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Hydrodynamical instabilities of thermocapillary flow in a half-zone

Published online by Cambridge University Press:  26 April 2006

Måarten Levenstam  and
Gustav Amberg
Måarten Levenstam
Affiliation:
Department of Mechanics, Royal Institute of Technology, S-100 44 Stockholm, Sweden Present address: Department of Mathematics, Chalmers University of Technology, S-411 22 Göteborg, Sweden.
Gustav Amberg
Affiliation:
Department of Mechanics, Royal Institute of Technology, S-100 44 Stockholm, Sweden
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Abstract

The stability of the flow in a half-zone configuration is analysed with the aid of direct numerical simulation. The work is concentrated on the small Prandtl numbers relevant for typical semiconductor melts. The axisymmetric thermocapillary flow is found to be unstable to a steady non-axisymmetric state with azimuthal wavenumber 2, for a zone with aspect ratio 1. The critical Reynolds number for this bifurcation is 1960. This three dimensional steady solution loses stability to an oscillatory state at a Reynolds number of 6250. For small Prandtl numbers, both bifurcations are seen to be quite insensitive to changes in the Prandtl number, and are thus hydrodynamic in nature. An analogy to the instability of thin vortex rings is made. This analogy suggests a physical mechanism behind the instability and also gives an explanation of how the azimuthal wavenumber of the bifurcated solution is selected. The implications of this for the floating-zone crystal growth process are discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 297 , 25 August 1995 , pp. 357 - 372
Copyright
© 1995 Cambridge University Press

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References

Ben Hadid, H. B. & Roux, B. 1990 Thermocapillary convection in long horizontal layers of low-Prandtl-number melts subject to a horizontal temperature gradient. J. Fluid Mech. 221, 77–103.Google Scholar
Chang, C. E. & Wilcox, W. R. 1976 Analysis of surface tension driven flow in floating zone melting. Int J. Heat Mass Transfer 19, 355–356.Google Scholar
Cröull, A., Müuller-Sebert, W. & Nitsche, R. 1989 The critical Marangoni number for the onset of time-dependent convection in silicon. Materials Res. Bull. 24, 995–1004.Google Scholar
Golub, G. H. & Loan, C. F. VAN 1989 Matrix Computations. The John Hopkins University Press.
Hansbo, P. & Szepessy, A. 1990 A velocity pressure streamline diffusion finite element method for the incompressible Navier-Stokes equations. Computer Meth. Appl. Mech. Engng 84, 175–192.Google Scholar
Kazarinoff, N. D. & Wilkowski, J. S. 1989 A numerical study of marangoni flows in zone-refined silicon crystals. Phys. Fluids A 1, 625–627.Google Scholar
Kazarinoff, N. D. & Wilkowski, J. S. 1990 Bifurcations of numerically simulated thermocapillary flows in axially symmetric float zones. Phys. Fluids A 2, 1797–1807.Google Scholar
Kuhlmann, H. C. & Rath, H. J. 1993 Hydrodynamic instabilities in cylindrical thermocapillary liquid bridges. J. Fluid Mech. 247, 247–274.Google Scholar
Neitzel, G. P., Chang, K. T., Jankowski D. F. & Mittelmann H. D. 1993 Linear stability theory of thermocapillary convection in a model of the float-zone crystal-growth process. Phys. Fluids A 5, 108–114.Google Scholar
Neitzel, G. P., Law, C. C., Jankowski, D. F. & Mittelmann, H. D. 1991 Energy stability of thermocapillary convection in a model of the float-zone crystal-growth process, ii: Nonaxisymmetric disturbances. Phys. Fluids A 3, 2841–2846.CrossRefGoogle Scholar
Preisser, F., Schwabe, D. & Scharmann, A. 1983 Steady and oscillatory thermocapillary convection in liquid columns with free cylindrical surface. J. Fluid Mech. 126, 545–567.Google Scholar
Rupp, R., Müuller, G. & Neumann, G. 1989 Three-dimensional time dependent modelling of the marangoni convection in zone melting configurations for gas. J. Cryst. Growth 97, 34–41.Google Scholar
Shen, Y., Neitzel, G. P., Jankowski, D. F. & Mittelmann, H. D. 1990 Energy stability of thermocapillary convection in a model of the float-zone crystal-growth process. J. Fluid Mech. 217, 639–660.Google Scholar
Shien, J. 1993 A remark on the projection-3 method. Intl J. Numer. Meth. Fluids 16, 249–253.Google Scholar
Smith, M. K. 1986 Instabilities mechanisms in dynamic thermocapillary liquid layers. Phys. Fluids 29, 3182–3186.Google Scholar
Smith, M. K. 1988 The nonlinear stability of dynamic thermocapillary liquid layers. J. Fluid Mech. 194, 391–415.Google Scholar
Smith, M. K. & Davis, S. H. 1983a Instabilities of dynamic thermocapillary liquid layers. Part 1. convective instabilities. J. Fluid Mech. 132, 119–144.Google Scholar
Smith, M. K. & Davis, S. H. 1983b Instabilities of dynamic thermocapillary liquid layers. Part 2. surface-wave instabilities. J. Fluid Mech. 132, 145–162.Google Scholar
Villers, D. & Platten, J. K. 1992 Coupled buoyancy and marangoni convection in acetone: experiments and comparison with numerical simulations. J. Fluid Mech. 234, 487–510.Google Scholar
Widnall, S. & Tsai, C.-Y. 1977 The instability of the thin vortex ring of constant vorticity. Proc. R. Soc. Lond. A 287, 273–305.Google Scholar
Xu, J.-J. & Davis, S. H. 1983 Liquid bridges with thermocapillarity. Phys. Fluids 26, 2880–2886.Google Scholar
Xu, J.-J. & Davis, S. H. 1984 Convective thermocapillary instabilities in liquid bridges. Phys. Fluids 27, 1102–1107.Google Scholar
Zebib, A., Homsy, G. & Meiburg, E. 1985 High marangoni number convection in a square cavity. Phys. Fluids 28, 3467–3476.Google Scholar