Inertio-viscous interactions between particles in oscillatory flow

We study the interaction between a pair of particles suspended in a uniform oscillatory flow. The time-averaged behavior of particles under these conditions, driven by inertial and viscous effects, is explored through a theoretical framework relying on small oscillation amplitude. We approximate the oscillatory flow in terms of dual multipole expansions, with which we compute time-averaged interaction forces using the Lorentz reciprocal theorem. The results demonstrate excellent agreement with existing numerical data. We then develop analytic approximations for the force, valid when Stokes layers surrounding the particles do not overlap. Finally, we show how the same formalism can be generalized to the situation where the particles are free to oscillate and drift in response to the applied flow. The theory thus provides an efficient means to calculate nonlinear particle interactions in oscillatory flows.