Pairwise scattering and bound states of spherical microorganisms

TL;DR

This paper develops analytical models for the pairwise scattering of spherical microorganisms, revealing bound states and trajectories, and compares well with numerical simulations, aiding mesoscale suspension modeling.

Contribution

It introduces an analytical approach to compute scattering dynamics of spherical squirmers, capturing complex behaviors like bound states and orbits, improving over purely numerical methods.

Findings

Identification of transient scattering and bound states

Analytical expressions match numerical boundary element results

Discovery of circular orbits and pairwise swimming behaviors

Abstract

The dynamic interactions between pairs of swimming microorganisms underpin the collective behaviour of larger suspensions, but accurately calculating pairwise collisions has typically required the use of numerical simulations in which hydrodynamic interactions are fully resolved. In this paper, we utilise analytical expressions for forces and torques acting on two closely separated spherical squirmers -- accurate to second order in the ratio of cell-cell spacing to squirmer radius -- in order to calculate their scattering dynamics. Attention is limited to squirmers whose orientation vectors lie in the same plane. We characterise the outgoing angles of pairs of bottom-heavy squirmers in terms of their incoming angles, the squirmer parameter $\beta$, and the strength of the external gravitational field, discovering transient scattering, stationary bound states, pairwise swimming motion, and circular orbits. These results compare well with full numerical solutions obtained using boundary element methods, highlighting the utility of lubrication theory. We expect these results will be useful for the foundations of mesoscale continuum models for suspensions of spherical microorganisms.