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Thermocapillary free boundaries in crystal growth

Published online by Cambridge University Press:  21 April 2006

C. Cuvelier  and
J. M. Driessen
C. Cuvelier
Affiliation:
Delft University of Technology, Department of Mathematics and Informatics, Delft, The Netherlands
J. M. Driessen
Affiliation:
Delft University of Technology, Department of Mathematics and Informatics, Delft, The Netherlands Present address: Koninklijke/Shell-Laboratorium, Amsterdam, Shell Research bv, Amsterdam, The Netherlands
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Abstract

In this paper a two-dimensional free boundary arising from the steady thermo-capillary flow in a viscous incompressible fluid is studied numerically. The problem is considered in the context of the open-boat crystal-growth technique. The motion of the fluid is governed by the Navier-Stokes equations coupled with the heat equation. The problem is solved numerically by a finite-element-method discretization. Three iterative methods are introduced for the computation of the free boundary. The non-dimensional form of the problem gives rise to the following characteristic parameters: Reynolds, Grashof, Prandtl, Marangoni, Bond, Ohnesorge, Biot numbers. The influence of these parameters on the flow field, the temperature distribution and the shape of the free boundary is studied.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 169 , August 1986 , pp. 1 - 26
Copyright
© 1986 Cambridge University Press

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References

Allain, G. 1984 Un problèms de Navier-Stokes avec surface libre et tension superficielle. In Proc. Colloque International Problèmes à Frontières Libres, Maubuisson (to appear).
Barbe, C. 1967 Etude de la tension superficielle des liquides purs et des mélanges. Synthèse des éléments bibliographiques. Air Liquide Rapport 57–67 CB/IV.
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Beale, J. T. 1984 Large-time regularity of viscous surface waves. Arch. Rat. Mech. Anal, 84, 307–352.Google Scholar
Brice, J. C. 1985 The Growth of Crystals from the Melt. North-Holland.
Chang, C. J. & Brown, R. A. 1983 Radial segregation induced by natural convection and melt/solid interface shape in vertical Bridgman growth. J. Cryst. Growth 63, 276–284.Google Scholar
Chin, L. Y. & Carlson, F. M. 1983 Finite element analysis of the control of interface shape in Bridgman crystal growth. J. Cryst. Growth 63, 561–567.Google Scholar
Crouzeix, M. & Raviart, P. A. 1973 Conforming and non-conforming finite element methods for solving the stationary Stokes equations. RAIRO 7, 33–76.Google Scholar
Curruthers, J. R. 1977 Crystal growth in a low gravity environment. J. Cryst. Growth 42, 379–385.Google Scholar
Cuvelier, C. 1982 On the solution of capillary free boundary problems governed by the Navier-Stokes equations. Rept. 82–09, Delft University of Technology.
Cuvelier, C. 1985a A time dependent free boundary governed by the Navier-Stokes equations. Lecture Notes in Physics, vol. 218, pp. 170–174. Springer.
Cuvelier, C. 1985b A capillary free boundary problem governed by the Navier-Stokes equations. Comput. Meths. Appl. Mech. Engng 48, 45–80.Google Scholar
Cuvelier, C., Segal, A. & van Steenhoven, A. A. 1986 Finite-Element Methods and Navier-Stokes Equations. Reidel.
Dennis, J. E. & More, J. 1977 Quasi-Newton methods, motivation and theory. SIAM Rev. 19, 46–89.Google Scholar
Derby, J. J. & Brown, R. A. 1983 Finite element analysis of a thermal-capillary model for liquid encapsulated Czochralski growth. J. Electrochem. Soc. February, 470–482.Google Scholar
