Abstract
During solidification of a binary alloy at constant velocity vertically upward, thermosolutal convection can occur if the solute rejected at the crystal-melt interface decreases the density of the melt. We assume that the crystal-melt interface remains planar and that the flow field is periodic in the horizontal direction. The time-dependent nonlinear differential equations for fluid flow, concentration, and temperature are solved numerically in two spatial dimensions for small Prandtl numbers and moderately large Schmidt numbers. For slow solidification velocities, the thermal field has an important stabilizing influence: near the onset of instability the flow is confined to the vicinity of the crystal-melt interface. Further, for slow velocities, as the concentration increases, the horizontal wavelength of the flow decreases rapidly — a phenomenon also indicated by linear stability analysis. The lateral in-homogeneity in solute concentration due to convection is obtained from the calculations. For a narrow range of solutal Rayleigh numbers and wavelengths, the flow is periodic in time.
Similar content being viewed by others
References
J. R. Carruthers: inPreparation and Properties of Solid State Materials, W. R. Wilcox and R. A. Lefever, eds., Dekker, New York, NY, 1977, vol. 3, p. 1.
D. T. J. Hurle:Current Topics in Materials Science, E. Kaldis and H. J. Scheel, eds., North-Holland, Amsterdam, 1977, vol. 2, p. 549.
S.M. Pimputkar and S. Ostrach:J. Crystal Growth, 1981, vol. 55, p. 614.
S.R. Coriell, M. R. Cordes, W.J. Boettinger, and R. F. Sekerka:J. Crystal Growth, 1980, vol. 49, p. 13.
S.R. Coriell, M. R. Cordes, W. J. Boettinger, and R. F. Sekerka:Adv. Space Res., 1981, vol. 1, p. 5.
S.R. Coriell and R.F. Sekerka:Physico Chemical Hydrodynamics, 1981, vol. 2, p. 281.
R. J. Schaefer and S. R. Coriell: inMaterials Processing in the Reduced Gravity Environment of Space, G.E. Rindone, ed., North- Holland, Amsterdam, 1982, p. 479.
D. T. J. Hurle, E. Jakeman, and A. A. Wheeler:J. Crystal Growth, 1982, vol. 58, p. 163.
D.T.J. Hurle, E. Jakeman, and A.A. Wheeler:Phys. Fluids, 1983, vol. 26, p. 624.
J. S. Turner:Buoyancy Effects in Fluids, Cambridge University Press, London, England, 1973, chap. 8.
J. Fromm:Methods of Computational Physics, Vol. 3, Fundamental Methods in Hydrodynamics, B. Alder, S. Fernbach, and M. Rotenberg, eds., Academic Press, New York, NY, 1964, p. 345.
E.C. DuFort and S.P. Frankel:Math. Tables and Other Aids to Computation, 1953, vol. 3, p. 135.
R. E. Richtmyer and K. W. Morton:Difference Methods for Initial Value Problems, Interscience Publishers, New York, NY, 1967, p. 211.
P. N. Swarztrauber and R. Sweet:ACM Trans. Math. Soft., 1979, vol. 5, p. 352.
V.G. Smith, W. A. Tiller, and J. W. Rutter:Can. J. Phys., 1955, vol. 33, p. 723.
C. J. Chang and R. A. Brown:J. Crystal Growth, 1983, vol. 63, p. 343.
Author information
Authors and Affiliations
Additional information
Formerly with the Mathematical Analysis Division, Center for Applied Mathematics, National Bureau of Standards, Washington, DC 20234.
This paper is based on a presentation made at the symposium “Fluid Flow at Solid-Liquid Interfaces” held at the fall meeting of the TMS-AIME in Philadelphia, PA on October 5, 1983 under the TMS-AIME Solidification Committee.
Rights and permissions
About this article
Cite this article
McFadden, G.B., Rehm, R.G., Coriell, S.R. et al. Thermosolutal convection during directional solidification. Metall Trans A 15, 2125–2137 (1984). https://doi.org/10.1007/BF02647095
Issue Date:
DOI: https://doi.org/10.1007/BF02647095