Spatiotemporal differential partitioning for one-dimensional fast acoustic streaming

Abstract

Historically, acoustofluidics modeling of streaming behaviors has relied heavily and almost exclusively upon formal asymptotic expansions in a relevant small parameter (e.g., the acoustic Mach number) in order to render tractable the highly nonlinear governing equations. However, the direction of modern acoustofluidics research dictates that no such order of magnitude separation between the acoustic and streaming fields can be generally relied upon—in extremal systems, formal asymptotic methods fail to properly extract the dynamics of interest. We outline progress toward the development of a theoretical approach that affords greater generality through its direct, explicit consideration and exploitation of all spatiotemporal scale disparities by partitioning differential operations. The method is applied to a one-dimensional problem of semiinfinite extent that, by convention, is classified as “fast streaming.” The compressible Navier-Stokes equations are solved in an approximate successive manner in order to obtain the acoustic and streaming fields. We show that the steady multi-temporal behavior of this result is characterized by a conservation of energy across acoustic and streaming scales.