Multiscale articulating differentials analysis of one-dimensional fast acoustic streaming*

Classical approaches to modeling acoustic streaming flows involve perturbative expansions that depend on the slowness of the streaming velocity relative to the driving acoustics. This "slow streaming" approach was first described by Rayleigh as a tractable means for extracting the streaming field from the governing nonlinear equations. In modern acoustofluidics applications, order of magnitude separation between the acoustics and the resulting streaming flow cannot be generally relied upon. In these extremal systems, classical methods fail to extract the essential dynamics. We describe a theoretical framework with greater flexibility via direct, explicit consideration and exploitation of drastic spatiotemporal scale disparities typifying microacoustofluidic systems. This is achieved with multiscale partitioning of differential operations. The framework is generally applicable to nonlinear continuous media and is particularly well suited to acoustofluidics modeling. We briefly demonstrate application of the framework to various media before delving more deeply into its application to a one-dimensional acoustofluidics problem of semiinfinite extent characterized by order of magnitude equivalence between the constituent flow fields—the "fast streaming" condition. The resulting Burgers-Riccati model is used to explain the fundamental characteristics of fast bulk acoustic streaming. We derive from the model a number of important flow properties including a remarkably simple non-constitutive upper bound on the energetic conversion efficiency of the driving acoustics to the resultant maximum streaming flow magnitude. The theory is confirmed with a broad survey of experimental data from the recent literature.