Oscillations, Waves, and Interactions - GWDG

176 R. Mettin
important quantity characterizing every bubble is the mass of non-condensable gas
inside, defining its equilibrium size. On a longer time scale, this size can change due
to diffusion (see below, Sect. 3.3).
The spherical symmetry renders the problem one-dimensional and facilitates calculations
significantly. Indeed, this assumption is surprisingly good in many situations
encountered in acoustic cavitation, and mainly violated for larger bubbles and for
very close boundaries imposed by other bubbles or objects. Instability of sphericity
is also discussed below (Sect. 3.2).
Spherical gas bubble models mainly vary in the Equation Of State (EOS) for
the liquid and the gas. The simplest assumption is an incompressible liquid and
an ideal gas, resulting in an equation frequently referred to as the Rayleigh-Plesset
equation [4,5]. It can be improved by incorporation of liquid compressibility and an
advanced EOS for the gas like a Van-der-Waals model. For the problems considered
here, a sufficient compromise between computational effort and accuracy is a model
with slight (first order) compressibility of the liquid, namely the Gilmore model [13]
or the Keller-Miksis model [14]. With an ideal gas law the Keller-Miksis model reads
�
1 − ˙ R
c
�
R ¨ R + 3
2 ˙ R 2
�
pl = p0 + 2σ
R0
�
1 − ˙ R
3c
�
=
�
1 + ˙ R
c
�
pl
ρ
R dpl
+
ρc dt ,
� � �3κ R0
− p0 −
R
2σ 4µ
−
R R ˙ R − pa(t) .
Here, R is the bubble radius, c and ρ the sound speed and the density of the liquid,
µ and σ the viscosity and the surface tension, and κ the polytropic exponent. The
acoustic pressure is denoted pa(t), while the static pressure is p0. R0 denotes the
bubble equilibrium radius, and vapour pressure is neglected.
In the following we focus on the volume oscillations, shape instabilities, and rectified
diffusion. Afterwards, acoustic forces are addressed which are induced by a cw
acoustic field on a gas bubble. The acoustic forces finally lead to spatial translation
and structure formation of the bubbles. For all calculations, the spherical bubble
model is used and thus it is the core of all results presented.
3.1 Bubble oscillations
A spherical gas-filled bubble under static conditions posseses an equilibrium radius
R0 (or an equilibrium volume V0 = 4πR 3 0/3) if the static pressure p0 is larger than
the Blake threshold pressure [4]
pB = p0 + 8σ
9
�
3σ
2[p0 + (2σ/R0)]R 3 0
� 1/2
For p0 < pB the bubble would expand infinitely. 5 A momentary excursion of the
bubble radius from the equilibrium results in damped oscillations around R0, the
5 However, in practice it is normally not possible to sustain a static negative pressure
for a longer time under the condition of bubble expansion, i. e., cavitation. Thus bubble
expansion will be stopped sooner or later in real systems.
.