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. The attenuation coefficient is α = 2 η ω 2 / ( 3 ρ 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 α − 1 ~10 cm at 1 MHz) than in water ( α − 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 δ = [ η / ( ρ ω ) ] 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 + 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:
ρ ¯ ∂ t u ¯ i + ρ ¯ u ¯ j ∂ j u ¯ i = − ∂ p ¯ i + η ∂ j j 2 u ¯ i − ∂ j ( ρ v i v j ¯ ) . The steady streaming originates from a steady body force f i = − ∂ ( ρ v i v j ¯ ) / ∂ x j that appears on the right hand side. This force is a function of what is known as the Reynolds stresses in turbulence − ρ 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 ϵ cos ( ω t ) , the quadratic non-linearity generates a steady force proportional to ϵ 2 cos 2 ( ω t ) ¯ = ϵ 2 / 2 .
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:
near a boundary (outside of the boundary layer): U ∼ − 3 / ( 4 ω ) × v 0 d v 0 / d x , 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 ϵ = δ r / a (or r = ϵ a sin ( ω t ) ), and whose center of mass also periodically translates with relative amplitude ϵ ′ = δ x / a (or x = ϵ ′ a sin ( ω t / ϕ ) ). with a phase shift ϕ U ∼ ϵ ϵ ′ a ω sin ϕ far from walls U ∼ α P / ( π μ c ) far from the origin of the flow ( with P the acoustic power, μ the dynamic viscosity and c the celerity of sound). Nearer from the origin of the flow, the velocity scales as the root of P .