A generalized theory of Brownian particle diffusion in shear flows

TL;DR

This paper develops a comprehensive mathematical theory for Brownian particle diffusion in shear flows, accurately describing particle behavior across all time scales and flow profiles, including polynomial and hyperbolic tangent flows.

Contribution

It introduces a new stochastic calculus-based formulation that generalizes diffusion theory to any polynomial shear flow at all time scales, including the particle relaxation time.

Findings

Particle MSD scales as n+2 at long times for polynomial order n.

The theory applies to Couette, plane-Poiseuille, and hyperbolic tangent flows.

Three stages of diffusion are identified with distinct physical mechanisms.

Abstract

This study presents a generalized theory for the diffusion of Brownian particles in shear flows. By solving the Langevin equations using stochastic instead of classical calculus, we propose a new mathematical formulation that resolves the particle MSD at all time scales for any two-dimensional parallel shear flow described by a polynomial velocity profile. We show that at long-time scales, the polynomial order of time in the particle MSD is n+2, where n is the polynomial order of the transverse coordinate of the velocity profile. We generalize the theory to resolve particle diffusion in any polynomial shear flow at all time scales, including the order of particle relaxation time scale, which is unresolved in current theories. Particle diffusion at all time scales is then studied for the cases of Couette and plane-Poiseuille flows and a polynomial expansion of a hyperbolic tangent flow while neglecting the boundary effects. We observe three main stages of particle diffusion along the timeline for which the particle MSD is distinctly different due to different dominated physical mechanisms. Thus, higher temporal and spatial resolution for diffusion processes in shear flows may be realized, suggesting a more accurate analytical approach for the diffusion of Brownian particles.