# Electrohydrodynamic Instability of Two Thin Viscous Leaky Dielectric Fluid Films in a Porous Medium

Authors: M. F. El-Sayed; M. H. M. Moussa; A. A. A. Hassan; N. M. Hafez
Journal: ISRN Applied Mathematics
 (2011)

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## Abstract

The effect of an applied electric field on the stability of the interface between two thin viscous leaky dielectric fluid films in porous medium is analyzed in the long-wave limit. A systematic asymptotic expansion is employed to derive coupled nonlinear evolution equations of the interface and interfacial free charge distribution. The linearized stability of these equations is determined and the effects of various parameters are examined in detail. For perfect-perfect dielectrics, the various parameters affect only for small wavenumber values. For dielectrics, the various parameters affect only for small wavenumber values. For effect for small wavenumbers, and a stabilizing effect afterwards, and for high wavenumber values for the other physical parameters, new regions of stability or instability appear. For leaky-leaky dielectrics, the conductivity of upper fluid has a destabilizing effect for small or high wavenumbers, while it has a dual role on the stability of the system in a wavenumber range between them. The effects of all other physical parameters behave 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-leaky dielectrics.


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## Body

## 1. Introduction

The effect of electric fields on the stability and dynamics of fluid-fluid interfaces has been an area of extensive research, beginning from the classic works of Taylor and McEwan [1] and Melcher and Smith [2]. These works and subsequent studies, see, for example, the reviews of Saville [3] and Griffiths [4], have amply demonstrated the role of electrical stresses on fluid interfaces and the associated electrohydrodynamic instabilities in such systems. One of the basic problems here is to understand the stability of the interface between two fluid layers bounded on the top and bottom by rigid plates, and this has been the subject of many previous studies. Mohamed et al. [5] concentrated on two superposed viscous fluids in a channel 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 viscosities, densities, conductivities and permittivities, in an unbounded domain subjected to a normal electric field. They studied, following Melcher [9], two special cases in detail: the electrohydrodynamic free-charge configuration (EH-If) and the electrohydrodynamic polarization charge configuration (EH-If).These studies have largely considered systems in which gravitational effects are important, and therefore, a critical applied voltage is required to cause the instability, very long waves are stabilized by interfacial tension, and waves of intermediate lengths become unstable. These earlier studies have focused on how the critical voltage required for instability is affected by the nature of the fluids, namely. whether they are perfect dielectrics, or whether they are leaky dielectrics in which there is the possibility of free charge conduction in the fluids, and also the possibility of accumulation and redistribution of charges on the interface between two fluids. Recently, there has been a renewed interest in this area, in part due to the relevance of such phenomena in the formation of well-controlled patterns using the application of electric fields to thin liquid films [10–12], which has demonstrated that the application of an external electric field to polymer-air or polymer-polymer interfaces enhances the spontaneous fluctuations at the interface leading to an instability. However, for leaky dielectrics with surface charges at the interface and two fluids that are not perfectly conducting, we must bear in mind that there is an electrical tangential shear stress at the interface, induced by the electric field, and hence, it changes the stability of the two-fluid layer system as previously investigated by Ozen et al. [13]. Wu and Chou [14] performed linear stability analysis of a leaky dielectric viscoelastic fluid whose constitutive behavior was 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 now well understood [16]. For recent developments of this topic, see the investigations of Papageorgiou and Petropoulos [17], Shankar and Sharma [18], Craster and Matar [19], Li et al. [20], Tomar et al. [21], and Supeene et al. [22].Theoretical studies that have considered the linear stability characteristics of thin fluid films subjected to electric fields were restricted to the following configurations: (i) the interface between a perfect dielectric liquid and air [10], (ii) the interface between two perfect dielectric liquids [11], (iii) the interface between a dielectric fluid of finite thickness and a conducting fluid of much larger thickness [23], and (iv) the interface between a leaky dielectric liquid and air [24]. Except the study of Pease III and Russel [24], all these studies have focused only on perfect dielectric systems. The presence of conductivity in one or both of the liquids could have a significant impact on the length scales and growth rates. It should be noted that in 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 of engineering, including material science, petroleum industry, chemical engineering, and soil mechanics as well as biomechanics. The flow through porous media has gained considerable interest in recent years, particularly among geophysical fluid dynamicists. It is well known that in Darcy's law, which relates the pressure gradient, the bulk viscous resistance, and the gravitational force in a porous medium, the usual viscous term in the equation of motion is replaced by the resistive force-(μ/k1)v, where μ is the fluid viscosity, k1 is the medium permeability, and v is the Darcian velocity of the fluid. Much of the recent studies on this topic are given by Ingham and Pop [25], Vafai [26], Del Rio and Whitaker [27], Pop and Ingham [28], and Nield and Bejan [29]. On the other hand, electrohydrodynamic instability studies 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 a growing need for original research in the updated electrohydrodynamic phenomena which have some physical and engineering applications.In this study, we consider the most general case of the effect of an external applied electric field on the stability of the interface between two thin leaky dielectric fluids with arbitrary viscosities and conductivities. We first use a systematic long-wave asymptotic analysis to derive the nonlinear evolution equations for the interface position and interfacial charge distribution, and we subsequently study the linearized stability of the nonlinear differential equations. This paper is organized in the following manner: Section2 discusses the relevant governing equations and boundary conditions, and in Section 3, we nondimensionalize these equations and conditions. In Sections 4 and 5, we outline the long-wave asymptotic analysis used to derive the nonlinear evolution equations for the interface position and charge, and in Section 6, we develop the linear stability analysis of the nonlinear equations, and discuss the important representative studies results obtained from the general case of two leaky dielectric interface and the limiting cases of both perfect-leaky dielectric, and perfect-perfect dielectric interfaces. 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 arbitrary viscosities occupying the regions-H<y<0 (fluid 2) and 0<y<βH (fluid 1) in the initial unperturbed state, see Figure 1, where β is the ratio of the thicknesses of top and bottom fluids. The perturbed interface between the two fluids is denoted by y=h(x). The two fluids are stationary in the initial state, with viscosities μi, dielectric constants ϵi, conductivities σi, porosity of porous medium ɛ and medium permeability k1, where i=1,2. Fluid 2 is bounded at the bottom y=-H by a rigid plate which is maintained at an electric potential ψ=ψb, while fluid 1 is bounded at the top y=βH by a rigid plate maintained at an electric potential ψ=0. In the ensuing analysis, we assume that the material properties of the fluid such as viscosities μi, dielectric constants ϵi, conductivities σi, porosity of porous medium ɛ, and medium permeability k1 are constants and independent of spatial position. Following the leaky dielectric model formulation of Saville [3], we assume that electroneutrality is valid in the bulk, while free charge is assumed to accumulate at the fluids interface. We also neglect the diffusion of free charge within the interface.Figure 1
