D Bubbles
643
∂ x
f x
(1 + f 2
x + f 2
y ) 1/2
+ ∂ y
f y
(1 + f 2
x + f 2
y ) 1/2
= 0 .
(D.9)
This reduces to the relation given Eq. (D.7).
Because the minimal area relation is a second order partial differential equation,
non-linear in the unknown function f (x, y), only a few exact solutions are known,
such as a catenoid between two separated rings and a helicoid. Other solutions can
be found numerically under conditions where the fluid surface boundary is known.
Solving Eq. (D.9) is called Plateau’s problem.
The minimal area surface equation when there is a pressure difference across the
surface is Eq. (3.14), is sometimes expressed as
σ ∇ · ˆ
n = σ
1
R 1
+
1
R 2
= p 2 − p 1 ,
(D.10)
where σ is the surface tension of the interface between the fluids on either side of
the interface, and R 1 , R 2 are the principle radii of curvature at the location on the
surface where ∇ · n is evaluated. In terms of the surface equation z = f (x, y), the
minimum area equation becomes
(1 + f
2
y )f xx + (1 + f
2
y )f yy − 2f x f y f xy =
p
σ
(1 + f
2
x + f
2
y )
3/2 .
(D.11)
D.3 Vibration of Bubbles
If a bubble occurs within a cell, it is subject to vibrational modes due to passing sonic
waves. Bubble vibrations can take energy from the passing wave, possibly leading
to excessive heat in the fluid surround the bubble and destruction of organelle or cell
walls.
For a gas or a cavitation bubble, the dominant mode of vibration is expansion
and contraction of the bubble radius. This case has been described by Rayleigh and
for traveling cavitation bubbles by Plesset. 2
The dynamics of bubble vibrations follows from the Navier-Stokes equation and
the proper boundary conditions, resulting in the Rayleigh-Plesset equation for the
radial motion of the bubble surface:
p = p o + ρ
r
d 2 r
dt 2 +
3
2
dr
dt
2
+
4η
r
dr
dt
+
σ
r
,
(D.12)
2 Lord Rayleigh, On the pressure developed in a liquid during the collapse of a spherical cavity
Phil. Mag. 34, 94–98 (1917); M.S. Plesset, The dynamics of cavitation bubbles, ASME J Appl
Mech 16, 228–231 (1949).
643
∂ x
f x
(1 + f 2
x + f 2
y ) 1/2
+ ∂ y
f y
(1 + f 2
x + f 2
y ) 1/2
= 0 .
(D.9)
This reduces to the relation given Eq. (D.7).
Because the minimal area relation is a second order partial differential equation,
non-linear in the unknown function f (x, y), only a few exact solutions are known,
such as a catenoid between two separated rings and a helicoid. Other solutions can
be found numerically under conditions where the fluid surface boundary is known.
Solving Eq. (D.9) is called Plateau’s problem.
The minimal area surface equation when there is a pressure difference across the
surface is Eq. (3.14), is sometimes expressed as
σ ∇ · ˆ
n = σ
1
R 1
+
1
R 2
= p 2 − p 1 ,
(D.10)
where σ is the surface tension of the interface between the fluids on either side of
the interface, and R 1 , R 2 are the principle radii of curvature at the location on the
surface where ∇ · n is evaluated. In terms of the surface equation z = f (x, y), the
minimum area equation becomes
(1 + f
2
y )f xx + (1 + f
2
y )f yy − 2f x f y f xy =
p
σ
(1 + f
2
x + f
2
y )
3/2 .
(D.11)
D.3 Vibration of Bubbles
If a bubble occurs within a cell, it is subject to vibrational modes due to passing sonic
waves. Bubble vibrations can take energy from the passing wave, possibly leading
to excessive heat in the fluid surround the bubble and destruction of organelle or cell
walls.
For a gas or a cavitation bubble, the dominant mode of vibration is expansion
and contraction of the bubble radius. This case has been described by Rayleigh and
for traveling cavitation bubbles by Plesset. 2
The dynamics of bubble vibrations follows from the Navier-Stokes equation and
the proper boundary conditions, resulting in the Rayleigh-Plesset equation for the
radial motion of the bubble surface:
p = p o + ρ
r
d 2 r
dt 2 +
3
2
dr
dt
2
+
4η
r
dr
dt
+
σ
r
,
(D.12)
2 Lord Rayleigh, On the pressure developed in a liquid during the collapse of a spherical cavity
Phil. Mag. 34, 94–98 (1917); M.S. Plesset, The dynamics of cavitation bubbles, ASME J Appl
Mech 16, 228–231 (1949).