# Three-Dimensional Stability Characteristics of Finite Electrified Conducting Fluids Streaming through a Porous Medium

Authors: T. M. N. Metwaly; Zakaria M. Gharsseldien
Journal: Journal of Mathematics
 (2020)

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## Abstract

A novel procedure is utilized to investigate the surface waves between two finite conducting fluids streaming through a porous medium in the presence of a horizontal electric field. Normal mode analysis is applied to study two- and three-dimension disturbances cases. The quadratic dispersion equation of complex coefficients representing the system is derived and discussed. It is noted that based on appropriate data selections, the stability criteria do not depend on the medium permeability. It is found that electrical conductivities, viscosities, medium porosity, and surface tension enhance the stability of the system while the dimension and the fluid velocities decrease the stability of the system. Finally, the fluid depths have a dual role (stabilizing as well as destabilizing effects) on the system.


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## Body

## 1. Introduction

The Kelvin–Helmholtz instability (KHI) appears if two superposed fluids streaming with a relative horizontal velocity. This instability is very important in describing more kinds of astrophysical and space phenomena involving sheared stream. As a consequence of its connection to geophysical, astrophysical, and laboratory areas [1], it has been examined by numerous contributors. The linear KHI is treated in Chandrasekhar’s monogragh [2]. The impact of streaming is destabilized in the linear access in the KHI problem.Electrohydrodynamics (EHD) can be treated as the branch of fluid mechanics interested in electrical force effects or as the part of EHD that is concerned with the influence of fluid motion on electric fields. Consequently, it is concerned with (electric fields polarized or free charges) interactions in fluids. The fluids may be very good insulators, slight conductors, or even high conductors. A survey on EHD subject with specific reference to several of the advancements in this area is provided by Melcher [3].In the surface wave phenomena through dielectric fluids when both surface tension and gravity are essential, the capillary waves and the gravity couple to create one wave distinguished by the two influences. By contrast, the effect of an external electric field during conducting fluid flows has many several characteristics and various valuable engineering and physical applications. When surface charges share in the motion of fluids, the arising waves are mentioned as EHD surface waves. The effect of an electric field on the linear stability of a sharp interface dividing two nonconducting dielectric fluids of unlimited range has been investigated by Melcher and Schwarz [4]. A review of the role of interfacial stresses on EHD has been presented by Melcher and Taylor [5] and later the same characteristics have been reviewed by many authors (see Baygents and Baldessare [6] and Rudraiah et al. [7]).In the linear EHD stability theory, it is well known that the horizontal electric field has a stabilizing effect while the vertical one always has a destabilizing influence [8–12]. EHD stability problems of the interface between conducting fluids in planar and cylindrical geometries have been as of late examined by many authors [13–18].Flows through a porous medium have been a topic of high interest for the latest many decades. This interest was driven by many engineering and geophysical applications in various majors [19]. Many recent works on this topic are investigated by many authors [20–23]. In most earlier investigations on porous media, many researchers considered the treatments based on Darcy’s law and Forschheimer-extended Darcy’s law models. Darcy’s law is an experimental formula relating the bulk viscous resistance, the gravitational force, and the pressure gradient in a porous medium.The KHI for flow in porous media has received great attention in the scientific literature. Sharma and Spanos [24] investigated the instability of the plane interface separating two uniform superposed fluids motion through porous media. El-Sayed [25] analyzed the instability of two superposed viscous streaming fluids through porous media in the presence of a horizontal electric field. For recent surveys regarding the developments of linear and nonlinear electromagnetic flows in porous media, see refs. [26–36].The goal of the present paper is to investigate the linear stability of two electrically conducting fluids of finite thickness moving through a porous medium under a uniform horizontal electric field. In addition, the uniform flow is taken through three dimensions to be more general and then using the normal modes analysis to obtain the solutions of system parameters. This problem, as far as we know, has not been studied yet. The stability results are presented in figures. All these flows are new and are presented for the first time in the literature.


