Added mass effect in coupled Brownian particles

TL;DR

This paper presents an analytical model demonstrating how the effective mass of a Brownian particle in a fluid depends on hydrodynamic interactions and relevant timescales, revealing conditions under which the added mass effect varies.

Contribution

The study introduces a solvable model of two coupled Brownian particles to analytically determine the effective mass influenced by hydrodynamics and measurement timescales.

Findings

Effective mass depends on three key timescales: $t_p$, $ au$, and $ riangle t$.

In different limits, the effective mass reduces to either the particle's mass or twice its mass.

The model generalizes to particles with unequal masses, showing similar behavior.

Abstract

The added mass effect is the contribution to a Brownian particle's effective mass arising from the hydrodynamic flow its motion induces. For a spherical particle in an incompressible fluid, the added mass is half the fluid's displaced mass, but in a compressible fluid its value depends on a competition between timescales. Here we illustrate this behavior with a solvable model of two harmonically coupled Brownian particles of mass $m$, one representing the sphere, the other the immediately surrounding fluid. The measured distribution of the Brownian particle's velocity, $P(\bar{v})$, follows a Maxwell-Boltzmann distribution with an effective mass $m^*$. Solving analytically for $m^*$, we find that its value is determined by three relevant timescales: the momentum relaxation time, $t_p$, the harmonic oscillation period, $\tau$, and the velocity measurement time resolution, $\Delta t$. In limiting cases $\Delta t \ll \tau,t_p$ and $\tau\ll\Delta t\ll t_p$, our expression for $m^*$ reduces to $m$ and $2m$, respectively. We find similar behavior upon generalizing the model to the case of unequal masses.