Time-averaged inertial particle dynamics in oscillatory flow

Oscillatory flows provide a powerful way to manipulate suspended particles in microfluidic settings. We present a comprehensive theory of the dynamics of a compressible spherical particle in oscillatory flow, spanning the range between classical acoustofluidics and streaming regimes. In particular, we study how the density and compressibility contrast between particle and fluid, and the Stokes layer thickness relative to the particle radius, influence the time-averaged motion of the particle. The particle produces an oscillatory disturbance flow, whose inertia drives a secondary disturbance flow. We show that the associated secondary force in turn drives steady particle motion through a Faxén-like relation. By applying the Lorentz Reciprocal Theorem to compressible flows with inertia, we obtain the secondary force analytically while circumventing the need to resolve the complicated secondary flow in detail. We find that the force depends on quadratic combinations of local moments of the ambient oscillatory flow, as well as the physical properties of the fluid and the particle. Our formulation recovers known results for secondary radiation forces in the limit of thin Stokes layers, while predicting qualitative differences at finite Stokes layer thicknesses. In particular, we show that the force's direction can be reversed for certain combinations of frequency, density and compressibility ratio. We demonstrate the application of the theory to efficiently compute time-averaged particle motion in oscillatory flows typical of microfluidic and acoustofluidic applications.