II-6

Alexander L. Frenkel and David Halpern

University of Alabama

Abstract

We study unsteady Couette-type flows of a viscous film under a heavier liquid, driven by various periodic motions of the tow horizontal plates sandwiching the two-layer system. The parametric conditions are such that the waves (appearing as the result of the Rayleigh-Taylor instability) are long, their slopes are small, and the Stokes approximation holds for the flow. The 3D Navier-Stokes problem for disturbances of the exact flat-film solution (which, in contrast to the previously studied cases, is a "closed flow") is reduced to a nonlinear evolution equation for the disturbance of the film thickness, which has a most tow independent spatial variables. This possible the numerical studies of the 3D wave patterns in spatially extended domains. In particular, we find coherent structures even in certain strange-attractor regimes of the dissipative film flow which is a high-dimensional dynamical system. We find certain Couettte-type flows for which, in contrast to the previously known cases, the growth on any possible disturbance is arrested at a small amplitude-which allows the film to persist indefinitely despite its flowing under the heavier fluid. We discuss the possibility of a similar "life-saving" result for a single film on a ceiling (the ultimate case of the Rayleigh-Taylor instability). Possible experiments to verify the theoretical predictions are discussed as well.