This study presents a numerical scenario for the effect of thermocapillary convection on the transient, two-dimensional molten pool shape during welding or melting. Tracing the melting process is necessary to achieve a better and more complete understanding of the physical mechanism of welding. This model is used to simulate a steady state, three-dimensional welding process, by introducing an incident flux with a Gaussian distribution with a time-dependent radius determined by scanning speed and distribution parameter. Aside from presenting the variations of peak surface velocities and temperature, and depth and width of the molten pool with time, the predicted results of this work show that surface velocity and temperature profiles for a high Prandtl number strongly deform in the course of melting. The velocity profile eventually exhibits two peaks, located near the edges of the incident flux and the pool, respectively. Conversely, only one peak velocity occurs near the pool edge for a small Prandtl number. In all cases, surface temperature can ultimately be divided into hot, intermediate, and cold regions. The pool becomes deep due to an induced secondary vortex cell near the bottom of the pool for a small Prandtl number. For a high Prandtl number, the pool edge is thin and shallow, as a result of penetration into the solid near the top surface. The predicted results agree with those obtained using a commercial computer code.