Oscillations, Waves, and Interactions - GWDG

Acoustic cavitation 179
λ =1 λ=10
Figure 6. Stability of a spherical bubble with respect to parametric surface mode instability
in the parameter plane of equilibrium radius R0 and driving pressure amplitude pa; driving
frequency 15 kHz. Colors indicate the stability with respect to the modes n = 2, 3, and 4.
Black: all modes stable, darkest blue: one mode unstable, blue: two modes unstable, and
light blue: all three modes unstable. Left: maximum allowed stable eigenvalue λ = 1, right:
λ = 10. At higher driving, all but very small bubbles (below the dynamic Blake threshold)
become unstable.
cally excited linear oscillator systems, which are for periodic radial oscillation of the
type of Hill’s differential equation. If am denotes the amplitude of the m-th surface
mode, its equation of motion reads
+(m − 1)
�
äm +
�
3 ˙ R
µ
+ 2(m + 2)(2m + 1)
R
ρR 2
(m + 1)(m + 2) σ
ρR 3 − ¨ R
R + 2(m + 2) µ ˙ R
ρR 3
�
�
˙am
am = 0 .
Here, the solution of the spherical bubble model, R(t), enters as the parametric driving.
The bubble is considered to be spherically unstable, if at least one surface mode
amplitude reaches the actual bubble radius value (Rayleigh-Taylor instability 7 ) or
grows indefinitely in time (parametric instability). For simplicity, only the latter case
is considered in the following. It is easily calculated numerically by integrating the
spherical bubble model together with the linear mode amplitude equations. Because
of linearity of the latter, it is sufficient for a periodic radial motion to integrate over
one period and calculate the Floquet multipliers (eigenvalues of the “monodromy
matrix”, see Ref. [26]). If the absolute value of at least one mode is larger than one,
formally that mode would grow and finally destroy the bubble. In reality, the linearization
breaks down from some point on, and mode coupling would lead to energy
redistribution, also to the fundamental mode. Without considering this issue here in
7 For this calculation, one has to assume an initial size of a disturbance of the spherical
form, and the outcome slightly depends on this choice.