Soap bubbles and the isoperimetric problem

bubble shot
bubble shot by Rhett Maxwell

During a moment of reflection last summer, I momentarily thought about the shape of balloons and bubbles in general. I wondered why of all the possible shapes a balloon could have, why the sphere? If there was a reason why the bubble converged to the shape of a sphere I wanted to know it. But, I forgot about this question amid other concerns until the thought occurred to me once again a couple weeks ago…I realized that it might have something to do with the isoperimetric inequality which in 2-dimensions states that for a closed curve C, if the length of its perimeter is a constant then the circle maximizes its area.

This time I posed the question on the Physics StackExchange. A person by the name of ‘Michiel’ gave a sufficiently detailed answer which I shall summarize here:

For a bubble to be spherical, the surface tension has to dominate other forces per unit length. This means that other factors such as gravity and the size of the balloon must be significantly less important. Now, if surface tension is indeed dominant then the pressure is mainly determined by the Laplace pressure jump across the bubble interface(\Delta P_{c})

 \gamma is the surface tension between the gas and liquid.

 \kappa = 1/R_{1} + 1/R_{2} is the local curvature of the surface.

\Delta P_{c} = \gamma(1/R_{1} + 1/R_{2})

The equilibrium situation is the case where the surface tension is dominant
and the bubble pressure is equal everywhere.   If that is not the case we get a
flow from one place to the other.  \gamma  is constant so  \kappa must be constant over the whole interface, and a sphere is the only closed shape for which this is possible.

Now, I was not entirely satisfied with this answer because I hoped to find
a connection with the isoperimetric problem. But, I found one answer that
mentioned that systems try to minimize their potential energy…This led me
to the following great insight into the nature of this problem:

If we assume that the bubble tries to minimize its potential energy then, for a given surface area, it will try to offer the maximum volume for the gas particles contained within it. This is equivalent to finding the minimum surface for a given volume.

Taking the isoperimetric inequality into account, the fact that a bubble
immersed in a fluid(air/gas) converges to the shape of a sphere isn’t surprising.

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