A numerical study on the exact onset of flow instabilities in thermo-solutal Marangoni convection driven by opposing forces in a half-zone liquid bridge under zero gravity

The theoretical onset of the Marangoni convection instability occurring in a half-zone liquid bridge of the floating zone (FZ) growth of Si_xGe_1-x has been investigated by numerical simulations. A half-zone model of the liquid bridge between a cold (bottom plane) and a hot (top plane) disk is considered. The highest Si concentration is on the top of the liquid bridge. Therefore, the driving forces for thermal and solutal Marangoni flows are in opposite directions in this configuration. The nonlinear equilibria and periodic orbits are obtained using the Newton-Krylov method, and their stability is analyzed with global linear stability analysis. A flow bifurcation analysis is conducted. The results show that when the solutal Marangoni number, Ma_C, ≤360, the primary flow bifurcation is from 2D axisymmetric to 3D steady flow. When Ma_C>360, the 2D axisymmetric flow becomes weakly periodic 2D axisymmetric, and eventually becomes chaotic. The critical thermal Marangoni number is monotonically increasing as Ma_C increases, and the flow is 2D axisymmetric when the solutal Marangoni forceis dominant. The stability evaluation for various thermal Marangoni number values at Ma_C=893 reveals that the onset of transition is from steady to periodic. However, the reverse transition does not occur at the same Marangoni number value. The simulation results show that the hysteresis behavior of the flow field exhibits an approximately 1% difference between two critical thermal Marangoni numbers. The present study suggests that the control parameters of FZ crystal growth should be smaller than the calculated critical values to avoid any flow-induced disturbances.