The effects of conformation-dependent friction and internal viscosity on the dynamics of the nonlinear dumbbell model for a dilute polymer solution

Abstract

The effects of a nonlinear spring, variable hydrodynamic friction and internal viscosity are examined in detail for the dumbbell model of dilute polymer solutions in a general two-dimensional flow. Both steady and transient startup flows are investigated and the response of the bulk stress is calculated. The existence of a hysteresis in the steady end-to-end length of the dumbbell, which arises from the variable friction factor, is found to depend strongly on both the flow type (i.e. the ratio of vorticity to the rate of strain) and the moleculecular weight. The influence of internal viscosity is examined in detail using two methods of solution for the problems of transient and steady two-dimensional flows. The first method uses a perturbation expansion valid for small values of the internal viscosity and leads largely to analytical results. The second technique uses the pre-averaging approximation of Cerf and is capable of predicting the model response when the internal viscosity becomes large. The most striking result for large values of the internal viscosity and a linear spring dumbbell is the appearance of multiple steady states in the dumbbell length for flows between simple shear and pure straining, and the removal of the well known singularity in end-to-end length which otherwise appears in the linear dumbbell model at a critical value of the strain rate.