Squirming motion in a Brinkman medium

Abstract

Micro-organisms encounter heterogeneous viscous environments consisting of networks of obstacles embedded in a viscous fluid medium. In this paper we analyse the characteristics of swimming in a porous medium modelled by the Brinkman equation via a spherical squirmer model. The idealized geometry allows an analytical and exact solution of the flow surrounding a squirmer. The propulsion speed obtained agrees with previous results using the Lorentz reciprocal theorem. Our analysis extends these results to calculate the power dissipation and hence the swimming efficiency of the squirmer in a Brinkman medium. The analytical solution enables a systematic analysis of the structure of the flow surrounding the squirmer, which can be represented in terms of singularities in Brinkman flows. We also discuss the spatial decay of flows due to squirming motion in a Brinkman medium in comparison with the decay in a purely viscous fluid. The results lay the foundation for subsequent studies on hydrodynamic interactions, nutrient transport and uptake by micro-organisms in heterogeneous viscous environments.