Nonlinear EHD Stability of the Interfacial Waves of Two Superposed Dielectric Fluids

Abstract

The slow modulation of the interfacial capillary–gravity waves of two superposed dielectric fluids with uniform depths and solid horizontal boundaries, under the influence of a normal electric field and in the absence of surface charges at their interface, is investigated by using the multiple-time scales method. It is found that the complex amplitude of quasi-monochromatic traveling waves can be described by a nonlinear Schrödinger equation in a frame of reference moving with the group velocity. The stability characteristics of a uniform wave train are examined analytically and numerically on the basis of the nonlinear Schrödinger equation, and some limiting cases are recovered. Three cases appear, depending on whether the depth of the lower fluid is equal to, greater than, or less than the depth of the upper fluid. The effect of the normal electric field is determined for the three stability regions of the pure hydrodynamic case. It is found that the normal electric field has a destabilizing influence in the first stability region and a stabilizing effect in the second and third stability regions. Moreover, one new unstable region or two new stable and unstable regions appear, all of which increase when the electric field increases. On the other hand, the complex amplitude of quasi-monochromatic standing waves near the cutoff wavenumber is governed by a similar type of nonlinear Schrödinger equation in which the roles of time and space are interchanged. This equation makes it possible to estimate the nonlinear effect on the cutoff wavenumber.