Oscillations, Waves, and Interactions - GWDG

140 W. Lauterborn et al.
liquid
κ
pi
Figure 1. Descriptors for a spherical bubble: bubble radius R(t), rest radius Rn, internal
pressure pi, external pressure pe. Parameters of the liquid: density ρ, viscosity µ, and
surface tension σ. The bubble medium is characterized by the polytropic exponent κ.
2.1 Bubble models
To formulate a bubble model, various parameters of the liquid surrounding the bubble
and of the gas and vapour contents inside the bubble need to be specified. For
a spherical bubble the dependent variables can be condensed to just one, the bubble
radius R whose variation with time t has to be determined, R(t). The model parameters
as used here are given in Fig. 1. Besides R these are the radius of the bubble at
rest, Rn, the external pressure in the liquid, pe, and the pressure inside the bubble,
pi, further the polytropic exponent of the gas in the bubble, κ, and ρ, µ, and σ, the
density, the viscosity, and the surface tension of the liquid, respectively.
The simplest model is the Rayleigh model [2]:
R
R
n
σ
p e
ρ, µ
ρR ¨ R + 3
2 ρ ˙ R 2 = pi − pe , (1)
where an overdot means differentiation with respect to time. The difference in pressure,
pi−pe, drives the bubble motion. The form of the inertial terms on the left-hand
side is due to the spherical three-dimensional geometry that is transformed to one
radial dimension in the differential equation. Both, pi and pe, become functions of
radius R and time t, when gas and vapour fill the bubble, and when surface tension
σ, liquid viscosity µ, and a sound field are taken into account. With these inclusions
the Rayleigh model takes the form [3–6]:
with
ρR ¨ R + 3
2 ρ ˙ R 2 = pgn
� �3κ Rn
+ pv − pstat −
R
2σ 4µ
−
R R ˙ R − p(t) , (2)
pgn = 2σ
Rn
+ pstat − pv , (3)
Rn being the equilibrium radius, pstat the static pressure, pv the (constant) vapour