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 three-dimensional 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.