Mechanism for the Faraday instability in viscous liquids

This report presents a discussion of the wave number selection mechanism in the Faraday instability, which arises when standing waves form on the free surface of a liquid subject to vertical sinusoidal oscillation. The focus is on the case where the viscous effects are strong, i.e., when the wavelength is of the same order of magnitude as the boundary layer thickness at the free surface. We investigate the relationship between the Faraday instability and the Rayleigh-Taylor instability by performing linear stability calculations. In the deep water limit (wavelength much less than liquid depth), our results indicate that the preferred wave number in the Faraday instability is primarily determined through a Rayleigh-Taylor instability. Tn the case of shallow water (wavelength similar to liquid depth), the agreement between the Rayleigh-Taylor and Faraday wave numbers does not appear to be as good, probably due to the interaction between the oscillatory motion of the standing waves and the bottom boundary.