Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-05-14T09:36:16.364Z Has data issue: false hasContentIssue false
  • English
  • Français

Natural convection in a mushy layer

Published online by Cambridge University Press:  26 April 2006

M. Grae Worster
M. Grae Worster
Affiliation:
Department of Engineering Sciences & Applied Mathematics and Department of Chemical Engineering, Northwestern University, Evanston, IL 60208, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Governing equations for a mushy layer are analysed in the asymptotic regime Rm [Gt ] 1, where Rm is an appropriately defined Rayleigh number. A model is proposed in which there is downward flow everywhere in the mushy layer except in and near localized chimneys, which are characterized by having zero solid fraction. Upward, convective flow within the chimneys is driven by compositional buoyancy. The radius of each chimney is determined locally by thermal balances within a boundary layer that surrounds it. Simple solutions are derived to determine the structure of the mushy layer away from the immediate vicinity of chimneys in order to demonstrate the gross effects of convection upon the solidification within the layer.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 224 , March 1991 , pp. 335 - 359
Copyright
© 1991 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.)

References

Batchelor, G. K.: 1974 Transport properties of two-phase materials with random structure. Ann. Rev. Fluid Mech. 6, 227–255.Google Scholar
Chen, F. & Chen, C. F., 1988 Onset of finger convection in a horizontal porous layer underlying a fluid layer. Trans. ASMEC: J. Heat Transfer 110, 403–407.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
Fearn, D. R., Loper, D. E. & Roberts, P. H., 1981 Structure of the Earth's inner core. Nature 292, 232–233.Google Scholar
Flemings, M. C.: 1981 Process modeling. In Modeling of Casting and Welding Processes (ed. H. D. Brody & D. Arpelian), pp. 533–548. Metallurgical Society, AIME.
Fowler, A. C.: 1985 The formation of freckles in binary alloys. IMA J. Appl. Maths 35, 159–174.Google Scholar
Hills, R. N., Lopeh, D. E. & Roberts, P. H., 1983 A thermodynamically consistent model of a mushy zone. Q. J. Mech. Appl. Maths 36, 505–539.Google Scholar
Howard, L. N.: 1966 Convection at high Rayleigh number. Proc. Eleventh Intl Congress Applied Mechanics, Munich (ed. H. Görtler), pp. 1109–1115. Springer.
Huppert, H. E. & Worsteb, M. G., 1985 Dynamic solidification of a binary melt. Nature 314, 703–707.Google Scholar
Hurle, D. T. J., Jakeman, E. & Wheeler, A. A., 1983 Hydrodynamic stability of the melt during solidification of a binary alloy. Phys. Fluids 26, 624–626.Google Scholar
Kerr, R. C., Woods, A. W., Worster, M. G. & Huppert, H. E., 1989 Disequilibrium and macrosegregation during solidification of a binary melt. Nature 340, 357–362.Google Scholar
Kerr, R. C., Woods, A. W., Worster, M. G. & Huppert, H. E., 1990a Solidification of an alloy cooled from above. Part 1. Equilibrium growth. J. Fluid Mech. 216, 323–342.Google Scholar
Kerr, R. C., Woods, A. W., Worster, M. G. & Huppert, H. E., 1990b Solidification of an alloy cooled from above. Part 2. Non-equilibrium interfacial kinetics. J. Fluid Mech. 217, 331–348.Google Scholar
Kerr, R. C., Woods, A. W., Worster, M. G. & Huppert, H. E., 1990a Solidification of an alloy cooled from above. Part 3. Compositional stratification within the solid. J. Fluid Mech. 218, 337–354.Google Scholar
Lighthill, M. J.: 1953 Theoretical considerations on free convection in tubes. Q. J. Mech. Appl. Maths 6, 398–439.Google Scholar
Loper, D. E.: 1983 Structure of the inner core boundary. Geophys. Astrophys. Fluid Dyn. 25, 139–155.Google Scholar
Roberts, P. H. & Loper, D. E., 1983 Towards a theory of the structure and evolution of a dendrite layer. In Stellar and Planetary Magnetism (ed. A. M. Soward), pp. 329–349. Gordon & Breach.
Sarazin, J. R. & Hellawell, A., 1988 Channel formation in Pb-Sn and Pb-Sn-Sb alloy ingots and comparison with the system NH4C1-H2 0. Metall. Trans. 19a, 1861–1871.Google Scholar
Tait, S. & Jaupart, C., 1989 Compositional convection in viscous melts. Nature 338, 571–574.Google Scholar
Turner, J. S.: 1979 Buoyancy Effects in Fluids. Cambridge University Press.
Turner, J. S., Huppert, H. E. & Sparks, R. S. J. 1986 Komatiites II: experimental and theoretical investigations of post-emplacement cooling and crystallization. J. Petrol. 27, 397–437.CrossRefGoogle Scholar
Woods, A. W. & Huppert, H. E., 1989 The growth of compositionally stratified solid by cooling a binary alloy from below. J. Fluid Mech. 199, 29–53.Google Scholar
Worster, M. G.: 1986 Solidification of an alloy from a cooled boundary. J. Fluid Mech. 167, 481–501.Google Scholar