Faraday's instability in viscous fluid

We find a quantitative approximation which explains the appearance and amplification of surface waves in a highly viscous fluid when it is subjected to vertical accelerations (Faraday's instability). Although stationary surface waves with frequency equal to half of the frequency of the excitation are observed in fluids of different kinematical viscosities we show here that the mechanism which produces the instability is very different for a highly viscous fluid as compared with a weakly viscous fluid. This is achieved by deriving an exact equation for the linear evolution of the surface which is non-local in time. We show that for a highly viscous fluid this equation becomes local and of second order and is then a Mathieu equation which is different from the one found for weak viscosity. Analysing the new equation we find an intimate relation with the Rayleigh-Taylor instability.