Pairwise scattering and bound states of spherical microorganisms

The dynamic interactions between pairs of swimming microorganisms underpin the collective behavior 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 utilize 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-to calculate their scattering dynamics. Attention is limited to squirmers whose orientation vectors lie in the same plane. We characterize 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.