International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Review Article | Open Access

Volume 2011 |Article ID 498718 | https://doi.org/10.5402/2011/498718

M. F. El-Sayed, M. H. M. Moussa, A. A. A. Hassan, N. M. Hafez, "Electrohydrodynamic Instability of Two Thin Viscous Leaky Dielectric Fluid Films in a Porous Medium", International Scholarly Research Notices, vol. 2011, Article ID 498718, 35 pages, 2011. https://doi.org/10.5402/2011/498718

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

Academic Editor: R. Cardoso
Received23 Mar 2011
Accepted04 May 2011
Published24 Aug 2011

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 βˆ’(πœ‡/π‘˜1)𝐯, where πœ‡ is the fluid viscosity, π‘˜1 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 βˆ’π»<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=β„Ž(π‘₯). The two fluids are stationary in the initial state, with viscosities πœ‡π‘–, dielectric constants πœ–π‘–, conductivities πœŽπ‘–, porosity of porous medium πœ€ and medium permeability π‘˜1, where 𝑖=1,2. Fluid 2 is bounded at the bottom 𝑦=βˆ’π» by a rigid plate which is maintained at an electric potential πœ“=πœ“π‘, while fluid 1 is bounded at the top 𝑦=𝛽𝐻 by a rigid plate maintained at an electric potential πœ“=0. In the ensuing analysis, we assume that the material properties of the fluid such as viscosities πœ‡π‘–, dielectric constants πœ–π‘–, conductivities πœŽπ‘–, porosity of porous medium πœ€, and medium permeability π‘˜1 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]βˆ‡β‹…π„π‘–=0,𝑖=1,2.(2.1) Since the electric fields are irrotational, 𝐄𝑖=βˆ’βˆ‡πœ“π‘–, where πœ“π‘– is the electric potential: in the fluid 𝑖. Substituting this into (2.1) gives the following Laplace's equation for the electric potential: πœ“π‘–βˆ‡2πœ“π‘–=0.(2.2) These governing equations are supplemented by the following boundary conditions. (1)The normal component of the electric field, at the interface 𝑦=β„Ž(π‘₯), satisfies πœ–2πœ–0ξ€·βˆ‡πœ“2ξ€Έβ‹…π§βˆ’πœ–1πœ–0ξ€·βˆ‡πœ“1⋅𝐧=π‘ž(π‘₯,𝑑)at𝑦=β„Ž(π‘₯),(2.3) where 𝐧 is the unit normal to the interface at 𝑦=β„Ž(π‘₯) (see Figure 1), πœ–π‘– (𝑖=1,2) is the dielectric constant in fluid 𝑖, πœ–0 is the permittivity of free space, and π‘ž(π‘₯,𝑑) 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 𝑦=β„Ž(π‘₯) translates to the continuity of the electric potentials; that is,πœ“1=πœ“2at𝑦=β„Ž(π‘₯).(2.4)(3)The electric potentials satisfy the following conditions at the rigid boundaries: πœ“2=πœ“π‘πœ“at𝑦=βˆ’π»,1=0at𝑦=𝛽𝐻.(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 balanceβˆ‡β‹…π•π‘–=0,βˆ‡β‹…π“π‘–=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 πœ‘π‘– such that 𝐕𝑖=βˆ’βˆ‡πœ‘π‘–. 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=βˆ’π‘π‘–πˆ+πœ‡π‘–ξ€·βˆ‡π•π‘–+βˆ‡π•π‘‡π‘–ξ€Έ+𝐦𝑖+πœ‡π‘–π‘˜1πœ‘π‘–πˆ,𝑖=1,2,(2.7) where 𝑝𝑖 is the pressure in fluid 𝑖, 𝐈 is the identity tensor, and the Maxwell stress tensor 𝐦𝑖 is given by Saville [3]𝐦𝑖=πœ–π‘–πœ–0ξ‚ƒπ„π‘–π„π‘–βˆ’12ξ€·π„π‘–β‹…π„π‘–ξ€Έπˆξ‚„.(2.8) The divergence of the Maxwell stress tensor βˆ‡β‹…π¦π‘–=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,βˆ‡β‹…π¦π‘–ξ‚†πœ–=βˆ‡β‹…π‘–πœ–0ξ‚ƒπ„π‘–π„π‘–βˆ’12ξ€·π„π‘–β‹…π„π‘–ξ€Έπˆπœ–ξ‚„ξ‚‡=βˆ’0𝐄𝑖2β‹…π„π‘–βˆ‡πœ–π‘–+πœŒπ‘“π„π‘–=0,(2.9) with πœŒπ‘“ 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, becomeπœŒπœ€π·π•π‘–π·π‘‘=βˆ’βˆ‡π‘π‘–+βˆ‡β‹…π¦π‘–+πœ‡π‘–πœ€βˆ‡2π•π‘–βˆ’πœ‡π‘–π‘˜1𝐕𝑖,𝑖=1,2.(2.10) Since there is no time scale, then 𝐷𝐯𝑖/𝐷𝑑=0, and we get from the above two equations thatβˆ‡π‘π‘–βˆ’πœ‡π‘–πœ€βˆ‡2𝐕𝑖+πœ‡π‘–π‘˜1𝐕𝑖=0,𝑖=1,2.(2.11) The fluid velocities satisfy no-slip and no-penetration conditions at the top and bottom plates𝐕1(𝑦=𝛽𝐻)=0,𝐕2(𝑦=βˆ’π»)=0.(2.12) At the interface 𝑦=β„Ž(π‘₯) 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+π›Ύπœ…,(2.15)(𝐭⋅𝐓⋅𝐧)2=(𝐭⋅𝐓⋅𝐧)1,(2.16) where 𝛾 is the interfacial tension between the two fluids, 𝐭 is the unit tangent to the interface (see Figure 1), and πœ…=πœ•2β„Ž(π‘₯)/πœ•π‘₯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 π‘₯ and 𝑦 directions by 𝑒 and 𝜐, respectively; that is, 𝐕𝑖=(𝑒𝑖,πœπ‘–). Upon substituting the expression for the Maxwell stress tensor (2.8) in (2.15) and (2.16), respectively, we get(𝐧⋅𝐓⋅𝐧)𝑖=βˆ’π‘π‘–+2πœ‡π‘–πœ•πœπ‘–πœ•π‘¦+πœ–π‘–πœ–0𝐸𝑁2π‘–βˆ’12𝐸2𝑖+πœ‡π‘–π‘˜1πœ‘π‘–,(𝐭⋅𝐓⋅𝐧)𝑖=πœ‡π‘–ξ‚΅πœ•π‘’π‘–+πœ•π‘¦πœ•πœπ‘–ξ‚Άπœ•π‘₯+πœ–π‘–πœ–0𝐸𝑇𝑖𝐸𝑁𝑖,(2.17) where 𝐸𝑇𝑖 and 𝐸𝑁𝑖 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=β„Ž(π‘₯):𝑝1βˆ’2πœ‡1πœ•πœ1ξ‚Ήβˆ’ξ‚Έπ‘πœ•π‘¦2βˆ’2πœ‡2πœ•πœ2ξ‚Ή+1πœ•π‘¦π‘˜1ξ€Ίπœ‡2πœ‘2βˆ’πœ‡1πœ‘1ξ€»πœ–=π›Ύπœ…+1πœ–02ξ‚ƒξ€·βˆ‡πœ“1⋅𝐧2βˆ’ξ€·βˆ‡πœ“1⋅𝐭2ξ‚„βˆ’πœ–2πœ–02ξ‚ƒξ€·βˆ‡πœ“2⋅𝐧2βˆ’ξ€·βˆ‡πœ“2⋅𝐭2ξ‚„,(2.18) for the normal stress continuity, andπœ‡2ξ‚Έπœ•π‘’2+πœ•π‘¦πœ•πœ2ξ‚Ήπœ•π‘₯βˆ’πœ‡1ξ‚Έπœ•π‘’1+πœ•π‘¦πœ•πœ1ξ‚Ήξ€·πœ•π‘₯=βˆ’βˆ‡πœ“1ξ€Έβ‹…π­π‘ž(π‘₯,𝑑),(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 β„Ž(π‘₯,𝑑),𝜐1(𝑦=β„Ž(π‘₯))=𝜐2(𝑦=β„Ž(π‘₯))=πœ€πœ•β„Žπœ•π‘‘+π•β‹…βˆ‡π‘ β„Ž,(2.20) where βˆ‡π‘  is the gradient operator along the interface 𝑦=β„Ž(π‘₯). Finally, the interfacial charge is governed by a conservation equationπœ€πœ•π‘žπœ•π‘‘+π•β‹…βˆ‡π‘ πœŽπ‘žβˆ’π‘žπ§β‹…(π§β‹…βˆ‡)𝐕=πœ€ξ€·ξ€Ί2𝐄2ξ€»βˆ’ξ€ΊπœŽβ‹…π§1𝐄1⋅𝐧,(2.21) where the terms on the left-hand side represent, respectively, the accumulation, convection, and variation of the charge due to dilation of the interface, while the right side represents 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 settingπœ“ξ…žπ‘–=πœ“π‘–πœ“π‘,π‘ξ…žπ‘–=𝑝𝑖𝐻2πœ–0πœ“2𝑏,π‘‡ξ…žπ‘–=𝑇𝑖𝐻2πœ–0πœ“2𝑏,π‘₯ξ…ž=π‘₯𝐻,π‘¦ξ…ž=𝑦𝐻,π„ξ…žπ‘–=π„π‘–π»πœ“π‘,π•ξ…žπ‘–=π•π‘–πœ‡2π»πœ–0πœ“2𝑏,π‘‘ξ…ž=π‘‘πœ–0πœ“2π‘πœ‡2𝐻2,πœ‘ξ…žπ‘–=πœ‡2πœ‘π‘–πœ–0πœ“2𝑏,π‘žξ…ž=π‘žπ»πœ–0πœ“π‘,πœ…ξ…ž=πœ…π»,π‘˜ξ…ž1=π‘˜1𝐻2,β„Žξ…ž=β„Žπ».(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 πœ“π‘– areβˆ‡2πœ“π‘–=0,𝑖=1,2,(3.2) with the following boundary conditions at the interface 𝑦=β„Ž(π‘₯),ξ€Ίπœ–2βˆ‡πœ“2ξ€»βˆ’ξ€Ίπœ–β‹…π§1βˆ‡πœ“1⋅𝐧=π‘ž,πœ“1=πœ“2,(3.3) and the following boundary conditions at the top and bottom boundariesπœ“1(𝑦=𝛽)=0,πœ“2(𝑦=βˆ’1)=1.(3.4) Similarly, the nondimensional equations governing the fluid motion areβˆ‡β‹…π•1=0,βˆ‡β‹…π•2=0,(3.5)βˆ‡π‘1βˆ’πœ‡π‘Ÿπœ€βˆ‡2𝐕1+πœ‡π‘Ÿπ‘˜1𝐕1=0,βˆ‡π‘2βˆ’1πœ€βˆ‡2𝐕2+1π‘˜1𝐕2=0.(3.6) with πœ‡π‘Ÿ=πœ‡1/πœ‡2 being the ratio of viscosities of the two fluids. The nondimensional normal and tangential stress continuity conditions at the interface become𝑝1βˆ’2πœ‡π‘Ÿπœ•πœ1ξ‚Ήβˆ’ξ‚Έπ‘πœ•π‘¦2βˆ’2πœ•πœ2ξ‚Ή+1πœ•π‘¦π‘˜1ξ€Ίπœ‘2βˆ’πœ‡π‘Ÿπœ‘1ξ€»=πœ–π›Ύπœ…+12ξ‚ƒξ€·βˆ‡πœ“1⋅𝐧2βˆ’ξ€·βˆ‡πœ“1⋅𝐭2ξ‚„βˆ’πœ–22ξ‚ƒξ€·βˆ‡πœ“2⋅𝐧2βˆ’ξ€·βˆ‡πœ“2⋅𝐭2ξ‚„,ξ‚Έ(3.7)πœ•π‘’2+πœ•π‘¦πœ•πœ2ξ‚Ήπœ•π‘₯βˆ’πœ‡π‘Ÿξ‚Έπœ•π‘’1+πœ•π‘¦πœ•πœ1ξ‚Ήξ€·πœ•π‘₯=βˆ’βˆ‡πœ“1ξ€Έβ‹…π­π‘ž(π‘₯,𝑑),(3.8) where 𝛾=𝛾𝐻/πœ–0πœ“2𝑏 is the nondimensional interfacial tension. The boundary conditions for the velocities at the top and bottom plates become𝑒1𝜐(𝑦=𝛽)=0,1𝑒(𝑦=𝛽)=0,2(𝑦=βˆ’1)=0,𝜐2(𝑦=βˆ’1)=0.(3.9) The nondimensional kinematic condition at the interface is𝜐1(𝑦=β„Ž(π‘₯))=𝜐2(𝑦=β„Ž(π‘₯))=πœ€πœ•β„Žπœ•π‘‘+π•β‹…βˆ‡π‘ β„Ž,(3.10) and the nondimensional charge conservation equation at the interface isπœ€πœ•π‘žπœ•π‘‘+π•β‹…βˆ‡π‘ π‘†π‘žβˆ’π‘žπ§β‹…(π§β‹…βˆ‡)𝐕=πœ€ξ€·ξ€Ί1βˆ‡πœ“1ξ€»βˆ’ξ€Ίπ‘†β‹…π§2βˆ‡πœ“2⋅𝐧,(3.11) where 𝑆𝑖=πœŽπ‘–πœ‡2𝐻2/πœ–20πœ“2𝑏, 𝑖=1,2 are the nondimensional conductivities in the two fluids.

