Oscillations, Waves, and Interactions - GWDG

190 R. Mettin
(a)
p a [Pa]
140
120
100
80
60
40
20
0
1 2 3 4
R0 [µm]
5 6 7
x [mm]
10
1
0.1
0.01
0.001
0.0001
(b)
0.10 bar
0.55 bar
1.00 bar
1.50 bar
0 2 4 6 8 10 12 14
R0 [µm]
Figure 14. (a) Bubble life trajectories (red) in the plane of rest radius R0 and pressure
amplitude pa for a harmonic plane standing wave at 20 kHz and 250 kPa antinode pressure.
Trajectories generally run from bottom to the top, i. e., bubbles travel to higher values
of pa. The black dotted line is the RD threshold, and the blue line marks the effective
border between dissolving and finally growing bubbles. The green, dashed line is the SI
line. (b) Trajectories in the parameter plane of R0 and x in a plane travelling wave of
20 kHz. Different colours (line styles) indicate different acoustic pressures, as given in the
plot. The bubble motion is generally towards the top, i. e., to larger x (from Ref. [50]).
levelling of the growing bubbles occurs: the longer the travelled distance, the more
the size distribution width of the bubble population is shrinking.
In a similar fashion, it is possible to discuss general features of bubble translation
and gas diffusion in other acoustic set-ups of different frequencies or geometries. It
should be kept in mind that all is based upon a single spherical bubble in a given
and fixed pressure field. This approximation excludes, for instance, interaction with
additional bubbles, and any influence of the bubble distribution on the sound field.
At higher bubble densities, however, both effects come into play, and the life cycle
inspection may turn out to be incomplete.
6 Structure formation
If we focus our interest on multibubble systems and their structures in acoustic fields,
one crucial point to consider is certainly the origin of the bubbles (compare Sect. 2).
However, it turns out not to be easy to assess or predict how many bubbles of what
size are created where in the liquid. For simulations, experimentally observed data
can be used as an input for this problem, and often this is still the most reasonable
approach. From this, heuristic bubble generation models can be derived.
Furthermore, bubble-bubble interaction is an essential issue. With help of the
secondary Bjerknes force (Sect. 4.2) and reasonable assumptions for bubble collision
and merging, it is possible to “close” the theoretical description of a multi-bubble
system. Between creation and destruction, each bubble follows its equations of motion,
derived by the balance of forces. We add the index i to indicate the number of