Schematic diagram showing the configuration and coordinate system.For leaky dielectric fluids of constant conductivities and with zero net charge in the bulk, the following governing equations are appropriate for the electric fieldE in the two fluids 1 and 2 [3](2.1)∇⋅Ei=0,i=1,2.
Since the electric fields are irrotational, Ei=-∇ψi, where ψi is the electric potential: in the fluid i. Substituting this into (2.1) gives the following Laplace's equation for the electric potential: ψi(2.2)∇2ψi=0.
These governing equations are supplemented by the following boundary conditions. (1)
The normal component of the electric field, at the interfacey=h(x), satisfies
(2.3)ϵ2ϵ0(∇ψ2⋅n)-ϵ1ϵ0(∇ψ1⋅n)=q(x,t)aty=h(x),
where n is the unit normal to the interface at y=h(x) (see Figure 1), ϵi (i=1,2) is the dielectric constant in fluid i, ϵ0 is the permittivity of free space, and q(x,t) is the surface charge density of free charges at the interface.(2)
The continuity of the tangential component of the electric field at the interfacey=h(x) translates to the continuity of the electric potentials; that is,(2.4)ψ1=ψ2aty=h(x).(3)
The electric potentials satisfy the following conditions at the rigid boundaries:(2.5)ψ2=ψbaty=-H,ψ1=0aty=βH.
We next turn to the equations governing the motion of the two fluids. Owing to the relatively small thicknesses of the fluids, we ignore inertial effects in both fluids, and hence, the governing equations are the Stokes equations for continuity and momentum balance(2.6)∇⋅Vi=0,∇⋅Ti=0,
where Vi and Ti are the velocity field and the total stress tensor, respectively, in fluid i. Both the fluids are assumed to be irrotational, then the fluid velocity Vi can be derived from a scalar velocity potential φi such that Vi=-∇φi. We have neglected the effects of gravity on the length scales of interest here. In addition, the effects of van der Waals dispersion forces are negligible for the films considered in the experimental studies [11]. The total stress tensor T 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 porous medium(2.7)Ti=-piI+μi(∇Vi+∇ViT)+mi+μik1φiI,i=1,2,
where pi is the pressure in fluid i, I is the identity tensor, and the Maxwell stress tensor mi is given by Saville [3](2.8)mi=ϵiϵ0[EiEi-12(Ei⋅Ei)I].
The divergence of the Maxwell stress tensor ∇·mi=0, because the bulk of the fluid is free of net charge, and the dielectric constants are independent of spatial position in the two fluids; that is,(2.9)∇⋅mi=∇⋅{ϵiϵ0[EiEi-12(Ei⋅Ei)I]}=-ϵ0Ei2⋅Ei∇ϵi+ρfEi=0,
with ρf is the bulk free charge. Thus, the Maxwell stress tensor will not appear in the momentum balance but will affect the flow only through the conditions at the interface. The governing momentum equations in the two fluids, therefore, become(2.10)ρɛDViDt=-∇pi+∇⋅mi+μiɛ∇2Vi-μik1Vi,i=1,2.
Since there is no time scale, then Dvi/Dt=0, and we get from the above two equations that(2.11)∇pi-μiɛ∇2Vi+μik1Vi=0,i=1,2.
The fluid velocities satisfy no-slip and no-penetration conditions at the top and bottom plates(2.12)V1(y=βH)=0,V2(y=-H)=0.
At the interface y=h(x) between the two fluids, continuity of velocities and stresses apply(2.13)(V⋅n)1=(V⋅n)2,(2.14)(V⋅t)1=(V⋅t)2,(2.15)(n⋅T⋅n)2=(n⋅T⋅n)1+γκ,(2.16)(t⋅T⋅n)2=(t⋅T⋅n)1,
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 our attention to two-dimensional systems which are invariant in the z direction and denote the velocity 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(2.17)(n⋅T⋅n)i=-pi+2μi∂υi∂y+ϵiϵ0[EiN2-12Ei2]+μik1φi,(t⋅T⋅n)i=μi(∂ui∂y+∂υi∂x)+ϵiϵ0EiTEiN,
where EiT and EiN are the tangential and normal components of the electric field Ei in the fluid i, respectively. Hence, we obtain the following conditions to be applied at the interface y=h(x):(2.18)[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],
for the normal stress continuity, and(2.19)μ2[∂u2∂y+∂υ2∂x]-μ1[∂u1∂y+∂υ1∂x]=-(∇ψ1⋅t)q(x,t),
for the tangential stress continuity. In the last equation, we have used the normal electric field continuity condition, (2.3), to simplify the right-hand side. The kinematic condition at the interface prescribes the evolution of the interface position h(x,t),(2.20)υ1(y=h(x))=υ2(y=h(x))=ɛ∂h∂t+V⋅∇sh,
where ∇s is the gradient operator along the interface y=h(x). Finally, the interfacial charge is governed by a conservation equation(2.21)ɛ∂q∂t+V⋅∇sq-qn⋅(n⋅∇)V=ɛ([σ2E2⋅n]-[σ1E1⋅n]),
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 the migration 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 conditions by setting(3.1)ψi′=ψiψb,pi′=piH2ϵ0ψb2,Ti′=TiH2ϵ0ψb2,x′=xH,y′=yH,Ei′=EiHψb,Vi′=Viμ2Hϵ0ψb2,t′=tϵ0ψb2μ2H2,φi′=μ2φiϵ0ψb2,q′=qHϵ0ψb,κ′=κH,k1′=k1H2,h′=hH.
Upon using these scales to nondimensionalize the above governing equations and boundary conditions, we end up with the following nondimensional set of equations. Without loss of clarity, and for the sake of brevity, we represent nondimensional variables with the same notation in the ensuing discussion by dropping dashes.The nondimensional governing equations for the electric potentialsψi are(3.2)∇2ψi=0,i=1,2,
with the following boundary conditions at the interface y=h(x),(3.3)[ϵ2∇ψ2⋅n]-[ϵ1∇ψ1⋅n]=q,ψ1=ψ2,
and the following boundary conditions at the top and bottom boundaries(3.4)ψ1(y=β)=0,ψ2(y=-1)=1.
Similarly, the nondimensional equations governing the fluid motion are(3.5)∇⋅V1=0,∇⋅V2=0,(3.6)∇p1-μrɛ∇2V1+μrk1V1=0,∇p2-1ɛ∇2V2+1k1V2=0.
with μr=μ1/μ2 being the ratio of viscosities of the two fluids. The nondimensional normal and tangential stress continuity conditions at the interface become(3.7)[p1-2μr∂υ1∂y]-[p2-2∂υ2∂y]+1k1[φ2-μrφ1]=γ¯κ+ϵ12[(∇ψ1⋅n)2-(∇ψ1⋅t)2]-ϵ22[(∇ψ2⋅n)2-(∇ψ2⋅t)2],(3.8)[∂u2∂y+∂υ2∂x]-μr[∂u1∂y+∂υ1∂x]=-(∇ψ1⋅t)q(x,t),
where γ¯=γH/ϵ0ψb2 is the nondimensional interfacial tension. The boundary conditions for the velocities at the top and bottom plates become(3.9)u1(y=β)=0,υ1(y=β)=0,u2(y=-1)=0,υ2(y=-1)=0.
The nondimensional kinematic condition at the interface is(3.10)υ1(y=h(x))=υ2(y=h(x))=ɛ∂h∂t+V⋅∇sh,
and the nondimensional charge conservation equation at the interface is(3.11)ɛ∂q∂t+V⋅∇sq-qn⋅(n⋅∇)V=ɛ([S1∇ψ1⋅n]-[S2∇ψ2⋅n]),
where Si=σiμ2H2/ϵ02ψb2, 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 interest here, the above system of equations undergoes a long-wave instability. We now carry out a long-wave asymptotic analysis to make the above system of equations tractable, and thereby derive coupled nonlinear evolution equations for the interface positionh(x,t) and charge q(x,t). While the main focus of this paper is to analyze the stability of the linearized equations, it is nonetheless useful to first derive the nonlinear evolution equations, since these equations can be used in future studies to understand (by numerical simulations) the nonlinear evolution processes that occur after the linear instability.


