Streamline finite‐element results are presented for the extrudate swell of a high density polyethylene (HDPE) melt flowing through straight, converging, and diverging annular dies. Viscometric data have been fitted using a spectrum of relaxation times and the K‐BKZ integral constitutive equation. For the elongational viscosity a reasonable strain‐thinning behavior has been assumed. Numerical calculations for three die designs show the dependence of diameter and thickness swell on the taper angle. It is found that the K‐BKZ equation is capable of capturing memory phenomena exhibited by polymer melts in such geometries, which are in sharp contrast with Newtonian or Maxwell fluid simulations. The present results show that the diameter swell is highest for the converging, followed by the straight and then the diverging dies, in qualitative agreement with experimental findings available in the literature. While the simulations for converging and diverging annular dies failed to converge at high flow rates, converged results were obtained for straight annular dies at high flow rates (very high shear rates on the wall upstream), and the predictions for the diameter swell are in satisfactory agreement with experimental data.