Nonlinear electrohydrodynamic Kelvin-Helmholtz instability conditions of an interface between two fluids under the effect of a tangential periodic electric field

Abstract

The electrohydrodynamic Kelvin-Helmholtz instability conditions of an interface separating two dielectric streaming fluids stressed by tangential electric field in absence of surface charges on the interface is studied. We use the method of multiple scales to solve nonlinear equations. In the first order problem we obtained the Mathieu's differential equation. For the second order, we obtain the nonhomogeneous Mathieu equation and we use the method of multiple scales to obtain a sequence of equations. In the third order, we obtain the second order differential equation of periodic coefficients. Also, we obtain a formula for surface elevation. The instability conditions are determined.