A modified heat transfer and material transfer model was investigated for friction stir welding of DH36 steel by considering the Eulerian framework in steady state. During this analysis, temperature-dependent properties of the workpiece and the tool material were used. The material viscosity was modelled as a non-Newtonian viscoplastic fluid depending on the temperature and flow stress. The heat generation at the tool workpiece interface incorporated the partial sticking and partial sliding condition. An asymmetric and skewed temperature distribution at the advancing trailing side was observed. Asymmetry of temperature distribution was increased with an increase in the tool traverse speed. It was observed that the temperature was maximum at the interface between the shoulder and tool, and the peak temperatures decreased non-uniformly along the thickness direction. The results of material flow analysis indicated that the hot plasticized material flew ahead the tool along the retreating side in counter-clockwise direction, passed the tool and got released behind the tool during the welding stage. There existed a swirl region on the advancing side which was highly prone to defect formation. The temperature field and plastic flow field of the computational model matched satisfactorily with the experiment results.