This completes the specification of the governing equations and boundary conditions, which are highly coupled. Due to the negligible effect of gravity at length scales of 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 β„Ž(π‘₯,𝑑) and charge π‘ž(π‘₯,𝑑). 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 𝐻 in the system, and it is useful to define a small parameter 𝛿=𝐻/𝐿β‰ͺ1. The lateral length scale 𝐿 is determined self-consistently in the following analysis to be 𝛾𝐻3/πœ–0πœ“2𝑏, and this is further estimated below to be much larger than 𝐻. Similarly, a slow time scale is necessary to describe the dynamics of the interface motion at such large length scales, and this is introduced a little later in (4.15). In the limit 𝐻β‰ͺ𝐿, the derivatives in the π‘₯ direction should be scaled with 𝐿. To this end, we define the slowly varying scale πœ’ in the following manner:πœ•πœ•πœ•π‘₯=π›Ώπœ•πœ’,(4.1) and πœ•/πœ•πœ’βˆΌπ‘‚(1). When we apply the above scalings, (4.1), to the Laplace equation, (3.2), for the electric potential πœ“π‘–, it simplifies in the limit 𝛿β‰ͺ1 toπœ•2πœ“π‘–πœ•π‘¦2=0,𝑖=1,2.(4.2)

The continuity condition for the normal component of the electric field at the interface 𝑦=β„Ž(πœ’), (3.3), is similarly simplified in the long-wave limit asπœ–2πœ•πœ“2πœ•π‘¦βˆ’πœ–1πœ•πœ“1πœ•π‘¦=π‘ž,(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 πœ‡π‘Ÿ,𝑖 such that πœ‡π‘Ÿ,𝑖=πœ‡π‘Ÿ for 𝑖=1 and πœ‡π‘Ÿ,𝑖=1 for 𝑖=2. The π‘₯-momentum equation can be simplified in the long-wave limit asπ›Ώπœ•π‘π‘–βˆ’πœ‡πœ•πœ’π‘Ÿ,π‘–πœ€πœ•2π‘’π‘–πœ•π‘¦2+πœ‡π‘Ÿ,π‘–π‘˜1𝑒𝑖=0.(4.4) This suggests that π‘’π‘–βˆΌπ‘‚(𝛿)𝑝𝑖. In order to make this explicit and to make ordering of various quantities simpler, we represent the pressure 𝑝𝑖 and the π‘₯-component velocity 𝑒𝑖 in the following manner:𝑝𝑖=𝑝𝑖(0),𝑒𝑖=𝛿𝑒𝑖(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 πœ•/πœ•π‘₯βˆΌπ›Ώπœ•/πœ•πœ’ and using the above expansion for 𝑒𝑖,𝛿2πœ•π‘’π‘–(0)+πœ•πœ’πœ•πœπ‘–πœ•π‘¦=0,(4.6) which suggests the following expansion for πœπ‘–:πœπ‘–=𝛿2πœπ‘–(0).(4.7) Upon using this expansion, the nondimensional 𝑦 component of the momentum equation yields to leading order in 𝛿, πœ•π‘π‘–(0)/πœ•π‘¦=0, implying that the pressure is constant in both films across the 𝑦 direction, and so 𝑝𝑖=𝑝𝑖(πœ’,𝑑). The simplified π‘₯-momentum equation, therefore, is given by𝑑𝑝𝑖(0)βˆ’πœ‡π‘‘πœ’π‘Ÿ,π‘–πœ€πœ•2𝑒𝑖(0)πœ•π‘¦2+πœ‡π‘Ÿ,π‘–π‘˜1𝑒𝑖(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)βˆ’π‘2(0)+1π‘˜1ξ€·πœ‘2βˆ’πœ‡π‘Ÿπœ‘1ξ€Έ=𝛾𝛿2πœ•2β„Žπœ•πœ’2+πœ–12ξ‚΅πœ•πœ“1ξ‚Άπœ•π‘¦2βˆ’πœ–22ξ‚΅πœ•πœ“2ξ‚Άπœ•π‘¦2.(4.9) In order for the interfacial tension to be of the same order as the other terms in the above equation, we require 𝛾𝛿2βˆΌπ‘‚(1), where 𝛿=𝐻/𝐿. We set 𝛾(𝐻/𝐿)2=1, and from this relation, we determine the lateral length scale 𝐿 to be √𝐿=𝛾𝐻2=𝛾𝐻3/πœ–0πœ“2𝑏. Upon using the relation 𝛾𝛿2=1, (4.9) becomes𝑝1(0)βˆ’π‘2(0)+1π‘˜1ξ€·πœ‘2βˆ’πœ‡π‘Ÿπœ‘1ξ€Έ=πœ•2β„Žπœ•πœ’2+πœ–12ξ‚΅πœ•πœ“1ξ‚Άπœ•π‘¦2βˆ’πœ–22ξ‚΅πœ•πœ“2ξ‚Άπœ•π‘¦2.(4.10) The tangential stress continuity, (3.8), simplifies to the following condition in the long-wave limit:πœ•π‘’2(0)πœ•π‘¦βˆ’πœ‡π‘Ÿπœ•π‘’1(0)πœ•π‘¦=βˆ’π‘žπœ•πœ“1πœ•πœ’.(4.11) The nondimensional kinematic condition at the interface, (3.10), after using the asymptotic expansion for πœπ‘– (4.7), yields𝛿2𝜐1(0)(𝑦=β„Ž(πœ’))=πœ€πœ•β„Žπœ•π‘‘+𝛿2𝑒1(0)πœ•β„Žπœ•πœ’.(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 thatπœ•πœ•π‘‘=𝛿2πœ•πœ•πœ,(4.13) where πœ•/πœ•πœ is 𝑂(1). The kinematic condition thus gives𝜐1(0)(𝑦=β„Ž(πœ’))=πœ€πœ•β„Žπœ•πœ+𝑒1(0)πœ•β„Žπœ•πœ’.(4.14) The dimensional slow time scale is obtained as follows:πœ•=πœ‡πœ•π‘‘2𝐻2πœ–0πœ“2π‘πœ•πœ•π‘‘dim=𝛿2πœ•,πœ•πœ(4.15) where 𝑑dim is the dimensional time. Thus, in order to nondimensionalize the dimensional time 𝑑dim in long-wave limit, the appropriate time scale is πœ‡2𝐻2/(πœ–0πœ“2𝑏𝛿2). After using 𝛾𝛿2=1 to eliminate 𝛿2, the time scale becomes πœ‡2𝐻3𝛾/(πœ–0πœ“2𝑏)2.

