Streaming Flow Effects in the Nearly Inviscid Faraday Instability

Abstract

We study the weakly nonlinear evolution of Faraday waves in a two dimensional container that is vertically vibrated. It is seen that the surface wave evolves to a drifting standing wave, namely a wave that is standing in a moving reference frame. In the small viscosity limit, the evolution of the surface waves is coupled to a non-oscillatory mean flow that develops in the bulk of the container. A system of equations is derived for the coupled slow evolution of the spatial phase of the surface wave and the streaming flow. These equations are numerically integrated to show that the simplest reflection symmetric steady state (the usual array of counter rotating eddies below the surface wave) becomes unstable for realistic values of the parameters. The new states include limit cycles, drifted standing waves and some more complex attractors.We also consider the effect of surface contamination, modelled by Marangoni elasticity with insoluble surfactant, in promoting drift instabilities in spatially uniform standing Faraday waves. It is seen that contamination enhances drift instabilities that lead to various steadily propagating and (both standing and propagating) oscillatory patterns. In particular, steadily propagating waves appear to be quite robust, as in the experiment by Douady et al., Europhysics Letters, pp. 309-315, 1989