Oscillations, Waves, and Interactions - GWDG

Acoustic cavitation 187
of the force between two oscillating bubbles. Even more, dissipative mechanisms like
viscosity and heat conduction can also play an important role. However, for many
aspects with respect to bubble structure formation, the spherical, nonlinear and time
averaged model is a good first choice. For a much more detailed discussion of the
Bjerknes forces see e. g. Ref. [41] and the references cited therein.
4.3 Added mass and viscous drag
The inertia of a moving bubble is dominated by the virtual or added mass of the liquid
circulating around it, and not by its gas mass (which is typically negligible). From
the assumption of potential flow, one can derive that the added mass Ma of a sphere
amounts to half of the mass of the displaced liquid, i. e., Ma = ρV/2 = 2πρR 3 /3 [6].
The force associated with a temporal change of this inertia reads
F M = − d
dt [Ma(t)U(t)] = − 2
3 πρ
�
3R 2 (t) ˙ R(t)U(t) + R 3 (t) ˙ �
U(t) , (5)
where U is the velocity of the bubble relative to the liquid. Note that the inertia is
changed both by acceleration of the bubble centroid and by a bubble volume change
caused by radial oscillation. This can lead for example to a “jerky” motion of a
bubble during its collapse [42] due to conservation of (liquid flow) momentum. A
direct observation of this phenomenon is shown in Fig. 12. Although the picture
demonstrates that the bubble motion needs not to be smooth, sometimes an averaging
approach is used to facilitate calculations. For example, the added mass is set
0.5 mm
Figure 12. Pseudo-streak image from a high-speed video sequence below a sonotrode
tip in water: vertical stripes from successive frames are printed next to each other, time
proceeding from left to right. The acoustic frequency is 20 kHz, and the framerate 50000 fps,
yielding an interframe time of 0.4 acoustic periods, which was also the exposure time. The
centroid positions of two bubbles that move upwards are marked by white lines. It can
clearly be observed that the bubble position frequently jumps upwards during collapse, and
stops during expansion. One such jumping event is magnified in the inset, and because of
the smearing due to the relatively long exposure time, the bubble shrinking simultaneous to
the upward translation is revealed. Note also the relatively fast net velocity of the bubbles
of 1 to 1.5 m/s.