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Acoustic radiation force

${\displaystyle |F^{\rm {rad}}|={\frac {2\alpha I}{c}}}$

where

• ${\displaystyle |F^{\rm {rad}}|}$ is a force per unit volume, here expressed in kg/(s²cm²);
• ${\displaystyle \alpha }$ is the absorption coefficientinNp/cm (nepers per cm);
• ${\displaystyle I}$ is the temporal average intensity of the acoustic wave at the given location in W/cm²; and
• ${\displaystyle c}$ is the speed of sound in the medium in cm/s.^[1][2]

The effect of frequency on acoustic radiation force is taken into account via intensity (higher pressures are more difficult to attain at higher frequencies) and absorption (higher frequencies have a higher absorption rate). As a reference, water has an acoustic absorption of 0.002 dB/(MHz²cm).^{[3](page number?)} Acoustic radiation forces on compressible particles such as bubbles are also known as Bjerknes forces, and are generated through a different mechanism, which does not require sound absorptionorreflection.^[4] Acoustic radiation forces can also be controlled through sub-wavelength patterning of the surface of the object.^[5]

When a particle is exposed to an acoustic standing wave it will experience a time-averaged force known as the primary acoustic radiation force (${\displaystyle F_{pr}}$)_.^[6] In a rectangular microfluidic channel with coplanar walls which acts as a resonance chamber, the incoming acoustic wave can be approximated as a resonant, standing pressure wave of the form:

${\displaystyle p_{1}=p_{a}\cos {kz}}$.

where ${\displaystyle k}$ is the wave number. For a compressible, spherical and micrometre-sized particle (of radius ${\displaystyle a}$) suspended in an inviscid fluid in a rectangular micro-channel with a 1D planar standing ultrasonic wave of wavelength ${\displaystyle \lambda }$, the expression for the primary radiation force (at the far-field region where ${\displaystyle a\ll \lambda }$）becomes then ^[7][8][9][6]:

${\displaystyle F_{pr}^{\rm {1D}}=4\pi \Phi ({\tilde {\kappa }},{\tilde {\rho }})a^{3}kE_{ac}\sin {2kz}}$

${\displaystyle \Phi ({\tilde {\kappa }},{\tilde {\rho }})={5{\tilde {\rho }}-2 \over 2{\tilde {\rho }}+1}-{\tilde {\kappa }}}$

${\displaystyle E_{\rm {ac}}={1 \over 4}\kappa _{f}p_{a}^{2}={p_{a}^{2} \over 4\rho _{f}c_{f}^{2}}}$

where

• ${\displaystyle \Phi }$ is the acoustic contrast factor
• ${\displaystyle {\tilde {\kappa }}}$ is relative compressibility between the particle ${\displaystyle \kappa _{p}}$ and the surrounding fluid ${\displaystyle \kappa _{f}}$: ${\displaystyle {\tilde {\kappa }}=\kappa _{p}/\kappa _{f}}$
• ${\displaystyle {\tilde {\rho }}}$ is relative density between the particle ${\displaystyle \rho _{p}}$ and the surrounding fluid ${\displaystyle \rho _{f}}$: ${\displaystyle {\tilde {\rho }}=\rho _{p}/\rho _{f}}$
• ${\displaystyle E_{\rm {ac}}}$ is the acoustic energy density
• The factor ${\displaystyle \sin {2kz}}$ makes the radiation force period doubled and phase shifted relative to the pressure wave ${\displaystyle p_{a}\cos {kz}}$
• ${\displaystyle c_{f}}$ is the speed of sound in the fluid

• Acoustic tweezers
• Radiation pressure