Partition function (mathematics)

Acoustic streaming is a steady current 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. The attenuation coefficient is $\alpha =2\eta \omega ^{2}/(3\rho c^{3})$, 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. 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 $\delta =[\eta /(\rho \omega )]^{1/2}$ 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. We can decompose the velocity field in a vibration part and a steady part ${u}=v+{\overline {u}}$. The vibration part $v$ 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:

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

The steady streaming originates from a steady body force $f_{i}=-{\partial }({\overline {\rho v_{i}v_{j}}})/{\partial x_{j}}$ that appears on the right hand side. This force is a function of what is known as the Reynolds stresses in turbulence $-{\overline {\rho v_{i}v_{j}}}$. 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 $\epsilon \cos(\omega t)$, the quadratic non-linearity generates a steady force proportional to $\scriptstyle {\overline {\epsilon ^{2}\cos ^{2}(\omega t)}}=\epsilon ^{2}/2$.

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.

The order of magnitude of streaming velocities are

• near a boundary (outside of the boundary layer):

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

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 of rest radius a, whose radius pulsates with relative amplitude $\epsilon =\delta r/a$ (or $r=\epsilon a\sin(\omega t)$), and whose center of mass also periodically translates with relative amplitude $\epsilon '=\delta x/a$ (or $x=\epsilon 'a\sin(\omega t/\phi )$). with a phase shift $\phi$

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

References

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