Numerical simulations have been undertaken for the flow of an IUPAC‐LDPE melt used previously in an international experimental study (Meissner, 1975). The flow geometry corresponds to an axisymmetric 4:1 contraction equipped with capillary dies of different length/radius (L/R) ratios, ranging from orifice dies (L/R=0) to infinitely long ones (L/R=∞). The constitutive equation used is an integral‐type K‐BKZ model with a relaxation spectrum, which fits well experimental data for the shear and elongational viscosities and the normal stresses as measured in shear flow. The simulations have been performed for the full range of experimental measurements in this system, where the apparent shear rates reach 10 s−1at 150 °C, but the viscoelastic character of the melt is very strong corresponding to a stress ratio of about 2 and a Trouton ratio of about 50. Stable solutions have been obtained for the whole range of experimental values. They show a dramatic vortex growth in the contraction with increasing flow rate, as expected for LDPE melts. New results concerning extrudate swell from orifice dies (L/R=0) are obtained and are in excellent agreement with the experimental data. Results for different ratiosL/R≥4 corroborate previous findings by Luo and Tanner (1988), who predicted a decrease in swelling with increasingL/Rvalues, also observed experimentally. The present simulations are also in good agreement with the experiments for the low range ofL/Rratios (0≤L/R≤4), which was not the case in the previous simulations. The pressures from the simulations have been used to compute the excess pressure losses in the system (end or Bagley correction). The results are also in good agreement with the experimental values at least in a qualitative sense.