Hostname: page-component-54dcc4c588-m259h Total loading time: 0 Render date: 2025-10-11T11:41:33.279Z Has data issue: false hasContentIssue false
  • English
  • Français

A new oscillatory instability in a mushy layer 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
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the solidification of a binary alloy in a mushy layer and analyse the linear stability of a quiescent state with specific interest in identifying an oscillatory convective instability. We employ a near-eutectic approximation and consider the limit of large far-field temperature. These asymptotic limits allow us to examine the 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. We consider also the limit of large Stefan number, which incorporates a key balance necessary for the existence of the oscillatory instability. The model we consider here contains no double-diffusive effects and no region in which a statically stable density gradient exists. The mechanism underlying the oscillatory instability we discover is instead associated with a complex interaction between heat transfer, convection and solidification.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 307 , 25 January 1996 , pp. 245 - 267
Copyright
© 1996 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.)

Article purchase

Temporarily unavailable

References

Amberg, G. & Homsy, G. M. 1993 Nonlinear analysis of buoyant convection in binary solidification with application to channel formation. J. Fluid Mech. 252, 79–98.Google Scholar
Anderson, D. & Worster, M. G. 1995 Weakly-nonlinear analysis of convection in a mushy layer during the solidification of binary alloys. J. Fluid Mech. 302, 307–331.Google Scholar
Bennon, W. D. & Incropera, F. P. 1987 A continuum model for momentum, heat and species transport in binary solid–liquid phase change systems–I. Model formulation. Intl J. Heat Mass Transfer 30, 2161–2170.Google Scholar
Chen, C. F. & Chen, F. 1991 Experimental study of directional solidification of aqueous ammonium chloride solution. J. Fluid Mech. 227, 567–586.Google Scholar
Chen, F., Lu, J. W. & Yang, T. L. 1994 Convective instability in ammonium chloride solution directionally solidified from below. J. Fluid Mech. 276, 163–187.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
Emms, P. W. & Fowler, A. C. 1994 Compositional convection in the solidification of binary alloys. J. Fluid Mech. 262, 111–139.Google Scholar
Fowler, A. C. 1985 The formation of freckles in binary alloys. IMA J. Appl. Maths 35, 159–174.Google Scholar
Hills, R. N., Loper, D. E. & Roberts, P. H. 1983 A thermodynamically consistent model of a mushy zone. Q. J. Mech. Appl. Maths 36, 505–539.Google Scholar
Kurz, W. & Fisher, D. J. 1989 Fundamentals of Solidification Trans Tech..Google Scholar
Palm, E., Weber, J. E. & Kvernvold, O. 1972 On steady convection in a porous medium. J. Fluid Mech. 54, 153–161.Google Scholar
Sample, A. K. & Hellawell, A. 1984 The mechanisms of formation and prevention of channel segregation during alloy solidification. Metall. Trans. 15A, 2163–2173.Google Scholar
Sarazin, J. R. & Hellawell, A. 1988 Channel formation in Pb-Sn, Pb-Sb, and Pb-Sn-Sb alloy ingots and comparison with the system NH4Cl-H2O. Metall. Trans. 19A, 1861–1871.Google Scholar
Tait, S., Jahrling, K. & Jaupart, C. 1992 The planform of compositional convection and chimney formation in a mushy layer. Nature 359, 406–408.Google Scholar
Tait, S. & Jaupart, C. 1992 Compositional convection in a reactive crystalline mush and melt differentiation. J. Geophys. Res. 97(B5), 6735–6756.Google Scholar
Voller, V. R. & Brent, A. D. 1989 The modelling of heat, mass and solute transport in solidification systems. Intl J. Heat Mass Transfer 32, 1719–1731.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. 1991 Natural convection in a mushy layer. J. Fluid Mech. 224, 335–359.Google Scholar
Worster, M. G. 1992a The dynamics of mushy layers. In Interactive Dynamics of Convection and Solidification (ed. S. H. Davis, H. E. Huppert, U. Müller & M. G. Worster), pp. 113–138. Kluwer.
Worster, M. G. 1992b Instabilities of the liquid and mushy regions during solidification of alloys. J. Fluid Mech. 237, 649–669.Google Scholar