Oscillations, Waves, and Interactions - GWDG

(Rmax - R0) / R0
30
25
20
15
10
5
0
140 kPa
130 kPa
150 kPa
20/1
|
20/1
|
1 2 5
R0 [ µm ]
10 20
Acoustic cavitation 177
10/1
|
20 kHz
10/1
|
(Rmax - R0) / R0
20
15
10
5
0
300 kPa
400 kPa
200 kPa
1/1
|
500 kHz
|
1/1
0.2 0.5 1
R0 [ µm ]
2 5
Figure 4. Bubble response at driving frequencies of 20 kHz (left) and 500 kHz (right). The
normalized maximum bubble radius during one driving period is plotted vs the equilibrium
radius for different driving pressures. If higher-periodic or chaotic solutions occur, more
than one point is visible. Certain resonances are indicated by numbers (from Ref. [17]).
restoring force being the static liquid pressure (for R > R0) or the gas pressure (for
R < R0). If the pressure oscillates around the static value, which is the case in a
sound wave, the bubble radius performs forced oscillations: the gas bubble behaves
as a driven oscillator system [5]. Small excitation allows to consider the linearized
model, which shows a resonance frequency of [15]
ω 2 res = (2πfres) 2 = 1
ρR2 �
0
3κp0 + 2σ
R0
�
(3κ − 1) .
In the case of water under normal conditions, we find the approximate relation
fresR0 ≈ 3 [m/s]. For monofrequent driving at the frequency f we can accordingly
find a resonant equilibrium radius Rres ≈ 3/f [m]. The linearized pulsation of the
gas bubble is important for analytical considerations of many aspects of bubble dynamics
and as a reference motion. It is discussed in textbooks [4–6]. However, one
has to keep in mind that it is only valid in the limit of small bubble volume changes,
and stronger excursions exhibit the nonlinear behaviour of the equation. In particular,
nonlinear resonances, hysteresis, and chaotic oscillations [16] occur for suitable
parameters. This has been shown for parameters relevant for acoustic cavitation
at lower ultrasonic frequencies in Ref. [17]. From there, Fig. 4 has been taken for
illustration of the rich behaviour.
A specific and important feature of the bubble response is the “dynamic” Blake
threshold: for sufficiently small bubbles, surface tension can not be overcome by the
negative acoustic pressure and the bubbles oscillate only weakly. From a certain
equilibrium radius on (or similar, from a certain stronger tension on), the bubble
grows to a multitude of its equilibrium size and shows subsequent strong collapse
and rebound events. The transition or border between small, “quiet” and “inactive”
bubbles and the larger, strongly expanding “loud” and “active” bubbles is relatively
sharp and equivalent to the standard static Blake threshold in the limit of an acous-