Finally, the nondimensional interfacial charge balance, (3.11) is simplified in the long-wave limit asπœ€π›Ώ2πœ•π‘žπœ•πœ+𝛿2𝑒1(0)πœ•π‘žπœ•πœ’βˆ’π›Ώ2π‘žπœ•πœ1(0)ξ‚Έπ‘†πœ•π‘¦=πœ€1πœ•πœ“1πœ•π‘¦βˆ’π‘†2πœ•πœ“2ξ‚Ή.πœ•π‘¦(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)=πœŽπ‘–πœ‡2𝛾𝐻3/(πœ–30πœ“4𝑏). Upon using these rescaled conductivities, the charge conservation equation becomes, after using the continuity equation, (4.6),πœ€πœ•π‘ž+πœ•ξ‚€π‘’πœ•πœ1(0)π‘žξ‚ξ‚Έπ‘†πœ•πœ’=πœ€1(0)πœ•πœ“1πœ•π‘¦βˆ’π‘†2(0)πœ•πœ“2ξ‚Ή.πœ•π‘¦(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 β„Ž(πœ’,𝜏) and surface charge density π‘ž(πœ’,𝜏). The simplified Laplacian for the potential πœ“π‘–, (4.2), is easily solved along with the boundary conditions, (4.3), to give the following expressions for the potentials πœ“π‘– (𝑖=1,2):πœ“1ξ€Ίπœ–(πœ’,𝑦)=(π›½βˆ’π‘¦)2+ξ€»(1+β„Ž(πœ’))π‘ž(πœ’)πœ–1+π›½πœ–2+ξ€·πœ–1βˆ’πœ–2ξ€Έ,πœ“β„Ž(πœ’)2(πœ’,𝑦)=π›½πœ–2βˆ’πœ–1ξ€Ίπœ–π‘¦+𝛽(1+𝑦)π‘ž(πœ’)+β„Ž(πœ’)1βˆ’πœ–2βˆ’ξ€»(1+𝑦)π‘ž(πœ’)πœ–1+π›½πœ–2+ξ€·πœ–1βˆ’πœ–2ξ€Έ,β„Ž(πœ’)(5.1) where the interfacial charge density π‘ž(πœ’,𝜏) is determined below by the interface charge conservation equation (4.17). The simplified π‘₯-momentum equation (4.8) can be integrated with respect to 𝑦, since 𝑑𝑝𝑖(0)/π‘‘πœ’ is independent of 𝑦 and the two constants of integration that arise are determined by the boundary conditions 𝑒1(0)(𝛽)=0,𝑒1(0)[]𝑦=β„Ž(πœ’)=𝑒(0)int𝑒(πœ’),2(0)(βˆ’1)=0,𝑒2(0)[]𝑦=β„Ž(πœ’)=𝑒(0)int(πœ’).(5.2) Here, 𝑒(0)int(πœ’) is the π‘₯ component of the velocity at the interface 𝑦=β„Ž(πœ’), 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 𝑒(0)int(πœ’) undetermined at present.

The solutions for the π‘₯-component velocities 𝑒𝑖(0) (𝑖=1,2) thus obtained are𝑒1(0)π‘˜(πœ’,𝑦)=1πœ‡π‘Ÿπ‘‘π‘1(0)βŽ‘βŽ’βŽ’βŽ£ξ‚ƒβˆšπ‘‘πœ’sinhπœ€/π‘˜1ξ‚„ξ‚ƒβˆš(π‘¦βˆ’π›½)βˆ’sinhπœ€/π‘˜1ξ‚„(π‘¦βˆ’β„Ž(πœ’))ξ‚ƒβˆšsinhπœ€/π‘˜1[]ξ‚„βŽ€βŽ₯βŽ₯βŽ¦β„Ž(πœ’)βˆ’π›½βˆ’1+𝑒(0)intξ‚ƒβˆš(πœ’)sinhπœ€/π‘˜1ξ‚„(π‘¦βˆ’π›½)ξ‚ƒβˆšsinhπœ€/π‘˜1[]ξ‚„,π‘’β„Ž(πœ’)βˆ’π›½(5.3)2(0)(πœ’,𝑦)=π‘˜1𝑑𝑝2(0)βŽ‘βŽ’βŽ’βŽ£ξ‚ƒβˆšπ‘‘πœ’sinhπœ€/π‘˜1ξ‚„ξ‚ƒβˆš(𝑦+1)βˆ’sinhπœ€/π‘˜1ξ‚„(π‘¦βˆ’β„Ž(πœ’))ξ‚ƒβˆšsinhπœ€/π‘˜1ξ‚„βŽ€βŽ₯βŽ₯⎦(β„Ž(πœ’)+1)βˆ’1+𝑒(0)intξ‚ƒβˆš(πœ’)sinhπœ€/π‘˜1ξ‚„(𝑦+1)ξ‚ƒβˆšsinhπœ€/π‘˜1ξ‚„,(β„Ž(πœ’)+1)(5.4) where the pressure gradients 𝑑𝑝𝑖(0)/π‘‘πœ’ (𝑖=1,2) are determined below.

