Computational studies of polyacrylamide based Boger fluids for the viscoelastic flow past a sphere using the FENE-CR model, have suggested that a small value of the dumbbell extensibility parameter and a large value of the concentration parameter gives reasonable agreement with experimental observations. However, it is not clear to what extent the FENE-CR model with these same parameter values can predict other flows. In this paper, the viscoelastic flow predicted by the FENE-CR model in a co-rotating two-roll mill is studied numerically for relatively small values of the dumbbell extensibility parameter and relatively large values of the concentration parameter. For the co-rotating two-roll mill, the kinematics of the flow are characterized in terms of the magnitude of thr rate of strain, the flow-type parameter, and a rotation or symmetry angle, all measured at the central stagnation point. For L-2 > 75, it is shown the flow-type parameter at the stagnation point decreases relative to the Newtonian value, in qualitative agreement with experimental observations on dilute solutions of high molecular weight, nearly monodisperse, polystyrene-based Boger fluids. For L-2=25, on the other hand, the flow-type parameter at the stagnation point increases relative to the Newtonian value. The increase in the flow-type parameter relative to the Newtonian value observed for L-2 = 25 is a clear signature of small extensibility which may be used for comparison with experiments for polyacrylamide based Boger fluids. It is hoped that the present study, coupled with future experiments using polyacrylamide based Boger fluids, will further our understanding of the behavior of these dilute, Boger-like fluids in extension dominated flows with large residence times. (C) 2000 Elsevier Science B.V. All rights reserved.