## 4. Long-Wave Asymptotic Analysis

In the long-wave limit, the wavelengthL of the fastest growing modes is much larger than the 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/ϵ0ψb2, 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 large length scales, and this is introduced a little later in (4.15). In the limit H≪L, the derivatives in the x direction should be scaled with L. To this end, we define the slowly varying scale χ in the following manner:(4.1)∂∂x=δ∂∂χ,
and ∂/∂χ~O(1). When we apply the above scalings, (4.1), to the Laplace equation, (3.2), for the electric potential ψi, it simplifies in the limit δ≪1 to(4.2)∂2ψi∂y2=0,i=1,2.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(4.3)ϵ2∂ψ2∂y-ϵ1∂ψ1∂y=q,
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 fluid motion in the long-wave limit. It is useful to define the variable μr,i such that μr,i=μr for i=1 and μr,i=1 for i=2. The x-momentum equation can be simplified in the long-wave limit as(4.4)δ∂pi∂χ-μr,iɛ∂2ui∂y2+μr,ik1ui=0.
This suggests that ui~O(δ)pi. In order to make this explicit and to make ordering of various quantities simpler, we represent the pressure pi and the x-component velocity ui in the following manner:(4.5)pi=pi(0),ui=δui(0).
The above variables are the leading order quantities in an asymptotic expansion in δ, and we will 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 for ui,(4.6)δ2∂ui(0)∂χ+∂υi∂y=0,
which suggests the following expansion for υi:(4.7)υi=δ2υi(0).
Upon using this expansion, the nondimensional y component of the momentum equation yields to leading order in δ, ∂pi(0)/∂y=0, implying that the pressure is constant in both films across the y direction, and so pi=pi(χ,t). The simplified x-momentum equation, therefore, is given by(4.8)dpi(0)dχ-μr,iɛ∂2ui(0)∂y2+μr,ik1ui(0)=0.
The normal stress condition at the interface, (3.7), simplifies to give the following equation in the long-wave limit:(4.9)p1(0)-p2(0)+1k1(φ2-μrφ1)=γ¯δ2∂2h∂χ2+ϵ12(∂ψ1∂y)2-ϵ22(∂ψ2∂y)2.
In order for the interfacial tension to be of the same order as the other terms in the above equation, we require γ¯δ2~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=γ¯H2=γH3/ϵ0ψb2. Upon using the relation γ¯δ2=1, (4.9) becomes(4.10)p1(0)-p2(0)+1k1(φ2-μrφ1)=∂2h∂χ2+ϵ12(∂ψ1∂y)2-ϵ22(∂ψ2∂y)2.
The tangential stress continuity, (3.8), simplifies to the following condition in the long-wave limit:(4.11)∂u2(0)∂y-μr∂u1(0)∂y=-q∂ψ1∂χ.
The nondimensional kinematic condition at the interface, (3.10), after using the asymptotic expansion for υi (4.7), yields(4.12)δ2υ1(0)(y=h(χ))=ɛ∂h∂t+δ2u1(0)∂h∂χ.
In order for the time derivative term in the above equation to be of the same order as the other two terms, it is necessary to stipulate a slow time scale in the long-wave limit such that(4.13)∂∂t=δ2∂∂τ,
where ∂/∂τ is O(1). The kinematic condition thus gives(4.14)υ1(0)(y=h(χ))=ɛ∂h∂τ+u1(0)∂h∂χ.
The dimensional slow time scale is obtained as follows:(4.15)∂∂t=μ2H2ϵ0ψb2∂∂tdim=δ2∂∂τ,
where tdim is the dimensional time. Thus, in order to nondimensionalize the dimensional time tdim in long-wave limit, the appropriate time scale is μ2H2/(ϵ0ψb2δ2). After using γ¯δ2=1 to eliminate δ2, the time scale becomes μ2H3γ/(ϵ0ψb2)2.Finally, the nondimensional interfacial charge balance, (3.11) is simplified in the long-wave limit as(4.16)ɛδ2∂q∂τ+δ2u1(0)∂q∂χ-δ2q∂υ1(0)∂y=ɛ[S1∂ψ1∂y-S2∂ψ2∂y].
The above equation suggests that the nondimensional conductivities S1 and S2 both should scale as δ2 in order to balance the left side of the equation. So, we let Si=δ2Si(0), i=1,2, where Si(0)=σiμ2γH3/(ϵ03ψb4). Upon using these rescaled conductivities, the charge conservation equation becomes, after using the continuity equation, (4.6),(4.17)ɛ∂q∂τ+∂(u1(0)q)∂χ=ɛ[S1(0)∂ψ1∂y-S2(0)∂ψ2∂y].
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 expressions for the potentials ψi (i=1,2):(5.1)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),
where the interfacial charge density q(χ,τ) is determined below by the interface charge conservation equation (4.17). The simplified x-momentum equation (4.8) can be integrated with respect to y, since dpi(0)/dχ is independent of y and the two constants of integration that arise are determined by the boundary conditions (5.2)u1(0)(β)=0,u1(0)[y=h(χ)]=uint⁡(0)(χ),u2(0)(-1)=0,u2(0)[y=h(χ)]=uint⁡(0)(χ).
Here, uint(0)(χ) is the x component of the velocity at the interface y=h(χ), and this quantity will 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 uint(0)(χ) undetermined at present.The solutions for thex-component velocities ui(0) (i=1,2) thus obtained are(5.3)u1(0)(χ,y)=k1μrdp1(0)dχ[sinh⁡[ɛ/k1(y-β)]-sinh⁡[ɛ/k1(y-h(χ))]sinh⁡[ɛ/k1[h(χ)-β]]-1]+uint⁡(0)(χ)sinh⁡[ɛ/k1(y-β)]sinh⁡[ɛ/k1[h(χ)-β]],(5.4)u2(0)(χ,y)=k1dp2(0)dχ[sinh⁡[ɛ/k1(y+1)]-sinh⁡[ɛ/k1(y-h(χ))]sinh⁡[ɛ/k1(h(χ)+1)]-1]+uint⁡(0)(χ)sinh⁡[ɛ/k1(y+1)]sinh⁡[ɛ/k1(h(χ)+1)],
