Description of Brownian motion including both kinetic and hydrodynamic effects

TL;DR

This paper derives a comprehensive analytical formula for Brownian motion that includes both kinetic and hydrodynamic effects in two and three dimensions, applicable to particles of various sizes.

Contribution

It provides the first general solution for Brownian motion incorporating both effects for particles of any size in 2D and 3D.

Findings

Analytical formulas match numerical simulations.

Applicable to particles smaller than fluid particles.

Extends previous models limited to large particles or specific limits.

Abstract

It is known that a full description of Brownian motion in the entire course of time should incorporate both kinetic and hydrodynamic effects, but a formula accounts for both effects has been established only in three dimension and only for the limiting case when Brownian particles are much bigger and heavier than fluid particles. Nevertheless, for applications, it is important to consider small Brownian particles (sometimes even smaller than fluid particles). In these cases, unfortunately, only formulae in the short (long) time limit considering exclusively kinetic (hydrodynamic) effect are available. Here we derive a general solution to both two- and three-dimensional Brownian motion. Our solution is applicable to a wide spectrum of suspended particles from being smaller than the fluid particles to usual Brownian particles. Our analytical results are well corroborated by the numerical simulations.