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ARTICLE

Three-dimensional Numerical Simulation of Unsteady Marangoni Convection in the CZ Method using GSMAC-FEM

by Haruhiko Kohno, Takahiko Tanahashi¹

1 Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama, 223-8522 JAPAN, E-mail: kohno@tana.mech.keio.ac.jp; taka@mech.keio.ac.jp

Computer Modeling in Engineering & Sciences 2001, 2(2), 155-170. https://doi.org/10.3970/cmes.2001.002.155

Abstract

Three-dimensional (3D) unsteady numerical simulations are carried out by means of the finite element method (FEM) with the generalized simplified marker and cell (GSMAC) method in silicon melt with a non-deformable free surface with Prandtl number Pr = 1.8534 × 10^-2, Marangoni number Ma = 0.0 - 6.2067 × 10², Grashof number Gr = 7.1104 × 10⁶, and the aspect ratio As = 1.0 in the Czochralski (CZ) method. The flow state becomes unstable earlier by increasing the absolute value of the thermal coefficient of surface tension in the range of σ_T =0.0 - 1.5 × 10^-5N/mK. Although the velocity distribution in the circumferential direction is isotropy in any direction first, its magnitude becomes periodic and has the wavelength equal to 1/8 of the circumference. Then the wavelength doubles, and the flow pattern becomes finally asymmetrical. Moreover, the oscillation of the velocity distribution is observed just under the single crystal, and the amplitude is found to depend on the value of σ_T . After imposing the vertical magnetic field more than 0.05T to the melt from 50s, the flow pattern becomes restored to symmetry. But the instability remains under the single crystal and it indicates that the influence of Marangoni convection can not be neglected in the crystal growing process.

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Keywords

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