Instability of two streaming conducting and dielectric bounded fluids in porous medium under time-varying electric field

In this paper, we consider the instability of the interface between two superposed streaming conducting and dielectric fluids of finite depths through porous medium in a vertical electric field varying periodically with time. A damped Mathieu equation with complex coefficients is obtained. The method of multiple scales is used to obtain an approximate solution of this equation, and then to analyze the stability criteria of the system. We distinguish between the non-resonance case, and the resonance case, respectively. It is found, in the first case, that both the porosity of porous medium, and the kinematic viscosities have stabilizing effects, and the medium permeability has a destabilizing effect on the system. While in the second case, it is found that each of the frequency of the electric field, and the fluid velocities, as well as the medium permeability, has a stabilizing effect, and decreases the value of the resonance point, while each of the porosity of the porous medium, and the kinematic viscosities has a destabilizing effect, and increases the value of the resonance point. In the absence of both streaming velocities and porous medium, we obtain the canonical form of the Mathieu equation. It is found that the fluid depth and the surface tension have a destabilizing effect on the system. This instability sets in for any value of the fluid depth, and by increasing the depth, the instability holds for higher values of the electric potential; while the surface tension has no effect on the instability region for small wavenumber values. Finally, the case of a steady electric field in the presence of a porous medium is also investigated, and the stability conditions show that each of the fluid depths and the porosity of the porous medium ɛ has a destabilizing effect, while the fluid velocities have stabilizing effect. The stability conditions for two limiting cases of interest, the case of purely fluids), and the case of absence of streaming, are also obtained and discussed in detail.     

Integration of linear and some model non-linear equations of motion of incompressible fluids

We suggest a new exact method that allows one to construct solutions to a wide class of linear and some model non-linear hydrodynamic-type systems. The method is based on splitting a system into a few simpler equations; two different representations of solutions (non-symmetric and symmetric) are given. We derive formulas that connect solutions to linear three-dimensional stationary and non-stationary systems (corresponding to different models of incompressible fluids in the absence of mass forces) with solutions to two independent equations, one of which being the Laplace equation and the other following from the equation of motion for any velocity component at zero pressure. To illustrate the potentials of the method, we consider the Stokes equations, describing slow flows of viscous incompressible fluids, as well as linearized equations corresponding to Maxwell's and some other viscoelastic models. We also suggest and analyze a differential-difference fluid model with a constant relaxation time. We give examples of integrable non-linear hydrodynamic-type systems. The results obtained can be suitable for the integration of linear hydrodynamic equations and for testing numerical methods designed to solve non-linear equations of continuum mechanics.     

Wilton ripples between two uniform streaming magnetic fluids

An analysis is made of the small-amplitude capillary-gravity waves which occur on the interface of two incompressible inviscid magnetic fluids of different densities. The waves arise as a result of second harmonic resonance. The fluids moving with uniform velocities parallel to their interface are stressed by an oblique magnetic field. The linear relations between the oblique magnetic field and the instability criteria of the linear waves are analyzed. At the stability region (away from the neutral curve) of the linear theory, a pair of coupled non-linear partial differential equations are presented. On the neutral curve, a pair of coupled non-linear partial differential equations are introduced. The last pair of equations may be regarded as the counterparts of the single Klein-Gordon equation which occurs in the non-resonant case. In all cases, the wave profile and its stability conditions are obtained. These conditions are discussed analytically and graphically.     

Non-linear analysis of creeping flow on the inclined permeable substrate plane subjected to an electric field

The effect of an externally applied electric field on the stability of a thin fluid film over an inclined porous plane is analyzed using linear and non-linear stability analysis in the long wave limit. The principle aim of this study is to illustrate the influence of electric field on the non-linear stability of a thin liquid layer flow down incline substrate when the plane is porous. The driving force for the instability under an electric field is an electrostatic force exerted on the free charges accumulated at the dividing interface. The coupled non-linear evolution equations for the local film thickness and the interfacial charge for two-dimensional disturbances are derived to analyze the effect of long-wave instabilities. The method of multiple scales is applied to obtain approximate solutions and analyze the stability criteria. Numerical simulations of this system of non-linear evolution equations are performed. It is found that the permeability parameter as well as the inclination of the plane plays a destabilizing role in the stability criteria, while the damping influence is observed for increasing of the electrical conductivity in both linear and non-linear behavior.     

