Brownian motion with geometry-dependent hydrodynamic memory

Abstract

Micron-sized particles moving through a fluid are subject to viscous resistance and thermal fluctuations. Beyond steady Stokes friction, colloids also exhibit hydrodynamic memory effects arising from conservation laws of the surrounding fluid. Although fluid-flow problems in complex geometries and confinements are often highly involved, numerical or approximate solutions can be obtained; translating these solutions into closed-form equations of motion for individual colloids, however, is rarely possible. Here, we develop a theoretical framework that provides both underdamped and overdamped colloidal dynamics from the solution of an arbitrary flow problem: (i) Using the Lorentz reciprocal theorem, we express the colloidal equation of motion in terms of the geometry's Green's function and the fluid-flow profile. This formulation yields both the corresponding Stokes drag as well as a Basset-like hydrodynamic memory contribution. (ii) As colloid inertia is often negligible, we furthermore derive the corresponding overdamped (colloid-inertia-less) limit, which inherits the Stokes and Basset-like resistances and additionally gives rise to a spurious drift. We propose applications of this framework to passive and active particles subject to involved confinements and problem geometries.