Steep capillary-gravity waves in oscillatory shear-driven flows

Abstract

We study steep capillary-gravity waves that form at the interface between two stably stratified layers of immiscible liquids in a horizontally oscillating vessel. The oscillatory nature of the external forcing prevents the waves from overturning, and thus enables the development of steep waves at large forcing. They arise through a supercritical pitchfork bifurcation, characterized by the square root dependence of the height of the wave on the excess vibrational Froude number (W, square root of the ratio of vibrational to gravitational forces). At a critical valueWc, a transition to a linear variation inWis observed. It is accompanied by sharp qualitative changes in the harmonic content of the wave shape, so that trochoidal waves characterize the weakly nonlinear regime, but ‘finger’-like waves form forW≥Wc. In this strongly nonlinear regime, the wavelength is a function of the product of amplitude and frequency of forcing, whereas forW<Wc, the wavelength exhibits an explicit dependence on the frequency of forcing that is due to the effect of viscosity. Most significantly, the radius of curvature of the wave crests decreases monotonically withWto reach the capillary length forW=Wc, i.e. the lengthscale for which surface tension forces balance gravitational forces. ForW<Wc, gravitational restoring forces dominate, but forW≥Wc, the wave development is increasingly defined by localized surface tension effects.