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 modelled 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 flow around the rotating tool is modelled as non-Newtonian fluid using partial-sticking/sliding boundary condition with a computed slip factor ( δ ) at the workpiece-tool material interfaces. 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, the effects of tool-pin profiles (cylindrical and conical) on thermal distribution, material flow, shear strain rates, thermal histories, and weldment zones were studied. 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 the advancing side are higher than those of the retreating side. Therefore, it is suggested to use the CFD model to design the FSW process and tool parameters in a cost-effective way in contrast to the tedious experimental route.