Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-23T12:54:20.508Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© 2014 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Beavers, G. S. & Joseph, D. D. 1967 Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30, 197–207.Google Scholar
Beckermann, C. & Wang, C. Y. 1995 Multiphase/-scale modeling of alloy solidification. Annual Reviews of Heat Transfer, 6. (ed. Tien, C. L.), pp. 115–198. Begell House.Google Scholar
Conroy, D. & Worster, M. G.2006 Mush–liquid interfaces with cross flow. Geophysical Fluid Dynamics Program Report, Woods Hole Oceanographic Institute.Google Scholar
Copley, S. M., Giamei, A. F., Johnson, S. M. & Hornbecker, M. F. 1970 The origin of freckles in unidirectionally solidified castings. Metall. Trans. 1, 2193–2204.Google Scholar
Le Bars, M. & Worster, M. G. 2006 Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification. J. Fluid Mech. 550, 149–173.Google Scholar
Peppin, S. S. L., Aussillous, P., Huppert, H. E. & Worster, M. G. 2007 Steady-state mushy layers: experiments and theory. J. Fluid Mech. 570, 69–77.CrossRefGoogle Scholar
Rees Jones, D. W. & Worster, M. G. 2013 Fluxes through steady chimneys in a mushy layer during binary alloy solidification. J. Fluid Mech. 714, 127–151.Google Scholar
Schulze, T. P. & Worster, M. G. 1999 Weak convection, liquid inclusions and the formation of chimneys in mushy layers. J. Fluid Mech. 388, 197–215.Google Scholar
Schulze, T. P. & Worster, M. G. 2005 A time-dependent formulation of the mushy-zone free-boundary problem. J. Fluid Mech. 541, 193–202.Google Scholar
Worster, M. G. 1986 Solidification of an alloy from a cooled boundary. J. Fluid Mech. 167, 481–501.Google Scholar
Worster, M. G. 1997 Convection in mushy layers. Ann. Rev. Fluid Mech. 29 (1), 91–122.Google Scholar
Worster, M. G. 2000 Solidification of fluids. Perspectives in Fluid Dynamics: a Collective Introduction to Current Research. (ed. Batchelor, G. K., Moffatt, H. K. & Worster, M. G.), pp. 393–446. Cambridge University Press.Google Scholar