielectric constant of upper fluid ε1, andthe ratio of viscosity of upper and lower fluids μr have stabilizing effects,

(ii) the medium permeability k1, the dielectric constant of lower fluid ε2, and theratio of thickness of upper and lower fluids β have destabilizing effects,

(iii) at any value of these physical parameters there are no modes of maximuminstability, that is. the system is always stable,

(iv) these physical parameters have no effect on the stability of the system for highwavenumber values.


ω

20

17.5

15

12.5

10

7.5

5

2.5

1 2

k

3

ɛ1 = 3ɛ1 = 5ɛ1 = 7

4 5



ω

25

20

15

10

5

ɛ2 = 4ɛ2 = 6ɛ2 = 10

1 2

k

3 4 5



(II) For perfect-leaky dielectric fluids, we found that

(i) the conductivity of lower fluid S(0)2 has a destabilizing effect for small wave-numbers and a stabilizing effect for high wavenumbers,

(ii) for small wavenumber values, the physical parameters ε, ε1, and μr havestabilizing effects, while the parameters, k1, ε2, and β have destabilizing effects,as in the case of perfect-perfect dielectrics,

(iii) for high wavenumber values, new regions of stability or instability appear; inother words, the physical parameters ε, ε1, and μr have destabilizing effects


for high wavenumbers, while the parameters ε2, and β have stabilizing effects,and k1 has no effect on the stability of the system for high wavenumber values,

(iv) there exists a mode of maximum instability for some of these physical param-eters which do not appear in the previous case of perfect-perfect dielectrics.

(III) For leaky-leaky dielectric fluids, we found that

(i) the conductivity of upper fluid S(0)1 has a destabilizing effect for small wave-numbers 0 < k < k1 and also a stabilizing effect for high wavenumbers k > k2,while it has a dual role on the stability of the considered system between themin the wavenumber range k1 < k < k2.

(ii) the effects of all other physical parameters on the stability of the consideredsystem behave in the same manner as their effects in the case of perfect-leakydielectrics, except that in the case of perfect-leaky dielectrics the stability orinstability regions occur more faster than the corresponding case of leaky-leaky dielectrics, and the maximum instability holds for more values of thephysical parameters included in the analysis.

It should be mentioned that the problem investigated in this article can be generalizedto study the linear electrohydrodynamic instabilities at the interface between two immisciblefluids, either perfect or leaky dielectrics, subjected to alternating electric fields and movingthrough a porous medium in the limit of the electrode spacing being large compared to thewavelength of the perturbation using the Floquet theory analysis [37], and this case is nowin a current research.

Appendix

B1 = coth(√

ε

k1

)+ μr coth

(β

√ε

k1

)+(μr − 1

)tanh

(12

√ε

k1

),

B2 =√

ε

k1

{−csch2

(√ε

k1

)+ μrcsch

2(β

√ε

k1

)+12(μr − 1

)sech2

(12

√ε

k1

)},

C1 = μr − 1 −√k1ε

[(2μr − 1

)tanh

(12

√ε

k1

)+ tanh

(12β

√ε

k1

)],

C2 = μr − 1 +12

{1 +(1 − 2μr

)sech2

(12

√ε

k1

)− tanh2

(12β

√ε

k1

)},

R1 = k1

⎧⎨

⎩

⎛

⎝C1

B1+ 2

√k1ε

⎞

⎠ tanh(12

√ε

k1

)− 1

⎫⎬

⎭,

R2 =1μr

⎧⎨

⎩β − 2

√k1εtanh

(β

2

√ε

k1

)− μr[R1

k1+C1

B1tanh

(β

2

√ε

k1

)]⎫⎬

⎭,

R3 = B1R2εμr{q0(S(0)1 + βS(0)

2

)− ε1S(0)

2 + ε2S(0)1

},


R4 = −(

−B2C1 + B1C2

B21

)[tanh

(12

√ε

k1

)+ tanh

(12β

√ε

k1

)],

R5 = R22(ε1 + βε2

)4,

R6 = εε1(1 − β) sinh

(12

√ε

k1

)+ β(ε1 + βε2

)√

ε

k1cosh

(12

√ε

k1

),

R7 =(ε1 + βε2

){ε(ε1 + β2ε2

)sinh(12

√ε

k1

)

+β(ε1 + βε2

)√

ε

k1cosh

(12

√ε

k1

)},

R8 = 2εε1(ε2 + βε1

)sinh(12

√ε

k1

)+ β(βε2 + ε1

)(ε2 − ε1)

