Eckart streaming with nonlinear high-order harmonics: an example at gigahertz

TL;DR

This paper derives a theoretical source term for Eckart streaming at gigahertz frequencies, accounting for high-order harmonics, and demonstrates that these harmonics significantly enhance streaming velocities.

Contribution

It provides the first exact expression for bulk streaming including high-order harmonics at GHz, improving upon empirical formulas and clarifying the role of nonlinear effects.

Findings

High-order harmonics increase streaming velocity by over 20%.

The derived force terms enable accurate evaluation of GHz acoustic streaming.

Nyborg's expression should be avoided in numerical simulations due to inaccuracies.

Abstract

Acoustic streaming shows great potential in applications such as bubble dynamics, cell aggregation, and nano-sized particle isolation in the biomedical and drug industries. As the acoustic shock distance decreases with the increase of incident frequency, the nonlinear propagation effect will play a role in acoustic streaming, e.g., Eckart (bulk) streaming at a few gigahertz (GHz). However, the theory of source terms of bulk streaming is still missing at this stage when high-order acoustic harmonics play a role. In this paper, we derive the source term including the contribution of higher-order harmonics. The streaming-induced hydrodynamic flow is assumed to be incompressible and no shock wave occurs during the nonlinear acoustic propagation as restricted by the traditional Goldberg number \Gamma < 1 or \Gamma \approx 1 which indicates the importance of nonlinearity relative to dissipation. The derived force terms allow evaluating bulk streaming with high-order harmonics at GHz and provide an exact expression compared to the existing empirical formulas. Numerical results show that the contribution of higher-order harmonics increases the streaming flow velocity by more than 20%. We show that the expression introduced by Nyborg should be avoided in numerical computations as it includes part of the acoustic radiation force that does not lead to acoustic streaming.