Oscillations, Waves, and Interactions - GWDG

178 R. Mettin
100 kPa 200 kPa
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Figure 5. Dynamic Blake threshold in the parameter plane of equilibrium radius R0 and
frequency f. The normalized maximum radius (Rmax − R0)/R0 is gray coded for the two
driving pressures 100 kPa (left) and 200 kPa (right). At 200 kPa, the sharp border of the
white area constitutes the threshold (arrows).
tic frequency of zero. 6 This phenomenon has been highlighted by Noltingk and
Neppiras [18,19]. In more detail it has been calculated and discussed later, for instance
in Refs. [17,20], and its relevance for trapped single bubbles and single bubble
sonoluminescence (SBSL) has been realized in Refs. [21,22]. Figure 5 from Ref. [20]
illustrates the sudden jump in parameter space of equilibrium radius and acoustic frequency.
The distinction between weakly and strongly oscillating bubbles, i. e., below
and beyond the dynamic Blake threshold, is very important. This is true obviously
for applications based on heavy bubble collapse, but also for structure formation of
acoustic cavitation bubbles, because acoustic forces, gas diffusion, and stability can
be changing dramatically when crossing this border. These issues are addressed in
the following sections.
3.2 Spherical shape stability
The spherical shape of a cavitation bubble is idealized. Due to surface tension, a static
bubble will always tend to form a sphere, which is also stable with respect to small
perturbations. This does not hold necessarily for a pulsating bubble. To estimate the
stability of a driven oscillating spherical bubble, usually the bubble shape is expanded
into spherical harmonics (modes), and the equations of motion of the corresponding
expansion coefficients are considered. For small excursions from the spherical form,
the equations can be linearized, and it turns out that then only unidirectional coupling
of the fundamental (radial) motion to the higher order surface modes persists
(no mutual coupling or back-coupling to the radial oscillation, see Refs. [4,5,23–25]).
Formally, in linear approximation all mode coefficients are described by parametri-
6 The notion “(in)active” relates to all cavitation effects due to a strong bubble collapse.
Nevertheless, bubbles below the dynamic Blake threshold oscillate, feel acoustic forces and
might be “active” from another point of view.