Mean flow corrections of secondary instabilities of Faraday waves


S. Ruediger, J. M. Vega, J. Vinals,

Florida State University



Parametrically driven surface waves  (also known as Faraday waves) can be excited on the free surface of  a  fluid layer that is periodically vibrated in the direction normal to the  surface. We built on  earlier research  to study Faraday waves well above onset. Specifically, we studied  traveling wave equations including mean flow effects, and used this  augmented  set of equations to calculate secondary instabilities of basic  periodic patterns, and the transition to spatio-temporal chaos. We focus on a completely phenomenological order parameter model of Faraday waves.  The advantage of the model, as compared to  other approaches  based on amplitude or phase equation, is that it retains the  original  rotational invariance of the fluid layer while remaining simple  enough  to  allow  an analytic determination  of  the stability  of  standing  waves  to  various perturbations. To improve the  previous  treatment  we explicitly allow for mean flows, long wavelength modes  that  can  be  excited non-linearly  by  surface vibration. We have determined the secondary instabilities and identified the usual Eckhaus and TAM lines. We have found that both lines are shifted and reduce the region of stable ideal solutions with increasing strength of the mean flow. We also  address  the transition to spatio-temporal chaos, and its relation to the secondary  instabilities will be described.