Brownian motion of finite-inertia particles in a simple shear flow

TL;DR

This paper analyzes how particle inertia influences diffusive transport of Brownian particles in shear flow, revealing non-Newtonian behavior and inertia-dependent diffusion coefficients through analytical methods.

Contribution

It provides a non-perturbative analytical evaluation of particle-phase pressure and derives a generalized convection-diffusion equation incorporating inertia effects.

Findings

Streamwise diffusion coefficient becomes negative with increasing Stokes number.

Total diffusion coefficients remain positive, ensuring stability.

Particle phase exhibits non-Newtonian fluid behavior.

Abstract

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. 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. The analytic, non-perturbative, 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 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.