Acoustic streaming

Consider a plane standing sound wave that corresponds to the velocity field $U(x,t)\; =\; v\_0\; \backslash cos\; kx\; \backslash cos\; \backslash omega\; t\; =\; \backslash varepsilon\; \backslash cos\; kx\; \backslash real(e^\{-i\backslash omega\; t\})$ where $k=2\backslash pi/\backslash lambda\; =\; \backslash omega/c$. 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 $\backslash delta\; =\; (2\backslash nu/\backslash omega)^\{1/2\}$. Rayleigh streaming is best visualized in the approximation $\backslash lambda\; \backslash gg\; l\; \backslash gg\; \backslash delta.$ 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 $\backslash delta$) 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

:$\backslash frac\{\backslash partial\; u\}\{\backslash partial\; t\}\; +\; u\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; x\}\; +\; v\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; y\}\; -\; \backslash nu\; \backslash frac\{\backslash partial^2\; u\}\{\backslash partial\; y^2\}\; =\; U\backslash frac\{\backslash partial\; U\}\{\backslash partial\; x\}\; +\; \backslash frac\{\backslash partial\; U\}\{\backslash partial\; t\}$

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 $\backslash psi$ that satisfies $u\; =\backslash partial\; \backslash psi/\backslash partial\; y$ and $v\; =\; -\backslash partial\; \backslash psi/\backslash partial\; x.$ Since by definition, 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 $\backslash varepsilon\; \backslash rightarrow\; 0$ as $u=\backslash varepsilon\; u\_1\; +\; \backslash varepsilon^2\; u\_2\; +\backslash cdots$, $\backslash psi=\; \backslash varepsilon\; \backslash psi\_1\; +\; \backslash varepsilon^2\; \backslash psi\_2\; \backslash cdots$ etc.

In the first approximation, one obtains

:$\backslash frac\{\backslash partial\; u\_1\}\{\backslash partial\; t\}\; -\; \backslash nu\; \backslash frac\{\backslash partial^2u\_1\}\{\backslash partial\; y^2\}\; =\; -\backslash omega\; \backslash cos\; kx\; \backslash real(ie^\{-i\backslash omega\; t\}).$

The solution that satisfies the no-slip condition at the wall $y/\backslash delta\; =0$ and approaches $U$ as $y/\backslash delta\backslash rightarrow\; \backslash infty$ is given by

:$u\_1\; =\; \backslash real\backslash left[\backslash cos\; kx\backslash ,\; (1-\; e^\{-\backslash kappa\; y\})\backslash ,\; e^\{-i\backslash omega\; t\}\; \backslash right],\; \backslash quad\; \backslash psi\_1\; =\; \backslash real\backslash left[\backslash cos\; kx\backslash ,\; \backslash zeta\_1(y)\backslash ,\; e^\{-i\backslash omega\; t\}\backslash right]$

where $\backslash kappa\; =\; (1-i)/\backslash delta$ and $\backslash zeta\_1\; =\; y+\; (e^\{-\backslash kappa\; y\}-1)/\backslash kappa.$

The equation at the next order is

:$\backslash frac\{\backslash partial\; u\_2\}\{\backslash partial\; t\}\; -\; \backslash nu\; \backslash frac\{\backslash partial^2u\_2\}\{\backslash partial\; y^2\}\; =\; U\; \backslash frac\{\backslash partial\; U\}\{\backslash partial\; x\}\; -\; u\_1\backslash frac\{\backslash partial\; u\_1\}\{\backslash partial\; x\}\; -\; v\_1\; \backslash frac\{\backslash partial\; u\_1\}\{\backslash partial\; y\}.$

Since each term on the right-hand side is quadratic, it will result in terms with frequencies $\backslash omega+\backslash omega=2\backslash omega$ and $\backslash omega-\backslash omega=0.$ The $\backslash omega=0$ terms correspond to time independent forcing for $u\_2$. Let us find solution that corresponds only to this time-independent part. This leads to $\backslash psi\_2\; =\; \backslash sin\; 2\; kx\backslash ,\; \backslash zeta\_2\; (y)/c$ where $\backslash zeta\_2$ satisfies the equationLandau, L. D., & Lifshitz, E. M. (2000). Fluid Mechanics (Course of Theoretical Physics, Volume 6).

:$2\backslash delta\; \backslash zeta\_2$' = 1 - |\zeta_1'|^2 + \real(\zeta_1 \zeta_1)

File:Rayleigh streaming.pdf

where prime denotes differentiation with respect to $y.$ The boundary condition at the wall implies that $\backslash zeta(0)=\backslash zeta\text{'}(0)=0.$ As $y/\backslash delta\backslash rightarrow\; \backslash infty$, $\backslash zeta\_2$ must be finite. Integrating the above equation twice gives

:$\backslash zeta\_2\text{'}\; =\; \backslash frac\{3\}\{8\}\; -\; \backslash frac\{1\}\{8\}e^\{-2y/\backslash delta\}\; -\; e^\{-y/\backslash delta\}\backslash left[\backslash sin\; \backslash frac\{y\}\{\backslash delta\}+\; \backslash frac\{1\}\{4\}\; \backslash cos\; \backslash frac\{y\}\{\backslash delta\}\; +\; \backslash frac\{y\}\{4\backslash delta\}\backslash left(\backslash sin\backslash frac\{y\}\{\backslash delta\}-\backslash cos\backslash frac\{y\}\{\backslash delta\}\backslash right)\; \backslash right].$

As $y/\backslash delta\; \backslash rightarrow\; \backslash infty$, $\backslash zeta\text{'}(\backslash infty)=3/8$ leading to the result that $v\_2(x,\backslash infty,t)\; =\; (3/8c)\; \backslash sin\; 2kx.$ 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 $v\_2(\backslash infty)$ is independent of $\backslash nu$, the steady streaming motion happening outside the boundary layer is also independent of viscosity, although its origin of existence 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

:$\backslash psi\_2\; =\; \backslash frac\{3\}\{16\; c\}\backslash sin\; 2kx\backslash ,\; [-(y-h)\; +\; (y-h)^3/h^2]$

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

Acoustic streaming is a non-linear effect.

