Numerical analysis of the flow regime within a capillary rheometer has been studied. The vector-based Remmelgas-Harrison-Leal (RHL), analogous to reptation-based polymer models, has been used to model the polymer dynamics. A series of computational methods for visualizing the flow characteristics of the whole region of the capillary rheometer are made including the area directly in front of the piston. Model predictions provide qualitative analysis of pressure drops observed at the entrance of the die, necessitating the use of the Bagley ends correction in practical rheology measurements. The results of the numerical code help reveal the complex flow kinematics that are involved in the entrance region to the capillary die as well as the barrel region in front of the rheometer piston.