The diffusion tensor for Hookean dumbbells in steady shear flow: Analytical approximation

Thermodynamic treatments of the phenomenon of polymer migration in inhomogeneous flows have so far assumed a flow rate independent coefficient of diffusion. Recent investigations have revealed that polymer diffusivity in simple shear flow is anisotropic and flow rate dependent. These properties arise because the average spatial extent of the macromolecule, from being spherical at equilibrium, becomes ellipsoidal on the imposition of a shear flow. Using this insight into the molecular shape, and assuming that there is only one physically important direction in the problem due to the flow field, we derive an approximate analytical expression for the diffusion tensor. This gives values within 1.2% of exact numerical ones, for a wide range of dimensionless shear rates. Clearly, it is important to refine the earlier thermodynamic theories of polymer migration by incorporating the shear rate dependence of polymer diffusivity. The expression derived in this paper is anticipated to be useful in a wide range of problems, where the assumption of “local homogeneity” is justified.