The continuity equation (4.6), after substituting the asymptotic expansion for πœπ‘–, (4.7), simplifies toπœ•π‘’π‘–(0)+πœ•πœ’πœ•πœπ‘–(0)πœ•π‘¦=0.(5.5) The above equation is integrated with respect to 𝑦 from 𝑦=𝛽 to 𝑦=β„Ž(πœ’) for fluid 1 and 𝑦=βˆ’1 to 𝑦=β„Ž(πœ’) for fluid 2, to yield the following expressions for the normal velocities at the interface πœπ‘–(0)[𝑦=β„Ž(πœ’)] 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=βˆ’1:𝜐1(0)[]ξ€œπ‘¦=β„Ž(πœ’)=βˆ’π›½β„Ž(πœ’)πœ•π‘’1(0)πœπœ•πœ’π‘‘π‘¦2(0)[]ξ€œπ‘¦=β„Ž(πœ’)=βˆ’β„Ž(πœ’)βˆ’1πœ•π‘’2(0)πœ•πœ’π‘‘π‘¦.(5.6) Note that the normal velocities of the two fluids are equal at the interface (normal velocity continuity condition), and so, 𝜐1(0)[]𝑦=β„Ž(πœ’)=𝜐2(0)[]𝑦=β„Ž(πœ’)=𝜐(0)int.(5.7) Therefore, we equate the two integrals in the above equation and apply Leibnitz ruleπœ•ξ€œπœ•πœ’π›½β„Ž(πœ’)𝑒1(0)π‘‘π‘¦βˆ’πœ•β„Žπ‘’πœ•πœ’1(0)β„Žπœ•(πœ’)=ξ€œπœ•πœ’β„Ž(πœ’)βˆ’1𝑒2(0)π‘‘π‘¦βˆ’πœ•β„Žπ‘’πœ•πœ’2(0)β„Ž(πœ’).(5.8) Noting that the π‘₯-component velocities are equal at the interface, 𝑒1(0)[β„Ž(πœ’)]=𝑒2(0)[β„Ž(πœ’)], the above equation is simplified toπœ•ξ‚Έξ€œπœ•πœ’π›½β„Ž(πœ’)𝑒1(0)ξ€œπ‘‘π‘¦βˆ’β„Ž(πœ’)βˆ’1𝑒2(0)𝑑𝑦=0.(5.9) We next integrate the above equation with respect to πœ’, 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)/π‘‘πœ’ 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 𝑒𝑖(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)/π‘‘πœ’ in terms of 𝑑𝑝1(0)/π‘‘πœ’. Prior to determining 𝑑𝑝1(0)/π‘‘πœ’, it is useful to determine the π‘₯-component of the fluid velocity at the interface 𝑒(0)int from the simplified tangential stress continuity condition, (4.11). Once 𝑒(0)int is determined, the pressure gradient 𝑑𝑝1(0)/π‘‘πœ’ 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)=ξƒ―π‘˜π‘‘πœ’1ξ€Ίβ„Ž(1+β„Ž(π‘₯))ξ…žξ…žξ…žξ€»ξ‚™(πœ’)βˆ’π‘€(πœ’)βˆ’πΊπΉπ‘˜1πœ€βˆ’π‘˜1ξ€Ίβ„Žξ…žξ…žξ…ž(ξ€»ξƒ¬ξ‚™πœ’)βˆ’π‘€(πœ’)𝐺+4π‘˜1πœ€ξƒ­ξ‚Έξ‚™tanhπœ€4π‘˜1[]ξ‚Ήξƒ°ξƒ―π‘˜1+β„Ž(πœ’)1ξ„”11+β„Ž(πœ’)+πœ‡π‘Ÿ[]ξ‚΅ξ‚Έξ‚™π›½βˆ’β„Ž(πœ’)βˆ’πΊtanhπœ€4π‘˜1[]ξ‚Ήξ‚Έξ‚™1+β„Ž(πœ’)βˆ’tanhπœ€4π‘˜1[]βˆ’ξ‚™β„Ž(πœ’)βˆ’π›½ξ‚Ήξ‚Ά4π‘˜1πœ€ξ‚΅ξ‚Έξ‚™tanhπœ€4π‘˜1[]ξ‚Ήβˆ’11+β„Ž(πœ’)πœ‡π‘Ÿξ‚Έξ‚™tanhπœ€4π‘˜1[]β„Ž(πœ’)βˆ’π›½ξ‚Ήξ‚Άξ„•ξƒ°βˆ’1,(5.10)𝑑𝑝2(0)=π‘‘πœ’π‘‘π‘1(0)(πœ’)βˆ’πœ•π‘‘πœ’3β„Ž(πœ’)πœ•πœ’3+𝑒(0)int(πœ’)π‘˜1ξ€·πœ‡π‘Ÿξ€Έπ‘’βˆ’1+𝑀(πœ’),(5.11)(0)intξƒ―ξ‚™(πœ’)=π‘˜1πœ€πΉβˆ’π‘˜1𝑑𝑝1(0)ξ‚΅ξ‚Έξ‚™π‘‘πœ’tanhπœ€4π‘˜1[]ξ‚Ήξ‚Έξ‚™1+β„Ž(πœ’)βˆ’tanhπœ€4π‘˜1[]β„Ž(πœ’)βˆ’π›½ξ‚Ήξ‚Ά+π‘˜1ξ‚Έξ‚™tanhπœ€4π‘˜1[]πœ•1+β„Ž(πœ’)ξ‚Ήξ‚΅3β„Ž(πœ’)πœ•πœ’3Γ—ξ‚»ξ‚Έξ‚™βˆ’π‘€ξ‚Άξ‚Όcothπœ€π‘˜1[]ξ‚Ή1+β„Ž(πœ’)βˆ’πœ‡π‘Ÿξ‚Έξ‚™cothπœ€π‘˜1[]ξ‚Ή+ξ€·πœ‡β„Ž(πœ’)βˆ’π›½π‘Ÿξ€Έξ‚Έξ‚™βˆ’1tanhπœ€4π‘˜1[]1+β„Ž(πœ’)ξ‚Ήξ‚Όβˆ’1,(5.12) where 𝐺, 𝑀, and 𝐹 are functions of πœ’, and they are defined as𝐺(πœ’)=cothπœ€π‘˜1[]ξ‚Ή1+β„Ž(πœ’)βˆ’πœ‡π‘Ÿξ‚Έξ‚™cothπœ€π‘˜1[]ξ‚Ή+ξ€·πœ‡β„Ž(πœ’)βˆ’π›½π‘Ÿξ€Έξ‚Έξ‚™βˆ’1tanhπœ€4π‘˜1[]1+β„Ž(πœ’)ξ‚Ήξ‚Όβˆ’1Γ—ξƒ―ξ€·πœ‡π‘Ÿξ€Έ[]+ξ‚™βˆ’11+β„Ž(πœ’)π‘˜1πœ€ξ‚΅ξ€·1βˆ’2πœ‡π‘Ÿξ€Έξ‚Έξ‚™tanhπœ€4π‘˜1[]ξ‚Ήξ‚Έξ‚™1+β„Ž(πœ’)+tanhπœ€4π‘˜1[]ξ€Ίπœ–β„Ž(πœ’)βˆ’π›½ξ‚Ήξ‚Άξ‚Όπ‘€(πœ’)=1+π›½πœ–2+ξ€·πœ–1βˆ’πœ–2ξ€Έξ€»β„Ž(πœ’)βˆ’3ξ€½πœ–1πœ–2β„Žξ…žξ€Ί(πœ’)π‘ž(πœ’)(1+𝛽)+πœ–2βˆ’πœ–1Γ—ξ€½πœ–ξ€»ξ€Ύ2+πœ–1[]ξ€Ύ+ξ€Ίπœ–+π‘ž(πœ’)1+2β„Ž(πœ’)βˆ’π›½1+π›½πœ–2+ξ€·πœ–1βˆ’πœ–2ξ€Έξ€»β„Ž(πœ’)βˆ’2Γ—ξ€·π‘ž(πœ’)π‘žξ…žξ€½πœ–(πœ’)1[]1+β„Ž(πœ’)2+πœ–2[]π›½βˆ’β„Ž(πœ’)2ξ€Ύ+πœ–1πœ–2π‘žξ…ž[]ξ€Έ,𝐹(πœ’)1+2β„Ž(πœ’)βˆ’π›½(πœ’)=βˆ’π‘ž(πœ’)β„Žξ…ž(πœ’)πœ–2ξ€Ί(π›½βˆ’β„Ž(πœ’))π‘ž(πœ’)(𝛽+1)βˆ’πœ–1+πœ–2ξ€»ξ€Ίπœ–1+π›½πœ–2+ξ€·πœ–1βˆ’πœ–2ξ€Έξ€»β„Ž(πœ’)2βˆ’π‘ž(πœ’)π‘žξ…ž(πœ’)(π›½βˆ’β„Ž(πœ’))(1+β„Ž(πœ’))ξ€Ίπœ–1+π›½πœ–2+ξ€·πœ–1βˆ’πœ–2ξ€Έξ€».