√ε

k1cosh

(12

√ε

k1

),

R9 = tanh(12

√ε

k1

)+ tanh

(β

2

√ε

k1

),

R10 =k2k1ε

R2(ε1 + βε2

)sech(12

√ε

k1

),

R11 = B1

[β√ε − 2

√k1 tanh

(β

2

√ε

k1

)],

R12 = R2μr√k1 tanh

(β

2

√ε

k1

),

R13 = R9R12 + R11R2,

R14 = R2R12k51 tanh

(12

√ε

k1

),

R15 = εR10R12 sinh(12

√ε

k1

),

R16 = R2

{4R3 + q0μrR2

[εB2

(S(0)1 − S(0)

2

)+ 4βk1k4ε2 tanh

(12

√ε

k1

)]},

R17 = R2

{4R3(2β − 1

)+ 3βq0μrR2

[εB1

(S(0)1 − S(0)

2

)+ 2βk1k4ε2 tanh

(12

√ε

k1

)]},

R18 =βq0k1k

4R2

εtanh

(12

√ε

k1

),

R19 = R2

{−4R3 + βεq0μrB1R2

(S(0)1 − S(0)

2

)},

R20 = R2

{4(β − 2

)R3 + 3βεq0μrB1R2

(S(0)1 − S(0)

2

)},

R21 = −k1k2(1 + β

)(ε1 + βε2

)q20ε2

{(β − 1

)εε1 +

(ε1 + βε2

)βC1

B1

√ε

k1

},


R22 = q0k1k2(ε1 + βε2

)2{ε(ε1 + β2ε2

)− (ε1 + βε2

)βC1

B1

√ε

k1

},

R23 = q0ε2(ε1 + βε2

)k1k

2{2εε1(βε1 + ε2

)+ (ε1 − ε2)

(ε1 + βε2

)βC1

B1

√ε

k1

},

R24 = ε(1 − β)k1k2ε1ε2

(ε1 + βε2

)2,

R25 = εk1k2(ε1 + βε2

){ε31

(k2 − ε2

)+ 3βk2ε21ε2 + β

3k2ε32 + ε1ε22

(3β2k2 + ε2

)},

R26 = −εq0k1k2⎡

⎣C1

B1+ 2

√k1ε

⎤

⎦ε21[ε1 + β

(2 + β

)ε2]tanh

(12

√ε

k1

),

R27 = εε2k1k2⎡

⎣C1

B1+ 2

√k1ε

⎤

⎦

×{(β − 1

)ε1(ε1 + βε2

)2 − β2q0ε2[β2ε2 +

(1 + 2β

)ε1]}

tanh(12

√ε

k1

),

R28 = εk1k2ε1ε2

⎧⎨

⎩

⎡

⎣C1

B1+ 2

√k1ε

⎤

⎦

×{β(β2 − 1

)q20ε2 +

(ε1 + βε2

)(ε21 − ε22

)

−2q0ε2((

1 + β2)ε1 + βε2

)}− 4βq0ε21

√k1ε

⎫⎬

⎭tanh

(12

√ε

k1

),

R29 = εk1k2

⎧⎨

⎩

⎡

⎣C1

B1+ 2

√k1ε

⎤

⎦

×{k2[ε41 + 4β3ε1ε32 + β

4ε42

]+ ε21ε2

[(1 − β2

)q20 + 2βk2

(2ε1 + 3βε2

)]}

+2β(C1

B1

)q0ε

31ε2

⎫⎬

⎭tanh

(12

√ε

k1

),

R30 = βq0k2ε2(ε1 + βε2

){(1 + β

)q0 − ε1 + ε2

},

R31 = βq0k2(ε1 + βε2

)2,

R32 = k2ε1ε2(ε1 + βε2

)[(β − 1

)q0 − ε1 − ε2

][(β + 1

)q0 − ε1 + ε2

],

R33 = k2(ε1 + βε2

)2[(β − 1

)ε1ε2 − q0

(ε1 + β2ε2

)],

R34 = εμr{q0(S(0)1 + βS(0)

2

)+(S(0)1 − 3βS(0)

2

)ε2},

R35 = βεμr{q0(S(0)1 + βS(0)

2

)+(S(0)1 − βS(0)

2

)ε2},


R36 = εk1R2 tanh(12

√ε

k1

),

R37 = R14 + k4R1R13,

R38 =

⎛

⎝R1(β − 1

) − C1

B1β

√k1ε

⎞

⎠ε1 − C1

B1β2

√k1εε2,

R39 = β

⎛

⎝2R1 − C1

B1

√k1ε

⎞

⎠ε21 +

⎛

⎝2R1 − C1

B1β(β − 1

)√k1ε

⎞

⎠ε1ε2 +C1

B1β2

√k1εε22,

R40 =

⎛

⎝R1 + βC1

B1

√k1ε

⎞

⎠ε1 + β2⎛

⎝R1 +C1

B1

√k1ε

⎞

⎠ε2,

R41 = −R37β4ε42 + ε

41

(R37 − k2R1R13ε2

)− ε21ε22

(6R37β

2 + k2R1R13ε2)

