Stokes traction on an active particle

TL;DR

This paper derives an exact analytical solution for the surface traction on a spherical active particle in viscous flow, generalizing Stokes laws and providing formulas for physical quantities and thermal fluctuations.

Contribution

It introduces a novel exact boundary integral solution for traction on active particles, extending classical Stokes laws and including thermal fluctuation effects.

Findings

Exact solution for traction using tensorial spherical harmonics

Generalized Stokes laws for force and torque on active particles

Explicit expressions for power dissipation and Brownian traction variance

Abstract

The mechanics and statistical mechanics of a suspension of active particles are determined by the traction (force per unit area) on their surfaces. Here we present an exact solution of the direct boundary integral equation for the traction on a spherical active particle in an imposed slow viscous flow. Both single- and double-layer integral operators can be simultaneously diagonalised in a basis of irreducible tensorial spherical harmonics and the solution, thus, can be presented as an infinite number of linear relations between the harmonic coefficients of the traction and the velocity at the boundary of the particle. These generalise Stokes laws for the force and torque. Using these relations we obtain simple expressions for physically relevant quantities such as the symmetric-irreducible dipole acting on, or the power dissipated by, an active particle in an arbitrary imposed flow. We further present an explicit expression for the variance of the Brownian contributions to the traction on an active colloid in a thermally fluctuating fluid.