where the pressure gradients dpi(0)/dχ (i=1,2) are determined below.The continuity equation (4.6), after substituting the asymptotic expansion for υi, (4.7), simplifies to(5.5)∂ui(0)∂χ+∂υi(0)∂y=0.
The above equation is integrated with respect to y from y=β to y=h(χ) for fluid 1 and y=-1 to y=h(χ) for fluid 2, to yield the following expressions for the normal velocities at the interface υi(0)[y=h(χ)] for i=1,2, where the integration constant is set to zero in order to satisfy the no-penetration condition at y=β and y=-1:(5.6)υ1(0)[y=h(χ)]=-∫βh(χ)∂u1(0)∂χdyυ2(0)[y=h(χ)]=-∫-1h(χ)∂u2(0)∂χdy.
Note that the normal velocities of the two fluids are equal at the interface (normal velocity continuity condition), and so, (5.7)υ1(0)[y=h(χ)]=υ2(0)[y=h(χ)]=υint(0).
Therefore, we equate the two integrals in the above equation and apply Leibnitz rule(5.8)∂∂χ∫βh(χ)u1(0)dy-∂h∂χu1(0)h(χ)=∂∂χ∫-1h(χ)u2(0)dy-∂h∂χu2(0)h(χ).
Noting that the x-component velocities are equal at the interface, u1(0)[h(χ)]=u2(0)[h(χ)], the above equation is simplified to(5.9)∂∂χ[∫βh(χ)u1(0)dy-∫-1h(χ)u2(0)dy]=0.
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, because the pressure gradients dpi(0)/dχ in the two fluids should be zero in the absence of electric field, and when the interfaces are flat. We then substitute the expressions for ui(0)(y), (5.3) and (5.4), in the above equation, carry out the integrations with respect to y, and substitute the simplified normal stress continuity condition (4.10) to eliminate dp2(0)/dχ in terms of dp1(0)/dχ. Prior to determining dp1(0)/dχ, it is useful to determine the x-component of the fluid velocity at the interface uint(0) from the simplified tangential stress continuity condition, (4.11). Once uint(0) is determined, the pressure gradient dp1(0)/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(5.10)dp1(0)dχ={k1(1+h(x))[h′′′(χ)-M(χ)]-GFk1ɛ-k1[h′′′(χ)-M(χ)][G+4k1ɛ]tanh[ɛ4k1[1+h(χ)]]}{k1〈[ɛ4k1[h(χ)-β]]∂1+h(χ)+1μr[β-h(χ)]-G(tanh[ɛ4k1[1+h(χ)]]-tanh[ɛ4k1[h(χ)-β]])-4k1ɛ(tanh[ɛ4k1[1+h(χ)]]-1μrtanh[ɛ4k1[h(χ)-β]])〉}-1,(5.11)dp2(0)dχ=dp1(0)(χ)dχ-∂3h(χ)∂χ3+uint(0)(χ)k1(μr-1)+M(χ),(5.12)uint⁡(0)(χ)={k1ɛF-k1dp1(0)dχ(tanh[ɛ4k1[1+h(χ)]]-tanh[ɛ4k1[h(χ)-β]])+k1tanh[ɛ4k1[1+h(χ)]](∂3h(χ)∂χ3-M)}×{coth⁡[ɛk1[1+h(χ)]]-μrcoth⁡[ɛk1[h(χ)-β]]+(μr-1)tanh[ɛ4k1[1+h(χ)]]}-1,
where G, M, and F are functions of χ, and they are defined as(5.13)G(χ)={coth⁡[ɛk1[1+h(χ)]]-μrcoth⁡[ɛk1[h(χ)-β]]+(μr-1)tanh[ɛ4k1[1+h(χ)]]}-1×{(μr-1)[1+h(χ)]+k1ɛ((1-2μr)tanh[ɛ4k1[1+h(χ)]]+tanh[ɛ4k1[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)].
Finally, the kinematic condition at the interface, (4.14), is used to derive the evolution equation for h(χ,τ), as follows. We first substitute the expressions for the normal fluid velocities at the interface, (5.6), after using Leibnitz rule on the integral(5.14)ɛ∂h∂τ+u1(0)[h(χ)]∂h∂χ=-∂∂χ∫βh(χ)u1(0)dy+u1(0)[h(χ)]∂h∂χ,
which is simplified to give (5.15)ɛ∂h∂τ+∂∂χ∫βh(χ)u1(0)dy=0,
where u1(0) is substituted from (5.3), after using the expressions for dp1(0)/dχ and uint(0) determined using the procedure outlined above. The evolution equation for the interfacial charge density q(χ,τ) is obtained from (4.17) after substituting the expressions for uint(0) and the gradients of the potential (from (5.1)) in that equation. The nonlinear evolution equations for h(χ,τ) and q(χ,τ), respectively, take the forms(5.16)ɛ∂h∂τ+1μr{(k1dp1(0)dχ+μr2uint(0)(χ))∂h∂χsech2[12ɛk1(β-h)]+k1[(β-h)d2p1(0)dχ2-dp1(0)dχ∂h∂χ]-k1ɛ(2k1d2p1(0)dχ2+μruint′(0)(χ))×tanh[12ɛk1(β-h)](k1dp1(0)dχ+μr2uint(0)(χ))}=0,ɛ∂q∂τ+uint⁡(0)(χ)∂q∂χ-q∂h∂χ{ɛk1μrdp1(0)dχtanh[12ɛk1(h-β)]+ɛk1uint⁡(0)(χ)coth⁡[ɛk1(h-β)]}+quint′(0)(χ)+ɛS1(0)[(1+h)q+ϵ2]ϵ1+βϵ2+(ϵ1-ϵ2)h+ɛS2(0)[(β-h)q-ϵ1]ϵ1+βϵ2+(ϵ1-ϵ2)h=0.
These coupled nonlinear equations can be solved numerically with appropriate initial conditions to determine the evolution of the interface in the presence of electric fields and porous medium. In the present work, however, we restrict ourselves to studying the linear stability 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 base state about which we perturb. The steady base state we consider is that of stationary fluids with a flat interfaceh(χ,τ)=0, and with a constant interfacial charge density q0 which is independent of χ and τ. This base state interfacial charge q0 is determined from (4.17) with the left side set to zero, since ∂/∂τ=0 and ui(0)=0 in the base state(6.1)S1(0)[∂ψ1∂y]y=0=S2(0)[∂ψ2∂y]y=0.
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:
(6.2)q0=ϵ1S2(0)-ϵ2S1(0)S1(0)+βS2(0).
The variables h and q are now perturbed about their base state values(6.3)h(χ,τ)=h1exp⁡⁡[ikχ+ωτ],q(χ,τ)=q0+q1exp⁡⁡[ikχ+ωτ],
