A semi-analytical approach to resolving hydrodynamic interactions between two model microswimmers

In an active suspension, hydrodynamic interactions between constituent microswimmers play an important role in their collective motion and rheological properties. To advance the understanding of these interactions, we introduce the boundary collocation method (BCM) and an easy-to-implement extension called the lubrication-theory-informed BCM to solve Stokes flow problems involving two spherical boundaries, where one or both spheres have squirming velocity boundary conditions, as a way to model motile micro-organisms. While the BCM has been widely used to calculate hydrodynamic interactions between two rigid spheres, we extend the application here to determine the hydrodynamic forces, torques and stresslets associated with a pair of equally sized spherical squirmers. This allows us to solve for the hydrodynamic interactions between squirmers at all orientations and separation distances. This fills in the gap between exact analytical solutions, which are currently only available for a limited set of configurations, and asymptotic approaches, which are only valid at extreme near-field and far-field limits. Comparison with analytical solutions for classic two-sphere lubrication problems validates BCM solutions, revealing better accuracy compared with the boundary element method. Our implementation of BCM uses twin multipole expansion solutions to the Stokes equations and takes advantage of the orthogonal basis functions, requiring only one-dimensional collocation points rather than a fully two-dimensional surface meshing for a three-dimensional problem. It is efficient, accurate and easy to implement. A database of the relevant forces and torques for a pair of interacting squirmers, together with the corresponding codes for these calculations, is provided in the supplementary material.