β„Ž(πœ’)(5.13) Finally, the kinematic condition at the interface, (4.14), is used to derive the evolution equation for β„Ž(πœ’,𝜏), as follows. We first substitute the expressions for the normal fluid velocities at the interface, (5.6), after using Leibnitz rule on the integralπœ€πœ•β„Žπœ•πœ+𝑒1(0)[]β„Ž(πœ’)πœ•β„Žπœ•πœ•πœ’=βˆ’ξ€œπœ•πœ’π›½β„Ž(πœ’)𝑒1(0)𝑑𝑦+𝑒1(0)[]β„Ž(πœ’)πœ•β„Ž,πœ•πœ’(5.14) which is simplified to give πœ€πœ•β„Ž+πœ•πœ•πœξ€œπœ•πœ’π›½β„Ž(πœ’)𝑒1(0)𝑑𝑦=0,(5.15) where 𝑒1(0) is substituted from (5.3), after using the expressions for 𝑑𝑝1(0)/π‘‘πœ’ and 𝑒(0)int determined using the procedure outlined above. The evolution equation for the interfacial charge density π‘ž(πœ’,𝜏) is obtained from (4.17) after substituting the expressions for 𝑒(0)int and the gradients of the potential (from (5.1)) in that equation. The nonlinear evolution equations for β„Ž(πœ’,𝜏) and π‘ž(πœ’,𝜏), respectively, take the formsπœ€πœ•β„Ž+1πœ•πœπœ‡π‘Ÿπ‘˜ξƒ―ξƒ©1𝑑𝑝1(0)+πœ‡π‘‘πœ’π‘Ÿ2𝑒(0)intξƒͺ(πœ’)πœ•β„Žπœ•πœ’sech2ξ‚Έ12ξ‚™πœ€π‘˜1ξ‚Ή(π›½βˆ’β„Ž)+π‘˜1𝑑(π›½βˆ’β„Ž)2𝑝1(0)π‘‘πœ’2βˆ’π‘‘π‘1(0)π‘‘πœ’πœ•β„Žξƒ­βˆ’ξ‚™πœ•πœ’π‘˜1πœ€ξƒ©2π‘˜1𝑑2𝑝1(0)π‘‘πœ’2+πœ‡π‘Ÿπ‘’ξ…ž(0)intξƒͺξ‚Έ1(πœ’)Γ—tanh2ξ‚™πœ€π‘˜1ξ‚Ήξƒ°πœ€(π›½βˆ’β„Ž)=0,πœ•π‘žπœ•πœ+𝑒(0)int(πœ’)πœ•π‘žπœ•πœ’βˆ’π‘žπœ•β„Žξƒ―βˆšπœ•πœ’πœ€π‘˜1πœ‡π‘Ÿπ‘‘π‘1(0)ξ‚Έ1π‘‘πœ’tanh2ξ‚™πœ€π‘˜1ξ‚Ή+ξ‚™(β„Žβˆ’π›½)πœ€π‘˜1𝑒(0)intξ‚Έξ‚™(πœ’)cothπœ€π‘˜1ξ‚Ήξƒ°(β„Žβˆ’π›½)+π‘žπ‘’ξ…ž(0)int+(πœ’)πœ€π‘†1(0)ξ€Ί(1+β„Ž)π‘ž+πœ–2ξ€»πœ–1+π›½πœ–2+ξ€·πœ–1βˆ’πœ–2ξ€Έβ„Ž+πœ€π‘†2(0)ξ€Ί(π›½βˆ’β„Ž)π‘žβˆ’πœ–1ξ€»πœ–1+π›½πœ–2+ξ€·πœ–1βˆ’πœ–2ξ€Έβ„Ž=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 β„Ž(πœ’,𝜏)=0, and with a constant interfacial charge density π‘ž0 which is independent of πœ’ and 𝜏. This base state interfacial charge π‘ž0 is determined from (4.17) with the left side set to zero, since πœ•/πœ•πœ=0 and 𝑒𝑖(0)=0 in the base state𝑆1(0)ξ‚Έπœ•πœ“1ξ‚Ήπœ•π‘¦π‘¦=0=𝑆2(0)ξ‚Έπœ•πœ“2ξ‚Ήπœ•π‘¦π‘¦=0.(6.1) The derivatives of the potentials in the above equation are calculated from (5.1), with β„Ž(πœ’) set to zero. This yields the following expression for the steady interfacial charge density: π‘ž0=πœ–1𝑆2(0)βˆ’πœ–2𝑆1(0)𝑆1(0)+𝛽𝑆2(0).(6.2) The variables β„Ž and π‘ž are now perturbed about their base state valuesβ„Ž(πœ’,𝜏)=β„Ž1[],π‘žexpπ‘–π‘˜πœ’+πœ”πœ(πœ’,𝜏)=π‘ž0+π‘ž1[],expπ‘–π‘˜πœ’+πœ”πœ(6.3) where β„Ž1 and π‘ž1 are the amplitudes of the perturbations which are independent of πœ’ and 𝜏, π‘˜ is the nondimensional wavenumber based on the lateral length scale 𝐿=𝛾𝐻3/πœ–0πœ“2𝑏 and πœ” is the nondimensional growth rate based on the time scale πœ‡2𝐻3𝛾/(πœ–0πœ“2𝑏)2. We substitute the expressions for 𝑑𝑝1(0)/π‘‘πœ’ and 𝑒(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 β„Ž(πœ’)=0. We then linearize the resulting coupled nonlinear evolution equations with respect to β„Ž1 and π‘ž1, to obtain a set linear homogeneous equations for β„Ž1 and π‘ž1 of the form𝐻1β„Ž1+𝑄1π‘ž1𝐻=0,2β„Ž1+𝑄2π‘ž1=0,(6.4) where𝐻1𝑅=βˆ’14+π‘˜4𝑅13𝑅1𝛽4πœ–42+πœ–41𝑅14+π‘˜4𝑅13𝑅1βˆ’π‘˜2𝑅13𝑅1πœ–2ξ€Έβˆ’πœ–21πœ–22ξ€Ί6𝑅14+π‘˜4𝑅13𝑅1𝛽2+π‘˜2𝑅13𝑅1πœ–2ξ€»+πœ–31πœ–2𝑅15ξ€·π‘…βˆ’4𝛽14+π‘˜4𝑅13𝑅1ξ€Έ+π‘˜2𝑅13𝑅1π›½πœ–2ξ€»βˆ’πœ–1πœ–32𝑅15+4𝛽3𝑅14+π‘˜4𝑅13R1ξ€Έ+π‘˜2𝑅13𝑅1π›½πœ–2ξ€»+π‘ž20(1+𝛽)πœ–2ξƒ―βˆ’π‘…6𝑅10𝑅12+π‘˜2𝑅13ξ€·πœ–1+π›½πœ–2×𝑅1𝐢(π›½βˆ’1)βˆ’1𝐡1π›½ξ‚™π‘˜1πœ€ξƒͺπœ–1βˆ’πΆ1𝐡1𝛽2ξ‚™π‘˜1πœ€πœ–2ξƒ­ξƒ°βˆ’π‘ž0πœ–2𝑅8𝑅10𝑅12+π‘˜2𝑅13ξ€·πœ–1+π›½πœ–2×𝛽2𝑅1βˆ’πΆ1𝐡1ξ‚™π‘˜1πœ€ξƒͺπœ–21+2𝑅1βˆ’πΆ1𝐡1𝛽(π›½βˆ’1)π‘˜1πœ€ξƒͺπœ–1πœ–2+𝐢1𝐡1𝛽2ξ‚™π‘˜1πœ€πœ–22ξƒ­ξƒ°+𝐡1𝑅5πœ”πœ€3/2πœ‡π‘Ÿ,𝑄1=(π›½βˆ’1)πœ–1πœ–2ξ€·πœ–1+π›½πœ–2𝑅15+π‘˜2𝑅1𝑅13ξ€·πœ–1+π›½πœ–2ξ€Έξ€»βˆ’π‘ž0𝑅7𝑅10𝑅12+π‘˜2𝑅13ξ€·πœ–1+π›½πœ–2ξ€Έ2×𝑅1𝐢+𝛽1𝐡1ξ‚™π‘˜1πœ€ξƒͺπœ–1+𝛽2𝑅1+𝐢1𝐡1ξ‚™π‘˜1πœ€ξƒͺπœ–2,𝐻2=ξ€·πœ–1+π›½πœ–2ξ€Έξƒ©πœ€ξƒ©πœ€π‘…2𝑅18βˆ’π΅1𝑅4𝑆2(0)ξ‚„πœ–41πœ‡r+πœ–31𝑅16+𝑅3𝑅4+𝐡1𝑅2𝑅4𝑅34ξ€Έ+πœ–21πœ–2𝑅17+3𝛽𝑅3𝑅4+3𝐡1𝑅2𝑅4𝑅35ξ€»+π›½πœ–2𝑅30𝑅22ξ‚™k1πœ€q0πœ‡r+π›½πœ–22𝑅19+𝛽𝑅3𝑅4+πœ€π›½R2×𝐡1𝑅4π‘ž0𝑆1(0)+𝛽𝑆2(0)ξ‚„+𝑅18+𝐡1𝑅4𝑆1(0)ξ‚„πœ–2ξ‚πœ‡π‘Ÿξ‚ξƒͺ+πœ–1𝑅30𝑅22ξ‚™π‘˜1πœ€π‘ž0πœ‡π‘Ÿ+π›½πœ–22𝑅20+3𝛽𝑅3𝑅4+πœ€π›½π‘…2ξ‚€3𝐡1𝑅4π‘ž0𝑆1(0)+𝛽𝑆2(0)ξ‚„+4𝛽𝑅18+3𝐡1𝑅4𝑆1(0)βˆ’π›½π΅1𝑅4𝑆2(0)ξ‚„πœ–2ξ‚πœ‡π‘Ÿξ‚ξƒͺξƒͺβˆ’π‘…2π‘ž0πœ‡π‘Ÿξ€·π‘…9𝑅21+𝑅23+𝑅25+𝑅28βˆ’π‘…29ξ€»+𝑅32𝑅36ξ€Έξƒͺ,𝑄2=𝑅2ξ€·πœ–1+π›½πœ–2ξ€Έπœ‡π‘Ÿξ‚†π΅1𝑅2πœ€2ξ€·πœ–1+π›½πœ–2ξ€Έ3𝑆1(0)+𝛽𝑆2(0)ξ€·πœ–+πœ”1+π›½πœ–2ξ€Έξ‚„βˆ’π‘ž0𝑅9𝑅22+𝑅24+𝑅26+𝑅27ξ€Έβˆ’π‘…31𝑅2βˆšπœ€π‘˜1ξ€·πœ–1+π›½πœ–2ξ€Έ+𝑅33𝑅36,(6.5) in which 𝐡1, 𝐡2, 𝐢1, 𝐢2, and 𝑅1βˆ’π‘…49 are given in the appendix.

