Oscillations, Waves, and Interactions - GWDG

Acoustic cavitation 183
up to some approximation by the following equation [33–35]:
dR0
dt = DRGT
�
c0
p0 +
R0
4σ
�−1 � � � �
R c∞
− 1 +
3R0 R0 c0
2σ
�
R0p0
R 3 0
〈R/R0〉
〈(R/R0) 4 �
. (1)
〉
Here the following symbols are used: D diffusion constant, RG gas constant, T
temperature, c0 and c∞ gas concentration in saturation and far from the bubble,
and 〈.〉 is indicating a time average.
In an extended cavitating field, rectified gas diffusion can play a role for the lifecycle
of individual bubbles and thus for the size distribution in the bubble population.
This depends, among other parameters, on the gas content of the liquid, spherical
shape stability of the bubbles, local pressure distribution and bubble translation.
This will be discussed later.
It should also be noted that the well-known degassing of liquids by ultrasound
is actually based on cavitation, and that rectified diffusion processes play here an
important role.
4 Acoustic forces and bubble motion
Bubbles in acoustic fields do not only perform volume oscillations, but they also
move in space. These translational motions are mainly caused by acoustic forces,
the Bjerknes forces, that are introduced in the following. To complete the picture of
bubble motion, its virtual mass and viscous drag are also briefly discussed.
4.1 Primary Bjerknes forces
It is an everyday experience that bubbles in quiet liquids rise to the surface. The
reason for this is the hydrostatic pressure, which is, roughly speaking, pushing with
a higher force on the bottom surface than on the top surface of the bubble. Mathematically,
the net force on the bubble is yielded by summing up the forces on all
surface elements, and by Gauss’ theorem, we find
� � � � �
F = p dS = ∇p dV .
If the pressure gradient is slowly varying over the scale of the bubble’s size, we can
substitute the right-hand side integral by V ∇p(x0), where x0 is the position of the
bubble center. This approximation is usually allowed, but care has to be taken in
cases where the bubbles reach wavelength extensions (like degassing processes at
higher frequencies).
The hydrostatic pressure turns out to be typically negligible compared to the effects
caused by the sound field: in water, gravity creates a gradient of about 10 mPa/µm,
while in a pressure wave of 1 bar amplitude and frequency of some MHz we find
gradients reaching hundreds of Pa/µm.
The sound pressure gradients and the bubble volumes are time varying, and the
net force on a bubble depends on the time average
F B1 = 〈V (t)∇p(x0, t)〉 , (2)