## 2. Problem Description

We consider the three-dimensional finite amplitude surface waves propagating at the interfacez=0 that represent the balance situation. The perturbed interface z=ζx,y,t separates two conducting incompressible fluids that have a uniform thickness. Fluid (1) occupies the lower region −h1<z<ζx,y,t while fluid (2) occupies the upper region ζx,y,t<z<h2. The two fluids are streaming with uniform velocities U1 and U2 along the x-axis through a porous medium. The system is influenced by a constant horizontal electric field E0 in the x-direction. The gravitational acceleration g directed normal to the interface. The surface tension forces T are taken into account between the two fluids. We shall denote ρj, μj, εj, αj, Ej, λ1, and m, j=1,2 to the fluids densities, viscosities coefficients, dielectric constants, electrical conductivities, electric field elements, permeability, and porosity of the medium. A sketch of the physical problem is given in Figure 1. The disturbed interface may be written as(1)Fx,y,z,t=z−ζx,y,t=0.Figure 1
A sketch of the physical problem.The unit outer perpendicular to the interface is(2)n=∇F∇F=1+ζx2+ζy2−1/2−ζx,−ζy,1.The equation of motion for an incompressible viscous fluid in a porous medium is(3)ρm∂v∂t+1mv⋅∇v=−∇p−ρgez−μλ1v.The continuity equation is(4)∇⋅v=0,where pandv are the pressure and the velocity, respectively. Considering irrotational fluid flows, then the velocity potentials is Φjx,y,z,t such that(5)vj=Ujex+∇Φj,j=1,2,where ex is the unit vector in the x-direction. The potentials Φj satisfy the Laplace’s equations:(6)∇2Φ1=0,in−h1<z<ζx,y,t,(7)∇2Φ2=0,inζx,y,t<z<h2.It is supposed that, in electrodynamics, the quasi-static approximation is valid. Hence, the electrical equations are(8)∇×E=0,∇⋅εE=q.The charge conservation equation is(9)∇·J+∂q∂t=0,where J=αE+qv is the free current density (stationary current), α is the electrical conductivity, and q is the free charge density. Therefore, the electric field can be represented as the gradient of electrostatic potentials Ψjx,y,z,t:(10)Ej=E0ex−∇Ψj,j=1,2.From equations (8) and (10), the scalar electric potentials Ψj to satisfy the Laplace’s equations are as follows:(11)∇2Ψ1=0,in−h1<z<ζx,y,t,(12)∇2Ψ2=0,inζx,y,t<z<h2.


## 3. The Boundary Conditions

(1)
The vanishing of the normal fluid velocities on both the lower and upper boundaries is(13)∂Φj∂z=0,atz=−1jhj,j=1,2.(2)
The vanishing of the normal components of the electric fields also on these boundaries is(14)∂Ψj∂z=0,atz=−1jhj,j=1,2.(3)
The kinematic boundary condition at the interfacez=ζx,y,t is given by(15)m∂ζ∂t−∂Φj∂z+∂ζ∂xUj+∂Φj∂x+∂ζ∂y∂Φj∂y=0,j=1,2.(4)
The tangential electric field component is supposed continuous at the interfacez=ζx,y,t:(16)Ψx+ζxΨz=0,wheref=f2−f1 describes the jump of the quantity f over the interface.(5)
When uniform conductivity of fluids is considered, the problem became more attractive yet difficult. Surface charges play an essential role in such cases; the conduction process in the interface region is a significant factor in many electrical systems. The continuity of stationary current normal to the interfacez=ζx,y,t must lead to charge accumulation on the interface [17]:(17)ζxαΨx+ζyαΨy−ζxE0α−αΨz=0.(6)
The interfacial normal stress component must be continuous at the interfacez=ζx,y,t:(18)1m2ρmΦt+UΦx+1λ1μΦ+ζx2E02ε+E0εΨx+2ζxE0εΨz−ζxεΨx−ζyεΨy+gζρ+Tζxx+ζyy=0.


