Multiscale differential analysis and modeling of one-dimensional fast acoustic streaming

Classical modeling of acoustically-driven flows relies almost exclusively on formal expansions about the smallness of the resulting streaming flow in relation to the driving particle velocity---a condition commonly referred to as ``slow streaming.'' This renders tractable the highly nonlinear governing equations in numerical and analytical settings. In contrast, the direction of modern microacoustofluidics research dictates that this order of magnitude separation assumption is not generally valid---in extremal systems, traditional perturbation approaches may fail to properly extract the dynamics of interest. We describe a theoretical approach that affords the user greater generality through its articulation and direct exploitation of the concomitant spatiotemporal scale disparities via multiscale differential operations. The method is applied to a one-dimensional problem of semiinfinite extent defined by particle and streaming velocities possessing similar magnitudes---the ``fast streaming'' condition. The compressible Navier-Stokes equations are solved in an approximate successive manner and the acoustic and streaming field equations are obtained. The steady state of the latter is solved analytically and a comparative analysis is undertaken with respect to the classical theory.