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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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References

Brattkus, K. & Davis, S. H. 1988 Cellular growth near absolute stability. Phys. Rev. B 38, 11452–11460.Google Scholar
Busse, F. H. 1967 The stability of finite amplitude cellular convection and its relation to an extremum principle. J. Fluid Mech. 30, 625–649.Google Scholar
Char, B. W., Geddes, K. O., Gonnet, G. H., Leong, B. L., Monagan, M. B. & Watt, S. M. 1991 Maple V Library Reference Manual. Springer.
Chen, C. F. & Chen, F. 1991 Experimental study of directional solidification of aqueous ammonium chloride solution. J. Fluid Mech. 227, 567–586.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, 2193–2205.Google Scholar
Felicelli, S. D., Heinrich, J. C. & Poirier, D. R. 1991 Simulation of freckles during vertical solidification of binary alloys. Metall. Trans. 22B, 847–859.Google Scholar
Fowler, A. C. 1985 The formation of freckles in binary alloys. IMA J. Appl. Maths 35, 159–174.Google Scholar
Glicksman, M. E., Coriell, S. R. & McFadden, G. B. 1986 Interaction of flows with the crystal—melt interface. Ann. Rev. Fluid Mech. 18, 307–335.Google Scholar
Hellawell, A. 1987 Local convective flows in partly solidified alloys. In Structure and Dynamics of Partially Solidified Systems (ed. D. E. Loper). Martinus Nijhoff.
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
Homsy, G. M. & Sherwood, A. E. 1976 Convective instabilities in porous media with through flow. AIChE J. 22, 168–174.Google Scholar
Huppert, H. 1990 The fluid mechanics of solidification. J. Fluid Mech. 212, 209–240.Google Scholar
Iooss, G. & Joseph, D. D. 1980 Elementary Stability and Bifurcation Theory. Springer.
Joseph, D. D. 1971 Stability of convection in containers of arbitrary shape. J. Fluid Mech. 47, 257–282.Google Scholar
Neilson, D. G. & Incropera, F. P. 1993 Effect of rotation on fluid motion and channel formation during unidirectional solidification of a binary alloy. Intl J. Heat Mass Transfer 36, 489–505.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. A 15, 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 in a mushy layer. Nature 359, 406.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
Wooding, R. 1960 Rayleigh instability of a thermal boundary layer in flow through a porous medium. J. Fluid Mech. 9, 183.Google Scholar
Worster, M. G. 1991 Natural convection in a mushy layer. J. Fluid Mech. 224, 335–359.Google Scholar
Worster, M. G. 1992 Instabilities of the liquid and mushy regions during solidification of alloys. J. Fluid Mech. 237, 649–669.Google Scholar