Faraday's Instability for Viscous Fluids

We derive an exact equation which is nonlocal in time for the linear evolution of the surface of a viscous fluid, and show that this equation becomes local and of second order in an interesting limit. We use our local equation to study Faraday's instability in a strongly dissipative regime and find a new scenario which is the analog of the Rayleigh-Taylor instability. Analytic and numerical calculations are presented for the threshold of the forcing and for the most unstable mode with impressive agreement with experiments and numerical work on the exact Navier-Stokes equations.