It is well known that the flow of a non-Newtonian fluid through a sudden contraction exhibits a vortex in the corner region bigger than the one observed in the corresponding flow of a Newtonian Fluid. This change of pattern of the flow affects significantly the heat transfer at the wall. It was investigated the case og a a 4:1 circular contraction, with uniform temperature distrubuition at the solid walls. This problem represents a first approach for the analysis of the polymeric liquids extrusion process. The flow and temperature field have been obtained from the numerical integration of the conservation equations. To account for the flow dependence of the stress tensor, a generalized Newtonian model and a convected Maxwell model have been employed. The creeping flow hypothesis has not been adopted, so it was possible to analyse the elastic effects and the inertial effects separately. The nuemerical solution have been obtained via a finite-volume method. The results show the importance of the non-Newtonian modeling and of the inclusion of inertial terms in the study of the flow and beat transfer in the polymeric liquids extrusion process.