+ ε31ε2(R15 − 4βR37 + βk2R1R13ε2

)− ε1ε32

(R15 + 4β3R37 + βk2R1R13ε2

)

+(1 + β

)q20ε2(−R6R10R12 + k2R13R38

(ε1 + βε2

))

− q0ε2(R8R10R12 + k2R13R39

(ε1 + βε2

)),

R42 = βε2

⎧⎨

⎩R30R

22

√k1εq0μr + βε22

{R19 + βR3R4 + εβR2

[B1R4q0

(S(0)1 + βS(0)

2

)+(R18 + B1R4S

(0)1

)ε2]μr}⎫⎬

⎭,

R43 = ε1

⎧⎨

⎩R30R

22

√k1εq0μr + βε22

×{R20 + 3βR3R4 + εβR2

[3B1R4q0

(S(0)1 + βS(0)

2

)

+(4βR18 + 3B1R4S

(0)1 − βB1R4S

(0)2

)ε2]μr}⎫⎬

⎭,

R44 = εR2

(R18 − B1R4S

(0)2

)ε41μr + ε

31[R16 + R3R4 + B1R2R4R34]

+ ε21ε2[R17 + 3βR3R4 + 3B1R2R4R35

],

R45 = −R2(ε1 + βε2

)μrq0{R9[R22 + R24 + R26 + R27]

−R31R2

√εk1(ε1 + βε2

)+ R33R36

},

R46 = B1R22ε

2μr(ε1 + βε2

)4,


R47 = R46

(S(0)1 + βS(0)

2

)+ R45,

R48 = R41R46(ε1 + βε2

)+ B1R5R47μrε

3/2,

R49 = 2B1R5R46μrε3/2(ε1 + βε2

).

(A.1)

References

[1] G. I. Taylor and A. D. McEwan, “The stability of a horizontal fluid interface in a vertical electric field,”The Journal of Fluid Mechanics, vol. 22, pp. 1–15, 1965.

[2] J. R. Melcher and C. V. Smith, “Electrohydrodynamic charge relaxation and interfacial perpendicular-field instability,” Physics of Fluids, vol. 12, no. 4, pp. 778–790, 1969.

[3] D. A. Saville, “Electrohydrodynamics: the Taylor-Melcher leaky dielectric model,” Annual Review ofFluid Mechanics, vol. 29, pp. 27–64, 1997.

[4] D. J. Griffiths, Introduction to Electrohydrodynamics, Pearson Education, Delhi, India, 3rd edition, 2006.[5] A. A. Mohamed, E. F. Elshehawey, and M. F. El-Sayed, “Electrohydrodynamic stability of two

superposed viscous fluids,” Journal of Colloid And Interface Science, vol. 169, no. 1, pp. 65–78, 1995.[6] N. T. M. El-Dabe, “Electrohydrodynamic stability of two superposed elasticoviscous liquids in plane

Couette flow,” Journal of Mathematical Physics, vol. 28, no. 11, pp. 2791–2800, 1987.[7] D. D. Joseph and Y. Y. Renardy, Fundamentals of Two-Fluid Dynamics—Part I: Mathematical Theory and

Application, vol. 3 of Interdisciplinary Applied Mathematics, Springer, New York, NY, USA, 1993.[8] K. Abdella and H. Rasmussen, “Electrohydrodynamic instability of two superposed fluids in normal

electric fields,” Journal of Computational and Applied Mathematics, vol. 78, no. 1, pp. 33–61, 1997.[9] J. R. Melcher, Continuum Electromechanics, MIT Press, Cambridge, UK, 1981.[10] E. Schaffer, T. Thurn-Albrecht, T. P. Russell, and U. Steiner, “Electrohydrodynamic instabilities in

polymer films,” Europhysics Letters, vol. 53, no. 4, pp. 518–523, 2001.[11] Z. Lin, T. Kerle, S. M. Baker et al., “Electric field induced instabilities at liquid/liquid interfaces,”

Journal of Chemical Physics, vol. 114, no. 5, pp. 2377–2381, 2001.[12] M. D. Morariu, N. E. Voicu, E. Schaffer, Z. Lin, T. P. Russell, and U. Steiner, “Hierarchical structure

formation and pattern replication induced by an electric field,” Nature Materials, vol. 2, no. 1, pp.48–52, 2003.

[13] O. Ozen, N. Aubry, D. T. Papageorgiou, and P. G. Petropoulos, “Electrohydrodynamic linear stabilityof a two immiscible fluids in channel flow,” Electrochimica Acta, vol. 51, pp. 5316–5323, 2006.

[14] L. Wu and S. Y. Chou, “Electrohydrodynamic instability of a thin film of viscoelastic polymerunderneath a lithographically manufactured mask,” Journal of Non-Newtonian Fluid Mechanics, vol.125, no. 2-3, pp. 91–99, 2005.

