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Steep capillary-gravity waves in oscillatory shear-driven flows

Published online by Cambridge University Press:  10 November 2009

SHREYAS V. JALIKOP  and
ANNE JUEL
SHREYAS V. JALIKOP*
Affiliation:
School of Mathematics and Manchester Centre for Nonlinear Dynamics, The University of Manchester, Manchester M13 9PL, UK
ANNE JUEL
Affiliation:
School of Mathematics and Manchester Centre for Nonlinear Dynamics, The University of Manchester, Manchester M13 9PL, UK
*
†Email address for correspondence: sjalikop@ma.man.ac.uk
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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 value Wc, a transition to a linear variation in W is 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 for W ≥ Wc. In this strongly nonlinear regime, the wavelength is a function of the product of amplitude and frequency of forcing, whereas for W < 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 with W to reach the capillary length for W = Wc, i.e. the lengthscale for which surface tension forces balance gravitational forces. For W < Wc, gravitational restoring forces dominate, but for W ≥ Wc, the wave development is increasingly defined by localized surface tension effects.

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Type
Papers
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Journal of Fluid Mechanics , Volume 640 , 10 December 2009 , pp. 131 - 150
Copyright
Copyright © Cambridge University Press 2009

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