Structure of the streaming flow generated by a sphere oscillating in a viscous fluid

A body undergoing oscillatory motion in a viscous fluid naturally produces a steady secondary flow due to convective inertia. This is embodied in the streaming flow generated by a sphere executing oscillations in an unbounded fluid. We review the literature on this canonical problem and summarise both exact and asymptotic formulae in the small amplitude limit. These analytical formulae are used to explore the characteristic counter-circulating structure of this flow and clarify some previously unreported features. A single vortex exists regardless of the oscillation frequency, which can drive a counter-circulating flow away from the sphere. The centre of this vortex moves monotonically away from the sphere with decreasing oscillation frequency and engulfs the entire flow domain for β ≡ ωR² / ν < 16.317, where ω is the oscillation frequency, R is the sphere radius, and ν is the fluid kinematic viscosity. This abrupt change in flow structure at the critical frequency, β_critical = 16.317, and its quantification appears to have not been reported previously. We perform a numerical simulation of the Navier-Stokes equations, which reveals a universal relationship between the critical frequency and oscillation amplitude, clarifying previous reports on the structure of this streaming flow.