Acoustic streaming

Short description: Phenomenon in physics

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. Acoustic streaming was explained first by Lord Rayleigh in 1884.[1] 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).[2] The attenuation coefficient is α=2ηω2/(3ρc3), 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 (Rayleigh streaming). Either when sound reaches a boundary, or when a boundary is vibrating in a still medium.[3] 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/(ρω)]1/2 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,[4] and even biological cells[5] are examples of boundary-driven acoustic streaming.

## Contents

* 1 Rayleigh streaming
* 2 Origin: a body force due to acoustic absorption in the fluid
* 3 Order of magnitude of acoustic streaming velocities
* 4 Applications
* 5 See also
* 6 References

## Rayleigh streaming

Consider a plane standing sound wave that corresponds to the velocity field U(x,t)=v0cos⁡kxcos⁡ωt=εcos⁡kxℜ(e−iωt) where k=2π/λ=ω/c. Let the characteristic (transverse) dimension of the problem be l. 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) δ=(2ν/ω)1/2. Rayleigh streaming is best visualized in the approximation λ≫l≫δ. As in U(x,t), the velocity components (u,v) are much less than c. In addition, the characteristic time scale within the boundary layer is very large (because of the smallness of δ) in comparison with the acoustic time scale l/c. These observations imply that the flow in the boundary layer may be regarded as incompressible.

The unsteady, incompressible boundary-layer equation is

∂u∂t+u∂u∂x+v∂u∂y−ν∂2u∂y2=U∂U∂x+∂U∂t

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 ψ that satisfies u=∂ψ/∂y and v=−∂ψ/∂x. Since by definition the velocity field U in the sound wave is very small, we can formally obtain the solution for the boundary-layer equation by introducing the asymptotic series for ε->0 as u=εu1+ε2u2+⋯, ψ=εψ1+ε2ψ2⋯ etc.

In the first approximation, one obtains

∂u1∂t−ν∂2u1∂y2=−ωcos⁡kxℜ(ie−iωt).

The solution that satisfies the no-slip condition at the wall y/δ=0 and approaches U as y/δ->∞ is given by

u1=ℜ[cos⁡kx(1−e−κy)e−iωt],ψ1=ℜ[cos⁡kxζ1(y)e−iωt]

where κ=(1−i)/δ and ζ1=y+(e−κy−1)/κ.

The equation at the next order is

∂u2∂t−ν∂2u2∂y2=U∂U∂x−u1∂u1∂x−v1∂u1∂y.

Since each term on the right-hand side is quadratic, it will result in terms with frequencies ω+ω=2ω and ω−ω=0. The ω=0 terms correspond to time-independent forcing for u2. Let us find a solution that corresponds only to this time-independent part. This leads to ψ2=sin⁡2kxζ2(y)/c where ζ2 satisfies the equation[6]

2δζ2‴=1−|ζ1′|2+ℜ(ζ1ζ1″),
File:Rayleigh streaming.pdfRayleigh streaming

where the primes denote differentiation with respect to y. The boundary condition at the wall implies that ζ(0)=ζ′(0)=0. As y/δ->∞, ζ2 must be finite. Integrating the above equation twice gives

ζ2′=38−18e−2y/δ−e−y/δ[sin⁡yδ+14cos⁡yδ+y4δ(sin⁡yδ−cos⁡yδ)].

As y/δ->∞, ζ′(∞)=3/8, leading to the result that v2(x,∞,t)=(3/8c)sin⁡2kx. Thus, at the edge of the boundary, there is a steady fluid motion superposed on the oscillating motion. This velocity forcing will drive a steady streaming motion outside the boundary layer. The interesting result is that since v2(∞) is independent of ν, the steady streaming motion happening outside the boundary layer is also independent of viscosity, although its origin is due to the viscous boundary layer.

The outer steady streaming incompressible motion will depend on the geometry of the problem. If there are two walls one at y=0 and y=2h, then the solution is

ψ2=316csin⁡2kx[−(y−h)+(y−h)3/h2],

which corresponds a periodic array of counter-rotating vortices, as shown in the figure.

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

Acoustic streaming is a non-linear effect.[7] 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‾∂tu‾i+ρ‾u‾j∂ju‾i=−∂p‾i+η∂j2u‾i−∂j(ρvivj‾/∂xj).

The steady streaming originates from a steady body force fi=−∂(ρvivj‾)/∂xj that appears on the right hand side. This force is a function of what is known as the Reynolds stresses in turbulence −ρvivj‾. 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 ϵcos⁡(ωt), then the quadratic non-linearity generates a steady force proportional to ϵ2cos2(ωt)‾=ϵ2/2.

## 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:[8]

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

U∼−3/(4ω)×v0dv0/dx,

with v0 the sound vibration velocity and x along the wall boundary. The flow is directed towards decreasing sound vibrations (vibration nodes).

