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

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.