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How do curved soap films remain in equilibrium, if surface tension tries to pull them taut?

What I understand:

  1. Surface tension acts tangentially on a surface.
  2. The potential is energy is proportional to surface area, hence the film tries to minimise it's surface area
  3. From this Libretexts article I learnt how to find the shape which minimises area, by solving the euler-lagrange equation. For a film between 2 parallel rings, it is a hyperboloid.
  4. For a soap bubble, the inward surface tension has a resultant pressure of $\frac{4S}{R}$. For equilibrium, this is countered by an outward excess pressure of $\frac{4S}{R}$ (S and R being the surface tension and radius of curvature respectively.)

For a soap film, there is atmospheric pressure on both sides. So, what counteracts the surface tension (which has a pressure of$\frac{4S}{R}$) trying to pull the surface flat?

I understand the energy-minimisation perspective gives a curved surface. I am asking from a force balance perspective.

See below diagram Hyperboloid Soap Film Diagram

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    $\begingroup$ The shape of the soap-film-stretched-between-coaxial-rings is referred to as catenoid, the surface of revolution of the catenary curve. Incidentally, the solution to the catenary problem is the hyperbolic cosine function. So a relation is there; but as we know: hyperbolic cosine and hyperbola are distinct. $\endgroup$
    – Cleonis
    Jan 20 at 17:26
  • $\begingroup$ My mistake, I confused hyperboloid with the cosh function. Thanks for the clarification. $\endgroup$ Jan 20 at 18:19

4 Answers 4

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Note that in your diagram you take a 2D slice in, let's say, the $xz$-plane where $z$ is the axis of symmetry and $x$ is the other axis you have chosen. This intersects the 2D soap film on a 1D line that is “concave-outward” and you consider this to imply an outward force.

If you instead use an $xy$-plane, $y$ being the remaining axis, you will discover that the resulting diagram is a perfect circle which is concave-inward, providing the corresponding balancing force.

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If we have the same pressure on both sides, like for an open soap film, then the surface is a minimal surface. That is, it has zero average curvature, which is equivalent to minimizing the local area. Now, zero average curvature means that along one direction the surface curves (if it is non-flat) maximally in one direction, and in another one it curves equally in the opposite direction. That means that the forces due to the tension from either balance each other perfectly. The same is obviously true for a flat surface. Meanwhile the atmospheric pressure is normal to the surface, so it cancels too.

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This image is just to complement the answer given by @CR Drost. Sometimes its easier when you visualise things, you know.

enter image description here

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After reading up some more I realise my mistake was in assuming $P_{excess}=\frac{4S}{R}$.

It is actually $2S(\frac{1}{R_1}+\frac{1}{R_2} )$ where $R_1$ and $R_2$ and the radii of curvature in 2 orthogonal directions.

As @anders-samberg said, the film will assume a shape in which $\frac{1}{R_1}+\frac{1}{R_2}=0$

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