Understanding the nonlinear behavior of Rayleigh–Taylor instability with a vertical electric field for perfect dielectric fluids

Abstract

It is well known that, influenced only by gravity, the fluid interface is unstable when a light fluid supports a heavy fluid and is stable when a heavy fluid supports a light fluid. The situation becomes much more complicated when a vertical electric field is externally applied to the dielectric fluids. We present a nonlinear perturbation solution for an unstable interface between two incompressible, inviscid, immiscible, and perfectly dielectric fluids in the presence of vertical electric fields and gravity in two dimensions. Our nonlinear stability analysis shows that even when the linear theory indicates that the interface is stable, this system is actually unstable. The destabilization effects of the vertical electric field always dominate when gravity provides stabilization effects. This is true even when the applied vertical electric field is very weak. Analytical expressions for the overall amplitude and velocity of the interface are derived up to an arbitrary order in terms of the initial perturbation amplitude and are displayed explicitly up to the fourth order. A comparison study between the predictions of the nonlinear perturbation solution and the numerical results shows that the derived solutions capture the primary nonlinear behavior of the unstable fluid interface. By analyzing the electrical force at the interface, we provide theoretical explanations for the nonlinear phenomena induced by the vertical electric field.