Brownian motion of finite-inertia particles in a simple shear flow.
Simultaneous diffusive and inertial motion of Brownian particles in laminar Couette flow is investigated via Lagrangian and Eulerian descriptions to determine the effect of particle inertia on diffusive transport in the long-time limit. The classical fluctuation dissipation theorem is used to calculate the amplitude of random-force correlations, thereby neglecting corrections of the order of the molecular relaxation time to the inverse shear rate. In the diffusive limit (time much greater than the particle relaxation time) the fluctuating particle-velocity autocorrelations functions are found to be stationary in time, the correlation in the streamwise direction being an exponential multiplied by an algebraic function and the cross correlation nonsymmetric in the time difference. The analytic, nonperturbative, evaluation of the particle-phase total pressure, which is calculated to be second order in the Stokes number (a dimensionless measure of particle inertia), shows that the particle phase behaves as a non-Newtonian fluid. The generalized Smoluchowski convective-diffusion equation, determined analytically from a combination of the particle-phase pressure tensor and the inertial acceleration term, contains a shear-dependent cross derivative term and an additional term along the streamwise direction, quadratic in the particle Stokes number. The long-time diffusion coefficients associated with the particle flux relative to the carrier flow are found to depend on particle inertia such that the streamwise diffusion coefficient becomes negative with increasing Stokes number, whereas one of the cross coefficients is always negative. The total diffusion coefficients measuring the rate of change of particle mean-square displacement are always positive as expected from general stability arguments.