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.