Secondary flows and forces around a sphere in a nonuniform oscillatory flow

Oscillatory flows provide a powerful means of harnessing inertial effects to direct the motion of suspended particles. This motion has varyingly been described using secondary radiation forces and streaming theory, with contrary results. In this work, we model the time-averaged secondary flow around a sphere suspended in a known, spatially varying, oscillatory flow. We first decompose the oscillatory flow into translating, dilating, and straining components and obtain the corresponding particle disturbance flow analytically. We then show that the time-averaged secondary flow is driven by a combination of an effective body force due to the inertia of the primary flow, and an effective slip condition arising from the Stokes drift of the primary flow. Assuming small oscillation amplitudes and axial symmetry, we solve for the secondary flow using a vorticity-stream function formulation over a wide range of oscillatory Stokes layer thicknesses. We then use the secondary flow to compute the secondary time-averaged hydrodynamic force felt by the sphere, and find it to be in excellent agreement with an analytical theory based on the Lorentz reciprocal theorem. In particular, the force changes sign when the Stokes layer thickness is comparable to the particle radius, and corresponds to a reversal in the secondary flow around the particle. We then show how the framework presented here smoothly connects the streaming-dominated and radiation-force-dominated pictures of time-averaged particle dynamics.