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In this work, the effect of transverse horizontal electric field on the stability of three layers of immiscible liquids is illustrated. The fluids are subjected to a uniform horizontal electric field. Analytical and numerical simulations of this system of linear evolution equations are performed. The solutions of the linearized equations of motion and the boundary conditions lead to deriving two simultaneous Mathieu equations of damping terms having complex coefficients. The effects of the streaming velocity, the permeability of the porous medium, and the electrical properties of the flow on the instability are investigated. In the case of uniform velocity, it is found that electric field has a stabilizing influence on the stability criteria. When the periodicity of the velocity is considered, the method of multiple scales is applied to obtain stability solution for the considered system. It is found that the phenomenon of the dual role is found for increasing the permeability parameter. In addition it is found that the velocity of the middle layer has a destabilizing effect whereas the dielectric constant ratio has an opposite influence to the stability of the fluid layers.

Electrohydrodynamics Instability of Three Periodic Streaming Fluids through Porous Media