Saturated Rayleigh–Taylor instability of an oscillating Couette film flow

Abstract

The nonlinear stability of a two-fluid system consisting of a viscous film bounded above by a heavier and thicker layer, between two horizontal plates, with one of the plates oscillating horizontally about a fixed position, is investigated. An evolution equation governing the thickness of the viscous film is derived. Numerical simulations of this equation on extended spatial intervals demonstrate nonlinear small-amplitude saturation of the Rayleigh–Taylor instability in certain parametric regimes. In the low-frequency time-asymptotic regimes, the averaged properties of the extensive spatio-temporal chaos are not steady, but rather oscillate in time. A quasi-equilibrium theory is proposed in which the low-frequency results are interpreted by building upon the notions developed earlier for the simpler case of a non-oscillatory film governed by the classical, constant-coefficient Kuramoto–Sivashinsky equation. In contrast, the higher-frequency solutions exhibit piecewise linear profiles that have never been encountered in simulations of non-oscillatory films. The amplitude as a function of frequency has a single minimum point which is of order one. Also, preliminary results of numerical simulations of film evolution are given for the large-amplitude parametric regimes. At some parameter values, rupture is observed, similar to the case with no base flow; in other regimes the basic flows succeeds in preventing rupture. The complete characterization of the factors responsible for the particular asymptotic fate of the film, rupture or no rupture, remains an open question.