* near a vibrating bubble[9] of rest radius a, whose radius pulsates with relative amplitude ϵ=δr/a (or r=ϵasin⁡(ωt)), and whose center of mass also periodically translates with relative amplitude ϵ′=δx/a (or x=ϵ′asin⁡(ωt/ϕ)). with a phase shift ϕ

U∼ϵϵ′aωsin⁡ϕ

* far from walls[10] U∼αP/(πμc) far from the origin of the flow ( with Pthe 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.

* it has been shown that even biological species, e.g., adherent cells, can also exhibit acoustic streaming flow when exposed to acoustic waves. Cells adhered to a surface can generate acoustic streaming flow in the order of mm/s without being detached from the surface.[11]

## Applications

Research around acoustic streaming shows many effective applications, especially around particle manipulation, although translation to commercial use is in early stages for most uses. In microfluidics, it can be used for cell manipulation and sorting.[12][13] These applications may include cell manipulation and cell sorting, drug delivery, and homogenizing reactants. Acoustic streaming is also relevant to sonoporation for increasing cell-membrane permeability. Acoustic streaming is also used in membrane processes, where it can control fouling and increase particle collection.[14] It can control biofilms in other applications as well.[15]

## See also

* Acoustic tweezers

## References

1. ↑ Rayleigh, L. (1884). On the circulation of air observed in Kundt's tubes, and on some allied acoustical problems. Philosophical Transactions of the Royal Society of London, 175, 1-21.
2. ↑ see video on http://lmfa.ec-lyon.fr/spip.php?article565&lang=en
3. ↑ 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.
4. ↑ Nama, N., Huang, P.H., Huang, T.J., and Costanzo, F., Investigation of acoustic streaming patterns around oscillating sharp edges, Lab on a Chip, Vol. 14, pp. 2824-2836, 2014
5. ↑ Salari, A.; Appak-Baskoy, S.; Ezzo, M.; Hinz, B.; Kolios, M.C.; Tsai, S.S.H. (2019) Dancing with the Cells: Acoustic Microflows Generated by Oscillating Cells. https://doi.org/10.1002/smll.201903788
6. ↑ Landau, L. D., & Lifshitz, E. M. (2000). Fluid Mechanics (Course of Theoretical Physics, Volume 6).
7. ↑ Sir James Lighthill (1978) "Acoustic streaming", 61, 391, Journal of Sound and Vibration
8. ↑ Squires, T. M. & Quake, S. R. (2005) Microfluidics: Fluid physics at the nanoliter scale, Review of Modern Physics, vol. 77, page 977
9. ↑ Longuet-Higgins, M. S. (1998). "Viscous streaming from an oscillating spherical bubble". Proc. R. Soc. Lond. A 454 (1970): 725–742. doi:10.1098/rspa.1998.0183. Bibcode: 1998RSPSA.454..725L.
10. ↑ 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. doi:10.1063/1.4895518. ISSN 1070-6631. Bibcode: 2014PhFl...26i3602M. https://hal.archives-ouvertes.fr/hal-00923712/file/Scaling_and_dimensional_analysis_of_acoustic_streaming_jets_Accepted_in_Physics_of_Fluids_HALv3.pdf.
11. ↑ Salari, A.; Appak-Baskoy, S.; Ezzo, M.; Hinz, B.; Kolios, M.C.; Tsai, S.S.H. (2019) Dancing with the Cells: Acoustic Microflows Generated by Oscillating Cells. https://doi.org/10.1002/smll.201903788
12. ↑ Nilsson, Andreas; Petersson, Filip; Jönsson, Henrik; Laurell, Thomas (2004). "Acoustic control of suspended particles in micro fluidic chips". Lab Chip 4 (2): 131–135. doi:10.1039/B313493H. ISSN 1473-0197.
13. ↑ Laurell, Thomas; Petersson, Filip; Nilsson, Andreas (2007). "Chip integrated strategies for acoustic separation and manipulation of cells and particles". Chem. Soc. Rev. 36 (3): 492–506. doi:10.1039/B601326K. ISSN 0306-0012.
14. ↑ Barrio-Zhang, Andres; Anandan, Sudharshan; Deolia, Akshay; Wagner, Ryan; Warsinger, David M.; Ardekani, Arezoo M. (2024). "Acoustically enhanced porous media enables dramatic improvements in filtration performance". Separation and Purification Technology 342. doi:10.1016/j.seppur.2024.126972.
15. ↑ Lin, Fangfei; Yuan, Songmei; Ji, Pengzhen; Xu, Weixian (2023). "Regulation of Bacterial Biofilm Formation by Ultrasound: Role of Autoinducer-2 and Finite-Element Analysis of Acoustic Streaming". Ultrasound in Medicine & Biology 49 (9): 2191–2198. doi:10.1016/j.ultrasmedbio.2023.06.016.

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