Oscillations, Waves, and Interactions - GWDG

180 R. Mettin
Figure 7. Trapped single bubble shedding microbubbles and recollecting them (high-speed
recording at 2250 fps in a 25 kHz standing wave; frame width 1.2 mm; from Ref. [29]).
detail, it is possible to replace the absolute value “1” for an unstable eigenvalue by a
higher number, e. g. 10 or 100, to model the larger damping of the mode. Indeed experimentally
observed surface instability of a single bubble could be better modelled
by this approach of an artificially higher stable eigenvalue [27,28]. An illustration is
given in Fig. 6, where the stability regions of several modes for a relevant parameter
range are plotted colour coded. Typically, the most unstable surface mode is n = 2,
and therefore it is often sufficient to calculate its behaviour.
If a bubble is spherically unstable, several things can happen: (1) It might shed
smaller bubbles to lose gas, reduce its equilibrium radius accordingly, and thus become
more stable (i. e., jump more to the bottom in Fig. 6). Afterwards, it might
(1a) regain spherical stability if the fragment disappears, or (1b) recollect the shedded
microbubble(s) in a continuing process and thus form a “jittering” or “dancing”
bubble, see Fig. 7 for an example. (2) Alternatively the bubble might split into
many fragments of similar size. The fragments can subsequently reunite again to
form one or more larger bubbles. If the destruction and merging process goes on,
a small cluster of fragment bubbles is observed, see Fig. 8. (3) The bubble might
get destroyed and vanish. This process is possibly an “atomization” into many very
small fragments that rapidly dissolve into the liquid. 8 As the examples have shown,
the spherical instability does not necessarily mean a destruction of a bubble, but may
at times give rise to a multitude of possibly interacting bubbles and thus to a bubble
structure. As an extreme example, Fig. 9 shows a quite large bubble attached to a
transducer surface, a sonotrode tip. This bubble is long-term existent, but shows
strong deviations from (semi)spherical shape. The high-speed sequence reveals that
it is also strongly collapsing. It continuously sheds smaller and larger daughter bubbles,
but keeps its size by remerging bubbles and possibly by diffusion processes.
From a spherical stability point of view, such bubbles should be very short-living,
but in some cases as the observed, they live quite long and serve as permanent bubble
source (cf. Sect. 2). Of course, the presence of a boundary is of importance here.
8 This happens typically in bubble trap experiments if the driving pressure is tuned too
high: the trapped bubble seems to disappear within a very short time [30].