The transport dynamics and stability limit of the axisymmetric steady flow driven by a surface tension variation in a liquid bridge configuration is studied numerically. The surface tension variation results from both thermal and solutal gradients, and varies linearly with both temperature and solute concentration. The pseudo-spectral method is employed to investigate the solutions under different combinations of Marangoni, Prandtl and Lewis numbers, and separation ratio. Steady bifurcations for low Prandtl numbers are observed. The Soret effect has important influence on the stability of the basic flow, as it can change the symmetry of the most destabilizing perturbations and affect the threshold for 3D convection. It is also shown that for large Prandtl number, the Hopf bifurcation observed for pure fluid is replaced by a steady one for negative separation ratio.

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