Electrohydrodynamic instability of two superposed fluids in normal electric fields

We consider the linear stability of two unbounded fluids that are separated by a plane interface, and stressed by initially perpendicular uniform electric field. On each side of the interface there is a Coquette flow. The fluids have different viscosities, densities, and electrical properties and surface tension acts at the interface. The linear stability of the flow is analyzed by deriving the exact dispersion relation in terms of the Airy functions and their integrals, and solving it numerically and asymptotically to find marginal stability curves. The stability of the system depends on ten parameters: the ratio of viscosities, ratio of the densities, surface tension, gravity, ratio of the permitivities, two conductivities, two equilibrium electric fields and velocity of the upper fluid in the unperturbed motion. We investigate the electric charge relaxation effects on the stability of the flow by considering various limiting cases. We also examine the effects of finite charge relaxation times.