# Acoustic streaming

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 ${\displaystyle \alpha ^{-1}}$~10 cm at 1 MHz) than in water (${\displaystyle \alpha ^{-1}}$~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.

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

Acoustic streaming is a non-linear effect. [3] We can decompose the velocity field in a vibration part and a steady part ${\displaystyle {u}=v+{\overline {u}}}$. The vibration part ${\displaystyle v}$ 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:

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

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:[4]

• near a boundary (outside of the boundary layer):
${\displaystyle U\sim -{3}/{(4\omega )}\times v_{0}dv_{0}/dx,}$

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

• near a vibrating bubble[5] 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 ${\displaystyle \phi }$
${\displaystyle \displaystyle U\sim \epsilon \epsilon 'a\omega \sin \phi }$
• far from walls[6] ${\displaystyle U\sim \alpha P/(\pi \mu c)}$ far from the origin of the flow ( with ${\displaystyle P}$the acoustic power, ${\displaystyle \mu }$ the dynamic viscosity and ${\displaystyle c}$ the celerity of sound). Nearer from the origin of the flow, the velocity scales as the root of ${\displaystyle P}$.

## References

1. ^
2. ^ 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. doi:10.1007/s10494-005-4132-4.
3. ^ Sir James Lighthill (1978) "Acoustic streaming", 61, 391, Journal of Sound and Vibration
4. ^ Squires, T. M. & Quake, S. R. (2005) Microfluidics: Fluid physics at the nanoliter scale, Review of Modern Physics, vol. 77, page 977
5. ^ 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.
6. ^ Moudjed, B.; V. Botton; D. Henry; Hamda Ben Hadid; J.-P. Garandet (2014-09-01). "Scaling and dimensional analysis of acoustic streaming jets". Physics of Fluids. 26 (9): 093602. Bibcode:2014PhFl...26i3602M. doi:10.1063/1.4895518. ISSN 1070-6631. Retrieved 2014-09-18.