Time-averaged Dynamics of Compressible Particles in Oscillatory Gradient Flows

TL;DR

This paper develops an analytic theory linking steady particle motion in oscillatory acoustic flows to the incident flow and forces, covering a broad range from inviscid to viscous regimes, and validates it with numerical simulations.

Contribution

It introduces a comprehensive analytic framework that relates acoustic flow parameters to steady particle motion, including nonlinear inertial effects and the influence of Stokes layer thickness.

Findings

The theory recovers secondary radiation forces for thin Stokes layers.

Predicts motion reversal when Stokes layer thickness is comparable to particle size.

Validated predictions with numerical simulations.

Abstract

Acoustic fields effect steady transport of suspended particles by rectifying the inertia of primary oscillations. We develop a fully analytic theory that relates this steady particle motion to incident oscillatory (acoustic) flow and the time-averaged force acting on the particle, systematically spanning the entire range between inviscid acoustofluidics and viscous particle hydrodynamics. By applying the Lorentz reciprocal theorem, we obtain a Fax\'{e}n-like relationship that includes nonlinear inertial forces, which depend on (i) the thickness of the oscillatory Stokes layer around the particle, and (ii) the density and compressibility contrast between the particle and the fluid. The framework recovers secondary radiation forces for thin Stokes layers, and predicts a reversal of the motion when the thickness of the Stokes layer is comparable to the particle size. We quantitatively validate the theory using numerical simulations of the timescale-separated hydrodynamics.