Author(s)

E. Martın & J.M. Vega

Abstract

We study the weakly nonlinear evolution of Faraday waves in a two dimensional version of a vertically vibrating annular container. 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, steadily travelling waves (which are standing in a moving reference frame), and some more complex attractors. We also consider the effect of surface contamination, modelled byMarangoni 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. (1989). Keywords: Faraday instability, mean flow, weakly nonlinear analysis, Marangoni elasticity. 1 Introduction We consider the parametric excitation of waves at the free surface of a horizontal liquid layer that is being vertically vibrated. If the forcing amplitude exceeds a threshold value, the system exhibits surface waves that are named after

Keywords

Faraday instability, mean flow, weakly nonlinear analysis, Marangoni elasticity.