Interactions and collective dynamics of inertial particles in oscillatory flow

Suspensions of particles exposed to oscillatory flows are known to exhibit complex collective behaviors. We develop a theoretical framework to study the dynamics of interacting particles in uniform oscillatory flow, focusing on time-averaged motion due to the interplay between viscous and inertial forces. Starting with a pair of particles, we utilize a dual multipole expansion to obtain the oscillatory disturbance flow created by each particle. We then use the Lorentz reciprocal theorem to evaluate time-averaged hydrodynamic interaction forces between the particles. The theory is in excellent agreement with previous numerical computations. It reveals regions of attraction and repulsion depending on the distance between the particles, the orientation between the particle line-of-centers and the imposed flow, and the oscillation frequency. We then demonstrate how the theory can be adapted to multiple interacting particles, and use it to explore collective motion and pattern formation in oscillating suspensions.