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.

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: Section 2 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 âˆ’đ»<𝑩<0 (fluid 2) and 0<𝑩<đ›œđ» (fluid 1) in the initial unperturbed state, see Figure 1, where đ›œ is the ratio of the thicknesses of top and bottom fluids. The perturbed interface between the two fluids is denoted by 𝑩=ℎ(đ‘„). The two fluids are stationary in the initial state, with viscosities 𝜇𝑖, dielectric constants 𝜖𝑖, conductivities 𝜎𝑖, porosity of porous medium 𝜀 and medium permeability 𝑘1, where 𝑖=1,2. Fluid 2 is bounded at the bottom 𝑩=âˆ’đ» by a rigid plate which is maintained at an electric potential 𝜓=𝜓𝑏, while fluid 1 is bounded at the top 𝑩=đ›œđ» 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 𝜀, 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.

For leaky dielectric fluids of constant conductivities and with zero net charge in the bulk, the following governing equations are appropriate for the electric field 𝐄 in the two fluids 1 and 2 [3]∇⋅𝐄𝑖=0,𝑖=1,2.(2.1) 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∇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 𝑩=ℎ(đ‘„), satisfies 𝜖2𝜖0î€·âˆ‡đœ“2⋅𝐧−𝜖1𝜖0î€·âˆ‡đœ“1⋅𝐧=𝑞(đ‘„,𝑡)at𝑩=ℎ(đ‘„),(2.3) where 𝐧 is the unit normal to the interface at 𝑩=ℎ(đ‘„) (see Figure 1), 𝜖𝑖 (𝑖=1,2) is the dielectric constant in fluid 𝑖, 𝜖0 is the permittivity of free space, and 𝑞(đ‘„,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 interface 𝑩=ℎ(đ‘„) translates to the continuity of the electric potentials; that is,𝜓1=𝜓2at𝑩=ℎ(đ‘„).(2.4)(3)The electric potentials satisfy the following conditions at the rigid boundaries: 𝜓2=𝜓𝑏𝜓at𝑩=âˆ’đ»,1=0at𝑩=đ›œđ».(2.5) 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∇⋅𝐕𝑖=0,∇⋅𝐓𝑖=0,(2.6) 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 medium𝐓𝑖=−𝑝𝑖𝐈+đœ‡đ‘–î€·âˆ‡đ•đ‘–+∇𝐕𝑇𝑖+𝐩𝑖+𝜇𝑖𝑘1𝜑𝑖𝐈,𝑖=1,2,(2.7) where 𝑝𝑖 is the pressure in fluid 𝑖, 𝐈 is the identity tensor, and the Maxwell stress tensor 𝐩𝑖 is given by Saville [3]𝐩𝑖=𝜖𝑖𝜖0𝐄𝑖𝐄𝑖−12î€·đ„đ‘–â‹…đ„đ‘–î€žđˆî‚„.(2.8) The divergence of the Maxwell stress tensor ∇⋅𝐩𝑖=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,∇⋅𝐩𝑖𝜖=∇⋅𝑖𝜖0𝐄𝑖𝐄𝑖−12î€·đ„đ‘–â‹…đ„đ‘–î€žđˆđœ–î‚„î‚‡=−0𝐄𝑖2⋅𝐄𝑖∇𝜖𝑖+𝜌𝑓𝐄𝑖=0,(2.9) 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, becomeđœŒđœ€đ·đ•đ‘–đ·đ‘Ą=−∇𝑝𝑖+∇⋅𝐩𝑖+𝜇𝑖𝜀∇2𝐕𝑖−𝜇𝑖𝑘1𝐕𝑖,𝑖=1,2.(2.10) Since there is no time scale, then đ·đŻđ‘–/đ·đ‘Ą=0, and we get from the above two equations that∇𝑝𝑖−𝜇𝑖𝜀∇2𝐕𝑖+𝜇𝑖𝑘1𝐕𝑖=0,𝑖=1,2.(2.11) The fluid velocities satisfy no-slip and no-penetration conditions at the top and bottom plates𝐕1(𝑩=đ›œđ»)=0,𝐕2(𝑩=âˆ’đ»)=0.