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Axisymmetric SoapBubbles
Consider an axisymmetric soapbubble whose surface takes the form in cylindrical coordinates. See Section C.3. The
unit normal to the surface is

(396) 
where
. Hence, from (1633), the mean curvature of the surface is given by

(397) 
The YoungLaplace equation, (353), then yields

(398) 
where

(399) 
Here, is the net surface tension, including the contributions from the internal and external soap/air interfaces. Moreover,
is the pressure difference between the interior and the exterior of the bubble. Equation (398)
can be integrated to give

(400) 
where is a constant.
Suppose that the bubble occupies the region
, where , and has a fixed
radius at its two endpoints, and .
This could most easily be achieved by supporting the
bubble on two rigid parallel coaxial rings located at and .
The net free energy required to create the bubble can be written

(401) 
where is area of the bubble surface, and the enclosed volume. The first term on the righthand side of
the above expression represents the work needed to overcome surface tension, whilst the second term
represents the work required to overcome the pressure difference, , between the exterior and
the interior of the bubble. Now, from the general principles of statics, we expect a stable equilibrium state
of a mechanical system to be such as to minimize the net free energy, subject to any dynamical
constraints. It follows that the equilibrium shape of the bubble
is such as to minimize

(402) 
subject to the constraint that the bubble radius, , be fixed at and .
Hence, we need to find the function that minimizes the integral

(403) 
where

(404) 
subject to the constraint that is fixed at the limits. This is a standard problem in the calculus of
variations. (See Appendix D.) In fact, since the functional
does not depend explicitly
on , the minimizing function is the solution of [see Equation (1687)]

(405) 
where is an arbitrary constant.
Thus, we obtain

(406) 
which can be rearranged to give Equation (400). Hence, we conclude that application of the YoungLaplace
equation does indeed lead to a bubble shape that minimizes the net free energy of the soap/air interfaces.
Consider the case , in which there is no pressure difference across the surface of the bubble.
In this situation, writing
, Equation (400) reduces to

(407) 
Moreover, according to the previous discussion, the bubble shape specified by (407) is such as to minimize the surface area of the bubble (since the only contribution to the free energy of the soap/air interfaces is directly proportional
to the bubble area).
The above equation can be rearranged to give

(408) 
which leads to

(409) 
or

(410) 
where is a constant.
This expression describes an axisymmetric surface known as a catenoid.
Figure 16:
Radius versus axial distance for a catenoid soap bubble supported by two parallel coaxial rings
of radius located at .

Suppose, for instance, that the soap bubble is
supported by identical rings of radius that are located a perpendicular
distance apart. Without loss of generality, we can specify that the rings lie at . It thus follows,
from (410), that , and

(411) 
Here, the parameter must be chosen so as to satisfy

(412) 
For example, if then , and the resulting bubble shape is illustrated in Figure 16.
Let and , in which case the above equation becomes

(413) 
Now, the function attains a maximum value

(414) 
when
. Moreover, if then Equation (413) possesses two roots. It turns
out that the root associated with the smaller value of minimizes the interface system energy, whereas
the other root maximizes the free energy. Hence, the former root corresponds to a stable
equilibrium state, whereas the latter corresponds to an unstable
equilibrium state. On the other hand, if then Equation (413)
possesses no roots, implying the absence of any equilibrium state. The critical case
corresponds to and , where
and
.
It is easily demonstrated that and
.
We conclude that a stable equilibrium state of a catenoid bubble only exists when
, which corresponds to
. If the relative ring spacing exceeds the critical value then the bubble presumably bursts.
Figure 17:
Radius versus axial distance for an unduloid soap bubble calculated with .

Consider the case , in which there is a pressure difference across the surface of the bubble.
In this situation, writing
Equation (400) becomes

(417) 
which can be rearranged to give

(418) 
We can assume, without loss of generality, that
.
It follows, from the above expression,
that
. Hence, we can write
where
and .
It follows that
and

(423) 
which can be integrated to give

(424) 
where and are incomplete elliptic integrals [see Equations (381)
and (382)]. Here, we have assumed that when .
There are three cases of interest.
Figure 18:
Radius versus axial distance for a positive pressure nodoid soap bubble calculated with .

In the first case, and . It follows that
for
, and
for , where
The axisymmetric curve parameterized by the above pair of equations is known as an unduloid. Note that an
unduloid bubble always has positive internal pressure (relative to the external pressure): i.e., .
An example unduloid soap bubble is illustrated in Figure 17
Figure 19:
Radius versus axial distance for a negative pressure nodoid soap bubble calculated with .

In the second case, and . It follows that
for
, and
for , where
, and
The axisymmetric curve parameterized by the above pair of equations is known as an nodoid. This
particular type of nodoid bubble has positive internal pressure: i.e., .
An example positive pressure nodoid soap bubble is illustrated in Figure 18.
In the third case, and . It follows that
for
(or
), and
for , where
The axisymmetric curve parameterized by the above pair of equations is again a nodoid. However, this
particular type of nodoid bubble has negative internal pressure: i.e., .
An example negative pressure nodoid soap bubble is illustrated in Figure 19.
Next: Exercises
Up: Surface Tension
Previous: Capillary Curves
Richard Fitzpatrick
20120427