We present some results of recent calculations of rigid rod-like particles in shear flow, based on the Doi model. This is an ideal model system for exhibiting the generic behavior of shear-thinning fluids (polymer solutions, wormlike micelles, surfactant solutions, liquid crystals) in shear flow. We present calculations of phase coexistence under shear among weakly-aligned (paranematic) and strongly-aligned phases, including alignment in the shear plane and in the vorticity direction (log-rolling). Phase coexistence is possible, in principle, under conditions of both common shear stress and common strain rate, corresponding to different orientations of the interface between phases. We discuss arguments for resolving this degeneracy. Calculation of phase coexistence relies on the presence of inhomogeneous terms in the dynamical equations of motion, which select the appropriate pair of coexisting states. We cast this condition in terms of an equivalent dynamical system, and explore some aspects of how this differs from equilibrium phase coexistence.
