Oscillations, Waves, and Interactions - GWDG

p a [kPa]
200
150
100
50
0
SI
BJ
RD 1.0
RD 0.1
20 kHz
2 4 6 8 10 12 14 16 18 20
R0 [µm]
Acoustic cavitation 189
p a [kPa]
300
250
200
150
100
1 MHz
50
0
SI
BJ
RD 1.0
RD 0.1
0.5 1 1.5 2 2.5 3 3.5 4
R0 [µm]
Figure 13. Phase diagrams for bubbles in a standing wave, indicating the lines of surface
instability (SI) and Bjerknes force inversion (BJ). Also, the thresholds of rectified diffusion
are given for water at gas saturation (RD 1.0) and degassed to 10% of saturation (RD
0.1). Left: f = 20 kHz, right: f = 1 MHz (water at normal conditions, κ = 1). Points of
accumulation are marked by a circle.
the SI line is forbidden, as bubbles there would be splitting into smaller ones soon
(and thus jump to the left). From such an analysis, one can find accumulation points
in the parameter plane: bubbles tend to gather at crossings of the BJ line with a
negative slope part of the SI line, as long as such a point lies above the RD line. In
the examples of Fig. 13, such a point is close to 175 kPa and 5 µm at 20 kHz, and
175 kPa and 2 µm at 1 MHz, respectively. The conclusion is that in standing waves
of the given frequencies “typical” or “frequently found” parameters of the bubbles
are those of the accumulation points.
If one considers the actual trajectories in the R0–pa plane, antinode pressure and
wavelength have to be fixed, and added mass and viscous drag have to be taken
into account as indicated in Sect. 4.3. An example is given in Fig. 14(a) where lines
are plotted that represent the way of test bubbles through that plane. The shown
case is interesting, because a trajectory separation along a line, similar to a water
shed, can be observed. Indeed it is this water shed line which separates growing from
dissolving bubbles, and not the RD threshold: bubbles moving fast enough towards
a high presure region can escape from the dissolution zone [50]. This leads to the
surprising result that, for the indicated parameters and irrespective of their location,
almost all bubbles of rest radius larger than about 6 µm finally do not dissolve, but
grow until surface instability sets in!
Now we briefly turn to plane travelling waves (compare also Ref. [50]). As the
pressure amplitude does not vary in space, the life diagrams look simpler: for a
given pressure value, it is sufficient to consider paths in the R0–x plane, where x is
the coordinate in direction of the wave vector. Examples are shown in Fig. 14(b) for
20 kHz. In the examined parameter range of small rest radii it is found that significant
(macroscopic) translation appears only for growing bubbles. On the other hand, at
elevated pressures the motion is rather fast, and the bubbles do not grow significantly
before they covered a wavelength’s distance. At intermediate pressure values, a size