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Instability of a dense seepage layer on a sloping boundary

Published online by Cambridge University Press:  14 January 2020

Lawrence K. Forbes Open the ORCID record for Lawrence K. Forbes [Opens in a new window] ,
Stephen J. Walters  and
Duncan E. Farrow Open the ORCID record for Duncan E. Farrow [Opens in a new window]
Lawrence K. Forbes*
Affiliation:
School of Mathematics and Physics, University of Tasmania, GPO Box 37, Hobart, Tasmania7001, Australia
Stephen J. Walters
Affiliation:
School of Mathematics and Physics, University of Tasmania, GPO Box 37, Hobart, Tasmania7001, Australia
Duncan E. Farrow
Affiliation:
Department of Mathematics and Statistics, Murdoch University, South Street, Perth, Western Australia6150, Australia
*
†Email address for correspondence: Larry.Forbes@utas.edu.au
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Abstract

When open-cut mines are eventually abandoned, they leave a large hole with sloping sides. The hole fills with rain water, and there is also contaminated run-off from surrounding land, that moves through the rock and eventually through the sloping sides of the abandoned mine. This paper considers a two-dimensional unsteady model motivated by this leaching flow through the rock and into the rain-water reservoir. The stability of the interface between the two fluids is analysed in the inviscid limit. A viscous Boussinesq model is also presented, and a closed-form solution is presented to this problem, after it has been linearized in a manner consistent with Boussinesq theory. That solution suggests that the interfacial zone is effectively neutrally stable as it evolves in time. However, an asymptotic theory in the interfacial region shows the interface to be unstable. In addition, the nonlinear Boussinesq model is solved using a spectral method. Interfacial travelling waves and roll-up are observed and discussed, and compared against the predictions of asymptotic Boussinesq theory.

JFM classification

Convection: Buoyancy-driven instability Geophysical and Geological Flows: Internal waves Mathematical Foundations: Computational methods
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 886 , 10 March 2020 , A21
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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