Both steady-state and time-dependent simulations of the two-dimensional Navier-Stokes equations were performed to investigate the influence of surface tension-driven forces on the structure of convective flow in an open crucible Cz oxide melt with a stationary crystal dummy deflected into the fluid. The internal radiative heat transport was ignored and, for computational purposes, the coefficients  and  were allowed to vary within some reasonable ranges of values. The simulation results for  were discussed in details.

For all cases with, the convective flow exhibits an undulating structure beyond the cold plume along the melt centerline and a small secondary vortex (RFV) nearby the crucible bottom. Increasing in  slightly up to its threshold value i.e., the steady-state behavior of the melt changes suddenly for so that the wavy pattern of the flow and the secondary vortex both disappear. This can be shown that the effect is associated with a jump-discontinuity in the magnitude of stream function , and the jump is larger for a higher intensity of the natural convection.

In the interior of the melt, the buoyancy frequency, N as well as the frequency of disturbances and more strongly, the local Richardson number, Ri are decreasing with , so that for the wave-induced shear instability is satisfied for. The internal Froude number,  increases exponentially with  and equals to its critical value at  for which the phase velocity,  of the internal waves is equal to the mean flow velocity, i.e. independent of. For  the subcritical state associated with shorter internal waves, changes abruptly to the super critical state with the waves of longer wavelength compared to the melt characteristic length [1]. In the critical state, the waves decay by a factor  for. The factor  found to be multiplied by ~10² for  implying a strong modification of the flow under the influence of the longitudinal Marangoni waves.

The unsteady-state simulations with for the case  display the effect of thermocapillary forces  on (i) the dynamics of the Kirchhoff-type secondary vortex (RFV) and (ii) its development against the Hadley cell circulation (HCC). During the time, the RFV core area  found to be highly (~3-4 times) increased. The ellipse orientation angle,  meets its maximum value,  at around  when the HCC disappears. In the course of time, the up-going wavy streamlines are more and more flattened towards the crucible hot wall. This can be shown that, the equation , with  and as the ratio of the ellipse semi-axes, is held at around  for which  [2]. The ratio between the straining part of the shear flow and the local vorticity (at the RFV center, found to be increasing with time at. However, the ratio remains smaller than unity suggesting that the fluid particles are entrained into the vertical region of coherent structure. The kinetic energy of the vortex ring found to be highly (~10 times) increased for a slight increment in  during the first steps of this irreversible process.

References

[1] C.P. Lee, Phys. Fluids 10 (1998) 2765.

[2] S. Kida, J. Phys. Soc. Japan 50 (1981) 3517.

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1. Internal Radiation Effect on the Buoyancy-driven Flow pattern of Czochralski Oxide Melt
2. Growth kinetics and crystallization front shape in Cz-process of Nd: YAG, Cr,Ca : YAG and Cr,Ca,Nd : YAG crystals