An inertial Hookean dumbbell model with Basset forces in steadyhyphen;state shear flow has been solved analytically to yield predictions for the dilute solution polymer contribution to viscosity and first normal stress coefficient. The generalization to the standard Hookean dumbbell model with only a Stokes law drag force is accomplished by utilizing the solution found by Landau and Lifshitz for the drag on a sphere undergoing arbitrary, timehyphen;dependent displacement assuming Stokes flow. The Basset forces depend upon the entire past history of the phasehyphen;space coordinates (an integral of a 1/(t)1/2memory kernel times the acceleration of the beads in the dumbbell); thus the resulting equation of motion for the internal phasehyphen;space coordinates of the polymer are no longer Markovian. These equations may still be solved in any steady flow using harmonic analysis. It is found that neither the viscosity nor the first normal stress coefficient are shear rate dependent. However, both constant results are weakly rescaled by the presence of the Basset forces in a complicated way.
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