### Double Bubble Details

Here are some details about the geometry of
the standard and nonstandard double bubbles
rendered by John Sullivan in 2000.
The left image is a cluster of three bubbles.
The central vertical bubble has volume approximately 6.10736,
while the thick waist bubble around it has volume 2.85446,
and the tiny outer belt bubble has volume 0.0382928.
The total surface area is 29.3233.
Considering the central and belt bubbles to be two components
of a single region, we can think of this as a double bubble
with volumes 6.146 and 2.854.

The right-hand image shows the standard double bubble with these
same volumes. Its Plateau border is a circle of radius 0.9, and
the three sheets meet that circle at angles 10deg, 130deg, and -110deg
from the horizontal. The three sheets are spheres of radii
5.18386, 1.17483, .95777, so the pressures in the two regions
are 1.702 and 2.088 above the ambient pressure.
The total surface area is 24.5459. It seems surprising that the
inner interface curves so little despite the fact that the volume
ratio is more than 2.

All the surfaces in the nonstandard bubble at the left are
CMC surfaces of revolution, that is unduloids or nodoids of Delaunay.
The two caps at the top and bottom are hemispheres of radius 1,
so the pressure in the central bubble is 2. At the Plateau borders
which are rings of radius 1, the sphere branches into an inner and
outer surface. These have force +-1.5, so the mean curvatures are
.5 and 2.5, with the pressure in the waist bubble being 2.5.
The outer surface becomes vertical sooner than the inner one
(at height .4855 from the first Plateau border instead of .5286),
leaving room for an even smaller belt bubble. The force/pressure
of its two surfaces are adjusted until they close up at the correct
height to fill this gap. This happens when the force is 32.3 for
the inner surface and hence -33.8 for the outer; the pressure
in the belt bubble is then about 21.7. Note that since the central
and belt bubbles are at (vastly) different pressures, this cluster
is not actually in equilibrium when considered as a double bubble.
(Also, since the two surfaces of the belt bubble don't quite close
up at the right place, it's not clear that there is even an equilibrium
triple bubble near to this picture.)

Both of these bubble clusters are surfaces of revolution.
Their generating curves are shown below.

Click on any image for a larger version