Exact coefficients for rigid dumbbell suspensions for steady shear flow material function expansions

Abstract

From kinetic molecular theory, we can attribute the elasticity of polymeric liquids to macromolecular orientation. For a suspension of rigid dumbbells, subject to a particular flow field, we must first solve the diffusion equation for the orientation distribution function. From this distribution, we then calculate physical properties such as the steady shear flow material functions. We thus arrive at power series expansions in the shear rate for both the orientation distribution function and for the steady shear flow material functions. Analytical work on many viscoelastic material functions must be checked for consistency, in their steady shear flow limits, against these power series. For instance, for large-amplitude oscillatory shear flow, we recover the coefficients of these expansions in the limits of low test frequency. The coefficients of the steady shear viscosity and the first normal stress coefficient functions are not known exactly beyond the fourth power. In this work, for both of these functions, we arrive at exact expressions for the first 20 coefficients. We close with five worked examples illustrating uses for our new coefficients.