(2.12) At the interface 𝑩=ℎ(đ‘„) 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+đ›Ÿđœ…,(2.15)(𝐭⋅𝐓⋅𝐧)2=(𝐭⋅𝐓⋅𝐧)1,(2.16) where đ›Ÿ is the interfacial tension between the two fluids, 𝐭 is the unit tangent to the interface (see Figure 1), and 𝜅=𝜕2ℎ(đ‘„)/đœ•đ‘„2 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 đ‘„ and 𝑩 directions by 𝑱 and 𝜐, respectively; that is, 𝐕𝑖=(𝑱𝑖,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𝜕𝑩+𝜖𝑖𝜖0𝐾𝑁2𝑖−12𝐾2𝑖+𝜇𝑖𝑘1𝜑𝑖,(𝐭⋅𝐓⋅𝐧)𝑖=đœ‡đ‘–î‚”đœ•đ‘ąđ‘–+đœ•đ‘Šđœ•đœđ‘–î‚¶đœ•đ‘„+𝜖𝑖𝜖0𝐾𝑇𝑖𝐾𝑁𝑖,(2.17) where 𝐾𝑇𝑖 and 𝐾𝑁𝑖 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 𝑩=ℎ(đ‘„):𝑝1−2𝜇1𝜕𝜐1î‚č−𝑝𝜕𝑩2−2𝜇2𝜕𝜐2î‚č+1𝜕𝑩𝑘1î€ș𝜇2𝜑2−𝜇1𝜑1î€»đœ–=đ›Ÿđœ…+1𝜖02î‚ƒî€·âˆ‡đœ“1⋅𝐧2âˆ’î€·âˆ‡đœ“1⋅𝐭2−𝜖2𝜖02î‚ƒî€·âˆ‡đœ“2⋅𝐧2âˆ’î€·âˆ‡đœ“2⋅𝐭2,(2.18) for the normal stress continuity, and𝜇2𝜕𝑱2+𝜕𝑩𝜕𝜐2î‚čđœ•đ‘„âˆ’đœ‡1𝜕𝑱1+𝜕𝑩𝜕𝜐1î‚čî€·đœ•đ‘„=−∇𝜓1⋅𝐭𝑞(đ‘„,𝑡),(2.19) 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(đ‘„,𝑡),𝜐1(𝑩=ℎ(đ‘„))=𝜐2(𝑩=ℎ(đ‘„))=𝜀𝜕ℎ𝜕𝑡+𝐕⋅∇𝑠ℎ,(2.20) where ∇𝑠 is the gradient operator along the interface 𝑩=ℎ(đ‘„). Finally, the interfacial charge is governed by a conservation equation𝜀𝜕𝑞𝜕𝑡+𝐕⋅∇𝑠𝜎𝑞−𝑞𝐧⋅(𝐧⋅∇)𝐕=đœ€î€·î€ș2𝐄2−î€ș𝜎⋅𝐧1𝐄1â‹…đ§î€»î€ž,(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 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 setting𝜓𝑖=𝜓𝑖𝜓𝑏,𝑝𝑖=đ‘đ‘–đ»2𝜖0𝜓2𝑏,𝑇𝑖=đ‘‡đ‘–đ»2𝜖0𝜓2𝑏,đ‘„î…ž=đ‘„đ»,𝑩=đ‘Šđ»,𝐄𝑖=đ„đ‘–đ»đœ“đ‘,𝐕𝑖=𝐕𝑖𝜇2đ»đœ–0𝜓2𝑏,𝑡=𝑡𝜖0𝜓2𝑏𝜇2đ»2,𝜑𝑖=𝜇2𝜑𝑖𝜖0𝜓2𝑏,𝑞=đ‘žđ»đœ–0𝜓𝑏,𝜅=đœ…đ»,𝑘1=𝑘1đ»2,ℎ=â„Žđ».(3.1) 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 potentials 𝜓𝑖 are∇2𝜓𝑖=0,𝑖=1,2,(3.2) with the following boundary conditions at the interface 𝑩=ℎ(đ‘„),î€ș𝜖2∇𝜓2−î€ș𝜖⋅𝐧1∇𝜓1î€»â‹…đ§=𝑞,𝜓1=𝜓2,(3.3) and the following boundary conditions at the top and bottom boundaries𝜓1(𝑩=đ›œ)=0,𝜓2(𝑩=−1)=1.(3.4) Similarly, the nondimensional equations governing the fluid motion are∇⋅𝐕1=0,∇⋅𝐕2=0,(3.5)∇𝑝1−𝜇𝑟𝜀∇2𝐕1+𝜇𝑟𝑘1𝐕1=0,∇𝑝2−1𝜀∇2𝐕2+1𝑘1𝐕2=0.(3.6) 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𝑝1−2𝜇𝑟𝜕𝜐1î‚č−𝑝𝜕𝑩2−2𝜕𝜐2î‚č+1𝜕𝑩𝑘1î€ș𝜑2−𝜇𝑟𝜑1=đœ–đ›Ÿđœ…+12î‚ƒî€·âˆ‡đœ“1⋅𝐧2âˆ’î€·âˆ‡đœ“1⋅𝐭2−𝜖22î‚ƒî€·âˆ‡đœ“2⋅𝐧2âˆ’î€·âˆ‡đœ“2⋅𝐭2,(3.7)𝜕𝑱2+𝜕𝑩𝜕𝜐2î‚čđœ•đ‘„âˆ’đœ‡đ‘Ÿî‚žđœ•đ‘ą1+𝜕𝑩𝜕𝜐1î‚čî€·đœ•đ‘„=−∇𝜓1⋅𝐭𝑞(đ‘„,𝑡),(3.8) where đ›Ÿ=đ›Ÿđ»/𝜖0𝜓2𝑏 is the nondimensional interfacial tension. The boundary conditions for the velocities at the top and bottom plates become𝑱1𝜐(𝑩=đ›œ)=0,1𝑱(𝑩=đ›œ)=0,2(𝑩=−1)=0,𝜐2(𝑩=−1)=0.(3.9) The nondimensional kinematic condition at the interface is𝜐1(𝑩=ℎ(đ‘„))=𝜐2(𝑩=ℎ(đ‘„))=𝜀𝜕ℎ𝜕𝑡+𝐕⋅∇𝑠ℎ,(3.10) and the nondimensional charge conservation equation at the interface is𝜀𝜕𝑞𝜕𝑡+𝐕⋅∇𝑠𝑆𝑞−𝑞𝐧⋅(𝐧⋅∇)𝐕=đœ€î€·î€ș1∇𝜓1−î€ș𝑆⋅𝐧2∇𝜓2â‹…đ§î€»î€ž,(3.11) where 𝑆𝑖=𝜎𝑖𝜇2đ»2/𝜖20𝜓2𝑏, 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 position ℎ(đ‘„,𝑡) and charge 𝑞(đ‘„,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 wavelength 𝐿 of the fastest growing modes is much larger than the transverse length scale đ» in the system, and it is useful to define a small parameter 𝛿=đ»/𝐿â‰Ș1. The lateral length scale 𝐿 is determined self-consistently in the following analysis to be î”đ›Ÿđ»3/𝜖0𝜓2𝑏, and this is further estimated below to be much larger than đ». 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 đ»â‰Ș𝐿, the derivatives in the đ‘„ direction should be scaled with 𝐿. To this end, we define the slowly varying scale 𝜒 in the following manner:đœ•đœ•đœ•đ‘„=𝛿𝜕𝜒,(4.1) and 𝜕/đœ•đœ’âˆŒđ‘‚(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 𝛿â‰Ș1 to𝜕2𝜓𝑖𝜕𝑩2=0,𝑖=1,2.(4.2)