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/ϵ0ψb2 and ω is the nondimensional growth rate based on the time scale μ2H3γ/(ϵ0ψb2)2. We substitute the expressions for dp1(0)/dχ and uint(0) from (5.10), (5.12), and (6.3) in the nonlinear evolution equations (5.16), and apply Taylor expansion on the hyperbolic functions about h(χ)=0. We then 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(6.4)H1̃h1+Q1̃q1=0,H2̃h1+Q2̃q1=0,
where(6.5)H1̃=-(R14+k4R13R1)β4ϵ24+ϵ14(R14+k4R13R1-k2R13R1ϵ2)-ϵ12ϵ22[6(R14+k4R13R1)β2+k2R13R1ϵ2]+ϵ13ϵ2[R15-4β(R14+k4R13R1)+k2R13R1βϵ2]-ϵ1ϵ23[R15+4β3(R14+k4R13R1)+k2R13R1βϵ2]+q02(1+β)ϵ2{[(R1(β-1)-C1B1βk1ɛ)ϵ1-C1B1β2k1ɛϵ2]-R6R10R12+k2R13(ϵ1+βϵ2)×[(R1(β-1)-C1B1βk1ɛ)ϵ1-C1B1β2k1ɛϵ2]}-q0ϵ2{k1ɛR8R10R12+k2R13(ϵ1+βϵ2)×[β(2R1-C1B1k1ɛ)ϵ12+(2R1-C1B1β(β-1)k1ɛ)ϵ1ϵ2+C1B1β2k1ɛϵ22]}+B1R5ωɛ3/2μr,Q1̃=(β-1)ϵ1ϵ2(ϵ1+βϵ2)[R15+k2R1R13(ϵ1+βϵ2)]-q0{k1ɛR7R10R12+k2R13(ϵ1+βϵ2)2×[(R1+βC1B1k1ɛ)ϵ1+β2(R1+C1B1k1ɛ)ϵ2]},H2̃=(ϵ1+βϵ2)(k1ɛɛ(k1ɛɛR2[R18-B1R4S2(0)]ϵ14μr+ϵ13(R16+R3R4+B1R2R4R34)+ϵ12ϵ2[R17+3βR3R4+3B1R2R4R35]+βϵ2(R30R22k1ɛq0μr+βϵ22(μrR19+βR3R4+ɛβR2×(B1R4q0[S1(0)+βS2(0)]+[R18+B1R4S1(0)]ϵ2)μr)k1ɛ)+ϵ1(R30R22k1ɛq0μr+βϵ22([S1(0)+βS2(0)]R20+3βR3R4+ɛβR2(3B1R4q0[S1(0)+βS2(0)]+[4βR18+3B1R4S1(0)-βB1R4S2(0)]ϵ2)μr[S1(0)+βS2(0)])k1ɛ))-R2q0μr(R9[R21+R23+R25+R28-R29]+R32R36)k1ɛ),Q2̃=R2(ϵ1+βϵ2)μr{B1R2ɛ2(ϵ1+βϵ2)3[S1(0)+βS2(0)+ω(ϵ1+βϵ2)]-q0[R9(R22+R24+R26+R27)-R31R2ɛk1(ϵ1+βϵ2)+R33R36]},
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 is set 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, S1(0), S2(0), ɛ, ϵ1, and ϵ2, is a quadratic equation for ω. The roots of the characteristic equation for ω can be written as (6.6)ω1=-R48R49-R482+2H2̃Q1̃R49-2R41R47R49R49,(6.7)ω2=-R48R49+R482+2H2̃Q1̃R49-2R41R47R49R49.
The roots ω1 and ω2 are always real, with one of them ω1 is always negative, and the other one ω2 can be positive or negative depending on the choice of the system parameters. We can take 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 rateω given by (6.7) as a function of the wavenumber k for different values of all physical parameters included in the analysis. These calculations are presented in Figures 2–7 for the general case of leaky-leaky dielectric fluids, Figures 8 and 9 for the limiting case of perfect-leaky dielectric fluids in which S1(0)=0, and Figures 10–13 for another limiting case of perfect-perfect dielectric fluids in which S1(0)=S2(0)=0, where we have given the growth rate ω against the wavenumber k for the porosity of porous medium ɛ, medium permeability k1, nondimensional conductivities S1(0), S2(0), dielectric constants ϵ1, ϵ2, the ratio of the thicknesses of top and bottom fluids β, and the ratio of viscosities of the two fluids μr=μ1/μ2, respectively.Variation of growth rateω with wavenumber k for various values of ɛ whenk1=3, S1(0)=103, S2(0)=104, ϵ1=3, ϵ2=4, β=0.5, μr=1, and ɛ=0.1,0.3,0.5, respectively, for the wavenumber ranges (a) 0≤k≤20 and (b) 0≤k≤5.
(a)(b)Figure 3
Variation of growth rateω with wavenumber k for various values of k1 when ɛ=0.2, S1(0)=103, S2(0)=104, ϵ1=3, ϵ2=4, β=0.5, μr=1, and k1=5,10,12, respectively.Variation of growth rateω with wavenumber k for various values of S1(0) when ɛ=0.4, k1=10, S2(0)=104, ϵ1=3, ϵ2=4, β=0.5, μr=1, and S1(0)=102, S1(0)=103,104, respectively, for the wavenumber ranges (a) 0≤k≤100 and (b) 0≤k≤10.
(a)(b)Variation of growth rateω with wavenumber k for various values of S2(0) when ɛ=0.4, k1=10, S1(0)=103, ϵ1=3, ϵ2=4, β=0.5, μr=1, and S2(0)=102,104,105, respectively, for the wavenumber ranges (a) 0≤k≤100 and (b) 0≤k≤3.
(a)(b)Figure 6
Variation of growth rateω with wavenumber k for various values of ϵ1 when ɛ=0.4, k1=5, S1(0)=103, S2(0)=104, ϵ2=4, β=0.5, μr=1, and ϵ1=3,5,7, respectively.Variation of growth rateω with wavenumber k for various values of μr when ɛ=0.4, k1=3, S1(0)=103, S2(0)=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.
(a)(b)Variation of growth rateω with wavenumber k for various values of ɛ whenk1=3, S1(0)=0, S2(0)=104, ϵ1=3, ϵ2=4, β=0.5, μr=1, and ɛ=0.1,0.3,0.5, respectively, for the wavenumber ranges (a) 0≤k≤20, and (b) 0≤k≤5.
(a)(b)Variation of growth rateω with wavenumber k for various values of μr whenɛ=0.4, k1=3, S1(0)=0, S2(0)=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.
(a)(b)Figure 10
Variation of growth rateω with wavenumber k for various values of ɛ when k1=3, S1(0)=0, S2(0)=0, ϵ1=3, ϵ2=4, β=0.5, μr=1, and ɛ=0.1,0.3,0.5, respectively.Figure 11
Variation of growth rateω with wavenumber k for various values of k1 when ɛ=0.2, S1(0)=0, S2(0)=0, ϵ1=3, ϵ2=4, β=0.5, μr=1, and k1=5,10,12, respectively.Figure 12
Variation of growth rateω with wavenumber k for various values of ϵ1 when ɛ=0.4, k1=5, S1(0)=0, S2(0)=0, ϵ2=4, β=0.5, μr=1, and ϵ1=3,5,7, respectively.Figure 13
Variation of growth rateω with wavenumber k for various values of ϵ2 when ɛ=0.4, k1=5, S1(0)=0, S2(0)=0, ϵ1=3, β=0.5, μr=1, and ϵ2=4,6,10, respectively.


