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Nonlinear analysis of buoyant convection in binary solidification with application to channel formation

Published online by Cambridge University Press:  26 April 2006

Gustav Amberg  and
G. M. Homsy
Gustav Amberg
Affiliation:
Department of Mechanics, Royal Institute of Technology, S-100 44 Stockholm, Sweden
G. M. Homsy
Affiliation:
Department of Chemical Engineering, Stanford University, Stanford, CA 94305, USA
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Abstract

We consider the problem of nonlinear thermal-solutal convection in the mushy zone accompanying unstable directional solidification of binary systems. Attention is focused on possible nonlinear mechanisms of chimney formation leading to the occurrence of freckles in solid castings, and in particular the coupling between the convection and the resulting porosity of the mush. We make analytical progress by considering the case of small growth Péclet number, δ, small departures from the eutectic point, and infinite Lewis number. Our linear stability results indicate a small O(δ) shift in the critical Darcy-Rayleigh number, in accord with previous analyses. We find that nonlinear two-dimensional rolls may be either sub- or supercritical, depending upon a single parameter combining the magnitude of the dependence of mush permeability on solids fraction and the variations in solids fraction owing to melting or freezing. A critical value of this combined parameter is given for the transition from supercritical to subcritical rolls. Three-dimensional hexagons are found to be transcritical, with branches corresponding to upflow and lower porosity in either the centres or boundaries of the cells. These general results are discussed in relation to experimental observations and are found to be in general qualitative agreement with them.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 252 , July 1993 , pp. 79 - 98
Copyright
© 1993 Cambridge University Press

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