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Stability of lubricated viscous gravity currents. Part 2. Global analysis and stabilisation by buoyancy forces

Published online by Cambridge University Press:  30 May 2019

Katarzyna N. Kowal Open the ORCID record for Katarzyna N. Kowal [Opens in a new window]  and
M. Grae Worster
Katarzyna N. Kowal*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK Trinity College, University of Cambridge, Cambridge CB2 1TQ, UK
M. Grae Worster
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK Trinity College, University of Cambridge, Cambridge CB2 1TQ, UK
*
†Email address for correspondence: [email protected]
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Abstract

The novel viscous fingering instability recently found in the experiments of Kowal & Worster (J. Fluid Mech., vol. 766, 2015, pp. 626–655), involving two superposed currents of viscous fluid, has been shown to originate at the lubrication front when the fluids are of equal density. However, when the densities are unequal, additional buoyancy forces associated with the underlying layer act to suppress this instability and are largest at the lubrication front, which is where the instability originates. In this paper, we investigate the interaction between the mechanism of the instability and the stabilising influence of these buoyancy forces by performing a global and fully time-dependent analysis, which does not use the frozen-time approximation. We determine a critical condition for instability in terms of the viscosity ratio and the density difference between the two layers. Consistently with the local analysis of the companion paper, instabilities occur when the jump in hydrostatic pressure gradient across the lubrication front is negative, or, equivalently, when the intruding fluid is less viscous than the overlying fluid, provided the two fluids are of equal densities. Once there is a non-zero density difference, these driving buoyancy forces suppress the instability for large wavelengths, giving rise to wavelength selection. As the density difference increases, the instability criterion requires higher viscosity ratios for any instability to occur, and the band of unstable wavenumbers becomes bounded. Large enough density differences suppress the instability completely.

JFM classification

Low-Reynolds-number flows: Lubrication theory Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 871 , 25 July 2019 , pp. 1007 - 1027
Copyright
© 2019 Cambridge University Press 

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