Nonlinear Kelvin-Helmholtz Instability of Two Superposed Dielectric Finite Fluids in Porous Medium Under Vertical Electric Fields

Abstract

A weakly nonlinear theory of wave propagation in two superposed dielectric fluids streaming through porous media in the presence of vertical electric field and in the absence of surface charges at their interface is investigated in three dimensions. The method of multiple scales is used to obtain a dispersion relation for the linear problem and a Ginzburg-Landau equation with complex coefficients for the nonlinear problem, describing the behavior of the system. The stability of the system is discussed both analytically and numerically in both cases, and the corresponding stability conditions are obtained. It is found, in the linear case, that the stability criterion is independent of the medium permeability and that the medium porosity, surface tension, and dimension all have stabilizing effects the fluid viscosities, velocities, and depths have destabilizing effects, and the electric field has a dual role in the stability of the system. In the nonlinear analysis, it is found that the electric field has a stabilizing effect in two-dimensional disturbances and destabilizing effect in three-dimensional disturbances cases. The surface tension, fluid depths, and medium porosity have stabilizing effects in both two-and three-dimensional disturbance cases and the fluid viscosities, velocities, and medium permeability have destabilizing effects in both cases, and this stability or instability occurs faster for threedimensional disturbance cases. It is found also that the system is unstable in the absence of fluid velocities or for nonporous media. Finally, the dimension was found to have a dual role (stabilizing as well as destabilizing) in the considered system, while it has a destabilizing effect in the case of nonporous media.

Nonlinear electroviscoelastic potential flow instability theory of two superposed streaming dielectric fluids

“…They found that the electric field plays a dual role in the stability criterion in the sense that it stabilized perturbations having relatively small wavenumbers if the dielectric constant of the upper fluid is smaller than that of the lower fluid and vice versa. For recent developments of this topic, see the investigations of El-Sayed and co-workers [7][8][9][10][11].…”

Wave breaking on the surface of a dielectric liquid in a horizontal electric field

“…The numerical scheme for integrating Equations ( 7) and ( 8) is based on the pseudo-spectral methods; it means that the functions ψ and η are approximated by the finite Fourier series. Consequently, the boundary conditions for Equations (7) and (8) are periodic in space with the period 2 . The total amount of the Fourier harmonics used in calculation presented was N = 2 15 .…”

“…For the numerical integration in time, we use the explicit fourth-order Runge-Kutta method with the step dt = 10 -6 . The initial conditions for (7) and (8) are taken in the form:…”

“…Currently, the nonlinear dynamics of the boundaries of non-conducting liquids in external fields is actively studied both experimentally [3][4][5] and theoretically [6,7]. Thus, the external electromagnetic fields can be used for controlling and stabilizing the dynamics of the fluid boundaries [8,9].…”

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

“…For recent surveys regarding the developments of linear and nonlinear electromagnetic flows in porous media, see refs. [26][27][28][29][30][31][32][33][34][35][36].…”