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On the thermodynamic boundary conditions of a solidifying mushy layer with outflow

Published online by Cambridge University Press:  27 November 2014

David W. Rees Jones Open the ORCID record for David W. Rees Jones [Opens in a new window]  and
M. Grae Worster
David W. Rees Jones*
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK Atmospheric, Oceanic and Planetary Physics, Clarendon Laboratory, Department of Physics, University of Oxford, Oxford OX1 3PU, UK
M. Grae Worster
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: reesjones@atm.ox.ac.uk
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Abstract

The free-boundary problem between a liquid region and a mushy layer (a reactive porous medium) must respect both thermodynamic and fluid dynamical considerations. We develop a steady two-dimensional forced-flow configuration to investigate the thermodynamic condition of marginal equilibrium that applies to a solidifying mushy layer with outflow and requires that streamlines are tangent to isotherms at the interface. We show that a ‘two-domain’ approach in which the mushy layer and liquid region are distinct domains is consistent with marginal equilibrium by extending the Stokes equations in a narrow transition region within the mushy layer. We show that the tangential fluid velocity changes rapidly in the transition region to satisfy marginal equilibrium. In convecting mushy layers with liquid channels, a buoyancy gradient can drive this tangential flow. We use asymptotic analysis in the limit of small Darcy number to derive a regime diagram for the existence of steady solutions. Thus we show that marginal equilibrium is a robust boundary condition and can be used without precise knowledge of the fluid flow near the interface.

JFM classification

Multiphase and Particle-laden Flows: Multiphase flow Low-Reynolds-number flows: Porous media Phase change: Solidification/melting
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 762 , 10 January 2015 , R1
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Copyright
© 2014 Cambridge University Press 

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