The continuity condition for the normal component of the electric field at the interface 𝑩=ℎ(𝜒), (3.3), is similarly simplified in the long-wave limit as𝜖2𝜕𝜓2𝜕𝑩−𝜖1𝜕𝜓1𝜕𝑩=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 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 đ‘„-momentum equation can be simplified in the long-wave limit as𝛿𝜕𝑝𝑖−𝜇𝜕𝜒𝑟,𝑖𝜀𝜕2𝑱𝑖𝜕𝑩2+𝜇𝑟,𝑖𝑘1𝑱𝑖=0.(4.4) This suggests that đ‘ąđ‘–âˆŒđ‘‚(𝛿)𝑝𝑖. In order to make this explicit and to make ordering of various quantities simpler, we represent the pressure 𝑝𝑖 and the đ‘„-component velocity 𝑱𝑖 in the following manner:𝑝𝑖=𝑝𝑖(0),𝑱𝑖=𝛿𝑱𝑖(0).(4.5) 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 𝜕/đœ•đ‘„âˆŒđ›żđœ•/𝜕𝜒 and using the above expansion for 𝑱𝑖,𝛿2𝜕𝑱𝑖(0)+𝜕𝜒𝜕𝜐𝑖𝜕𝑩=0,(4.6) which suggests the following expansion for 𝜐𝑖:𝜐𝑖=𝛿2𝜐𝑖(0).(4.7) Upon using this expansion, the nondimensional 𝑩 component of the momentum equation yields to leading order in 𝛿, 𝜕𝑝𝑖(0)/𝜕𝑩=0, implying that the pressure is constant in both films across the 𝑩 direction, and so 𝑝𝑖=𝑝𝑖(𝜒,𝑡). The simplified đ‘„-momentum equation, therefore, is given by𝑑𝑝𝑖(0)−𝜇𝑑𝜒𝑟,𝑖𝜀𝜕2𝑱𝑖(0)𝜕𝑩2+𝜇𝑟,𝑖𝑘1𝑱𝑖(0)=0.(4.8) The normal stress condition at the interface, (3.7), simplifies to give the following equation in the long-wave limit:𝑝1(0)−𝑝2(0)+1𝑘1î€·đœ‘2−𝜇𝑟𝜑1=đ›Ÿđ›ż2𝜕2ℎ𝜕𝜒2+𝜖12î‚”đœ•đœ“1î‚¶đœ•đ‘Š2−𝜖22î‚”đœ•đœ“2î‚¶đœ•đ‘Š2.(4.9) In order for the interfacial tension to be of the same order as the other terms in the above equation, we require đ›Ÿđ›ż2âˆŒđ‘‚(1), where 𝛿=đ»/𝐿. We set đ›Ÿ(đ»/𝐿)2=1, and from this relation, we determine the lateral length scale 𝐿 to be √𝐿=đ›Ÿđ»2=î”đ›Ÿđ»3/𝜖0𝜓2𝑏. Upon using the relation đ›Ÿđ›ż2=1, (4.9) becomes𝑝1(0)−𝑝2(0)+1𝑘1î€·đœ‘2−𝜇𝑟𝜑1=𝜕2ℎ𝜕𝜒2+𝜖12î‚”đœ•đœ“1î‚¶đœ•đ‘Š2−𝜖22î‚”đœ•đœ“2î‚¶đœ•đ‘Š2.(4.10) The tangential stress continuity, (3.8), simplifies to the following condition in the long-wave limit:𝜕𝑱2(0)𝜕𝑩−𝜇𝑟𝜕𝑱1(0)𝜕𝑩=−𝑞𝜕𝜓1𝜕𝜒.(4.11) The nondimensional kinematic condition at the interface, (3.10), after using the asymptotic expansion for 𝜐𝑖 (4.7), yields𝛿2𝜐1(0)(𝑩=ℎ(𝜒))=𝜀𝜕ℎ𝜕𝑡+𝛿2𝑱1(0)𝜕ℎ𝜕𝜒.(4.12) 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𝜕𝜕𝑡=𝛿2𝜕𝜕𝜏,(4.13) where 𝜕/𝜕𝜏 is 𝑂(1). The kinematic condition thus gives𝜐1(0)(𝑩=ℎ(𝜒))=𝜀𝜕ℎ𝜕𝜏+𝑱1(0)𝜕ℎ𝜕𝜒.(4.14) The dimensional slow time scale is obtained as follows:𝜕=𝜇𝜕𝑡2đ»2𝜖0𝜓2𝑏𝜕𝜕𝑡dim=𝛿2𝜕,𝜕𝜏(4.15) where 𝑡dim is the dimensional time. Thus, in order to nondimensionalize the dimensional time 𝑡dim in long-wave limit, the appropriate time scale is 𝜇2đ»2/(𝜖0𝜓2𝑏𝛿2). After using đ›Ÿđ›ż2=1 to eliminate 𝛿2, the time scale becomes 𝜇2đ»3đ›Ÿ/(𝜖0𝜓2𝑏)2.

