In order to understand the possible initiating mechanism of the sharkskin defect, a numerical study of flow conditions at the exit of an axisymmetric die has been carried out. Using the finite element method, the flow of a viscoelastic fluid along the reservoir, the convergent and the die land, and the free surface of the swelling extrudate, have been computed. The constitutive equation is a multimode Phan-Thien and Tanner model with five relaxation times, whose parameters were derived from rheological measurements in shear and elongation. The tangential stresses and deformations supported by the molten polymer flowing along peripheral streamlines close to the free surface have been analyzed. The results put in evidence the existence of a small traction zone, located at the periphery of the free surface of the extrudate, close to the die exit. With an increase of flow rate, the dimensions and the intensity of this peripheral tensile area grows. Whatever the flow rate, the depth of this zone remains limited, and of the order of magnitude of sharkskin amplitude. If a critical value of tensile stress is assumed, the numerical results can support the interpretation of a rupture of the extrudate skin. On the other hand, whereas differences on sharkskin amplitude and periodicity are relevant between a long die and an orifice die, neither the area nor the intensity of the peripheral traction zone varies with die geometry. As a consequence, tangential stress can not be the unique parameter able to explain the dynamic of the surface defect. The history of deformation experienced by the polymer before the die exit seems to be also implied in the mechanism of sharkskin formation.