Abstract
In order to understand the characteristics of pure solutocapillary flow in a shallow annular pool subjected to a constant radial solutal gradient, a series of three-dimensional numerical simulations were performed. The annular pool was filled with the toluene/n-hexane mixture fluid with the Schmidt number of 142.8. The inner and outer cylinders were respectively maintained at low and high solutal concentrations. Aspect ratio of the annular pool is fixed at ε = 0.15 or 0.05. Results indicate that the solutocapillary flow is steady and axisymmetric at a small solutal capillary Reynolds number. The surface fluid flows radially from the inner cylinder toward the outer cylinder and a return flow exists near the bottom. With the increase of the solutal capillary Reynolds number, an axisymmetric oscillatory flow firstly appears and then becomes a three-dimensional oscillatory flow at ε = 0.15. Whereas at ε = 0.05 a direct transition from the steady and axisymmetric flow to the three-dimensional oscillatory flow is observed. Three types of the flow instabilities are the standing wave, hydrosolutal wave and source/sink type wave instabilities. Furthermore, the physical mechanism of the flow destabilization is analyzed.
Similar content being viewed by others
References
Abbasoglu, S., Sezai, I.: Three-dimensional modelling of melt flow and segregation during Czochralski growth of GexSi1−x single crystals. Int. J. Therm. Sci. 46(6), 561–572 (2007)
Arafune, K., Hirata, A.: Thermal and solutal Marangoni convection in In-Ga-Sb system. J. Cryst. Growth 197(4), 811–817 (1999)
Arafune, K., Yamamoto, K., Hirata, A.: Interactive thermal and solutal Marangoni convection during compound semiconductor growth in a rectangular open boat. Int. J. Heat Mass Transf. 44(13), 2405–2411 (2011)
Cröll, A., Mitric, A., Aniol, O., Schütt, S., Simon, P.: Solutocapillary convection in germanium-silicon melts. Cryst. Res. Technol. 44(10), 1101–1108 (2009)
Duffar, T. (ed.): Crystal Growth Processes Based on Capillarity: Czochralski, Floating Zone, Shaping and Crucible Techniques. Wiley (2010a)
Duffar, T.: Marangoni convection in crystal growth. In: Crystal Growth Processes Based on Capillarity: Czochralski, Floating Zone, Shaping and Crucible Techniques, pp 413–464. Wiley (2010b)
Ezersky, A.B., Garcimartin, A., Burguete, J., Mancini, H.L., Perez-Garcia, C.: Hydrothermal waves in Marangoni convection in a cylindrical container. Phys. Rev. E 47(2), 1126 (1993)
Garnier, N., Chiffaudel, A., Daviaud, F.: Hydrothermal waves in a disk of fluid. In: Dynamics of Spatio-Temporal Cellular Structures. Springer, New York (2006)
Köllner, T., Schwarzenberger, K., Eckert, K., Boeck, T.: Multiscale structures in solutal Marangoni convection: three-dimensional simulations and supporting experiments. Phys. Fluid 25(9), 092109 (2013)
Lappa, M.: Thermocapillary Convection. In: Thermal convection: patterns, evolution and stability. Wiley, Chchester (2009)
Li, Y.R., Peng, L., Akiyama, Y., Imaishi, N.: Three-dimensional numerical simulation of thermocapillary flow of moderate Prandtl number fluid in an annular pool. J. Cryst. Growth 259(4), 374–387 (2003)
Li, Y.R., Imaishi, N., Azami, T., Hibiya, T.: Three-dimensional numerical oscillatory flow in a thin annular pool of silicon melt. J. Cryst. Growth 260(1), 28–42 (2004)
Li, Y.R., Peng, L., Shi, W.Y., Imaishi, N.: Convective instability in annular pools. FDMP: Fluid Dyn. Mater. Process. 2(3), 153–166 (2006)
Li, Y.R., Peng, L., Shi, W.Y., Imaishi, N.: Bifurcation of thermocapillary convection in a shallow annular pool of silicon melt. Acta Mech. Sin. 23(1), 43–48 (2007)