Finally, the nondimensional interfacial charge balance, (3.11) is simplified in the long-wave limit as𝜀𝛿2𝜕𝑞𝜕𝜏+𝛿2𝑱1(0)𝜕𝑞𝜕𝜒−𝛿2𝑞𝜕𝜐1(0)𝑆𝜕𝑩=𝜀1𝜕𝜓1𝜕𝑩−𝑆2𝜕𝜓2î‚č.𝜕𝑩(4.16) 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)=𝜎𝑖𝜇2đ›Ÿđ»3/(𝜖30𝜓4𝑏). Upon using these rescaled conductivities, the charge conservation equation becomes, after using the continuity equation, (4.6),𝜀𝜕𝑞+𝜕𝑱𝜕𝜏1(0)𝑞𝑆𝜕𝜒=𝜀1(0)𝜕𝜓1𝜕𝑩−𝑆2(0)𝜕𝜓2î‚č.𝜕𝑩(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 position ℎ(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):𝜓1î€ș𝜖(𝜒,𝑩)=(đ›œâˆ’đ‘Š)2+(1+ℎ(𝜒))𝑞(𝜒)𝜖1+đ›œđœ–2+î€·đœ–1−𝜖2,𝜓ℎ(𝜒)2(𝜒,𝑩)=đ›œđœ–2−𝜖1î€ș𝜖𝑩+đ›œ(1+𝑩)𝑞(𝜒)+ℎ(𝜒)1−𝜖2−(1+𝑩)𝑞(𝜒)𝜖1+đ›œđœ–2+î€·đœ–1−𝜖2,ℎ(𝜒)(5.1) where the interfacial charge density 𝑞(𝜒,𝜏) is determined below by the interface charge conservation equation (4.17). The simplified đ‘„-momentum equation (4.8) can be integrated with respect to 𝑩, since 𝑑𝑝𝑖(0)/𝑑𝜒 is independent of 𝑩 and the two constants of integration that arise are determined by the boundary conditions 𝑱1(0)(đ›œ)=0,𝑱1(0)[]𝑩=ℎ(𝜒)=𝑱(0)int𝑱(𝜒),2(0)(−1)=0,𝑱2(0)[]𝑩=ℎ(𝜒)=𝑱(0)int(𝜒).(5.2) Here, 𝑱(0)int(𝜒) is the đ‘„ component of the velocity at the interface 𝑩=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 𝑱(0)int(𝜒) undetermined at present.

The solutions for the đ‘„-component velocities 𝑱𝑖(0) (𝑖=1,2) thus obtained are𝑱1(0)𝑘(𝜒,𝑩)=1𝜇𝑟𝑑𝑝1(0)⎡⎱⎱⎣√𝑑𝜒sinh𝜀/𝑘1√(đ‘Šâˆ’đ›œ)−sinh𝜀/𝑘1(𝑩−ℎ(𝜒))√sinh𝜀/𝑘1[]⎀⎄⎄⎊ℎ(𝜒)âˆ’đ›œâˆ’1+𝑱(0)int√(𝜒)sinh𝜀/𝑘1(đ‘Šâˆ’đ›œ)√sinh𝜀/𝑘1[],𝑱ℎ(𝜒)âˆ’đ›œ(5.3)2(0)(𝜒,𝑩)=𝑘1𝑑𝑝2(0)⎡⎱⎱⎣√𝑑𝜒sinh𝜀/𝑘1√(𝑩+1)−sinh𝜀/𝑘1(𝑩−ℎ(𝜒))√sinh𝜀/𝑘1⎀⎄⎄⎊(ℎ(𝜒)+1)−1+𝑱(0)int√(𝜒)sinh𝜀/𝑘1(𝑩+1)√sinh𝜀/𝑘1,(ℎ(𝜒)+1)(5.4) 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𝜕𝑱𝑖(0)+𝜕𝜒𝜕𝜐𝑖(0)𝜕𝑩=0.(5.5) The above equation is integrated with respect to 𝑩 from 𝑩=đ›œ to 𝑩=ℎ(𝜒) for fluid 1 and 𝑩=−1 to 𝑩=ℎ(𝜒) for fluid 2, to yield the following expressions for the normal velocities at the interface 𝜐𝑖(0)[𝑩=ℎ(𝜒)] for 𝑖=1,2, where the integration constant is set to zero in order to satisfy the no-penetration condition at 𝑩=đ›œ and 𝑩=−1:𝜐1(0)[]𝑩=ℎ(𝜒)=âˆ’đ›œâ„Ž(𝜒)𝜕𝑱1(0)𝜐𝜕𝜒𝑑𝑩2(0)[]𝑩=ℎ(𝜒)=−ℎ(𝜒)−1𝜕𝑱2(0)𝜕𝜒𝑑𝑩.(5.6) Note that the normal velocities of the two fluids are equal at the interface (normal velocity continuity condition), and so, 𝜐1(0)[]𝑩=ℎ(𝜒)=𝜐2(0)[]𝑩=ℎ(𝜒)=𝜐(0)int.