Nearly inviscid Faraday waves in containers with broken symmetry

Abstract

In the weakly inviscid regime parametrically driven surface gravity-capillary waves generate oscillatory viscous boundary layers along the container walls and the free surface. Through nonlinear rectification these generate Reynolds stresses which drive a streaming flow in the nominally inviscid bulk; this flow in turn advects the waves responsible for the boundary layers. The resulting system is described by amplitude equations coupled to a Navier-Stokes-like equation for the bulk streaming flow, with boundary conditions obtained by matching to the boundary layers, and represents a novel type of pattern-forming system. The coupling to the streaming flow is responsible for various types of drift instabilities of standing waves, and in appropriate regimes can lead to the presence of relaxations oscillations. These are present because in the nearly inviscid regime the streaming flow decays much more slowly than the waves. Two model systems, obtained by projection of the Navier-Stokes-like equation onto the slowest mode of the domain, are examined to clarify the origin of this behavior. In the first the domain is an elliptically distorted cylinder while in the second it is an almost square rectangle. In both cases the forced symmetry breaking results in a nonlinear competition between two nearly degenerate oscillatory modes. This interaction destabilizes standing waves at small amplitudes and amplifies the role played by the streaming flow. In both systems the coupling to the streaming flow triggered by these instabilities leads to slow drifts along slow manifolds of fixed points or periodic orbits of the fast system, and generates behavior that resembles bursting in excitable systems. The results are compared to experiments. © 2005 IOP Publishing Ltd.