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On small-time similarity-solution behaviour in the solidification shrinkage of binary alloys

Part of: Basic methods in fluid mechanics Phase transformations in solids

Published online by Cambridge University Press:  17 April 2020

M. ASSUNÇÃO Open the ORCID record for M. ASSUNÇÃO [Opens in a new window] ,
M. VYNNYCKY Open the ORCID record for M. VYNNYCKY [Opens in a new window]  and
S.L. MITCHELL
M. ASSUNÇÃO
Affiliation:
Department of Applied Mathematics and Statistics, Institute of Mathematical and Computer Sciences, University of São Paulo at São Carlos, PO Box 668, São Carlos, 13560-970SP, Brazil, email: [email protected]
M. VYNNYCKY
Affiliation:
Department of Mathematics and Statistics, University of Limerick, LimerickV94 T9PX, Ireland
S.L. MITCHELL
Affiliation:
Department of Mathematics and Statistics, University of Limerick, LimerickV94 T9PX, Ireland
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Abstract

In the one-dimensional solidification of a binary alloy undergoing shrinkage, there is a relative motion between solid and liquid phases in the mushy zone, leading to the possibility of macrosegregation; thus, the problem constitutes an invaluable benchmark for the testing of numerical codes that model these phenomena. Here, we revisit an earlier obtained solution for this problem, that was posed on a semi-infinite spatial domain and valid for the case of low superheat, with a view to extending it to the more general situation of a finite spatial domain, arbitrarily large superheat and both eutectic and non-eutectic solidification. We find that a similarity solution is available for short times which contains a boundary layer on the liquid side of the mush–liquid interface; this solution is believed to constitute the correct initial condition for the subsequent numerical solution of the full non-similar problem, which is deferred to future work.

Keywords

Moving boundary problemsdynamics of phase boundariesdimensional analysis and similarity

MSC classification

Primary: 76M55: Dimensional analysis and similarity
Secondary: 74N20: Dynamics of phase boundaries
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Issue 2 , April 2021 , pp. 199 - 225
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

†

The first author acknowledges the financial support of Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) for the researcher grant (grant number 2016/12678-0). The second author acknowledges the financial support of FAPESP for the award of a visiting researcher grant (grant number 2018/07643-8).

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