The joint influence of normal vibrations and electric field on the stability of a two-layer fluid system

Abstract

The article focuses on the joint influence of normal vibrations and normal electric field on the stability of a system consisting of two flat horizontal fluid layers. Two cases are considered. In the first one, the system is formed by two layers of dielectric fluids (both layers have the same thickness) and, in the second, the system includes a dielectric fluid layer and an ideal electrically conductive fluid layer. The results obtained for these two cases are qualitatively similar. We study two approximations: the low-viscosity approximation and high-frequency vibrations. For the low-viscosity approximation, the so-called electric instability mode and the resonant mode are analyzed. In this case, the short-wavelength and long-wavelength instability modes are considered. In the low viscosity approximation, there occurs a viscous boundary layer, which is analyzed using a fast coordinate. The problem is studied by applying a multiscale method. It is demonstrated that vibrations increase the critical value of the electric field strength for short-wave instability in the presence of an electric mode. In the case of the long-wave instability, vibrations increase the curvature of the neutral curve near the zero value of the wave number. As for the influence of the electric field on resonant modes, it can lead to the splitting of the first resonance into two or three modes, depending on the values of the Weber number. When splitting is absent, the electric field reduces the critical amplitude of the first resonant mode. Otherwise, the effect of the electric field is more complex. In the high-frequency limit, the expressions describing the critical value of the electric field strength are similar to those for the electric mode considered in the low-viscosity approximation for finite vibration frequencies, and no resonant perturbations occur in this case.