Lyubimova, T.P., Skuridin, R.V., Faizrakhmanova, I.S.: Thermo- and soluto-capillary convection in the floating zone process in zero gravity conditions. J. Cryst. Growth 303(1), 274–278 (2007)
Lyubimova, T.P., Scuridyn, R.V.: Numerical modelling of three-dimensional thermo- and solutocapillary-induced flows in a floating zone during crystal growth. Eur. Phys. J. Special Topics 192(1), 41–46 (2011)
Minakuchi, H., Takagi, Y., Okano, Y., Gima, S., Dost, S.: A Numerical study of oscillatory thermo-solutal Marangoni convection in a floating half zone. doi:10.1615/ICHMT.2012.ProcSevIntSympTurbHeatTransfPal.2230 10.1615/ICHMT.2012.ProcSevIntSympTurbHeatTransfPal.2230 (2012)
Minakuchi, H., Takagi, Y., Okano, Y., Gima, S., Dost, S.: The relative contributions of thermo-solutal Marangoni convections on flow patterns in a liquid bridge. J. Cryst. Growth 385, 61–65 (2014)
Sauvage, E., Jacoutot, L., Fautrelle, Y., Gagnoud, A., Blumenfeld, L., Favre, E., Daviaud, F.: Comparison between numerical and experimental results on thermoconvective instabilities of a high-Prandtl-number liquid. Phys. Rev. E 76(6), 066307 (2007)
Schwabe, D., Da, X., Scharmann, A.: Unstable flow and solidification speed due to the interaction of thermocapillary and solutocapillary forces in directional thermocapillary and solutocapillary forces in directional solidification. J. Cryst. Growth 166(1), 483–488 (1996)
Schwabe, D., Zebib, A., Sim, B.: Oscillatory thermocapillary convection in open cylindrical annuli. Part 1. Experiments under microgravity. J. Fluid Mech. 491, 239–258 (2003)
Schwabe, D.: Convective instabilities in complex systems with partly free surface. J. Phys.: Conf. Ser. 64(1), 012001 (2007)
Schwarzenberger, K., Köllner, T., Linde, H., Boeck, T., Odenbach, S., Eckert, K.: Pattern formation and mass transfer under stationary solutal Marangoni instability. Adv. Colloid Interface Sci. 206, 344–371 (2014)
Semenov, S.N., Schimpf, M.E.: Molecular thermodiffusion (thermophoresis) in liquid mixtures. Phys. Rev. E 72(4), 041202 (2005)
Shi, W., Imaishi, N.: Hydrothermal waves in differentially heated shallow annular pools of silicone oil. J. Crys. Growth 290(1), 280–291 (2006)
Shi, W.Y., Ermakov, M.K., Li, Y.R., Peng, L., Imaishi, N.: Influence of buoyancy force on thermocapillary convection instability in the differentially heated annular pools of silicon melt. Microgravity Sci. Technol. 21(1), 289–297 (2009)
Singh, R.P., Sinha, C.P.: V Viscosities and activation energies of viscous flow of the binary mixtures of n-hexane with toluene, chlorobenzene and 1-hexanol. J. Chem. Eng. Data 29(2), 132–135 (1984)
Smith, M.K., Davis, S.H.: Instabilities of dynamic thermocapillary liquid layers. Part 1. Convective instabilities. J. Fluid Mech. 132, 119–114 (1983)
Tyn, M.T., Calus, W.F.: Diffusion coefficients in dilute binary liquid mixtures. J. Chem. Eng. Data 20(1), 106–109 (1975)
Witkowski, L.M., Walker, J.S.: Solutocapillary instabilities in liquid bridges. Phys. Fluids 14(8), 2647–2656 (2002)
Zhan, J.M., Chen, Z.W., Li, Y.S., Nie, Y.H.: Three-dimensional double-diffusive Marangoni convection in a cubic cavity with horizontal temperature and concentration gradients. Phys. Rev. E 82(6), 066305 (2010)
Acknowledgments
This work is supported by National Natural Science Foundation of China (grant number 51176209).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, JC., Zhang, L., Li, YR. et al. Three-Dimensional Numerical Simulation of Pure Solutocapillary Flow in a Shallow Annular Pool for Mixture Fluid with High Schmidt Number. Microgravity Sci. Technol. 28, 49–57 (2016). https://doi.org/10.1007/s12217-015-9476-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12217-015-9476-3