The heat transfer and fluid flow in the floating-zone crystal-growth process for molybdenum (Mo) materials is studied numerically. The input power induced by the heat source is assumed to be a Gaussian distribution. The steady, axisymmetric flow and temperature fields are solved using a finite difference method, employing a boundary-fitted curvilinear coordinate system. The shape of the molten zone and the temperature and velocity fields in the melt are coupled and are strongly dependent on the magnitude of the input power. A four-cell flow structure in the melt is obtained. The effect of thermocapillary convection is much more significant than that of buoyancy-driven convection. The results show that the steady, axisymmetric flow may not exist when the strength of the flow reaches a certain magnitude, which varies insignificantly with the change of the heating region and the rod diameter. The present results are in good agreement with experimental results.