## 4. Linear Theory and Characteristic Equation

To study the stability of the problem, first we shall analyze the perturbation quantities into normal modes, and we suppose that all the perturbed quantities take the form(19)frexpiθ,where fr is any function of r, θ=kx+ly−ωt is the carrier wave phase, K=k2+l2, k and l represent the wave number elements in x− and y− directions, respectively, while ω, A, i, and c.c denote the angular frequency, the complex amplitude of the surface wave, the imaginary unit, and the complex conjugate of the previous terms, respectively.Here, using equation (19) together with equations (6), (7), (11), and (12) with the suitable boundary conditions, the potential function solutions may be written as(20)ζ=Aexpiθ+c⋅c,Φ1=ikU1−mωKsinhKh1coshKz+h1Aexpiθ+c⋅c,Φ2=−ikU2−mωKsinhKh2coshKz−h2Aexpiθ+c⋅c,Ψ1=ikE0α2−α1Kα1σ1+α2σ2coshKz+h1coshKh1Aexpiθ+c⋅c,Ψ2=ikE0α2−α1Kα1σ1+α2σ2coshKz−h2coshKh2Aexpiθ+c⋅c.Now,σj=tanhKhjj=1,2, and to obtain a nontrivial starting solution, the frequency ω and the wave number K should satisfy the next characteristic equation:(21)Sω,K=1m2Kρ1mω−kU12σ1+ρ2mω−kU22σ2+iKλ1μ1mω−kU1σ1+μ2mω−kU2σ2−k2E02ε2−ε1α2−α1Kα1σ1+α2σ2+gρ2−ρ1−TK2=0.Equation (21) is simplified in the dispersion relation(22)a0ω2+a1+ib1ω+a2+ib2=0,where(23)a0=ρ1σ1+ρ2σ2,a1=−2kmρ1U1σ1+ρ2U2σ2,b1=mλ1μ1σ1+μ2σ2,a2=k2m2ρ1U12σ1+ρ2U22σ2−k2E02ε2−ε1α2−α1α1σ1+α2σ2+gKρ2−ρ1−TK3,b2=−kλ1μ1U1σ1+μ2U2σ2.We have the following special cases:(1)
In two semiinfinite nonconducting fluids, equation (22) reduces to the corresponding dispersion relation obtained first by El-Sayed [25](2)
In two-dimensional nonporous case, equation (22) reduces to the same equation derived by Melcher [8](3)
In the absence of the electric field and the porosity, equation (22) reduces to the corresponding relation presented by Chandrasekhar [2]


