Abstract
Marangoni convection is induced by the variation of surface tension along a free surface, which depends not only on temperature but also concentration. However, the onset of thermo-solutal Marangoni convection in a liquid bridge system is still unknown. Here, we perform a global linear stability analysis to determine the theoretical onset of the Marangoni convection occurring in the half-zone liquid bridge of the floating zone method of SixGe1−x crystal growth. The cylindrical liquid bridge is heated from the bottom and the highest Silicon concentration is on the top. The thermal and solutal Marangoni forces are in the same direction in this configuration. The stability diagram of the axisymmetric base flow is obtained by solving the large-scale eigenvalue problem using a Jacobian-free Arnoldi method. Oscillatory disturbance patterns appear with different azimuthal wavenumbers for unstable eigenmodes. The present linear stability analysis results explain our previous numerical simulation results.
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References
Arnoldi, WE: The principle of minimized iterations in the solution of the matrix eigenvalue problem. Q. Appl. Math. 9, 17–29 (1951). https://doi.org/10.1090/qam/42792
Carotenuto, L, Castagnolo, D, Albanese, C, Monti, R: Instability of thermocapillary convection in liquid bridges. Phys Fluids 10, 555–565 (1998). https://doi.org/10.1063/1.869583
Chen, G, Lizée, A, Roux, B: Bifurcation analysis of the thermocapillary convection in cylindrical liquid bridges. J Cryst Growth 180, 638–647 (1997). https://doi.org/10.1016/S0022-0248(97)00259-5
Gómez, F, Gómez, R, Theofilis, V: On three-dimensional global linear instability analysis of flows with standard aerodynamics codes. Aerospace Sci Technol 32, 223–234 (2014). https://doi.org/10.1016/j.ast.2013.10.006
Imaishi, N, Yasuhiro, S, Akiyama, Y, Yoda, S: Numerical simulation of oscillatory marangoni flow in half-zone liquid bridge of low Prandtl number fluid. J Cryst Growth 230, 164–171 (2001). https://doi.org/10.1016/S0022-0248(01)01332-X
Leypoldt, J, Kuhlmann, HC, Rath, HJ: Three-dimensional numerical simulation of thermocapillary flows in cylindrical liquid bridges. J Fluid Mech 414, 285–314 (2000). https://doi.org/10.1017/S0022112000008570
Minakuchi, H, Okano, Y, Dost, S: A three-dimensional numerical simulation study of the Marangoni convection occurring in the crystal growth of SixGe1−x by the float-zone technique in zero gravity. J Cryst Growth 266, 140–144 (2004). https://doi.org/10.1016/j.jcrysgro.2004.02.038
Minakuchi, H, Takagi, Y, Okano, Y, Mizoguchi, K, Gima, S, Dost, S: A grid refinement study of half-zone configuration of the Floating Zone growth system. Journal of Advanced Research in Physics 3(1), 011201 (2012)
Minakuchi, H, Yoshino, T, Okano, Y: The relative contributions and control of thermo-solutal Marangoni convections on flow patterns in a liquid bridge (in Japanese), Proc of The 63rd Nat Cong of Theoretical & Applied Mechanics, pp 055–25 (2014)
Minakuchi, H, Okano, Y, Dost, S: Effect of thermo-solutal Marangoni convection on the azimuthal wave number in a liquid bridge. J Cryst Growth 468, 502–505 (2017). https://doi.org/10.1016/j.jcrysgro.2016.09.028
Minakuchi, H, Okano, Y, Dost, S: The hysteresis phenomena of flow patterns due to thermal and solutal Marangoni convections in a liquid bridge under zero gravity. Fluid Mechanics Research International Journal 2(1), 00018 (2018). https://doi.org/10.15406/fmrij.2018.02.00018
Mo, DM, Ruan, DF: Linear-Stability Analysis of Thermocapillary-Buoyancy Convection in an Annular Two-Layer System with Upper Rigid Wall Subjected to a Radial Temperature Gradient. Microgravity Science and Technology 31, 293–304 (2019). https://doi.org/10.1007/s12217-019-9692-3
Rupp, R, Müller, G, Neumann, G: Three-dimensional time dependent modelling of the marangoni convection in zone melting configurations for GaAs. J Cryst Growth 97, 34–41 (1989). https://doi.org/10.1016/0022-0248(89)90244-3
Saad, Y: Variations on Arnoldi’s method for computing eigenelements of large unsymmetric matrices. Linear Algebra and its Applications 34, 269–295 (1980). https://doi.org/10.1016/0024-3795(80)90169-X
Schwabe, D: Thermocapillary liquid bridges and Marangoni convection under microgravity - Results and lessons learned. Microgravity Science and Technology 26(1), 1–10 (2014). https://doi.org/10.1007/s12217-014-9358-0
Theofilis, V: Global linear instability. Ann Rev Fluid Mech 43, 319–352 (2011). https://doi.org/10.1146/annurev-fluid-122109-160705
Wanschura, M, Shevtsova, VM, Kuhlmann, HC, Rath, HJ: Convective instability mechanisms in thermocapillary liquid bridges. Phys Fluids 7, 912–925 (1995). https://doi.org/10.1063/1.868567
Zou, Y, Huang, H, Zhu, G, Zhou, X: Effect of Rotating Magnetic Field on Thermal Convection and Dopant Transport in Floating-Zone Crystal Growth. Microgravity Science and Technology. https://doi.org/10.1007/s12217-019-09776-w (2020)
Acknowledgements
This research partly used computational resources of Research Institute for Information Technology, Kyushu University. This work was partially supported by JSPS KAKENHI Grant Number JP19K22015
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Mendis, R.L.A., Sekimoto, A., Okano, Y. et al. Global Linear Stability Analysis of Thermo-solutal Marangoni Convection in a Liquid Bridge Under Zero Gravity. Microgravity Sci. Technol. 32, 729–735 (2020). https://doi.org/10.1007/s12217-020-09798-9
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DOI: https://doi.org/10.1007/s12217-020-09798-9