Nonlinear Stability of an Electrified Interface Supporting Surface Charges between Two Viscous Fluids

The charged boundary between two viscous fluid media is taken to be under the influence of a vertical constant electric field producing surface charges. We illustrate the solution of the nonlinear perturbation using the method of multiple scales. In the neutral stability the relation between surface charge density and surface viscous force is obtained. Because the viscosity ratio has the same value as the dielectric constant ratio a singularity in the nonlinear coefficient of Schrödinger equation is observed and gradually creates a growth in the instability. Previous works have failed to recognize this singularity by not including the effects of viscosity. The absence of surface viscous force leads to decay in the surface charge density and a growth in the stability occurs. Global stability and local instability are observed in the stability diagrams. The duality mechanism is observed for the electric field, the surface charge density, and the stratified kinematic viscosity. Further, the behavior away from neutral stability is explained by examination of the dynamical system. It is observed that the stabilizing or destabilizing role for the surface charge density has been related to the destabilizing or stabilizing role of the density influence. Copyright 1999 Academic Press.