Sir James Lighthill (1978) "Acoustic streaming", 61, 391, Journal of Sound and Vibration

We can decompose the velocity field in a vibration part and a steady part $\{u\}=v+\backslash overline\{u\}$.

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^2_{j} \overline{u}_i}-{\partial_j}(\overline{\rho v_i v_j}/{\partial x_j} ).

The steady streaming originates from a steady body force $f\_i=-\{\backslash partial\}(\backslash overline\{\backslash rho\; v\_i\; v\_j\}\; )/\{\backslash partial\; x\_j\}$ that appears on the right hand side. This force is a function of what is known as the Reynolds stresses in turbulence $-\backslash overline\{\backslash rho\; v\_i\; v\_j\}$. 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 $\backslash epsilon\backslash cos(\backslash omega\; t)$, the quadratic non-linearity generates a steady force proportional to

$\backslash scriptstyle\; \backslash overline\{\backslash epsilon^2\backslash cos^2(\backslash omega\; t)\}=\backslash epsilon^2/2$.

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:Squires, T. M. & Quake, S. R. (2005) Microfluidics: Fluid physics at the nanoliter scale, Review of Modern Physics, vol. 77, page 977

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

:$U\; \backslash sim\; -\{3\}/\{(4\backslash omega)\}\; \backslash times\; v\_0\; dv\_0/dx,$

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{{cite journal | last=Longuet-Higgins | first=M. S. | authorlink=Michael S. Longuet-Higgins | title=Viscous streaming from an oscillating spherical bubble | journal=Proc. R. Soc. Lond. A | year=1998 | volume=454 | pages=725–742 | doi=10.1098/rspa.1998.0183 |bibcode = 1998RSPSA.454..725L | issue=1970 | s2cid=123104032 }} of rest radius a, whose radius pulsates with relative amplitude $\backslash epsilon=\backslash delta\; r/a$ (or $r=\backslash epsilon\; a\; \backslash sin(\; \backslash omega\; t)$), and whose center of mass also periodically translates with relative amplitude $\backslash epsilon\text{'}=\backslash delta\; x/a$ (or $x=\backslash epsilon\text{'}\; a\; \backslash sin(\; \backslash omega\; t/\backslash phi)$). with a phase shift $\backslash phi$

:$\backslash displaystyle\; U\; \backslash sim\; \backslash epsilon\; \backslash epsilon\text{'}\; a\; \backslash omega\; \backslash sin\; \backslash phi$

• far from walls{{Cite journal

| doi = 10.1063/1.4895518

| issn = 1070-6631

| volume = 26

| issue = 9

| pages = 093602

| last = Moudjed

| first = B.

| author2 = V. Botton |author3=D. Henry |author4=Hamda Ben Hadid |author5=J.-P. Garandet

| title = Scaling and dimensional analysis of acoustic streaming jets

| journal = Physics of Fluids

| date = 2014-09-01

|bibcode = 2014PhFl...26i3602M | url = https://hal.archives-ouvertes.fr/hal-00923712/file/Scaling_and_dimensional_analysis_of_acoustic_streaming_jets_Accepted_in_Physics_of_Fluids_HALv3.pdf

}} $U\; \backslash sim\; \backslash alpha\; P/(\backslash pi\; \backslash mu\; c)$ far from the origin of the flow ( with $P$the acoustic power, $\backslash mu$ 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. 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

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.{{cite journal | last=Nilsson | first=Andreas | last2=Petersson | first2=Filip | last3=Jönsson | first3=Henrik | last4=Laurell | first4=Thomas | title=Acoustic control of suspended particles in micro fluidic chips | journal=Lab Chip | volume=4 | issue=2 | date=2004 | issn=1473-0197 | doi=10.1039/B313493H | pages=131–135}}{{cite journal | last=Laurell | first=Thomas | last2=Petersson | first2=Filip | last3=Nilsson | first3=Andreas | title=Chip integrated strategies for acoustic separation and manipulation of cells and particles | journal=Chem. Soc. Rev. | volume=36 | issue=3 | date=2007 | issn=0306-0012 | doi=10.1039/B601326K | pages=492–506}} These applications may include cell manipulation and cell sorting, drug delivery, 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.{{cite journal | last=Barrio-Zhang | first=Andres | last2=Anandan | first2=Sudharshan | last3=Deolia | first3=Akshay | last4=Wagner | first4=Ryan | last5=Warsinger | first5=David M. | last6=Ardekani | first6=Arezoo M. | title=Acoustically enhanced porous media enables dramatic improvements in filtration performance | journal=Separation and Purification Technology | volume=342 | date=2024 | doi=10.1016/j.seppur.2024.126972 | page=126972}} It can control biofilms in other applications as well.{{cite journal | last=Lin | first=Fangfei | last2=Yuan | first2=Songmei | last3=Ji | first3=Pengzhen | last4=Xu | first4=Weixian | title=Regulation of Bacterial Biofilm Formation by Ultrasound: Role of Autoinducer-2 and Finite-Element Analysis of Acoustic Streaming | journal=Ultrasound in Medicine & Biology | volume=49 | issue=9 | date=2023 | doi=10.1016/j.ultrasmedbio.2023.06.016 | pages=2191–2198}}

• Acoustic tweezers

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Category:Acoustics

Category:Sound