Nonlinear electrohydrodynamic Rayleigh–Taylor instability. Part 1. A perpendicular field in the absence of surface charges

Abstract

Nonlinear electrohydrodynamic Rayleigh-Taylor instability is investigated. A charge-free surface separating two semi-infinite dielectric fluids influenced by a normal electric field is subjected to nonlinear deformations. We use the method of multiple-scale perturbations in order to obtain uniformly valid expansions near the cutoff wavenumber separating stable from unstable flows. We obtain two nonlinear Schrödinger equations by means of which we can deduce the cutoff wavenumber and analyse the stability of the system. It is found that if a finite-amplitude wave exists then its small modulation is stable. We also obtain the surface elevation for such waves. The electric field plays a dual role in the stability criterion and the dielectric constant plays a distinctive role in this analysis. If the dielectric constant of the upper fluid is smaller than that of the lower fluid the field has a destabilizing effect for large wavenumbers. For relatively smaller wavenumbers the electric field stabilizes considerable parts of the first and second subharmonic regions in the stability diagrams; a result which is in contrast with the linear theory. If the dielectric constant of the upper fluid is larger than that of the lower fluid, then the field is stabilizing for larger values of the wavenumber K′ when ρ is small (ρ is the density ratio) and destabilizing for smaller values of K′.