Electrohydrodynamic instability of dielectric fluid layer between two semi-infinite identical conducting fluids in porous medium

The electrohydrodynamic instability of a plane layer of dielectric fluid which is in hydrostatic equilibrium between two semi-infinite conducting fluids with surface charges in porous media is investigated. The dispersion relation is derived for the general case of Brinkman model for both high medium viscosity and permeability, and by considering the fluid layer is very thin. The limiting case of Darcy model in which the viscous convective term is absent is also investigated. The dispersion relations for both antisymmetrical and symmetrical disturbance cases are derived, respectively. For non-porous medium and absence of the electric field, we found that all waves, in both models, are damped, while for porous medium and sufficiently large electric field values, there will be stability (in Brinkman model), and instability (in Darcy model). It is found (in Brinkman model) that each of the electric fields, porosity of the porous medium, surface tension, and the fluid layer depth have a stabilizing effect, while the fluid viscosity has a destabilizing influence, and the medium permeability has a dual role (stabilizing and then destabilizing) separated by a critical wavenumber value that increases with increasing medium permeability values. It is found (in Darcy model) that the electric field has a destabilizing effect, and both the surface tension and the liquid depth have stabilizing effects, while the fluid viscosity has a dual role (stabilizing and then destabilizing), whereas both the porosity of porous medium and the medium permeability have dual roles (destabilizing and then stabilizing) separated by a constant critical wavenumber value. The effects of these parameters hold, for the symmetrical disturbance case, in both models, more faster than their effects in the corresponding antisymmetrical disturbance ones. Finally, consideration is given to the relevance of the results in explaining the mechanism by which the presence of an electric field in porous medium promotes more readily the coalescence of water droplets on a water surface.