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

2. Order of magnitude of acoustic streaming velocities

3. References

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 , 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 ~10 cm at 1 MHz) than in water (~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 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 .

The vibration part 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:

The steady streaming originates from a steady body force that appears on the right hand side. This force is a function of what is known as the Reynolds stresses in turbulence . 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 , the quadratic non-linearity generates a steady force proportional to

.

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):

with the sound vibration velocity and 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 (or ), and whose center of mass also periodically translates with relative amplitude (or ). with a phase shift

• far from walls^[6] far from the origin of the flow ( with the acoustic power, the dynamic viscosity and the celerity of sound). Nearer from the origin of the flow, the velocity scales as the root of .

References

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• Chaim Azriel Weizmann
• Chaim Bar Lev
• Chaim Ben Pesach
• Chaim Berlin
• Chaim Berlin (rabbi)
• Chaim Berlin Yeshiva
• Chaim Bialik
• Chaim Elazar Shapiro
• Chaim Goldberg
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