This work focuses on mitigating the gross melt fracture defect of polymer flowing through axisymmetrical and two-dimensional dies. The die entrance angle is considered as well as the influence of the converging wall roughness. Singular results are obtained with a random styrene butadiene rubber (SBR) copolymer, as the gross melt fracture defect cannot be eliminated or mitigated by reducing the die entrance angle. Other experiments carried out with rough converging dies do not give better results. Indeed, the polymer essentially slips along the walls, as shown from capillary rheometer and birefringence experiments. Thus, these results point out the importance of elongational stresses and interfacial conditions in the die entrance region on flow instabilities and the gross melt fracture defect.