The electrohydrodynamics of superimposed fluids subjected to a nonuniform transverse electric field

This study aims to investigate electrohydrodynamics of two superimposed fluids that are confined between a pair of two-dimensional flat plates and are exposed to a sinusoidal electric field in zero gravity. The goal is to identify the parameters that affect the flow structure and interface deformation using a simple closed form solution. The governing electrohydrodynamic equations are solved analytically for Newtonian and immiscible fluids in the framework of leaky-dielectric theory and in the limit of small electric field and fluid inertia. A detailed analysis of the electric and flow fields is presented and it is shown that the electric field induces sinusoidal electrical stresses at the interface, which lead to periodic convection cells. The parameters affecting the sense of flow circulation and strength are investigated and it is shown that the former depends on the relative magnitude of the electric permittivity and conductivity ratios while the latter is controlled by the relative thicknesses of the fluid layers and the ratio of the electric conductivities and viscosities of the fluids. The maximum flow strength is achieved at a relative thickness that is set by the competition between the electric and hydrodynamic effects. For small deformation, the distortion of the interface is examined using a normal stress balance at the interface, and it is shown that the degree of interface deformation scales with the square of the amplitude of the electric potential nonuniformity, while its wavenumber is twice that of the imposed potential nonuniformity. Furthermore, a zero-deformation curve is found, which delineates the region in the permittivity-conductivity space according to the sense of interface deformation. The results show that for certain ranges of fluid layer thicknesses and permittivity ratios, the interface will remain flat, despite the action of the nonuiform field.