Gravitational instability of thin gas layer between two thick liquid layers

Abstract

We consider the problem of gravitational instability (Rayleigh-Taylor instability) of a horizontal thin gas layer between two liquid half-spaces (or thick layers), where the light liquid overlies the heavy one. This study is motivated by the phenomenon of boiling at the surface of direct contact between two immiscible liquids, where the rate of the “break-away” of the vapour layer growing at the contact interface due to development of the Rayleigh-Taylor instability on the upper liquid-gas interface is of interest. The problem is solved analytically under the assumptions of inviscid liquids and viscous weightless vapour. These assumptions correspond well to the processes in real systems, e.g., they are relevant for the case of interfacial boiling in the system water- n -heptane. In order to verify the results, the limiting cases of infinitely thin and infinitely thick gas layers were considered, for which the results can be obviously deduced from the classical problem of the Rayleigh-Taylor instability. These limiting cases are completely identical to the well-studied cases of gravity waves at the liquid-liquid and liquid-gas interfaces. When the horizontal extent of the system is long enough, the wavenumber of perturbations is not limited from below, and the system is always unstable.The wavelength of the most dangerous perturbations and the rate of their exponential growth are derived as a function of the layer thickness. The dependence of the exponential growth rate on the gas layer thickness is cubic.