The set of linear homogeneous equations (6.4) for β„Ž1 and π‘ž1 is written in the matrix form 𝑀⋅𝐢𝑇=0, where the vector 𝐢=[β„Ž1, π‘ž1] and the determinant of this matrix 𝑀 is set to zero for nontrivial solutions in order to obtain the characteristic equation for πœ”. This characteristic equation, which gives the growth rate πœ” as a function of π‘˜, π‘˜1, 𝛽, πœ‡π‘Ÿ, 𝑆1(0), 𝑆2(0), πœ€, πœ–1, and πœ–2, is a quadratic equation for πœ”. The roots of the characteristic equation for πœ” can be written as πœ”1𝑅=βˆ’48𝑅49βˆ’ξ”π‘…248𝐻+22𝑄1𝑅49βˆ’2𝑅41𝑅47𝑅49𝑅49,πœ”(6.6)2𝑅=βˆ’48𝑅49+𝑅248𝐻+22𝑄1𝑅49βˆ’2𝑅41𝑅47𝑅49𝑅49.(6.7) The roots πœ”1 and πœ”2 are always real, with one of them πœ”1 is always negative, and the other one πœ”2 can be positive or negative depending on the choice of the system parameters. We can take πœ”=πœ”2 as a function of π‘˜ 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 πœ” given by (6.7) as a function of the wavenumber π‘˜ 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 πœ” against the wavenumber π‘˜ for the porosity of porous medium πœ€, medium permeability π‘˜1, nondimensional conductivities 𝑆1(0), 𝑆2(0), dielectric constants πœ–1, πœ–2, the ratio of the thicknesses of top and bottom fluids 𝛽, and the ratio of viscosities of the two fluids πœ‡π‘Ÿ=πœ‡1/πœ‡2, respectively.