### 6.1. Leaky Dielectric-Leaky Dielectric Interface

This is the general case in whichS1(0)≠0 and S2(0)≠0, which is discussed in Figures 2–7, when the conductivities of the upper and lower fluids are present in the analysis. Figures 2(a) and 2(b) shows the variation of the growth rate ω versus the wavenumber k for various values of the porosity of porous medium ɛ. It is clear from Figure 2(b) that for small values of the porosity (e.g., ɛ=0.1), the growth rate ω increases by increasing the wavenumber k till a maximum value of ɛ after which ω decreases by increasing k; that is, there exists a maximum mode of instability only for small values of the porosity ɛ, while for any other value of ɛ, we found that ω decreases by increasing k, that is. the system is always stable in this case. It is clear 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 porosity of porous medium has a dual role on the stability of the considered system (stabilizing and then destabilizing) depends on the wavenumber range (less than or hifher than a critical wavenumber value k=15, resp.). Figure 3 shows the variation of growth rate ω with the wavenumber k for different values of the medium permeability k1. We conclude from this figure that for any wavenumber value k≤20, the growth rate ω increases by increasing the medium permeability k1 values, while for wavenumber values k>20, all the curves correspond to different values of medium permeability k1 coincide. This means that the medium permeability k1 has a destabilizing effect for the wavenumber range 0<k≤20 and it has no effect on the stability of the considered system for higher wavenumber values k>20.Figures4(a) and 4(b) shows the variation of the growth rate ω versus the wavenumber k for various values of the conductivity of upper fluid S1(0). It is clear from this figure that the conductivity S1(0) has a stabilizing effect for small wavenumber values and a destabilizing effect for high wavenumber values, as the growth rates ω decrease and increase by increasing the conductivity S1(0) values, respectively. It is also seen that between these two wavenumber ranges, the conductivity of upper fluid S1(0) has a dual role on the stability of the considered system; that is, it has a destabilizing effect for S1(0) values greater than 102, while it has a stabilizing effect for S1(0) values greater than 104. Therefore, we conclude that the conductivity of upper fluid S1(0) 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 the wavenumber k for different values of the conductivity of lower fluid S2(0). It indicates that the conductivity S1(0) has a destabilizing effect for the wavenumber range 0<k<10, while it has a stabilizing effect for higher wavenumber values k≥10, since the growth rate ω increases in the first wavenumber range and decreases in the second wavenumber range by increasing the conductivity of lower fluid S2(0). We conclude also from Figure 5(b) that there exists a mode of maximum instability for small values of the conductivity S2(0) which disappears for high values of the conductivity of lower fluid S2(0)≥104.Figure6 shows the variation of growth rate ω with the wavenumber k for different values of the dielectric constant of upper fluid ϵ1. It is seen from this figure that there exists a critical wavenumber value k=4.6, before which the growth rates decrease by increasing the dielectric constant ϵ1, and after which they increase by increasing ϵ1 values. Thus, the dielectric constant of upper fluid ϵ1 has a stabilizing as well as a destabilizing effects, for wavenumber 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 upper and lower fluids β on the stability of the considered system are found to have opposite effects to the effect of the dielectric constant ϵ1 given by Figure 6, but the corresponding figures are not given here. In other words, we conclude that the dielectric constant of lower fluid ϵ2 has a destabilizing as well as a stabilizing effects for wavenumber ranges before and after the same critical wavenumber value k=4.6, respectively, while the ratio of thicknesses of upper and lower fluids β has also a destabilizing as well as a stabilizing effects for wavenumber ranges before 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 of the ratio of viscosity of upper and lower fluids μr. It indicates that the viscosity ratio μr has a stabilizing effect for small wavenumber values, and it has also a destabilizing effect for higher wavenumber values, since the growth rate ω decreases and increases by increasing the increase of μr, respectively. It is clear also from Figure 7(b) that for μr>1, that is, when the viscosity of upper fluid is larger than the viscosity of lower fluid, there exists a mode of maximum instability which disappear for μr≤1.


