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Weakly nonlinear analysis of convection in mushy layers during the solidification of binary alloys

Published online by Cambridge University Press:  26 April 2006

D. M. Anderson  and
M. Grae Worster
D. M. Anderson
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK Present address : Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA.
M. Grae Worster
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
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Abstract

We consider the solidification of a binary alloy in a mushy layer and analyse the system near the onset of buoyancy-driven convection in the layer. We employ a neareutectic approximation and consider the limit of large far-field temperature. These asymptotic limits allow us to examine the rich dynamics of the mushy layer in the form of small deviations from the classical case of convection in a horizontal porous layer of uniform permeability. Of particular interest are the effects of the asymmetries in the basic state and the non-uniform permeability in the mushy layer, which lead to transcritically bifurcating convection with hexagonal planform. We obtain a set of three coupled amplitude equations describing the evolution of small-amplitude convecting states in the mushy layer. These equations are analysed to determine the stability of and competition between two-dimensional roll and hexagonal convection patterns. We find that either rolls or hexagons can be stable. Furthermore, hexagons with either upflow or downflow at the centres can be stable, depending on the relative strengths of different physical mechanisms. We determine how to adjust the control parameters to minimize the degree of subcriticality of the bifurcation and hence render the system globally more stable. Finally, the amplitude equations reveal the presence of a new oscillatory instability.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 302 , 10 November 1995 , pp. 307 - 331
Copyright
© 1995 Cambridge University Press

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