Kinetic theory and rheology of dilute, nonhomogeneous polymer solutions

Abstract

A phase-space kinetic theory of dilute polymer solutions is developed to account for the effects of nonhomogeneous velocity and stress fields. The theory allows the configuration distribution function to depend on spatial location and explicitly treats the polymer molecule as an extended object in space. Constitutive equations for the mass flux vector and stress tensor are derived that predict polymer migration induced by stress gradients and nonuniform velocity gradients. In addition, the constitutive equation for stress contains a diffusive term in stress, and hence the model does not fall within the class of simple fluids. Simple shear flow between parallel plates is solved to illustrate the features of the constitutive equations. Asymptotic analysis and numerical calculations show the formation of boundary layers in stress, velocity gradient, and polymer concentration that arise near solid walls as a result of preferential orientation of the polymer molecules there. The thickness of these layers scales as λHDtr/L2, where λH is the relaxation time of the macromolecule modeled as a Hookean dumbbell, Dtr is its translational diffusivity in solution, and L is the characteristic length scale of the macroscopic flow. The presence of these layers causes only a small change in the shear stress measured in typical rheometers, but can have a profound effect on the macroscale flow of polymer solutions in complex geometries by causing apparent fluid slip near solid boundaries.