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The influence of viscosity on the frozen wave instability: theory and experiment

Published online by Cambridge University Press:  25 July 2007

EMMA TALIB ,
SHREYAS V. JALIKOP  and
ANNE JUEL
EMMA TALIB
Affiliation:
Manchester Centre for Nonlinear Dynamics and School of Mathematics, The University of Manchester, Manchester M13 9PL, UK, [email protected].
SHREYAS V. JALIKOP
Affiliation:
Manchester Centre for Nonlinear Dynamics and School of Mathematics, The University of Manchester, Manchester M13 9PL, UK, [email protected].
ANNE JUEL
Affiliation:
Manchester Centre for Nonlinear Dynamics and School of Mathematics, The University of Manchester, Manchester M13 9PL, UK, [email protected].
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Abstract

We present the results of an experimental and linear stability study of the influence of viscosity on the frozen wave (FW) instability, which arises when a vessel containing stably stratified layers of immiscible liquids is oscillated horizontally. Our linear stability model consists of two superposed fluid layers of arbitrary viscosities and infinite lateral extent, subject to horizontal oscillation. The effect of the endwalls of the experimental vessel is simulated by enforcing the conservation of horizontal volume flux, so that the base flow consists of counterflowing layers.

We perform experiments with four pairs of fluids, keeping the viscosity of the lower layer (ν1) constant, and increasing the viscosity of the upper layer (ν2), so that 1.02 × 102 ≤ N1 = ν2/ν1 ≤ 1.21 × 104. We find excellent quantitative agreement between theory and experiment despite the simple model geometry, for both the critical onset parameter and wavenumber of the FW. We show that the model of lyubimov:1987 (Fluid Dyn. vol. 86, 1987, p. 849), which is valid in the limit of inviscid fluids, consistently underestimates the instability threshold for fluids of equal viscosity, but generally overestimates the threshold for fluids of unequal viscosity. We extend the experimental parameter range numerically to viscosity contrasts 1 ≤ N1 ≤ 6 × 104 and identify four regions of N1 where qualitatively different dynamics occur, which are reflected in the non-monotonic dependence of the most unstable wavenumber and the critical amplitude on N1. In particular, we find that increasing the viscosity contrast between the layers leads to destabilization over a wide range of N1, 10 ≤ N1 ≤ 8 × 103. The intricate dependence of the instability on viscosity contrast is due to considerable changes in the time-averaged perturbation vorticity distribution near the interface.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 584 , 10 August 2007 , pp. 45 - 68
Copyright
Copyright © Cambridge University Press 2007

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