A solvable weak-potential model of a non-Newtonian fluid

A theoretical model is developed for a non-Newtonian fluid of spherical molecules interacting with a weak potential. The Kirkwood-Smoluchowski equation for planar Couette flow reduces in leading order in potential strength to a shear-diffusion equation with an inhomogeneous source term. The pressure tensor elements are calculated and, for a Gaussian potential, reduce to one-dimensional integrals which are evaluated numerically. The model reproduces several qualitative features of non-Newtonian liquids and the computer simulations of Evans and Hanley. These features include shear thinning, shear dilatancy, normal pressure differences, and dependence on shear rate to a half-integer power.