Using the Brownian dynamics technique, we have simulated the fracture process of flexible polymer chains subjected to convergent (sink) flow, where the elongational rate is strongly position dependent. The system studied has been a very dilute solution of polystyrene with molecular weight 2 x 10(6) in cyclohexane at a temperature of 35 degrees C. The polymer was modeled as a bead-spring chain, adequately parametrized as to reproduce real polymer/solvent conditions. The fracture yield varies with the flow rate and is seen to depend to a high degree on the instrumental setup. The distribution of fragments at the end of the convergent flow region is found to be centered around half the initial molecular weight, and we find that there is no fracture below a certain critical flow rate. These results are in qualitative agreement with the experimental results of Reese and Zimm(1) on DNA and Nguyen and Kausch(2) on polystyrene, a finding that confirms the applicability of the Brownian dynamics simulation technique for studying polymer chain fracture.