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Acoustic streaming is a steady flow in a fluid driven by the absorption of high amplitude acoustic oscillations. This phenomenon can be observed near sound emitters, or in the standing waves within a Kundt's tube. It is the less-known opposite of sound generation by a flow.

There are two situations where sound is absorbed in its medium of propagation:

during propagation.[1] The attenuation coefficient is $${\displaystyle \alpha =2\eta \omega ^{2}/(3\rho c^{3})}$$, following Stokes' law (sound attenuation). This effect is more intense at elevated frequencies and is much greater in air (where attenuation occurs on a characteristic distance $$\alpha ^{-1}$$~10 cm at 1 MHz) than in water ( $$\alpha ^{-1}$$~100 m at 1 MHz). In air it is known as the Quartz wind.
near a boundary. Either when sound reaches a boundary, or when a boundary is vibrating in a still medium.[2] A wall vibrating parallel to itself generates a shear wave, of attenuated amplitude within the Stokes oscillating boundary layer. This effect is localised on an attenuation length of characteristic size $${\displaystyle \delta =[\eta /(\rho \omega )]^{1/2}}$$ whose order of magnitude is a few micrometres in both air and water at 1 MHz. The streaming flow generated due to the interaction of sound waves and microbubbles, elastic polymers,[3] and even biological cells[4] are examples of boundary driven acoustic streaming.

Origin: a body force due to acoustic absorption in the fluid

Acoustic streaming is a non-linear effect. [5] We can decompose the velocity field in a vibration part and a steady part $${\displaystyle {u}=v+{\overline {u}}}$$. The vibration part v {\displaystyle v} v is due to sound, while the steady part is the acoustic streaming velocity (average velocity). The Navier–Stokes equations implies for the acoustic streaming velocity:

$${\displaystyle {\overline {\rho }}{\partial _{t}{\overline {u}}_{i}}+{\overline {\rho }}{\overline {u}}_{j}{\partial _{j}{\overline {u}}_{i}}=-{\partial {\overline {p}}_{i}}+\eta {\partial _{j}^{2}{\overline {u}}_{i}}-{\partial _{j}}({\overline {\rho v_{i}v_{j}}}/{\partial x_{j}}).}$$

The steady streaming originates from a steady body force $${\displaystyle f_{i}=-{\partial }({\overline {\rho v_{i}v_{j}}})/{\partial x_{j}}}$$ that appears on the right hand side. This force is a function of what is known as the Reynolds stresses in turbulence $${\displaystyle -{\overline {\rho v_{i}v_{j}}}}.$$ The Reynolds stress depends on the amplitude of sound vibrations, and the body force reflects diminutions in this sound amplitude.

We see that this stress is non-linear (quadratic) in the velocity amplitude. It is non-vanishing only where the velocity amplitude varies. If the velocity of the fluid oscillates because of sound as $${\displaystyle \epsilon \cos(\omega t)}$$, the quadratic non-linearity generates a steady force proportional to $${\displaystyle \scriptstyle {\overline {\epsilon ^{2}\cos ^{2}(\omega t)}}=\epsilon ^{2}/2}.$$
Order of magnitude of acoustic streaming velocities

Even if viscosity is responsible for acoustic streaming, the value of viscosity disappears from the resulting streaming velocities in the case of near-boundary acoustic steaming.

The order of magnitude of streaming velocities are:[6]

near a boundary (outside of the boundary layer):

$${\displaystyle U\sim -{3}/{(4\omega )}\times v_{0}dv_{0}/dx,}$$

with $$v_{0}$$ the sound vibration velocity and x along the wall boundary. The flow is directed towards decreasing sound vibrations (vibration nodes).

near a vibrating bubble[7] of rest radius a, whose radius pulsates with relative amplitude $${\displaystyle \epsilon =\delta r/a}$$ (or $${\displaystyle r=\epsilon a\sin(\omega t)})$$, and whose center of mass also periodically translates with relative amplitude $${\displaystyle \epsilon '=\delta x/a}$$ (or $${\displaystyle x=\epsilon 'a\sin(\omega t/\phi )})$$. with a phase shift $$\phi$$

$${\displaystyle \displaystyle U\sim \epsilon \epsilon 'a\omega \sin \phi }$$

far from walls[8] $${\displaystyle U\sim \alpha P/(\pi \mu c)}$$ far from the origin of the flow ( with Pthe acoustic power, $$\mu$$ the dynamic viscosity and c {\displaystyle c} c the celerity of sound). Nearer from the origin of the flow, the velocity scales as the root of P.

it has been shown that even biological species, e.g., adherent cells, can also exhibit acoustic streaming flow when exposed to acoustic waves. Cells adhered to a surface can generate acoustic streaming flow in the order of mm/s without being detached from the surface.[9]

References

see video on http://lmfa.ec-lyon.fr/spip.php?article565&lang=en
Wan, Qun; Wu, Tao; Chastain, John; Roberts, William L.; Kuznetsov, Andrey V.; Ro, Paul I. (2005). "Forced Convective Cooling via Acoustic Streaming in a Narrow Channel Established by a Vibrating Piezoelectric Bimorph". Flow, Turbulence and Combustion. 74 (2): 195–206. CiteSeerX 10.1.1.471.6679. doi:10.1007/s10494-005-4132-4. S2CID 54043789.
Nama, N., Huang, P.H., Huang, T.J., and Costanzo, F., Investigation of acoustic streaming patterns around oscillating sharp edges, Lab on a Chip, Vol. 14, pp. 2824-2836, 2014
Salari, A.; Appak-Baskoy, S.; Ezzo, M.; Hinz, B.; Kolios, M.C.; Tsai, S.S.H. (2019) Dancing with the Cells: Acoustic Microflows Generated by Oscillating Cells. https://doi.org/10.1002/smll.201903788
Sir James Lighthill (1978) "Acoustic streaming", 61, 391, Journal of Sound and Vibration
Squires, T. M. & Quake, S. R. (2005) Microfluidics: Fluid physics at the nanoliter scale, Review of Modern Physics, vol. 77, page 977
Longuet-Higgins, M. S. (1998). "Viscous streaming from an oscillating spherical bubble". Proc. R. Soc. Lond. A. 454 (1970): 725–742. Bibcode:1998RSPSA.454..725L. doi:10.1098/rspa.1998.0183. S2CID 123104032.
Moudjed, B.; V. Botton; D. Henry; Hamda Ben Hadid; J.-P. Garandet (2014-09-01). "Scaling and dimensional analysis of acoustic streaming jets" (PDF). Physics of Fluids. 26 (9): 093602. Bibcode:2014PhFl...26i3602M. doi:10.1063/1.4895518. ISSN 1070-6631.
Salari, A.; Appak-Baskoy, S.; Ezzo, M.; Hinz, B.; Kolios, M.C.; Tsai, S.S.H. (2019) Dancing with the Cells: Acoustic Microflows Generated by Oscillating Cells. https://doi.org/10.1002/smll.201903788

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