Acoustic streaming

Acoustic streaming is a non-linear effect. [5] We can decompose the velocity field in a vibration part and a steady part ${\displaystyle {u}=v+{\overline {u}}}$. The vibration part ${\displaystyle v}$ 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:

${\displaystyle {\overline {\rho }}{\partial _{t}{\overline {u}}_{i}}+{\overline {\rho }}{\overline {u}}_{j}{\partial _{j}{\overline {u}}_{i}}=-{\partial {\overline {p}}_{i}}+\eta {\partial _{j}^{2}{\overline {u}}_{i}}-{\partial _{j}}({\overline {\rho v_{i}v_{j}}}/{\partial x_{j}}).}$

The steady streaming originates from a steady body force ${\displaystyle f_{i}=-{\partial }({\overline {\rho v_{i}v_{j}}})/{\partial x_{j}}}$ that appears on the right hand side. This force is a function of what is known as the Reynolds stresses in turbulence ${\displaystyle -{\overline {\rho v_{i}v_{j}}}}$. 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 ${\displaystyle \epsilon \cos(\omega t)}$, the quadratic non-linearity generates a steady force proportional to ${\displaystyle \scriptstyle {\overline {\epsilon ^{2}\cos ^{2}(\omega t)}}=\epsilon ^{2}/2}$.

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:[6]

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

${\displaystyle U\sim -{3}/{(4\omega )}\times v_{0}dv_{0}/dx,}$

with ${\displaystyle v_{0}}$ the sound vibration velocity and ${\displaystyle x}$ along the wall boundary. The flow is directed towards decreasing sound vibrations (vibration nodes).

• near a vibrating bubble[7] of rest radius a, whose radius pulsates with relative amplitude ${\displaystyle \epsilon =\delta r/a}$ (or ${\displaystyle r=\epsilon a\sin(\omega t)}$), and whose center of mass also periodically translates with relative amplitude ${\displaystyle \epsilon '=\delta x/a}$ (or ${\displaystyle x=\epsilon 'a\sin(\omega t/\phi )}$). with a phase shift ${\displaystyle \phi }$

${\displaystyle \displaystyle U\sim \epsilon \epsilon 'a\omega \sin \phi }$

• far from walls[8] ${\displaystyle U\sim \alpha P/(\pi \mu c)}$ far from the origin of the flow ( with ${\displaystyle P}$the acoustic power, ${\displaystyle \mu }$ the dynamic viscosity and ${\displaystyle c}$ the celerity of sound). Nearer from the origin of the flow, the velocity scales as the root of ${\displaystyle P}$.

• 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.[9]