Abstract: Floating zone method is an important technology for growth of high-integrity and high-uniformity single crystal materials due to its free of crucible contamination. However, the capillary convection in the melt brings a great challenge to the floating zone crystal growth. This is because the instability of convection will cause the formation of some crystal defects such as microscopic imperfections and macroscopic stripes. Therefore, it is very important to investigate the behaviors of the capillary flow and control its instability in order to improve the quality of the produced single crystal materials. In this paper, numerical simulations are performed to investigate all kind of the capillary convection in the half floating liquid bridge on the Si_xGe_1-x system. And the impact of the external rotating magnetic field is also investigated on the stability of capillary convection. The results show that the purely solute capillary convection is a two-dimensional axisymmetric model, and the temperature field is mainly determined by thermal diffusion while the concentration field is dominated by convection and solute diffusion together. On the other hand, the purely thermo-capillary convection presents three-dimensional unsteady axisymmetric flow. The concentration distribution is closely related to the flow direction of thermo-capillary convection. The isotherms bend in the region with strong convection. The coupled solute and thermocapillary convection is a three-dimensional periodic rotating oscillatory flow. When the rotating magnetic field is applied, the circumferential velocity of the melt increases with increasing radius. Both the concentration field and the flow field in the melt show a two-dimensional axisymmetric distribution.