We calculate the extrusion of a viscoelastic fluid through a circular die. At the inflow, we assume that the velocity and stress fields are fully developed. We suppose that the fluid sticks to the wall. No essential boundary conditions are imposed at the exit section, which is taken long enough to ensure fully developed profiles. The length of the entry section is equal to 6 die radii while the length of the jet is equal to 24 die radii.
Viscoelastic fluids exhibit a large swelling.
Due to normal stresses, the diameter of an extruded jet usually increases at the exit of the die. However, Joseph et al. observed at high rate of extrusion that the swelling effect can start several radii downstream the die exit. This delayed die-swell effect is therefore a critical phenomenon, as it occurs beyond a critical flow rate. Using an Oldroyd-B fluid with a small retardation time, Delvaux et al. have shown that this phenomena could be explained by a change of type of the system of governing equations, when inertia was taken into account. Numerical difficulties associated with this problem can be attributed both to the discontinuity arising at the die exit and to the strong coupling between inertia and viscoelasticity.