Analysis of Transient Acoustic Streaming

Acoustic streaming has traditionally been assumed to be a steady-state, relatively slow fluid response to passing acoustic waves. This slow streaming assumption was first made by Lord Rayleigh a century and a half ago. From intractable nonlinear governing equations, the method produces useful solutions by first solving the acoustic field independently of the streaming it generates and other hydrodynamic phenomena. The resolved acoustic field is then used to determine the streaming through a time average of the nonlinear phenomenon. However, in modern use, acoustic streaming exhibits large velocities comparable to the acoustic field, rendering the traditional approach flawed if not outright invalid. However, the method remains widely used today, as there is no suitable alternative. We provide a novel approach to supplant this method, seeking to properly treat the spatiotemporal scale disparities present between the acoustics and remaining fluid dynamics. 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.