6.1. Leaky Dielectric-Leaky Dielectric Interface

This is the general case in which 𝑆1(0)β‰ 0 and 𝑆2(0)β‰ 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 πœ” versus the wavenumber π‘˜ for various values of the porosity of porous medium πœ€. It is clear from Figure 2(b) that for small values of the porosity (e.g., πœ€=0.1), the growth rate πœ” increases by increasing the wavenumber π‘˜ till a maximum value of πœ€ after which πœ” decreases by increasing π‘˜; that is, there exists a maximum mode of instability only for small values of the porosity πœ€, while for any other value of πœ€, we found that πœ” decreases by increasing π‘˜, 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 0β‰€π‘˜β‰€15, and it has a destabilizing effect for higher wavenumber values π‘˜>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 π‘˜=15, resp.). Figure 3 shows the variation of growth rate πœ” with the wavenumber π‘˜ for different values of the medium permeability π‘˜1. We conclude from this figure that for any wavenumber value π‘˜β‰€20, the growth rate πœ” increases by increasing the medium permeability π‘˜1 values, while for wavenumber values π‘˜>20, all the curves correspond to different values of medium permeability π‘˜1 coincide. This means that the medium permeability π‘˜1 has a destabilizing effect for the wavenumber range 0<π‘˜β‰€20 and it has no effect on the stability of the considered system for higher wavenumber values π‘˜>20.