Driessen, J. M. 1984 Effects of non-uniform surface tension in fluid flow: Marangoni effect. Master's thesis, Delft University of Technology.
Engelman, M. S., Strang, G. & Bathe, K. J. 1981 The application of quasi-Newton methods in fluid mechanics. Intl J. Num. Meth. Engng 17, 707–718.Google Scholar
Engelman, M. S. 1984 Fluid Dynamics Analysis Package FIDAP. Fluid Dynamics Int.
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.
Guibert, J. P., Huynh, H. T. & Marce, J. L. 1976 Etude générale de l'influence des liquides contenus dans les satellites Rep. ESTEC 2720.
Hardy, S. C. 1985 The surface tension for liquid gallium. J. Cryst. Growth 71, 602–606.Google Scholar
Hartman, P. (ed.) 1973 Crystal Growth: An Introduction. North-Holland.
Haynes, J. M. 1979 Fundamental and applied aspects of fluid physics under microgravity. In Proc. 3rd Symp. on Materials in Space, Grenoble, ESA SP-142 pp. 275–279.
Hurle, D. T. J. & Jakeman, E. 1981 Introduction to the techniques of crystal growth. PCH: Physico Chemical Hydrodynamics 2, 237–244.Google Scholar
Jones, I. P. & Thomson, C. P. (ed.) 1981 Contribution in numerical solutions for a comparison problem in natural convection in an enclosed cavity. AERE Harwell.
Kistler, S. F. & Scriven, L. E. 1984 Coating flow theory by finite element and asymptotic analysis of the Navier-Stokes system. Intl J. Numer. Meth. Fluids 4, 207–229.Google Scholar
Landau, L. D. & Lifchitz, E. M. 1963 Mécanique de Fluides. Editions MIR Moscou.
Langlois, W. E. 1981 Convection in Czochralski growth melts. PCH: Physico Chemical Hydrodynamics 2, 245–261.Google Scholar
Langlois, W. E. 1982 A parameter sensitivity study for Czochralski bulk flow of silicon. J. Cryst. Growth 56, 15–19.Google Scholar
Lions, J. L. 1973 Perturbations singulières dans les problèmes aux limites et en contrôle optimal. Lecture Notes in Mathematics, vol. 323. Springer.
Moiseev, N. N. & Rumyantsev, V. V. 1968 Dynamic Stability of Bodies Containing Fluid. Springer.
Ostrach, S. 1976 Convection phenomena of importance for materials processing in space. COSPAR Symposium on Materials Sciences in Space, Philadelphia, pp. 2–32.
Ostrach, S. 1977 Motion induced by capillarity, PCH: Physico Chemical Hydrodynamics 2, 571–589.Google Scholar
Ostrach, S. 1979 Convection due to surface-tension gradients. Space Res. 19, 563–570.Google Scholar
Ostrach, S. 1982 Low-gravity fluid flows. Ann. Rev. Fluid Mech. 14, 313–345.Google Scholar
Pimputkar, S. H. & Ostrach, S. 1981 Convective effects in crystals grown from melt. J. Cryst. Growth 55, 614–646.Google Scholar
Pukhnachev, V. V. 1974 Problems with free boundary for the Navier-Stokes equations. Ph.D. thesis, Novosibirsk.
Ruschak, K. 1980 A method for incorporating free boundaries with surface tension in finite element fluid-flow simulations. Intl J. Num. Meth. Engng 15, 639–648.Google Scholar
Saito, H. & Scriven, L. E. 1981 Study of coating flow by the finite element method. J. Comp. Phys. 42, 53–76.Google Scholar
Schwabe, D. 1981 Marangoni effects in crystal growth melts. PCH: Physico Chemical Hydrodynamics 2, 263–280.Google Scholar
Schwabe, D. & Scharmann, A. 1981 Marangoni convection in open boat and crucible, J. Cryst. Growth 52, 435–449.Google Scholar
Segal, A. 1979 A Finite Element Package AFEP. Rep. Delft University of Technology.
Solonnikov, V. A. 1979 Solvability of the boundary value problem describing the motion of a viscous incompressible capillary fluid in an open vessel in the two dimensional case. Izv. Akad. Nauk. SSR 43, 203–236.Google Scholar
Witomski, P. 1977 Modélisation et étude numérique d'une expérience de croissance cristalline, Thèse Mathématiques Appliquées Grenoble.