Inertio-viscous hydrodynamic interactions between particles in oscillatory flow

Particles suspended in oscillatory flows and acoustic fields are known to exhibit complex behavior, including collective dynamics due to interactions. Here, we study how particles in a uniform oscillatory flow interact with each other by generating oscillatory disturbances, and how the inertia of the flow is rectified into time-averaged particle motion. We present a comprehensive theory of the dynamics of a pair of spherical particles. Utilizing multipole expansions around each particle, we apply the method of reflections to calculate oscillatory hydrodynamic interactions between the particles. We then use these interacting oscillatory flows in an analytic framework to evaluate time-averaged secondary hydrodynamic force on the particles. The forces produce either attraction or repulsion of the particles depending on their orientation relative to the flow, their relative separation distance, density contrast relative to the fluid, and the Stokes layer thickness. The theory is found to be in good agreement with previous direct numerical simulations.