Hostname: page-component-7dd5485656-glrdx Total loading time: 0 Render date: 2025-10-31T00:27:00.727Z Has data issue: false hasContentIssue false
  • English
  • Français

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
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 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.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 302 , 10 November 1995 , pp. 307 - 331
Copyright
© 1995 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 & 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 A new oscillatory instability in the mushy layer during solidification of binary allows. J. Fluid Mech. (submitted).Google Scholar
Bennon, W. D. & Incropera, F. P. 1987 A continuum model for momentum, heat and species transport I n binary solid-liquid phase change systems–I. Model formulation. Intl. J. Heat Mass Transfer 30, 2161–2170.Google Scholar
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 principal. J. Fluid Mech. 30, 625–649.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
Davis, S. H., Müller, U. & Dietsche, C. 1984 Pattern selection in single-component systems coupling Bénard convection and solidification. J. Fluid Mech. 144, 133–151.Google Scholar
Davis, S. H. & Segel, L. A. 1968 Effects of surface curvature and property variation on cellular convection.. Phys. Fluids 11 470–476.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
Golubitsky, M., Swift, J. W. & Knobloch, E. 1984 Symmetries and pattern selection in Rayleigh-Bénard convection. Physica D 10, 249–276.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
Karcher, C. & Müller, U. 1995 Convection in a porous medium with solidification. Fluid Dyn. Res. 15, 25–42.Google Scholar
Krishnamurti, R. 1968 Finite amplitude convection with changing mean temperature. Part. 1. Theory. J. Fluid Mech. 33, 445–455.Google Scholar
Kurz, W. & Fisher D. J. 1989 Fundamentals of Solidification. Trans Tech.
Kuske, R. & Matkowsky, B. J. 1994 On roll, square and hexagonal cellular flames. Eur. J. Appl. Maths. 5, 65–93.Google Scholar
Palm, E. 1960 On the tendency towards hexagonal cells in steady convection.. J. Fluid Mech. 8183–192.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
Roppo, M. N., Davis, S. H. & Rosenblat, S. 1984 Bénard convection with time periodic heating. Phys. Fluids 27, 796–803.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
Scanlon, J. W. & Segel, L. A. 1967 Finite amplitude cellular convection by surface tension. J. Fluid Mech. 30, 149–162.Google Scholar
Segel, L. A. 1965 The non-linear interaction of a finite number of disturbances to a layer of fluid heated from below. J. Fluid Mech. 21, 359–384.Google Scholar
Segel, L. A. 1969 Distant side-walls cause slow amplitude modulation of cellular convection. J. Fluid Mech. 38, 203–224.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 the 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