Double Bubble Pipe
The applet above shows a bubble pipe (black outline) with several openings
(green), each with a bubble above it (red). The total volume of the bubbles
may be controlled through the inlet at the bottom using the slider.
The applet shows either a 2D bubble pipe or a cross-section of a
3D bubble pipe. The openings are the same size, and circular in the 3D case.
This example illustrates several points:
- Symmetric problems don't always have symmetric solutions.
- Joining two stable systems may result in an unstable system.
- For bubbles there is a simple rule for telling when joined
systems are stable: If the pressure increases as the volume increases,
then the joined system is stable.
The bubbles are 2D circular arcs or 3D spherical caps, since those shapes
minimize the surface length or surface area for a given interior area or
volume. The pressure inside a bubble is proportional to the curvature,
so inversely proportional to the radius. Since the bubbles are all
connected through the pipe, all the bubbles thus have the same radius.
For a bubble radius greater than the opening radius, there are two
possible cap sizes: a small cap less than a hemisphere, and a large
cap greater than a hemisphere. So equilibrium configurations consist
of any mixture of large and small caps.
If there are two large caps, then the configuration is unstable. For
a large cap, note that increasing the volume of a cap increases its
radius and thus decreases it's pressure. If there are two large caps,
then a perturbation that increases the volume of one and decreases the
volume of the other will decrease the pressure in the first and increase
the pressure in the second, thus pumping air from the second to the first
through the pipe, making the perturbation grow. Thus the only stable
equilibria are those with no large caps or just one large cap. Note
that this instability argument does not apply to small caps,
since increasing the volume of a small cap decreases its radius and
increases its pressure, thus opposing the perturbation.
- The applet first appears with two small-cap bubbles. Drag the
volume slider with your left mouse button to the right to increase
the volume. The stable configuration remains two small caps until
the caps become hemispheres. After that, one bubble grows to a
large cap and the other shrinks. The applet chooses at random which
cap grows. Note that the transition happens smoothly; there is no
sudden jump between small and large caps.
- Having one large and one small cap, decrease the volume. The
two bubbles become hemispheres and then become two small caps. Again,
the transition is smooth.
- Use the "Bubbles" choice box to pick 3 bubbles. It starts as three
small caps. Increase the volume. As it reaches hemispheres, there is
a jump to one large cap (chosen at random by the applet) and two small
caps. Unlike the two-bubble case, this is a discontinuous jump.
- If start with three bubbles with one large cap and reduce the
volume, then there will be a discontinuous jump to small caps. The
minimum volume for a one large cap configuration is below the volume
for all hemispheres, and the one-large-cap configuration is stable
all the way down, so there is a discontinuous jump to small caps
when the minimum volume is reached.
- Higher numbers of bubbles act like three bubbles, with discontinuous
jumps at critical volumes.
- The left-most choice box can be used to set the applet to calculate
two-dimensional bubbles, circles instead of spheres. The phenomena are
the same, continuous changes for two bubbles, jumps for three or more,
and you really can't tell the difference visually.
Ken Brakke's home page.