Oscillations, Waves, and Interactions - GWDG

184 R. Mettin
which is called the primary Bjerknes force.
The standard discussion of this average considers weak bubble oscillations and a
plane standing or travelling pressure wave [4,5]. Then the results can be captured by
the relative phase of pressure and bubble oscillation, which depends on the bubble
size R0 in relation to the linear resonance radius Rres: larger bubbles are attracted by
pressure nodes, and smaller by the pressure antinodes. In propagating waves, bubbles
around Rres feel a strong force in the wave’s direction, and the force diminishes for
bubbles much smaller or much larger than the linear resonance size.
In the case of strongly driven bubbles, which is typical for cavitating liquids, F B1
becomes more complicated because of the anharmonic bubble oscillation. In particular,
the collapse time of stronger and stronger expanding bubbles shifts to later
instances, and thus the primary Bjerknes force in standing waves can change its sign
at high pressure zones for small bubbles. This effect was first noted in the context of
SBSL [36,37], and it is very important for cavitation fields: also bubbles and bubble
clusters much smaller than the linear resonance radius can get expelled from pressure
antinodes, and the high pressure regions may get depleted.
Figure 10 shows for the two typical ultrasonic frequencies 20 kHz and 1 MHz the
regions in R0 − pa parameter space where the bubbles are driven away from the
pressure antinode due to the primary Bjerknes force. Here, pa is the local driving
amplitude at the bubble’s position and can usually be identified with a spatial position
in the standing wave. In the case of a traveling wave, the changes caused by
nonlinear bubble oscillations are not so dramatic, although the forces can get much
larger than for linear approximation, and also parameter combinations exist where
the sign reverses, i. e., the bubbles feel a primary Bjerknes force against the wave
direction [38].
p a [kPa]
200
180
160
140
120
100
80
60
40
20
20 kHz
0
0 20 40 60 80 100 120 140 160 180
R0 [µm]
p a [kPa]
400
350
300
250
200
150
100
50
1 MHz
0
0 0.5 1 1.5 2
R0 [µm]
2.5 3 3.5 4
Figure 10. Sign of the primary Bjerknes force in a standing sound wave in the parameter
space of bubble equilibrium radius R0 and pressure amplitude pa. White areas indicate a
force towards the high pressure regions, dark areas a force towards the pressure nodes. Left:
f = 20 kHz, Rres = 138 µm; right: f = 1 MHz, Rres = 3.15 µm (water at normal conditions,
κ = 1, σ = 0.07275 N/m).