A three-dimensional coupled model in a Eulerian framework has been developed in COMSOL Multiphysics software and used to study the complex phenomena of thermal and material flow during the Friction Stir Welding (FSW) process. The moving heat source (tool) effect is modeled using a coordinate transformation. The frictional heat as a function of temperature dependent yield strength of AA2219-T87 material, and the deformation energy of plasticized material flow are considered. Further, the plasticized material is modeled as non-Newtonian fluid and a no-slip boundary condition at the workpiece-tool material interfaces is used. The coupled Eulerian model prediction accuracy has been validated against the experimental weldment zones, and found a good agreement in terms of the shape and size. Subsequently, studied the effects of tool-pin profiles (cylindrical and conical) on thermal distribution, material flow, shear strain rates, thermal histories and weldment zones. It is found that the maximum temperatures, material flow velocities and shear strain rates are low with the conical tool-pin in contrast to the cylindrical one, and it is partly attributed to increased mixing of shoulder and pin driven material flow around the rotating tool, which in turn decreased the size of weldment zones. Also, the maximum temperatures, material flow velocities and shear strain rates on advancing side are higher than that of retreating side. Therefore, it is suggested to use the coupled Eulerian model to design the FSW process and tool parameters in a cost effective way in contrast to the tedious experimental route.