### 6.2. Perfect Dielectric-Leaky Dielectric Interface

This is the limiting case in whichS1(0)=0 and S2(0)≠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) and 8(b) show the variation of the growth rate ω against the wavenumber k for various values of the porosity of porous medium ɛ. In comparison with Figures 2(a) and 2(b), we conclude that the porosity of porous medium ɛ behaves in the same manner as in the previous case of leaky-leaky dielectric fluids, except that the values of growth rates ω are higher than their values in the previous case and the corresponding curves intersect the k-axis at larger wavenumber values than in the previous case. We conclude also that for small wavenumber values, there exists a mode of maximum instability for porosity values ɛ≤0.3. Similarly, the effects of medium permeability k1, the conductivity of lower fluid S2(0), the dielectric constants ϵ1, ϵ2, and the ratio of thicknesses of upper and lower fluids β on the stability of the considered 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 the effect of k1, the growth rate values are higher than their values in the previous case; for the effect of S2(0), the obtained curves intersect the k-axis at larger wavenumber values than in the previous case; for the effect of ϵ1, there exists a mode of maximum instability for all values of ϵ1; for the effect of ϵ2, the obtained curves are exactly similar to those obtained in the previous case; finally, for the effect of β, there is a mode of maximum instability and the obtained curves intersect 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 for various values of the viscosity ratio of upper and lower fluids μr. In comparison with Figures 7(a) and 7(b), we conclude that the viscosity ratio μr behaves in the same manner as in the previous case of leaky-leaky dielectric fluids with the only difference that in this case, for all values of μr⪋1, there exists a mode of maximum instability and not only for μr>1 shown in the previous case.


### 6.3. Perfect Dielectric-Perfect Dielectric Interface

This is the limiting case in whichS1(0)=S2(0)=0, which is discussed in Figures 10–13, when the conductivities of the upper and lower fluids are absent in the analysis. Figure 10 shows the variation of the growth rate ω versus the wavenumber k for different values of the porosity of porous medium ɛ. In view of the above discussion, we conclude from this figure that the porosity ɛ has a slightly stabilizing effect for small wavenumber values k≤2.5 and that it has no effect on the stability of the considered system afterwards for higher wavenumber values. Figure 11 shows the variation of growth rate ω with the wavenumber k for various values of the medium permeability k1. It is clear from this figure that the permeability k1 has a destabilizing effect, since the growth rate ω increases by increasing the medium permeability values at any fixed wavenumber value. Similarly, the effect of dielectric constant of the upper fluid ϵ1 on the stability of the system is illustrated in Figure 12, and it shows that the dielectric constant ϵ1 has a stabilizing effect since the growth rate ω decreases by increasing the dielectric constant ϵ1 at any wavenumber value. Figure 13 shows the variation of growth rate ω with the wavenumber k for different values of the dielectric constant of lower fluid ϵ2, and from which we conclude thatthe dielectric constant ϵ1, and after which they increase by increasing ϵ2 has a destabilizing effect in the wavenumber range 0<k≤4.6, and it has no effect on the stability of the considered system afterwards for higher wavenumber values k>4.6. Similarly, the effects of both the ratio of thicknesses of upper and lower fluids β, and the ratio of viscosities of upper and lower fluids μr, respectively, on the stability of the system are found to has the same and the opposite effects as the effect of the dielectric constant ϵ2 shown in Figure 13, but the corresponding figures are not given here to avoid any kind of repitation; that is, the parameters β and μr have destabilizing and stabilizing effects for small wavenumber values, respectively.


## 6.1. Leaky Dielectric-Leaky Dielectric Interface

This is the general case in whichS1(0)≠0 and S2(0)≠0, which is discussed in Figures 2–7, when the conductivities of the upper and lower fluids are present in the analysis. Figures 2(a) and 2(b) shows the variation of the growth rate ω versus the wavenumber k for various values of the porosity of porous medium ɛ. It is clear from Figure 2(b) that for small values of the porosity (e.g., ɛ=0.1), the growth rate ω increases by increasing the wavenumber k till a maximum value of ɛ after which ω decreases by increasing k; that is, there exists a maximum mode of instability only for small values of the porosity ɛ, while for any other value of ɛ, we found that ω decreases by increasing k, that is. the system is always stable in this case. It is clear 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 porosity of porous medium has a dual role on the stability of the considered system (stabilizing and then destabilizing) depends on the wavenumber range (less than or hifher than a critical wavenumber value k=15, resp.). Figure 3 shows the variation of growth rate ω with the wavenumber k for different values of the medium permeability k1. We conclude from this figure that for any wavenumber value k≤20, the growth rate ω increases by increasing the medium permeability k1 values, while for wavenumber values k>20, all the curves correspond to different values of medium permeability k1 coincide. This means that the medium permeability k1 has a destabilizing effect for the wavenumber range 0<k≤20 and it has no effect on the stability of the considered system for higher wavenumber values k>20.Figures4(a) and 4(b) shows the variation of the growth rate ω versus the wavenumber k for various values of the conductivity of upper fluid S1(0). It is clear from this figure that the conductivity S1(0) has a stabilizing effect for small wavenumber values and a destabilizing effect for high wavenumber values, as the growth rates ω decrease and increase by increasing the conductivity S1(0) values, respectively. It is also seen that between these two wavenumber ranges, the conductivity of upper fluid S1(0) has a dual role on the stability of the considered system; that is, it has a destabilizing effect for S1(0) values greater than 102, while it has a stabilizing effect for S1(0) values greater than 104. Therefore, we conclude that the conductivity of upper fluid S1(0) 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 the wavenumber k for different values of the conductivity of lower fluid S2(0). It indicates that the conductivity S1(0) has a destabilizing effect for the wavenumber range 0<k<10, while it has a stabilizing effect for higher wavenumber values k≥10, since the growth rate ω increases in the first wavenumber range and decreases in the second wavenumber range by increasing the conductivity of lower fluid S2(0). We conclude also from Figure 5(b) that there exists a mode of maximum instability for small values of the conductivity S2(0) which disappears for high values of the conductivity of lower fluid S2(0)≥104.Figure6 shows the variation of growth rate ω with the wavenumber k for different values of the dielectric constant of upper fluid ϵ1. It is seen from this figure that there exists a critical wavenumber value k=4.6, before which the growth rates decrease by increasing the dielectric constant ϵ1, and after which they increase by increasing ϵ1 values. Thus, the dielectric constant of upper fluid ϵ1 has a stabilizing as well as a destabilizing effects, for wavenumber 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 upper and lower fluids β on the stability of the considered system are found to have opposite effects to the effect of the dielectric constant ϵ1 given by Figure 6, but the corresponding figures are not given here. In other words, we conclude that the dielectric constant of lower fluid ϵ2 has a destabilizing as well as a stabilizing effects for wavenumber ranges before and after the same critical wavenumber value k=4.6, respectively, while the ratio of thicknesses of upper and lower fluids β has also a destabilizing as well as a stabilizing effects for wavenumber ranges before 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 of the ratio of viscosity of upper and lower fluids μr. It indicates that the viscosity ratio μr has a stabilizing effect for small wavenumber values, and it has also a destabilizing effect for higher wavenumber values, since the growth rate ω decreases and increases by increasing the increase of μr, respectively. It is clear also from Figure 7(b) that for μr>1, that is, when the viscosity of upper fluid is larger than the viscosity of lower fluid, there exists a mode of maximum instability which disappear for μr≤1.


