Macroscopic description of a heavy particle immersed within a flow of light particles

Abstract

A micro-hydrodynamics model based on elastic collisions of light point solvent particles with a heavy solute particle is investigated in the setting where the light particles have velocity distribution corresponding to a background flow. Considering a range of stationary background flows and distributions for the solvent particle velocities, the macroscopic Langevin-type description of the behaviour of the heavy particle is derived in the form of a generalized Ornstein-Uhlenbeck process. At leading order, the drift term in this process depends upon both the geometric structure of the background flow and the size of the heavy particle, while both drift and diffusion terms scale with moments of the light particle velocity distribution. Computational methods for simulating the micro-hydrodynamics model are then designed to confirm the theoretical results. To enable long-time stochastic simulations, simulations are performed in a frame co-moving with the heavy particle. Efficient methods for sampling the position and velocity distributions of incoming solvent particles at the boundaries of the co-moving frame are derived for a range of distributions of solvent particles. The simulations show good agreement with the theoretical results.