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
Kundt's tube is an experimental acoustical apparatus invented in 1866 by German physicist August Kundt for the measurement of the speed of sound in a gas or a solid rod. The experiment is still taught today due to its ability to demonstrate l ...
. Acoustic streaming was explained first by
Lord Rayleigh
John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. Amo ...
in 1884.
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 in bulk flow ('Eckart streaming'). The attenuation coefficient is
, following
Stokes' law (sound attenuation) Stokes's law of sound attenuation is a formula for the attenuation of sound in a Newtonian fluid, such as water or air, due to the fluid's viscosity. It states that the amplitude of a plane wave decreases exponentially with distance traveled, at a ...
. 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 ('Rayleigh streaming'). 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
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, and even biological cells are examples of boundary driven acoustic streaming.
Rayleigh streaming
Consider a plane standing sound wave that corresponds to the velocity field
where
. Let the characteristic (transverse) dimension of the problem be
. The flow field just described corresponds to inviscid flow. However viscous effects will be important close to a solid wall; there then exists a boundary layer of thickness or, penetration depth
. Rayleigh streaming is best visualized in the approximation
As in
, the velocity components
are much less than
. In addition, the characteristic time scale within the boundary layer is very large (because of the smallness of
) in comparison with the acoustic time scale
. These observations imply that the flow in the boundary layer may be regarded as incompressible.
The unsteady, incompressible
boundary-layer
In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary condi ...
equation is
:
where the right-hand side terms correspond to the pressure gradient imposed on the boundary layer. The problem can be solved using the
stream function
The stream function is defined for incompressible flow, incompressible (divergence-free) fluid flow, flows in two dimensions – as well as in three dimensions with axisymmetry. The flow velocity components can be expressed as the derivatives of t ...
that satisfies
and
Since by definition, velocity field
in the sound wave is very small, we can formally obtain the solution for the boundary layer equation by introducing the asymptotic series for
as
,
etc.
In the first approximation, one obtains
:
The solution that satisfies the no-slip condition at the wall
and approaches
as
is given by
: