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Thermal convection during the directional solidification of a pure liquid with variable viscosity

Published online by Cambridge University Press:  21 April 2006

Marc K. Smith
Affiliation:
Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA

Abstract

The onset of a buoyancy-driven instability during the directional solidification of a pure liquid with a strongly temperature-dependent viscosity and an arbitrary Prandtl number is investigated using linear stability theory. The Rayleigh number for this system contains the lengthscale Ls defined as the ratio of the thermal diffusivity of the liquid and the solidification velocity times the density ratio of the two phases. It is independent of the actual depth of the liquid and it reflects the fact that increasing the solidification velocity stabilizes the system. The theory also shows that the difference in material properties between the two phases and the properties of the solidifying interface itself cause the interface to look like a boundary of finite conductivity measured by a wavenumber-dependent Biot number. For large viscosity variations, convection occurs below a stagnant layer which forms just beneath the interface where the liquid is immobilized by its very large viscosity. The thickness of this layer is measured by the natural logarithm of the viscosity contrast in the liquid times the lengthscale Ls. In this limit, the influence of the solidifying boundary is shielded from the bulk liquid by the stagnant layer and so the effect of the Biot number on the critical Rayleigh number is small. However, inertial effects, being associated with the bulk liquid, are very important for small Prandtl numbers of the fluid far from the interface. The model has applications to the solidification of magma chambers or lava lakes and to the material processing of polymeric liquids.

Type
Research Article
Copyright
© 1988 Cambridge University Press

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References

Booker, J. R. 1976 Thermal convection with strongly temperature-dependent viscosity. J. Fluid Mech. 76, 741754.Google Scholar
Caroli, B., Caroli, C., Misbah, C. & Roulet, B. 1985a Solutal convection and morphological instability in directional solidification of binary alloys. J. Phys. Paris 46, 401413.Google Scholar
Caroli, B., Caroli, C., Misbah, C. & Roulet, B. 1985b Solutal convection and morphological instability in directional solidification of binary alloys. II. Effect of the density difference between the two phases. J. Phys. Paris 46, 16571665.Google Scholar
Carslaw, H. S. & Jaeger, J. C. 1959 Conduction of Heat in Solids. Oxford University Press.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Dover.
Coriell, S. R., Cordes, M. R., Boettinger, W. J. & Sekerka, R. F. 1980 Convective and interfacial instabilities during unidirectional solidification of a binary alloy. J. Cryst. Growth 49, 1328.Google Scholar
Currie, I. G. 1967 The effect of heating rate on the stability of stationary fluids. J. Fluid Mech. 29, 337347.Google Scholar
Foster, T. D. 1965 Stability of a homogeneous fluid cooled uniformly from above. Phys. Fluids 8, 12491257.Google Scholar
Gershuni, G. Z. & Zhukhovitskii, E. M. 1976 Convective Stability of Incompressible Fluids. Jerusalem: Keter.
Homsy, G. M. 1973 Global stability of time-dependent flows: impulsively heated or cooled fluid layers. J. Fluid Mech. 60, 129139.Google Scholar
Hurle, D. T. J., Jakeman, E. & Wheeler, A. A. 1982 Effect of solutal convection on the morphological stability of a binary alloy. J. Cryst. Growth 58, 163179.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, 624626.Google Scholar
Jaupart, C. & Parsons, B. 1985 Convective instabilities in a variable viscosity fluid cooled from above. Phys. Earth Planet. Inter. 39, 1432.Google Scholar
Jhaveri, B. S. & Homsy, G. M. 1982 The onset of convection in fluid layers heated rapidly in a time-dependent manner. J. Fluid Mech. 114, 251260.Google Scholar
Mahler, E. G., Schechter, R. S. & Wissler, E. H. 1968 Stability of a fluid layer with time-dependent density gradients. Phys. Fluids 11, 19011912.Google Scholar
McFadden, G. B., Rehm, R. G., Coriell, S. R., Chuck, W. & Morrish, K. A. 1984 Thermosolutal convection during directional solidification. Metall. Trans. A 15, 21252137.Google Scholar
Morris, S. & Canright, D. 1984 A boundary-layer analysis of Bénard convection in a fluid of strongly temperature-dependent viscosity. Phys. Earth Planet. Inter. 36, 355373.Google Scholar
Mullins, W. W. & Sekerka, R. F. 1964 Stability of a planar interface during solidification of a dilute binary alloy. J. Appl. Phys. 35, 444451.Google Scholar
Neitzel, G. P. 1982 Onset of convection in impulsively heated or cooled fluid layers. Phys. Fluids 25, 210211.Google Scholar
Oliver, D. S. & Booker, J. R. 1983 Planform of convection with strongly temperature-dependent viscosity. Geophys. Astrophys. Fluid Dyn. 27, 7385.Google Scholar
Quareni, F., Yuen, D. A., Sewell, G. & Christensen, U. R. 1985 High Rayleigh number convection with strongly variable viscosity: a comprison between mean field and two dimensional solutions. J. Geophys. Res. 90, 1263312644.Google Scholar
Rayleigh, Lord 1916 On convective currents in a horizontal layer of fluid when the higher temperature is on the under side. Phil. Mag. 32, 529546.Google Scholar
Richter, F. M., Nataf, H.-C. & Daly, S. F. 1983 Heat transfer and horizontally averaged temperature of convection with large viscosity variations. J. Fluid Mech. 129, 173192.Google Scholar
Scott, M. R. & Watts, H. A. 1975 SUPORT - A computer code for two-point boundary-value problems via orthonormalization. Sandia Labs, Albuquerque, NM, Rep. SAND 75-0198.Google Scholar
Scott, M. R. & Watts, H. A. 1977 Computational solution of linear two-point boundary-value problems via orthonormalization. SIAM J. Numer. Anal. 14, 4070.Google Scholar
Shaw, H. R., Wright, T. L., Peck, D. L. & Okamura, R. 1968 The viscosity of basaltic magma: an analysis of field measurements in Makaopuhi lava lake, Hawaii. Am. J. Sci. 266, 225264.Google Scholar
Stengel, K. C., Oliver, D. S. & Booker, J. R. 1982 Onset of convection in a variable-viscosity fluid. J. Fluid Mech. 120, 411431.Google Scholar
Young, G. W. & Davis, S. H. 1986 Directional solidification with buoyancy in systems with small segregation coefficient. Phys. Rev. B 34, 33883396.Google Scholar