(5.7) Therefore, we equate the two integrals in the above equation and apply Leibnitz ruleđœ•î€œđœ•đœ’đ›œâ„Ž(𝜒)𝑱1(0)𝑑𝑩−𝜕ℎ𝑱𝜕𝜒1(0)ℎ𝜕(𝜒)=𝜕𝜒ℎ(𝜒)−1𝑱2(0)𝑑𝑩−𝜕ℎ𝑱𝜕𝜒2(0)ℎ(𝜒).(5.8) Noting that the đ‘„-component velocities are equal at the interface, 𝑱1(0)[ℎ(𝜒)]=𝑱2(0)[ℎ(𝜒)], the above equation is simplified tođœ•î‚žî€œđœ•đœ’đ›œâ„Ž(𝜒)𝑱1(0)𝑑𝑩−ℎ(𝜒)−1𝑱2(0)î‚č𝑑𝑩=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, 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 𝑱𝑖(0)(𝑩), (5.3) and (5.4), in the above equation, carry out the integrations with respect to 𝑩, 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 đ‘„-component of the fluid velocity at the interface 𝑱(0)int from the simplified tangential stress continuity condition, (4.11). Once 𝑱(0)int 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 get𝑑𝑝1(0)=𝑘𝑑𝜒1î€șℎ(1+ℎ(đ‘„))(𝜒)−𝑀(𝜒)−đșđč𝑘1𝜀−𝑘1î€șℎ(î€»îƒŹî‚™đœ’)−𝑀(𝜒)đș+4𝑘1𝜀tanh𝜀4𝑘1[]î‚č𝑘1+ℎ(𝜒)111+ℎ(𝜒)+𝜇𝑟[]î‚”î‚žî‚™đ›œâˆ’â„Ž(𝜒)−đștanh𝜀4𝑘1[]î‚č1+ℎ(𝜒)−tanh𝜀4𝑘1[]−ℎ(𝜒)âˆ’đ›œî‚č4𝑘1đœ€î‚”î‚žî‚™tanh𝜀4𝑘1[]î‚č−11+ℎ(𝜒)𝜇𝑟tanh𝜀4𝑘1[]ℎ(𝜒)âˆ’đ›œî‚č−1,(5.10)𝑑𝑝2(0)=𝑑𝜒𝑑𝑝1(0)(𝜒)−𝜕𝑑𝜒3ℎ(𝜒)𝜕𝜒3+𝑱(0)int(𝜒)𝑘1î€·đœ‡đ‘Ÿî€žđ‘ąâˆ’1+𝑀(𝜒),(5.11)(0)int(𝜒)=𝑘1𝜀đč−𝑘1𝑑𝑝1(0)î‚”î‚žî‚™đ‘‘đœ’tanh𝜀4𝑘1[]î‚č1+ℎ(𝜒)−tanh𝜀4𝑘1[]ℎ(𝜒)âˆ’đ›œî‚č+𝑘1tanh𝜀4𝑘1[]𝜕1+ℎ(𝜒)î‚čî‚”3ℎ(𝜒)𝜕𝜒3Ă—î‚»î‚žî‚™âˆ’đ‘€î‚¶î‚Œcoth𝜀𝑘1[]î‚č1+ℎ(𝜒)−𝜇𝑟coth𝜀𝑘1[]î‚č+î€·đœ‡â„Ž(𝜒)âˆ’đ›œđ‘Ÿî€žî‚žî‚™âˆ’1tanh𝜀4𝑘1[]1+ℎ(𝜒)î‚č−1,(5.12) where đș, 𝑀, and đč are functions of 𝜒, and they are defined asđș(𝜒)=coth𝜀𝑘1[]î‚č1+ℎ(𝜒)−𝜇𝑟coth𝜀𝑘1[]î‚č+î€·đœ‡â„Ž(𝜒)âˆ’đ›œđ‘Ÿî€žî‚žî‚™âˆ’1tanh𝜀4𝑘1[]1+ℎ(𝜒)î‚č−1Ă—îƒŻî€·đœ‡đ‘Ÿî€ž[]+−11+ℎ(𝜒)𝑘1đœ€î‚”î€·1−2𝜇𝑟tanh𝜀4𝑘1[]î‚č1+ℎ(𝜒)+tanh𝜀4𝑘1[]î€ș𝜖ℎ(𝜒)âˆ’đ›œî‚čî‚¶î‚Œđ‘€(𝜒)=1+đ›œđœ–2+î€·đœ–1−𝜖2ℎ(𝜒)−3î€œđœ–1𝜖2ℎî€ș(𝜒)𝑞(𝜒)(1+đ›œ)+𝜖2−𝜖1Ă—î€œđœ–î€»î€Ÿ2+𝜖1[]+î€ș𝜖+𝑞(𝜒)1+2ℎ(𝜒)âˆ’đ›œ1+đ›œđœ–2+î€·đœ–1−𝜖2ℎ(𝜒)−2Ă—î€·đ‘ž(𝜒)đ‘žî…žî€œđœ–(𝜒)1[]1+ℎ(𝜒)2+𝜖2[]đ›œâˆ’â„Ž(𝜒)2+𝜖1𝜖2𝑞[],đč(𝜒)1+2ℎ(𝜒)âˆ’đ›œ(𝜒)=−𝑞(𝜒)ℎ(𝜒)𝜖2î€ș(đ›œâˆ’â„Ž(𝜒))𝑞(𝜒)(đ›œ+1)−𝜖1+𝜖2î€ș𝜖1+đ›œđœ–2+î€·đœ–1−𝜖2ℎ(𝜒)2−𝑞(𝜒)𝑞(𝜒)(đ›œâˆ’â„Ž(𝜒))(1+ℎ(𝜒))î€ș𝜖1+đ›œđœ–2+î€·đœ–1−𝜖2.