Numerical modeling of free burning arcs and their electrodes is useful for clarifying the heat transfer phenomena in the welding process and to elucidate those effects which determine the weld penetration. This paper presents predictions for a stationary welding process by the free-burning argon arc. The whole region of the welding process, namely, tungsten cathode, arc plasma and stainless steel anode is treated in a unified numerical model to take into account the close interaction between the arc plasma and the molten anode. The time dependent development of two-dimensional distributions of temperature and velocity, in the whole region of the welding process, are predicted at a current of 150 A. The weld penetration geometry as a function of time is thus predicted. It is shown also that different surface tension properties can change the direction of re-circulatory flow in the molten anode and dramatically vary the weld penetration geometry.