## 6.2. Perfect Dielectric-Leaky Dielectric Interface

This is the limiting case in whichS1(0)=0 and S2(0)≠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) and 8(b) show the variation of the growth rate ω against the wavenumber k for various values of the porosity of porous medium ɛ. In comparison with Figures 2(a) and 2(b), we conclude that the porosity of porous medium ɛ behaves in the same manner as in the previous case of leaky-leaky dielectric fluids, except that the values of growth rates ω are higher than their values in the previous case and the corresponding curves intersect the k-axis at larger wavenumber values than in the previous case. We conclude also that for small wavenumber values, there exists a mode of maximum instability for porosity values ɛ≤0.3. Similarly, the effects of medium permeability k1, the conductivity of lower fluid S2(0), the dielectric constants ϵ1, ϵ2, and the ratio of thicknesses of upper and lower fluids β on the stability of the considered 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 the effect of k1, the growth rate values are higher than their values in the previous case; for the effect of S2(0), the obtained curves intersect the k-axis at larger wavenumber values than in the previous case; for the effect of ϵ1, there exists a mode of maximum instability for all values of ϵ1; for the effect of ϵ2, the obtained curves are exactly similar to those obtained in the previous case; finally, for the effect of β, there is a mode of maximum instability and the obtained curves intersect 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 for various values of the viscosity ratio of upper and lower fluids μr. In comparison with Figures 7(a) and 7(b), we conclude that the viscosity ratio μr behaves in the same manner as in the previous case of leaky-leaky dielectric fluids with the only difference that in this case, for all values of μr⪋1, there exists a mode of maximum instability and not only for μr>1 shown in the previous case.


## 6.3. Perfect Dielectric-Perfect Dielectric Interface

This is the limiting case in whichS1(0)=S2(0)=0, which is discussed in Figures 10–13, when the conductivities of the upper and lower fluids are absent in the analysis. Figure 10 shows the variation of the growth rate ω versus the wavenumber k for different values of the porosity of porous medium ɛ. In view of the above discussion, we conclude from this figure that the porosity ɛ has a slightly stabilizing effect for small wavenumber values k≤2.5 and that it has no effect on the stability of the considered system afterwards for higher wavenumber values. Figure 11 shows the variation of growth rate ω with the wavenumber k for various values of the medium permeability k1. It is clear from this figure that the permeability k1 has a destabilizing effect, since the growth rate ω increases by increasing the medium permeability values at any fixed wavenumber value. Similarly, the effect of dielectric constant of the upper fluid ϵ1 on the stability of the system is illustrated in Figure 12, and it shows that the dielectric constant ϵ1 has a stabilizing effect since the growth rate ω decreases by increasing the dielectric constant ϵ1 at any wavenumber value. Figure 13 shows the variation of growth rate ω with the wavenumber k for different values of the dielectric constant of lower fluid ϵ2, and from which we conclude thatthe dielectric constant ϵ1, and after which they increase by increasing ϵ2 has a destabilizing effect in the wavenumber range 0<k≤4.6, and it has no effect on the stability of the considered system afterwards for higher wavenumber values k>4.6. Similarly, the effects of both the ratio of thicknesses of upper and lower fluids β, and the ratio of viscosities of upper and lower fluids μr, respectively, on the stability of the system are found to has the same and the opposite effects as the effect of the dielectric constant ϵ2 shown in Figure 13, but the corresponding figures are not given here to avoid any kind of repitation; that is, the parameters β and μr have destabilizing and stabilizing effects for small wavenumber values, respectively.


## 7. Concluding Remarks

In conclusion, we have provided a general formulation for analyzing the effect of an externally applied electric field on the stability and dynamics of the interface between two leaky dielectric fluids of arbitrary viscosities and conductivities in porous medium. A systematic long-wave asymptotic analysis was used to derive coupled nonlinear evolution equations for the position of the interface and free charge density at the interface. Attention was restricted to linearized stability of the coupled nonlinear equations and the effect of a variety of system parameters 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 in these limiting cases and the general case of two leaky dielectric fluids can be summarized as follows:(I)
For perfect-perfect dielectric fluids, we conclude for small wavenumbers that
(i)
the porosity of porous mediumɛ, the dielectric constant of upper fluid ϵ1, and the ratio of viscosity of upper and lower fluids μr have stabilizing effects,(ii)
the medium permeabilityk1, the dielectric constant of lower fluid ϵ2, and the ratio of thickness of upper and lower fluids β have destabilizing effects,(iii)
at any value of these physical parameters there are no modes of maximum instability, that is. the system is always stable,(iv)
these physical parameters have no effect on the stability of the system for high wavenumber values.(II)
For perfect-leaky dielectric fluids, we found that
(i)
the conductivity of lower fluidS2(0) has a destabilizing effect for small wavenumbers and a stabilizing effect for high wavenumbers,(ii)
for small wavenumber values, the physical parametersɛ, ϵ1, and μr have stabilizing 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; in other words, the physical parametersɛ, 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 parameters 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 fluidS1(0) has a destabilizing effect for small wavenumbers 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 them in the wavenumber range k1<k<k2.(ii)
the effects of all other physical parameters on the stability of the considered system behave in the same manner as their effects in the case of perfect-leaky dielectrics, except that in the case of perfect-leaky dielectrics the stability or instability regions occur more faster than the corresponding case of leaky-leaky dielectrics, and the maximum instability holds for more values of the physical parameters included in the analysis.It should be mentioned that the problem investigated in this article can be generalized to study the linear electrohydrodynamic instabilities at the interface between two immiscible fluids, either perfect or leaky dielectrics, subjected to alternating electric fields and moving through a porous medium in the limit of the electrode spacing being large compared to the wavelength of the perturbation using the Floquet theory analysis [37], and this case is now in a current research.



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*Source: 498718-2011-08-24.xml*