Inhomogeneous suspensions of rigid rods in flow

An expression for the divergence of the stress tensor is derived for inhomogeneous suspensions of very long and thin, rigid rods. The stress tensor is expressed in terms of the suspension flow velocity and the probability density function for the position and orientation of a rod. The expression for the stress tensor includes stresses arising from possibly very large spatial gradients in the shear rate, concentration, and orientational order parameter. The resulting Navier–Stokes equation couples to the equation of motion for the probability density function of the position and orientation of a rod. The equation of motion for this probability density function is derived from the N-particle Smoluchowski equation, including contributions from inhomogeneities. It is argued that for very long and thin rods, hydrodynamic interactions are of minor importance, and are therefore neglected, both in the expression for the stress tensor and in the equation of motion for the above-mentioned probability density function. The thus obtained complete set of equations of motion can be applied to describe phenomena where possibly very large spatial gradients occur, such as phase coexistence under shear flow conditions, including shear-banding, and phase separation kinetics.