Oscillations, Waves, and Interactions - GWDG

pressure, and
The single bubble – a hot microlaboratory 141
p(t) = −ˆpa sin ωt , (4)
the acoustic pressure, taken here to vary sinusoidally, with angular frequency ω and
pressure amplitude ˆpa. This model and some variants are called Rayleigh–Plesset
models.
In this work we widely use the Gilmore model [7] that incorporates sound radiation
into the liquid from the oscillating bubble, whose surface acts like the membrane of a
spherical loudspeaker. It is further augmented by a van der Waals law [8] to account
for a noncompressible volume of the inert gas inside the bubble. This bubble model
reads:
�
where
1 − ˙ R
C
�
R ¨ R + 3
�
1 −
2
˙ �
R
3C
H =
� p|r=R
p|r→∞
�n �
ρ
p(ρ) = A
ρ0
˙R 2 =
�
1 + ˙ R
C
�
H + ˙ �
R
1 −
C
˙ �
R
R
C
dH
, (5)
dR
dp(ρ)
, (6)
ρ
− B , (7)
p|r=R =
�
pstat + 2σ
� � 3 Rn − bR
Rn
3 n
R3 − bR3 �κ
−
n
2σ 4µ
−
R R ˙ R , (8)
p|r→∞ = pstat + p(t) , (9)
C = � c0 2 + (n − 1)H . (10)
The additional parameters and variables in this model are the sound velocity in the
liquid at normal conditions, c0, the sound velocity at the wall of the bubble, C, the
enthalpy, H, the parameters of the equation of state, where b is the van der Waals
constant, and the Tait equation (7) is chosen for the liquid with its parameters A,
B, and n [9].
2.2 Oscillation properties
Both models described so far consist of just one ordinary differential equation of
first order, however a strongly nonlinear one. They can be viewed as describing a
nonlinear oscillator, but one with really peculiar properties, for instance a varying
mass during the oscillation. This leads to special oscillation features, in particular
extremely strong collapses of the bubble at elevated acoustic forcing.
The equations of motion for the bubble radius as a function of time can be solved
after initial conditions for the radius and for the velocity of the bubble wall have been
specified. This has been done for a substantial part of the parameter space [6,10,11].
The main parameters of interest are the bubble radius at rest, Rn, and the sound
field parameters pressure amplitude, ˆpa, and sound field (circular) frequency, ω. The
other parameters are connected with the properties of the liquid outside the bubble
and the gas or gases and the vapour of the liquid inside the bubble.