Instability through porous media of three layers superposed conducting fluids

The instability properties of streaming superposed conducting fluids through porous media under the influence of uniform magnetic field have been investigated. The system is composed of a middle fluid sheet of finite thickness embedded between two semi-infinite fluids. The fluids are assumed to be incompressible, perfectly conducting and there are weak viscous stresses on the interfaces. The Rayleigh–Taylor and Kelvin–Helmholtz problems have been studied. Such configurations are of relevance in a variety of astrophysical and space configurations. The solutions of the linearized equations of motion together with the boundary conditions lead to deriving the dispersion equation with complex coefficients. The limiting case of the stability of one interface between two fluids has been discussed. The stability criteria are discussed theoretically and numerically in which stability diagrams are obtained. It has been found that the increase of the viscosity coefficient as well as the porosity plays a regular stabilizing role in the stability behavior, while the increase of the fluid velocity plays a destabilizing influence in the stability criteria.     

The effect of normal electric field on the evolution of immiscible Rayleigh-Taylor instability

Manipulation of the Rayleigh-Taylor instability using an external electric field has been the subject of many studies. However, most of these studies are focused on early stages of the evolution. In this work, the long-term evolution of the instability is investigated, focusing on the forces acting on the interface between the two fluids. To this end, numerical simulations are carried out at various electric permittivity and conductivity ratios as well as electric field intensities using Smoothed Particle Hydrodynamics method. The electric field is applied in parallel to gravity to maintain unstable evolution. The results show that increasing top-to-bottom permittivity ratio increases the rising velocity of the bubble while hindering the spike descent. The opposite trend is observed for increasing top-to-bottom conductivity ratio. These effects are amplified at larger electric field intensities, resulting in narrower structures as the response to the excitation is non-uniform along the interface.     

Electrohydrodynamic instability of two superposed Walters B´ viscoelastic fluids in relative motion through porous medium

Summary  The electrohydrodynamic Kelvin–Helmholtz instability of the interface between two uniform superposed viscoelastic (B′ model) dielectric fluids streaming through a porous medium is investigated. The considered system is influenced by applied electric fields acting normally to the interface between the two media, at which there are no surface charges present. In the absence of surface tension, perturbations transverse to the direction of streaming are found to be unaffected by either streaming and applied electric fields for the potentially unstable configuration, or streaming only for the potentially stable configuration, as long as perturbations in the direction of streaming are ignored. For perturbations in all other directions, there exists instability for a certain wavenumber range. The instability of this system can be enhanced (increased) by normal electric fields. In the presence of surface tension, it is found also that the normal electric fields have destabilizing effects, and that the surface tension is able to suppress the Kelvin–Helmholtz instability for small wavelength perturbations, and the medium porosity reduces the stability range given in terms of the velocities difference and the electric fields effect. Finally, it is shown that the presence of surface tension enhances the stabilizing effect played by the fluid velocities, and that the kinematic viscoelasticity has a stabilizing as well as a destabilizing effect on the considered system under certain conditions. Graphics have been plotted by giving numerical values to the parameters, to depict the stability characteristics. Received 27 March 2000; accepted for publication 3 May 2001     

Stability of conducting liquid flowing down an inclined plane at moderate Reynolds number in the presence of constant electromagnetic field

The stability of a conducting viscous film flowing down an inclined plane at moderate Reynolds number in the presence of electromagnetic field is investigated under induction-free approximation. Using momentum integral method a non-linear evolution equation for the development of the free surface is derived. The linear stability analysis of the evolution equation shows that the magnetic field stabilizes the flow whereas the electric field stabilizes or destabilizes the flow depending on its orientation with the flow. The weakly non-linear study reveals that both the supercritical stability and subcritical instability are possible for this type of thin film flow. The influence of magnetic field on the different zones is very significant, while the impact of electric field is very feeble in comparison.