Multiple scales analysis of the Fokker–Planck equation for simple shear flow

We extend the multiple time scales formalism originally introduced for the Fokker–Planck equation by Wycoff and Balazs (Physica A 146 (1987) 175) to the case of simple shear flow. The analysis is carried out for small values of the Stokes number, St, a dimensionless measure of the inertia of a Brownian particle. The probability density is expanded in terms of Hermite functions of the fluctuation velocity, the difference between the actual particle velocity and the unperturbed flow velocity at its location, and consequently, the structure of the multiple scales hierarchy differs from the analogous equilibrium problem with a conservative forcing. The multiple scales solutions are compared to a small St expansion of the exact solutions for two canonical initial conditions. We also obtain the form of the Smoluchowski equation correct to O(St) for the effects of particle inertia. Modifications to the scheme for hydrodynamically interacting particles are discussed.