Interfacial Hydrodynamic Mulitpoles, their Comparison to Experiment, and Open Issues

While hydrodynamic multipoles have proven indispensable for describing the fluid mechanics of active (self-propelled), driven (externally forced) colloids in bulk fluids, a suitable description for colloids adsorbed to fluid interfaces was lacking. Here, we derive hydrodynamic multipoles for the typical stress conditions encountered in colloidal systems. Owing to the small scale of the colloidal agents, the Marangoni number is large even for scant surfactant, and fluid interfaces act as incompressible layers. For active colloids in particular, we find that an infinitesimally separated pairs of point forces at the interface do not fully describe the leading-order hydrodynamics as is true in the bulk, and "new" modes of flow arise from these pronounced Marangoni effects. The predicted modes capture the detailed structure of flows measured experimentally by correlated displacement velocimetry, including those around Brownian (thermally-driven) colloids and bacteria at fluid interfaces. Finally, we address open issues regarding our assumption of interfacial incompressibility and regimes where it may break down.