Theoretical Study of Brownian Particle Diffusion in Generalized Shear flows

Brownian particle diffusion in shear flows is of broad scientific and engineering interest, with applications ranging from drug delivery and diffusion-based separation to sedimentation and soot dispersion in aerospace propulsion. In this study, we present a mathematical framework for predicting the diffusion of Brownian particles in parallel 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 timescales, both with and without external forcing, in generalized two-dimensional parallel shear flow.
In the absence of constant forcing, we observe three main stages in the evolution of particle diffusion for Couette and plane Poiseuille flows, and four main stages for a polynomial approximation of a hyperbolic tangent flow. The particle MSD is distinctly different across these stages due to different dominant physical mechanisms. In the presence of constant forcing, the analysis reveals that the influence of the external body force emerges at long timescales and is primarily dependent on the flow velocity gradient. Notably, the variance of particle displacement undergoes a decline followed by a rise when particles are drifted crossing the region where the velocity gradient reverses sign.
For unsteady flows, we investigate the particle diffusion in time-periodic shear flows under the influence of a constant forcing in the transverse direction. The results further demonstrate that particle diffusion in the streamwise direction is markedly enhanced due to the combined effects of Brownian motion in the transverse direction, the forcing term, and flow velocity gradients.

Nan Wang is a Ph.D. candidate under the supervision of Professor Yuval Dagan in the faculty of Aerospace Engineering. Her research focuses on the theoretical investigation of Brownian particle diffusion in shear flows.