Modeling fast acoustic streaming: steady state and transient flow solutions

TL;DR

This paper introduces a new mathematical approach to model acoustic streaming, capturing both steady and unsteady flows by separating fast acoustic and slow hydrodynamic scales, providing new physical insights and bounds on energy efficiency.

Contribution

The novel method naturally derives scale separation from fluid properties, enabling accurate modeling of unsteady streaming and deriving equations that reveal nonlinearity and efficiency limits.

Findings

Derived a Burgers equation for unsteady flow

Obtained a Riccati equation for steady flow

Established a 50% upper bound on energy transduction efficiency

Abstract

We describe a novel mathematical method to supplant the classic approach and properly treat the spatiotemporal scale disparities present between the acoustics and remaining fluid dynamics. The method is applied in this work to well-known problems of semi-infinite extent defined by the Navier-Stokes equations, and preserves unsteady fluid behavior driven by the acoustic wave. The separation of the governing equations between the fast (acoustic) and slow (hydrodynamic) spatiotemporal scales are shown to naturally arise from the intrinsic properties of the fluid under forcing, not by arbitrary assumption beforehand. Solution of the unsteady streaming field equations provides physical insight into observed temporal evolution of bulk streaming flows that, to date, have not been modeled. A Burgers equation is derived from the new method to represent unsteady flow. By then assuming steady flow, a Riccati equation is found to represent it. Solving these equations produces direct, concise insight into the nonlinearity of the acoustic streaming phenomenon alongside an absolute, universal upper bound of 50\% for the energy efficiency in transducing acoustic energy input to the acoustic streaming energy output. Rigorous validation with respect to experimental and theoretical results from the classic literature is presented to connect this work to past efforts by many authors.