Weakly-nonlinear analysis of the Rayleigh–Taylor instability in a vertically vibrated, large aspect ratio container

Abstract

We consider a horizontal liquid layer supported by air in a wide (as compared to depth) container, which is vertically vibrated with an appropriately large frequency, intending to counterbalance the Rayleigh-Taylor instability of the fíat, rigid-body vibrating state. We apply a long-wave, weakly-nonlinear analysis that yields a generalized Cahn-Hilliard equation for the evolution of the fluid interface, with appropriate boundary conditions obtained by a boundary layer analysis. This equation shows that the stabilizing effect of vibration is like that of surface tensión, and is used to analyze the linear stability of the fíat state, and the local bifurcation at the instability threshold.