(9b) Hydrodynamic Interactions between Microswimmers Trapped at Interfaces

A fluid interface can strongly influence the behavior of a nearby self-propelled colloid (microswimmer), allowing for boundary guidance or directed self assembly. However, such swimmers are often not present alone; they can interact with other swimmers or material on the interface resulting in collective motion and/or mixing enhancement. At present, hydrodynamic interactions of microswimmers at an interface are not well understood. Known results for a bulk fluid do not directly apply because the interface alters the trajectory of the swimmer as well as the fluid flow. Motivated by this gap in knowledge, we theoretically quantify the pairwise hydrodynamic interactions of microswimmers at an interface. The Reynolds and capillary numbers are assumed small, in line with many colloidal systems involving an air- or oil-water interface. We assume the swimmer is trapped at the interface either by physical adhesion or hydrodynamic interaction. Thus, the trajectory of a lone swimmer is prescribed to be circular and parallel to the interface. Swimmers are modeled as appropriate flow singularities at the interface. Thus, our analysis is generally valid in the far field for any active colloid, regardless of propulsion mode. We first consider a clean interface where the Marangoni number (Ma) is small and the surface tension is constant. We then address an incompressible interface (Ma >> 1), as is often the case if surfactant is sufficiently concentrated on the interface. Finally, we consider the effect of a finite surface viscosity over a range of Boussinesq numbers (Bo), which increases the range of hydrodynamic interactions, especially in the interfacial plane. Our results will help guide future work regarding the collective motion of interfacially guided colloidal swimmers and enhanced mixing in the vicinity of a microswimmer-activated interface.