Figures 4(a) and 4(b) shows the variation of the growth rate πœ” versus the wavenumber π‘˜ 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 πœ” 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 πœ” with the wavenumber π‘˜ 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<π‘˜<10, while it has a stabilizing effect for higher wavenumber values π‘˜β‰₯10, since the growth rate πœ” 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)β‰₯104.

Figure 6 shows the variation of growth rate πœ” with the wavenumber π‘˜ for different values of the dielectric constant of upper fluid πœ–1. It is seen from this figure that there exists a critical wavenumber value π‘˜=4.6, before which the growth rates decrease by increasing the dielectric constant πœ–1, and after which they increase by increasing πœ–1 values. Thus, the dielectric constant of upper fluid πœ–1 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 πœ–2, 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 πœ–1 given by Figure 6, but the corresponding figures are not given here. In other words, we conclude that the dielectric constant of lower fluid πœ–2 has a destabilizing as well as a stabilizing effects for wavenumber ranges before and after the same critical wavenumber value π‘˜=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 π‘˜=2.5, respectively. Figures 7(a) and 7(b) shows the variation of the growth rate πœ” versus the wavenumber π‘˜ for various values of the ratio of viscosity of upper and lower fluids πœ‡π‘Ÿ. It indicates that the viscosity ratio πœ‡π‘Ÿ has a stabilizing effect for small wavenumber values, and it has also a destabilizing effect for higher wavenumber values, since the growth rate πœ” decreases and increases by increasing the increase of πœ‡π‘Ÿ, respectively. It is clear also from Figure 7(b) that for πœ‡π‘Ÿ>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 πœ‡π‘Ÿβ‰€1.

6.2. Perfect Dielectric-Leaky Dielectric Interface

This is the limiting case in which 𝑆1(0)=0 and 𝑆2(0)β‰ 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 πœ” against the wavenumber π‘˜ for various values of the porosity of porous medium πœ€. In comparison with Figures 2(a) and 2(b), we conclude that the porosity of porous medium πœ€ behaves in the same manner as in the previous case of leaky-leaky dielectric fluids, except that the values of growth rates πœ” are higher than their values in the previous case and the corresponding curves intersect the π‘˜-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 πœ€β‰€0.3. Similarly, the effects of medium permeability π‘˜1, the conductivity of lower fluid 𝑆2(0), the dielectric constants πœ–1, πœ–2, 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 π‘˜1, the growth rate values are higher than their values in the previous case; for the effect of 𝑆2(0), the obtained curves intersect the π‘˜-axis at larger wavenumber values than in the previous case; for the effect of πœ–1, there exists a mode of maximum instability for all values of πœ–1; for the effect of πœ–2, 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 π‘˜-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 πœ” versus the wavenumber π‘˜ for various values of the viscosity ratio of upper and lower fluids πœ‡r. In comparison with Figures 7(a) and 7(b), we conclude that the viscosity ratio πœ‡π‘Ÿ 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 πœ‡π‘Ÿβͺ‹1, there exists a mode of maximum instability and not only for πœ‡π‘Ÿ>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 πœ” versus the wavenumber π‘˜ for different values of the porosity of porous medium πœ€. In view of the above discussion, we conclude from this figure that the porosity πœ€ has a slightly stabilizing effect for small wavenumber values π‘˜β‰€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 πœ” with the wavenumber π‘˜ for various values of the medium permeability π‘˜1. It is clear from this figure that the permeability π‘˜1 has a destabilizing effect, since the growth rate πœ” increases by increasing the medium permeability values at any fixed wavenumber value. Similarly, the effect of dielectric constant of the upper fluid πœ–1 on the stability of the system is illustrated in Figure 12, and it shows that the dielectric constant πœ–1 has a stabilizing effect since the growth rate πœ” decreases by increasing the dielectric constant πœ–1 at any wavenumber value. Figure 13 shows the variation of growth rate πœ” with the wavenumber π‘˜ for different values of the dielectric constant of lower fluid πœ–2, and from which we conclude thatthe dielectric constant πœ–1, and after which they increase by increasing πœ–2 has a destabilizing effect in the wavenumber range 0<π‘˜β‰€4.6, and it has no effect on the stability of the considered system afterwards for higher wavenumber values π‘˜>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 πœ‡π‘Ÿ, respectively, on the stability of the system are found to has the same and the opposite effects as the effect of the dielectric constant πœ–2 shown in Figure 13, but the corresponding figures are not given here to avoid any kind of repitation; that is, the parameters 𝛽 and πœ‡π‘Ÿ 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 πœ€, the dielectric constant of upper fluid πœ–1, and the ratio of viscosity of upper and lower fluids πœ‡π‘Ÿ have stabilizing effects,(ii)the medium permeability π‘˜1, the dielectric constant of lower fluid πœ–2, 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 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 πœ€, πœ–1, and πœ‡π‘Ÿ have stabilizing effects, while the parameters, π‘˜1, πœ–2, 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 parameters πœ€, πœ–1, and πœ‡r have destabilizing effects for high wavenumbers, while the parameters πœ–2, 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 fluid S1(0) has a destabilizing effect for small wavenumbers 0<π‘˜<π‘˜1 and also a stabilizing effect for high wavenumbers π‘˜>π‘˜2, while it has a dual role on the stability of the considered system between them in the wavenumber range π‘˜1<π‘˜<π‘˜2.(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

𝐡1ξ‚΅ξ‚™=cothπœ€π‘˜1ξ‚Ά+πœ‡π‘Ÿξ‚΅π›½ξ‚™cothπœ€π‘˜1ξ‚Ά+ξ€·πœ‡π‘Ÿξ€Έξ‚΅1βˆ’1tanh2ξ‚™πœ€π‘˜1ξ‚Ά,𝐡2=ξ‚™πœ€π‘˜1ξ‚»βˆ’csch2ξ‚΅ξ‚™πœ€π‘˜1ξ‚Ά+πœ‡π‘Ÿcsch2ξ‚΅π›½ξ‚™πœ€π‘˜1ξ‚Ά+12ξ€·πœ‡π‘Ÿξ€Έβˆ’1sech2ξ‚΅12ξ‚™πœ€π‘˜1,𝐢1=πœ‡π‘Ÿξ‚™βˆ’1βˆ’π‘˜1πœ€ξ‚Έξ€·2πœ‡π‘Ÿξ€Έξ‚΅1βˆ’1tanh2ξ‚™πœ€π‘˜1ξ‚Άξ‚΅1+tanh2π›½ξ‚™πœ€π‘˜1,𝐢2=πœ‡π‘Ÿ1βˆ’1+2ξ‚»ξ€·1+1βˆ’2πœ‡π‘Ÿξ€Έsech2ξ‚΅12ξ‚™πœ€π‘˜1ξ‚Άβˆ’tanh2ξ‚΅12π›½ξ‚™πœ€π‘˜1,𝑅1=π‘˜1𝐢1𝐡1ξ‚™+2π‘˜1πœ€ξƒͺξ‚΅1tanh2ξ‚™πœ€π‘˜1ξ‚Άξƒ°,π‘…βˆ’12=1πœ‡π‘Ÿξƒ―