Abstract

We study a fifty-year-old problem of fast acoustic streaming, that is, the generation of moderate or large hydrodynamic Reynolds number ( $\textit {Re}$ ) acoustic streaming (or steady flow) by the convection of momentum in an acoustic wave (or another periodic flow), while the latter is simultaneously altered by the former. The intrinsic disparity of length and time scales makes a brute-force solution of the full Navier–Stokes and continuity equations a formidable problem. Circumventing this difficulty, we split the problem into a time-averaged system of equations for the steady flow component and a dynamic system of equations for its quasi-periodic flow counterpart. The latter system of equations is obtained by subtracting the time-averaged Navier–Stokes equation from its original dynamic form, and is rendered a nonlinear wave equation using the continuity equation and an adiabatic connection between density and pressure. The resulting equations are compatible with the theory by Eckart for small $\textit {Re}$ flow, and capture large- $\textit {Re}$ effects. Scaling analysis and a case study show that acoustic streaming is weak and does not contribute to the acoustic wave close to the wave source, relevant to many microfluidic systems. At small $\textit {Re}$ , the streaming magnitude is proportional to an inverse Strouhal number, a small quantity in experiments. Moderate and large $\textit {Re}$ render the streaming magnitude comparable to the pre-attenuating periodic flow (or particle velocity of the wave) at approximately a wave attenuation length away from the wave source or further; the wave is altered by the streaming that it generates, and the streaming dominates the flow far from the wave source.

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