February 4, 2011
Title: Numerical investigation of Faraday instability
Abstract: In 1831, Farday discovered that when two immiscible fluid layers were subjected to vertical vibration, standing waves would appear at the interface if the oscillation amplitude was sufficiently large. Recent work has shown an astonishing variety of patterns, which has contributed to the current interest in the Faraday problem. We have carried out the first complete three-dimensional simulation of Faraday waves. The algorithm we use combines a projection method to solve the Navier-Stokes equations with a Front-Tracking method to treat the interface. \r\n\r\nThe neutral curves and temporal modes agree with Floquet theory. We then simulated the Faraday problem under the conditions in which square and hexagonal patterns appear. The spatio-temporal spectra of the simulated patterns agree with experiment. Nevertheless, the hexagons destabilize in favour of a an alternating sequence of patterns with different symmetries, suggesting the presence of a homoclinic orbit. We have developed an algorithm which forces the hexagonal symmetry in order to calculate the fixed point of this orbit. Finally, we have carried out a numerical study of the drift instability in the Faraday experiment. A horizontal displacement of initially stationary patterns has been experimentally observed when the oscillation amplitude exceeds a secondary threshold. Our numerical simulations have confirmed this result. Bifurcation diagrams displaying additional instabilities have been constructed, as well as a complementary spatio-temporal spectral analysis.