[15] R. G. Larson, Constitutive Equations for Polymer Melts and Solutions, Butterworth, Stoneham, Mass,USA, 1988.

[16] R. Verma, A. Sharma, K. Kargupta, and J. Bhaumik, “Electric field induced instability and patternformation in thin liquid films,” Langmuir, vol. 21, no. 8, pp. 3710–3721, 2005.

[17] D. T. Papageorgiou and P. G. Petropoulos, “Generation of interfacial instabilities in charged electrifiedviscous liquid films,” Journal of Engineering Mathematics, vol. 50, no. 2-3, pp. 223–240, 2004.

[18] V. Shankar and A. Sharma, “Instability of the interface between thin fluid films subjected to electricfields,” Journal of Colloid and Interface Science, vol. 274, pp. 294–308, 2004.

[19] R. V. Craster and O. K. Matar, “Electrically induced pattern formation in thin leaky dielectric films,”Physics of Fluids, vol. 17, no. 3, Article ID 032104, 2005.

[20] F. Li, O. Ozen, N. Aubry, D. T. Papageorgiou, and P. G. Petropoulos, “Linear stability of a two-fluidinterface for electrohydrodynamic mixing in a channel,” Journal of Fluid Mechanics, vol. 583, pp. 347–377, 2007.

[21] G. Tomar, V. Shankar, A. Sharma, and G. Biswas, “Electrohydrodynamic instability of a confinedviscoelastic liquid film,” Journal of Non-Newtonian Fluid Mechanics, vol. 143, no. 2-3, pp. 120–130, 2007.

[22] G. Supeene, C. R. Koch, and S. Bhattacharjee, “Deformation of a droplet in an electric field: nonlineartransient response in perfect and leaky dielectric media,” Journal of Colloid and Interface Science, vol.318, no. 2, pp. 463–476, 2008.

[23] S. Herminghaus, “Dynamical instability of thin liquid films between conducting media,” PhysicalReview Letters, vol. 83, pp. 2539–2542, 1999.


[24] L. F. Pease III and W. B. Russel, “Linear stability analysis of thin leaky dielectric films subjected toelectric fields,” Journal of Non-Newtonian Fluid Mechanics, vol. 102, no. 2, pp. 233–250, 2002.

[25] D. B. Ingham and I. Pop, Transport Phenomena in Porous Media, Pergamon Press, Oxford UK, 1998.[26] K. Vafai, Handbook of Porous Media, Marcel Dekker, New York, NY, USA, 2000.[27] J. A. Del Rio and S. Whitaker, “Electrohydrodynamics in porous media,” Transport in Porous Media,

vol. 44, no. 2, pp. 385–405, 2001.[28] I. Pop and D. B. Ingham, Convevtive Heat Transfer: Mathematical and Computational Modeling of Viscous

Fluids and Porous Media, Pergamon Press, Oxford, UK, 2001.[29] D. A. Nield and A. Bejan, Convection in Porous Media, Springer, New York, NY, USA, 3rd edition, 2006.[30] M. F. El-Sayed, “Electrohydrodynamic instability of two superposed viscous streaming fluids through

porous media,” Canadian Journal of Physics, vol. 75, no. 7, pp. 499–508, 1997.[31] M. F. El-Sayed, “Effect of normal electric fields on Kelvin-Helmholtz instability for porous media with

Darcian and Forchheimer flows,” Physica A, vol. 255, no. 1-2, pp. 1–14, 1998.[32] M. F. El-Sayed, “Electrohydrodynamic instability of dielectric fluid layer between two semi-infinite

identical conducting fluids in porous medium,” Physica A, vol. 367, pp. 25–41, 2006.[33] M. F. El-Sayed, “Instability of two streaming conducting and dielectric bounded fluids in porous

medium under time-varying electric field,” Archive of Applied Mechanics, vol. 79, no. 1, pp. 19–39,2009.

[34] M. F. El-Sayed, “Thermohydrodynamic instability of viscous rotating dielectric fluid layer in porousmediumwith vertical ac electric field,” Special Topics & Reviews in Porous Media, vol. 1, pp. 15–29, 2010.

[35] M. F. El-Sayed, “Electrohydrodynamic instability conditions for two superposed dielectric boundedfluids streaming with fine dust in a porous medium,” Journal of Porous Media, vol. 13, pp. 457–473,2010.

[36] M. F. El-Sayed, G. M. Moatimid, and T. M. N. Metwaly, “Nonlinear electrohydrodynamic stability oftwo superposed streaming finite dielectric fluids in porous mediumwith interfacial surface charges,”Transport in Porous Media, vol. 86, pp. 559–578, 2011.

[37] P. Gambhire and R. M. Thaokar, “Electrohydrodynamic instabilities at interfaces subjected toalternating electric field,” Physics of Fluids, vol. 22, Article ID 064103, 2010.

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