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Extended Stefan problem for the solidification of binary alloys in a sphere

Part of: Asymptotic theory Thermodynamics and heat transfer Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  04 May 2020

FERRAN BROSA PLANELLA ,
COLIN P. PLEASE  and
ROBERT A. VAN GORDER Open the ORCID record for ROBERT A. VAN GORDER [Opens in a new window]
FERRAN BROSA PLANELLA
Affiliation:
1WMG, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK
COLIN P. PLEASE
Affiliation:
2Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
ROBERT A. VAN GORDER
Affiliation:
3Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin 9054, New Zealand email: rvangorder@maths.otago.ac.nz
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Abstract

We study the extended Stefan problem which includes constitutional supercooling for the solidification of a binary alloy in a finite spherical domain. We perform an asymptotic analysis in the limits of large Lewis number and small Stefan number which allows us to identify a number of spatio-temporal regimes signifying distinct behaviours in the solidification process, resulting in an intricate boundary layer structure. Our results generalise those present in the literature by considering all time regimes for the Stefan problem while also accounting for impurities and constitutional supercooling. These results also generalise recent work on the extended Stefan problem for finite planar domains to spherical domains, and we shall highlight key differences in the asymptotic solutions and the underlying boundary layer structure which result from this change in geometry. We compare our asymptotic solutions with both numerical simulations and real experimental data arising from the casting of molten metallurgical grade silicon through the water granulation process, with our analysis highlighting the role played by supercooling in the solidification of binary alloys appearing in such applications.

Keywords

Two-phase Stefan problemsolidification of binary alloysmatched asymptotic analysis

MSC classification

Primary: 80A22: Stefan problems, phase changes, etc. 35R37: Moving boundary problems
Secondary: 34E10: Perturbations, asymptotics 80A20: Heat and mass transfer, heat flow

Information

Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Issue 2 , April 2021 , pp. 242 - 279
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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