Modeling fast acoustic streaming: Steady-state and transient flow solutions.

Abstract

Traditionally, acoustic streaming is assumed to be a steady-state, relatively slow fluid response to passing acoustic waves. This assumption, the so-called slow streaming assumption, was made over a century ago by Lord Rayleigh. It produces a tractable asymptotic perturbation analysis from the nonlinear governing equations, separating the acoustic field from the acoustic streaming that it generates. Unfortunately, this assumption is often invalid in the modern microacoustofluidics context, where the fluid flow and acoustic particle velocities are comparable. Despite this issue, the assumption is still widely used today, as there is no suitable alternative. We describe a 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 equationsbetween 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 equationsprovides physical insight into observed temporal evolution of bulk streaming flows that, to date, have not been modeled. A Burgers equationis derived from our method to represent unsteady flow. By then assuming steady flow, a Riccati equationis found to represent it. Solving these equationsproduces 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.