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The effect of solutal Marangoni convection on flow instabilities in the presence of thermal Marangoni convection in a Si-Ge liquid bridge with different aspect ratios As has been investigated by three-dimensional (3D) numerical simulations under zero gravity. We consider a half-zone model of a liquid bridge between a cold (top plane) and a hot (bottom plane) disks. The highest Si concentration is on the top of the liquid bridge. The aspect ratio (As) drastically affects the critical Marangoni numbers: the critical solutal Marangoni number (under small thermal Marangoni numbers (MaTAs≲1800)) has the same dependence on As as the critical thermal Marangoni number (under small solutal Marangoni numbers (400≲MaCAs≲800)), i.e., it decreases with increasing As. The azimuthal wavenumber of the traveling wave mode increases as decreasing As, i.e., larger azimuthal wavenumbers (m=6,7,11,12, and 13) appear for As=0.25, and only m=2 appears when As is one and larger. The oscillatory modes of the hydro waves have been extracted as the spatiotemporal structures by using dynamic mode decomposition (DMD). The present study suggests a proper parameter region of quiescent steady flow suitable for crystal growth for smaller aspect ratios of the liquid bridge.

The theoretical onset of the Marangoni convection instability occurring in a half-zone liquid bridge of the oating zone (FZ) growth of Si x Ge 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 ows are in opposite directions in this con guration. The nonlinear equilibria and periodic orbits are obtained using the Newton-Krylov method, and their stability is analyzed with global linear stability analysis. A ow bifurcation analysis is conducted. The results show that when the solutal Marangoni number, Ma C , ≤360, the primary ow bifurcation is from 2D axisymmetric to 3D steady ow. When Ma C >360, the 2D axisymmetric ow becomes weakly periodic 2D axisymmetric, and eventually becomes chaotic. The critical thermal Marangoni number is monotonically increasing as Ma C increases, and the ow is 2D axisymmetric when the solutal Marangoni forceis dominant. The stability evaluation for various thermal Marangoni number values atMa 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 ow eld exhibits an approximately 1% di erence 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 ow-induced disturbances.

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