## 5. Stability Analysis and Discussion

Applying here the Routh–Hurwitz stability criterion [37] to equation (22), the system will be linearly stable according to(24)b1>0,a2b12−a1b1b2+a0b22≤0.Sincem and λ1 are positive parameters, the first condition in (24) is trivially satisfied. The second inequality is satisfied if(25)E02≥Ec2,where(26)E02=α1σ1+α2σ2k2ε2−ε1α2−α1k2U2−U12ρ2σ2μ12+ρ1σ1μ22m2μ1σ2+μ2σ12+gKρ2−ρ1−TK3.From the previous stability criteria, it is obvious that the permeabilityλ1 of the meduim has no effect on the linear stability of the system. The horizontal electric field has a dual role according to the sign of the term ε2−ε1α2−α1 in contrast with the pure dielectric fluids case, and this was well studied by Moatimid [38].The surface tension has stabilizing effects, while the uniform streaming velocity is strictly destabilizing. Next, we shall make a numerical discussion to clarify the stability of the problem by graphing the transition curves represented by equation (26) in E02−k plane when E02=Ec2. These transition curves that represent the marginal stability state separate the upper stable S regions from the lower U unstable ones. The calculations are made in that case, for a system having ρ1=0.000365 gr/cm3, ρ2=0.597 gr/cm3, ε1=1.007 farad/cm, ε2=1.7 farad/cm, α1=0.2 mho/cm, α2=0.6 mho/cm, h1=1.5 cm, h2=3.5 cm, μ1=0.03 cm2/sec, μ2=0.7 cm2/sec, U1=4 cm/sec, U2=20 cm/sec, g=981 cm/sec2, T=0.05 dyn/cm, and m=0.06 sec/cm. In Figures 2–8, the first dimension expresses disturbance wave number k, the second expresses the critical electric field E02, and the third expresses one of the previous parameters.Figure 2
Variation ofE02 with k for different values of the electrical conductivites α1andα2.Figure 3
Variation ofE02 with k for different values of the porosity of a porous medium m.Figure 4
Variation of withk for different values of the fluid viscosities μ1 and μ2.Figure 5
Variation ofE02 with k for different values of the surface tension T.Figure 6
Variation ofE02 with k for different values of the fluid velocities U1 and U2.Figure 7
Variation ofE02 with k for small values of the fluid velocities h1 and h2.Figure 8
Variation ofE02 with k for large values of the fluid depths h1 and h2.Figure2 shows the variation of the electric field E02 versus k for different electrical conductivity values α1andα2 in the two-dimension l=0 and three-dimension l=1,2 cases. It was clear that, for l=0, the stable region increases by increasing α1andα2, while it decreases by increasing the dimension ll=1,2 for small wave number range k≤6 after that the stability does not depend on l since the curves coincide. Also, for any value l and small E02 values, the system is always unstable, while for high E02, the electric field has a stabilizing effect since stable region increases by increasing the electric field E02 and wave number value k. Then, both the electrical conductivites α1andα2 and the electric field E02 have stabilizing effects, while the dimension l has a destabilizing effect only for small wave number value k.Also the effects of the porositym of the medium (see Figure 3) as well as the fluid viscosities μ1 and μ2 (see Figure 4) on the stability are found to be similar to the effect of the electrical conductivites α1andα2 given in Figure 2. Thus, each of the porosity m of the medium and the fluid viscosities μ1 and μ2 has stabilizing effects on the system.In Figure5, we draw E02 versus k for different surface tension values T in two- and three-dimension disturbances cases. It was clear that, when l=0 and for small values of T, the stability increases by increasing T while it decreases by increasing the dimension l for small wave numbers k≤6 just like the previous figures. In contrast, for high values of T, t is seen that the stability increases by increasing T and the dimension l for wave numbers range k≽1. We conclude that, the dimension l plays a dual role (stablizing and destablizing) when there are large variation in surface tension force between the fluids.Figure6 shows the variation of E02 against k for different fluid velocities values U1 and U2 in two- and three-dimension cases. It is shown that, when l=0,1,2, the instability increases by increasing the fluid velocities U1 and U2, and for any fixed values of U1 and U2, the instability also increases by increasing l for small wave number k≤6 values after which l has no effect on the stability. Therefore, both the fluid velocities U1 and U2 and the dimension l have destabilizing effects, while for the high wave number value k, the dimension l has no effect on the stability.Finally, Figures7 and 8 show the effects of the fluid depths h1 and h2 accompanied with the dimension l on the stability, stabilizing the system for small values of h1 and h2 (see Figure 7) while destabilizing it for large values of h1 and h2 (see Figure 8). Thus, the fluid depths and the dimension have a dual role in the system.


## 6. Conclusions

In this paper, new results of bounded flows through porous medium have been presented and analyzed for weakly electrically conducting fluids. Exact analytical solutions have been given for all parameters.Using the normal mode analysis, we obtain quadratic dispersion relation of complex coefficients characterizing the behaviour of the disturbed system. Based on appropriate data selections, we conclude the following:(1)
The permeabilityλ1 of the medium has no effect on the linear stability of the system(2)
The horizontal electric fieldE02, porosity of the medium m, surface tension T, fluid viscosities μ1 and μ2, and electrical conductivites α1andα2 have stabilizing effects or enhance the stability(3)
The fluid velocitiesU1 and U2 (including the case of absence of fluid velocities U1=0 and U2=0 or Rayleigh–Taylor instability) have destabilizing effects or tend to reduce the stability(4)
Both the fluid depthh1 and h2 and the dimension l have a dual role in the system, i.e., stabilizing the system for small values of h1 and h2, while destabilizing it for large values of h1 and h2 (including the case of two semiinfinite fluids).According to the importance of nonlinear effects in EHD phenomena, the describing equations of EHD flow are nonlinear. Therefore, we will discuss the nonlinear stability characteristics for this problem in a subsequent article.



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