Diffusion in shear flow

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Abstract
Diffusion of a tagged particle in a fluid with uniform shear flow is described. An expression for the diffusion tensor is obtained in a form similar to the Green-Kubo result for the equilibrium self-diffusion coefficient, but which is applicable even far from equilibrium. In one form the diffusion tensor is determined from a type of Einstein relationship to the nonequilibrium mean-square displacement in the Lagrangian frame for the fluid. Alternatively, the diffusion tensor is expressed as a time integral of the autocorrelation function for the velocity of the tagged particle in a local rest frame. A frequency- and shear-rate-dependent diffusion tensor is defined from these results and evaluated in two limits. The first case is the Boltzmann limit for a low-density gas, and the diffusion tensor is found to be an analytic function of both frequency and shear rate. Consequently, nonlinear transport coefficients of arbitrary order are well defined and finite. In the second case the mode-coupling contributions for a general fluid are obtained by an extension of the equilibrium mode-coupling phenomenology to shear flow. The hydrodynamic modes are obtained as a function of time and shear rate and lead to a nonanalytic dependence of the diffusion tensor on both frequency and shear rate. A universal function is obtained to describe the crossover from asymptotic frequency to asymptotic shear-rate dependence. This function is compared with a similar result predicted by a phenomenological model from nonlinear continuum mechanics.