644
D Bubbles
where p is the pressure in the bubble, p 0 the ambient pressure in the fluid, r the
radius of the bubble, ρ the density of the fluid, η the viscosity of the fluid, σ the
surface tension of the fluid in contact with the vapor in the bubble, and t is time.
A large bubble (with adiabatic behavior of the enclosed vapor) will naturally
vibrate in radial mode near the resonance frequency
f R =
1
2πr
γ
3p o
ρ o
,
(D.13)
where γ = c p /c v is the ratio of specific heats at constant pressure to constant
volume for the vapor in the bubble.
Small bubbles (with isothermal behavior of the enclosed vapor) naturally vibrate
in radial mode near
f R =
1
2πr
3p o + 4σ/r
ρ o
.
(D.14)
For example, if the cavitation bubble in interstitial fluid had a radius 1 μm, then at
body temperature (with σ ∼ 50 × 10 −3 N/m) f R is about 3 MHz. A radius of a
tenth of a micron gives a resonant cavitation frequency of about 70 MHz. These are
ultrasonic frequencies.
Inversely, the most likely cavitation bubble size will be those bubbles near the
resonant frequency. When surface tension dominates over ambient pressure times
radius, i.e. very small bubbles, we will have, from Eq. (D.14),
r ∼
4σ
ρ(2πf ) 2
1/3
∼ 1.7 μm
1 MHz
f
2/3
.
Higher frequency modes of vibration (harmonics) are also expected. As an
example of such modes, we give the results of an early investigation Sir Horace
Lamb, 3 who calculated the vibrational frequencies of a bubble formed by the
immersion of one fluid of density ρ i into another with density ρ o . For simplicity,
he assumed the fluids were incompressible and non-viscous. His result is
f l =
1
2π
σ
r 3
(l − 1)l(l + 1)(l + 2)
(l + 1)ρ i + lρ o
.
(D.15)
The index l takes values 2, 3, · · · corresponding to harmonics on the surface of the
bubble. 4
3 Horace Lamb, Hydrodynamics [Cambridge Univ. Press] (1879).
4 The radial mode (l = 0) does not occur for an incompressible bubble. The l = 1 is excluded as
the motion corresponds to an oscillation of the center-of-mass.
D Bubbles
where p is the pressure in the bubble, p 0 the ambient pressure in the fluid, r the
radius of the bubble, ρ the density of the fluid, η the viscosity of the fluid, σ the
surface tension of the fluid in contact with the vapor in the bubble, and t is time.
A large bubble (with adiabatic behavior of the enclosed vapor) will naturally
vibrate in radial mode near the resonance frequency
f R =
1
2πr
γ
3p o
ρ o
,
(D.13)
where γ = c p /c v is the ratio of specific heats at constant pressure to constant
volume for the vapor in the bubble.
Small bubbles (with isothermal behavior of the enclosed vapor) naturally vibrate
in radial mode near
f R =
1
2πr
3p o + 4σ/r
ρ o
.
(D.14)
For example, if the cavitation bubble in interstitial fluid had a radius 1 μm, then at
body temperature (with σ ∼ 50 × 10 −3 N/m) f R is about 3 MHz. A radius of a
tenth of a micron gives a resonant cavitation frequency of about 70 MHz. These are
ultrasonic frequencies.
Inversely, the most likely cavitation bubble size will be those bubbles near the
resonant frequency. When surface tension dominates over ambient pressure times
radius, i.e. very small bubbles, we will have, from Eq. (D.14),
r ∼
4σ
ρ(2πf ) 2
1/3
∼ 1.7 μm
1 MHz
f
2/3
.
Higher frequency modes of vibration (harmonics) are also expected. As an
example of such modes, we give the results of an early investigation Sir Horace
Lamb, 3 who calculated the vibrational frequencies of a bubble formed by the
immersion of one fluid of density ρ i into another with density ρ o . For simplicity,
he assumed the fluids were incompressible and non-viscous. His result is
f l =
1
2π
σ
r 3
(l − 1)l(l + 1)(l + 2)
(l + 1)ρ i + lρ o
.
(D.15)
The index l takes values 2, 3, · · · corresponding to harmonics on the surface of the
bubble. 4
3 Horace Lamb, Hydrodynamics [Cambridge Univ. Press] (1879).
4 The radial mode (l = 0) does not occur for an incompressible bubble. The l = 1 is excluded as
the motion corresponds to an oscillation of the center-of-mass.