Abstract
The influence of buoyancy force on the thermocapillary convection instability in the annular pools (R i = 20 mm, R o = 40 mm, and depth d ranging from 1 to 10 mm) of silicon melt (Pr = 0.011), differentially heated at the outer wall and cooled at the inner wall, is investigated numerically. The critical Marangoni numbers (Ma c) for the incipience of oscillatory flow are determined by linear stability analysis (LSA) under both microgravity and normal gravity conditions. The results indicate that the buoyancy force destabilizes the thermocapillary convection under different liquid layer depths from 3 to 10 mm. With increasing the layer depth, the critical Ma number, critical azimuthal wave number and critical phase velocity decrease. Some of 3-D simulation results are compared with those of LSA. 3-D results are found consistent with the LSA results except for a case of D = 0.05 where 3-D simulation gives a stationary 3-D flow under a large Ma.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Azami, T., Nakamura, S., Eguchi, M., Hibiya, T.: The role of surface-tension-driven flow in the formation of a surface pattern on a Czochralski silicon melt. J. Cryst. Growth 233, 99–107 (2001)
Ermakov, M.K., Ermakova, M.S.: Linear-stability analysis off thermocapillary convection in liquid bridges with highly deformed free surface. J. Cryst. Growth 266, 160–166 (2004)
Hoyas, S., Herrero, H., Mancho, A.M.: Bifurcation diversity of dynamic thermocapillary liquid layers. Phys. Rev., E 66, 057301 (2002a)
Hoyas, S., Herrero, H., Mancho, A.M.: Thermal convection in a cylindrical annulus heated laterally. J. Phys. A: Math. Gen. 35, 4067–4083 (2002b)
Hoyas, S., Mancho, A.M., Herrero, H., Garnier, N., Chiffaudel, A.: Bénard–Marangoni convection in a differentially heated cylindrical cavity. Phys. Fluids 17, 054104 (2005)
Kakimoto, K., Ozoe, H.: Heat and mass transfer during crystal growth. Comput. Mater. Sci. 10, 127–133 (1998)
Li, Y.-R., Imaishi, N., Azami, T., Hibiya, T.: Three-dimensional oscillatory flow in a thin annular pool of silicon melt. J. Cryst. Growth 260, 28–42 (2004)
Miller, D.C., Pernell, T.L.: Fluid flow patterns in a simulated garnet melt. J. Cryst. Growth 58, 253–260 (1982)
Peng, L., Li, Y.-R., Shi, W.Y., Imaishi, N.: Three-dimensional thermocapillary-buoyancy flow of silicone oil in a differentially heated annular pool. Int. J. Heat Mass Transfer 50, 872–880 (2007)
Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)
Schwabe, D.: Buoyant-thermocapillary-and-pure thermocapillary convective instabilities in Czochralski systems. J. Cryst. Growth 237–239, 1849–1853 (2002)
Schwabe, D., Benz, S.: Thermocapillary flow instabilities in an annulus under microgravity—results of the experiment magia. Adv. Space Rec. 29, 629–638 (2002)
Schwabe, D., Zebib, A., Sim, B.-C.: Oscillatory thermocapillary convection in open cylindrical annuli. Part 1. Experiments under microgravity. J. Fluid Mech. 491, 239–258 (2003)
Shevtsova, V.: Thermal convection in liquid bridges with curved free surfaces: benchmark of numerical solutions. J. Cryst. Growth 280, 632–651 (2005)
Shi, W.Y., Imaishi, N.: Hydrothermal waves in differentially heated shallow annular pools of silicone oil. J. Cryst. Growth 290, 280–291 (2006)
Shi, W.Y., Ermakov, M.K., Imaishi, N.: Effect of pool rotation on thermocapillary convection in shallow annular pool of silicone oil. J. Cryst. Growth 294, 474–485 (2006)
Sim, B.-C., Zebib, A.: Thermocapillary convection in cylindrical liquid bridges and annuli. Comptes Rendus Mecanique 332, 473–486 (2004)
Sim, B.C., Zebib, A., Schwabe, D.: Oscillatory thermocapillary convection in open cylindrical annuli, part 2 simulations. J. Fluid Mech. 491, 259–274 (2003)
Wanschura, M., Shevtsova, V.M., Kuhlmann, H.C., Rath, H.J.: Convective instability mechanisms in thermocapillary liquid bridges. Phys. Fluids 7, 912–925 (1995)
Yi, K.-W., Kakimoto, K., Eguchi, M., Watanabe, M., Shyo, T., Hibiya, T.: Spoke patterns on molten silicon in Czochralski system. J. Cryst. Growth 144, 20–28 (1994)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shi, W., Ermakov, M.K., Li, YR. et al. Influence of Buoyancy Force on Thermocapillary Convection Instability in the Differentially Heated Annular Pools of Silicon Melt. Microgravity Sci. Technol. 21 (Suppl 1), 289–297 (2009). https://doi.org/10.1007/s12217-009-9118-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12217-009-9118-8