ℎ(𝜒)(5.13) 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 integral𝜀𝜕ℎ𝜕𝜏+𝑱1(0)[]ℎ(𝜒)𝜕ℎ𝜕𝜕𝜒=âˆ’î€œđœ•đœ’đ›œâ„Ž(𝜒)𝑱1(0)𝑑𝑩+𝑱1(0)[]ℎ(𝜒)𝜕ℎ,𝜕𝜒(5.14) which is simplified to give 𝜀𝜕ℎ+đœ•đœ•đœî€œđœ•đœ’đ›œâ„Ž(𝜒)𝑱1(0)𝑑𝑩=0,(5.15) where 𝑱1(0) is substituted from (5.3), after using the expressions for 𝑑𝑝1(0)/𝑑𝜒 and 𝑱(0)int 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 𝑱(0)int 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 forms𝜀𝜕ℎ+1đœ•đœđœ‡đ‘Ÿđ‘˜îƒŻîƒ©1𝑑𝑝1(0)+𝜇𝑑𝜒𝑟2𝑱(0)intîƒȘ(𝜒)𝜕ℎ𝜕𝜒sech212𝜀𝑘1î‚č(đ›œâˆ’â„Ž)+𝑘1𝑑(đ›œâˆ’â„Ž)2𝑝1(0)𝑑𝜒2−𝑑𝑝1(0)𝑑𝜒𝜕ℎ−𝜕𝜒𝑘1đœ€îƒ©2𝑘1𝑑2𝑝1(0)𝑑𝜒2+𝜇𝑟𝑱(0)intîƒȘ1(𝜒)×tanh2𝜀𝑘1î‚č𝜀(đ›œâˆ’â„Ž)=0,𝜕𝑞𝜕𝜏+𝑱(0)int(𝜒)𝜕𝑞𝜕𝜒−𝑞𝜕ℎ√𝜕𝜒𝜀𝑘1𝜇𝑟𝑑𝑝1(0)1𝑑𝜒tanh2𝜀𝑘1î‚č+(â„Žâˆ’đ›œ)𝜀𝑘1𝑱(0)int(𝜒)coth𝜀𝑘1î‚č(â„Žâˆ’đ›œ)+𝑞𝑱(0)int+(𝜒)𝜀𝑆1(0)î€ș(1+ℎ)𝑞+𝜖2î€»đœ–1+đ›œđœ–2+î€·đœ–1−𝜖2ℎ+𝜀𝑆2(0)î€ș(đ›œâˆ’â„Ž)𝑞−𝜖1î€»đœ–1+đ›œđœ–2+î€·đœ–1−𝜖2ℎ=0.(5.16) 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 interface ℎ(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 𝑱𝑖(0)=0 in the base state𝑆1(0)𝜕𝜓1î‚č𝜕𝑩𝑩=0=𝑆2(0)𝜕𝜓2î‚č𝜕𝑩𝑩=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=𝜖1𝑆2(0)−𝜖2𝑆1(0)𝑆1(0)+đ›œđ‘†2(0).(6.2) The variables ℎ and 𝑞 are now perturbed about their base state valuesℎ(𝜒,𝜏)=ℎ1[],𝑞exp𝑖𝑘𝜒+𝜔𝜏(𝜒,𝜏)=𝑞0+𝑞1[],exp𝑖𝑘𝜒+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=đ›Ÿđ»3/𝜖0𝜓2𝑏 and 𝜔 is the nondimensional growth rate based on the time scale 𝜇2đ»3đ›Ÿ/(𝜖0𝜓2𝑏)2. We substitute the expressions for 𝑑𝑝1(0)/𝑑𝜒 and 𝑱(0)int 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 formî‚‹đ»1ℎ1+𝑄1𝑞1î‚‹đ»=0,2ℎ1+𝑄2𝑞1=0,(6.4) whereî‚‹đ»1î€·đ‘…=−14+𝑘4𝑅13𝑅1î€žđ›œ4𝜖42+𝜖41î€·đ‘…14+𝑘4𝑅13𝑅1−𝑘2𝑅13𝑅1𝜖2−𝜖21𝜖22î€ș6î€·đ‘…14+𝑘4𝑅13𝑅1î€žđ›œ2+𝑘2𝑅13𝑅1𝜖2+𝜖31𝜖2î€ș𝑅15î€·đ‘…âˆ’4đ›œ14+𝑘4𝑅13𝑅1+𝑘2𝑅13𝑅1đ›œđœ–2î€»âˆ’đœ–1𝜖32î€ș𝑅15+4đ›œ3î€·đ‘…14+𝑘4𝑅13R1+𝑘2𝑅13𝑅1đ›œđœ–2+𝑞20(1+đ›œ)𝜖2−𝑅6𝑅10𝑅12+𝑘2𝑅13î€·đœ–1+đ›œđœ–2î€žĂ—đ‘…îƒŹîƒ©1đ¶(đ›œâˆ’1)−1đ”1đ›œî‚™đ‘˜1𝜀îƒȘ𝜖1âˆ’đ¶1đ”1đ›œ2𝑘1𝜀𝜖2−𝑞0𝜖2𝑅8𝑅10𝑅12+𝑘2𝑅13î€·đœ–1+đ›œđœ–2î€žĂ—îƒŹđ›œîƒ©2𝑅1âˆ’đ¶1đ”1𝑘1𝜀îƒȘ𝜖21+2𝑅1âˆ’đ¶1đ”1î‚™đ›œ(đ›œâˆ’1)𝑘1𝜀îƒȘ𝜖1𝜖2+đ¶1đ”1đ›œ2𝑘1𝜀𝜖22+đ”1𝑅5𝜔𝜀3/2𝜇𝑟,𝑄1=(đ›œâˆ’1)𝜖1𝜖2î€·đœ–1+đ›œđœ–2𝑅î€ș15+𝑘2𝑅1𝑅13î€·đœ–1+đ›œđœ–2î€žî€»âˆ’đ‘ž0𝑅7𝑅10𝑅12+𝑘2𝑅13î€·đœ–1+đ›œđœ–22Ă—đ‘…îƒŹîƒ©1đ¶+đ›œ1đ”1𝑘1𝜀îƒȘ𝜖1+đ›œ2îƒ©đ‘…1+đ¶1đ”1𝑘1𝜀îƒȘ𝜖2,î‚‹đ»îƒ­îƒ°2=î€·đœ–1+đ›œđœ–2î€žîƒ©đœ€îƒ©đœ€đ‘…2𝑅18âˆ’đ”1𝑅4𝑆2(0)𝜖41𝜇r+𝜖31î€·đ‘…16+𝑅3𝑅4+đ”1𝑅2𝑅4𝑅34+𝜖21𝜖2î€ș𝑅17+3đ›œđ‘…3𝑅4+3đ”1𝑅2𝑅4𝑅35+đ›œđœ–2îƒ©đ‘…30𝑅22k1𝜀q0𝜇r+đ›œđœ–22î€·đ‘…19+đ›œđ‘…3𝑅4+đœ€đ›œR2Ă—î‚€đ”1𝑅4𝑞0𝑆1(0)+đ›œđ‘†2(0)+𝑅18+đ”1𝑅4𝑆1(0)𝜖2𝜇𝑟îƒȘ+𝜖1îƒ©đ‘…30𝑅22𝑘1𝜀𝑞0𝜇𝑟+đ›œđœ–22𝑅20+3đ›œđ‘…3𝑅4+đœ€đ›œđ‘…23đ”1𝑅4𝑞0𝑆1(0)+đ›œđ‘†2(0)+4đ›œđ‘…18+3đ”1𝑅4𝑆1(0)âˆ’đ›œđ”1𝑅4𝑆2(0)𝜖2𝜇𝑟îƒȘîƒȘ−𝑅2𝑞0đœ‡đ‘Ÿî€·đ‘…9î€ș𝑅21+𝑅23+𝑅25+𝑅28−𝑅29+𝑅32𝑅36îƒȘ,𝑄2=𝑅2î€·đœ–1+đ›œđœ–2î€žđœ‡đ‘Ÿî‚†đ”1𝑅2𝜀2î€·đœ–1+đ›œđœ–23𝑆1(0)+đ›œđ‘†2(0)î€·đœ–+𝜔1+đ›œđœ–2−𝑞0𝑅9î€·đ‘…22+𝑅24+𝑅26+𝑅27−𝑅31𝑅2√𝜀𝑘1î€·đœ–1+đ›œđœ–2+𝑅33𝑅36,(6.5) in which đ”1, đ”2, đ¶1, đ¶2, 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 đ‘€â‹…đ¶đ‘‡=0, where the vector đ¶=[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, đ›œ, 𝜇𝑟, 𝑆1(0), 𝑆2(0), 𝜀, 𝜖1, and 𝜖2, is a quadratic equation for 𝜔. The roots of the characteristic equation for 𝜔 can be written as 𝜔1𝑅=−48𝑅49−𝑅248î‚‹đ»+22𝑄1𝑅49−2𝑅41𝑅47𝑅49𝑅49,𝜔(6.6)2𝑅=−48𝑅49+𝑅248î‚‹đ»+22𝑄1𝑅49−2𝑅41𝑅47𝑅49𝑅49.(6.7) 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 rate 𝜔 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 𝜀, medium permeability 𝑘1, nondimensional conductivities 𝑆1(0), 𝑆2(0), dielectric constants 𝜖1, 𝜖2, the ratio of the thicknesses of top and bottom fluids đ›œ, and the ratio of viscosities of the two fluids 𝜇𝑟=𝜇1/𝜇2, respectively.

6.1. Leaky Dielectric-Leaky Dielectric Interface

This is the general case in which 𝑆1(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 𝜀. It is clear from Figure 2(b) that for small values of the porosity (e.g., 𝜀=0.1), the growth rate 𝜔 increases by increasing the wavenumber 𝑘 till a maximum value of 𝜀 after which 𝜔 decreases by increasing 𝑘; that is, there exists a maximum mode of instability only for small values of the porosity 𝜀, while for any other value of 𝜀, 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 0â‰€đ‘˜â‰€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 đ‘˜â‰€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<đ‘˜â‰€20 and it has no effect on the stability of the considered system for higher wavenumber values 𝑘>20.

Figures 4(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 đ‘˜â‰„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.

Figure 6 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 đ›œ 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 đ›œ 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 đœ‡đ‘Ÿâ‰€1.

