This paper examines the predicted flow behavior of a recently published constitutive model (the RHL model) for entangled polymeric fluids in a co-rotating two-roll mill. The constitutive model is an approximation of the Doi-Edwards-Marrucci-Grizzuti (DEMG) model. The DEMG model incorporates an explicit description of chain stretching into the basic Doi-Edwards (DE) theory. Thus, the DEMG and the approximate RHL models both exhibit strain-rate softening (or thinning) behavior similar to the DE model for strain rates that are less than the inverse of the largest Rouse relaxation time. This behavior includes the well-known but controversial prediction of a non-monotonic dependence of the shear stress on the shear rate in steady, viscometric flows. The objectives of the present work are two-fold. First, we explore whether the flow simulations capture the decrease in the strength of the extensional flow at the stagnation point of the two-roll mill that is observed experimentally for entangled polymer solutions. A similar decrease in the extensional strength of the flow is both observed and predicted for dilute polymer solutions, but for these systems this behavior is a consequence of the fact that the extensional viscosity increases monotonically with increasing shear rate. On the other hand, for entangled polymer solutions, as well. as the RHL approximation to the DE theory, the viscosity decreases with increasing strain rate for strain rates smaller than the inverse of the longest Rouse relaxation time. The second objective is to explore how the non-monotonicity of the stress is manifested in a complex flow system where the flow-type is generally not close to viscometric, and the shear rate varies with position along streamlines in regions where the flow is nearly viscometric. In order to understand the signatures of the shear stress maximum for the complex flow in the co-rotating two-roll mill, we also briefly consider the numerically predicted behavior for two-dimensional channel flow and for Couette flow, both of which have been studied earlier by other authors using different models with a non-monotonic shear stress. (C)2000 Elsevier Science B.V. All rights reserved.