6.2. Perfect Dielectric-Leaky Dielectric Interface

This is the limiting case in which 𝑆1(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 𝜀. In comparison with Figures 2(a) and 2(b), we conclude that the porosity of porous medium 𝜀 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 đœ€â‰€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 đ›œ 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 đ›œ, 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 𝜇𝑟âȘ‹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 which 𝑆1(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 𝜀. In view of the above discussion, we conclude from this figure that the porosity 𝜀 has a slightly stabilizing effect for small wavenumber values đ‘˜â‰€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<đ‘˜â‰€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 đ›œ, 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 đ›œ 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 medium 𝜀, the dielectric constant of upper fluid 𝜖1, and the 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 the ratio of thickness of upper and lower fluids đ›œ 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 fluid S2(0) has a destabilizing effect for small wavenumbers and a stabilizing effect for high wavenumbers,(ii)for small wavenumber values, the physical parameters 𝜀, 𝜖1, and 𝜇𝑟 have stabilizing effects, while the parameters, 𝑘1, 𝜖2, and đ›œ 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 parameters 𝜀, 𝜖1, and 𝜇r have destabilizing effects for high wavenumbers, while the parameters 𝜖2, and đ›œ 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 fluid S1(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.

Appendix

đ”1=coth𝜀𝑘1+đœ‡đ‘Ÿî‚”đ›œî‚™coth𝜀𝑘1+î€·đœ‡đ‘Ÿî€žî‚”1−1tanh2𝜀𝑘1,đ”2=𝜀𝑘1−csch2î‚”î‚™đœ€đ‘˜1+𝜇𝑟csch2î‚”đ›œî‚™đœ€đ‘˜1+12î€·đœ‡đ‘Ÿî€žâˆ’1sech2î‚”12𝜀𝑘1,đ¶î‚¶î‚Œ1=𝜇𝑟−1−𝑘1đœ€î‚žî€·2đœ‡đ‘Ÿî€žî‚”1−1tanh2𝜀𝑘11+tanh2đ›œî‚™đœ€đ‘˜1,đ¶î‚¶î‚č2=𝜇𝑟1−1+21+1−2𝜇𝑟sech2î‚”12𝜀𝑘1−tanh2î‚”12đ›œî‚™đœ€đ‘˜1,đ‘…î‚¶î‚Œ1=𝑘1đ¶îƒŻîƒ©1đ”1+2𝑘1𝜀îƒȘî‚”1tanh2𝜀𝑘1,𝑅−12=1đœ‡đ‘ŸîƒŻî‚™đ›œâˆ’2𝑘1đœ€î‚”đ›œtanh2𝜀𝑘1î‚¶âˆ’đœ‡đ‘Ÿî‚žđ‘…1𝑘1+đ¶1đ”1î‚”đ›œtanh2𝜀𝑘1,đ‘…î‚¶î‚č3=đ”1𝑅2𝜀𝜇𝑟𝑞0𝑆1(0)+đ›œđ‘†2(0)−𝜖1𝑆2(0)+𝜖2𝑆1(0),𝑅4=âˆ’âˆ’đ”2đ¶1+đ”1đ¶2đ”21îƒȘî‚žî‚”1tanh2𝜀𝑘11+tanh2đ›œî‚™đœ€đ‘˜1,đ‘…î‚¶î‚č5=𝑅22î€·đœ–1+đ›œđœ–24,𝑅6=𝜀𝜖1î‚”1(1âˆ’đ›œ)sinh2𝜀𝑘1î‚¶î€·đœ–+đ›œ1+đ›œđœ–2𝜀𝑘1î‚”1cosh2𝜀𝑘1,𝑅7=î€·đœ–1+đ›œđœ–2î€žî‚»đœ€î€·đœ–1+đ›œ2𝜖21sinh2𝜀𝑘1î‚¶î€·đœ–+đ›œ1+đ›œđœ–2𝜀𝑘1î‚”1cosh2𝜀𝑘1,đ‘…î‚¶î‚Œ8=2𝜀𝜖1î€·đœ–2+đ›œđœ–11sinh2𝜀𝑘1+đ›œđ›œđœ–2+𝜖1đœ–î€žî€·2−𝜖1𝜀𝑘1î‚”1cosh2𝜀𝑘1,𝑅9î‚”1=tanh2𝜀𝑘1î‚¶î‚”đ›œ+tanh2𝜀𝑘1,𝑅10=𝑘2𝑘1𝜀𝑅2î€·đœ–1+đ›œđœ–21sech2𝜀𝑘1,𝑅11=đ”1î‚žđ›œâˆšâˆšđœ€âˆ’2𝑘1î‚”đ›œtanh2𝜀𝑘1,đ‘…î‚¶î‚č12=𝑅2𝜇𝑟√𝑘1î‚”đ›œtanh2𝜀𝑘1,𝑅13=𝑅9𝑅12+𝑅11𝑅2,𝑅14=𝑅2𝑅12𝑘51î‚”1tanh2𝜀𝑘1,𝑅15=𝜀𝑅10𝑅12î‚”1sinh2𝜀𝑘1,𝑅16=𝑅24𝑅3+𝑞0𝜇𝑟𝑅2î‚žđœ€đ”2𝑆1(0)−𝑆2(0)+4đ›œđ‘˜1𝑘4𝜖2î‚”1tanh2𝜀𝑘1,đ‘…î‚¶î‚čî‚Œ17=𝑅24𝑅3(2đ›œâˆ’1)+3đ›œq0𝜇𝑟𝑅2î‚žđœ€đ”1𝑆1(0)−𝑆2(0)+2đ›œđ‘˜1𝑘4𝜖2î‚”1tanh2𝜀𝑘1,đ‘…î‚¶î‚čî‚Œ18=đ›œđ‘ž0𝑘1𝑘