How do curved soap films remain in equilibrium, if surface tension tries to pull them taut?
What I understand:
- Surface tension acts tangentially on a surface.
- The potential is energy is proportional to surface area, hence the film tries to minimise it's surface area
- 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.
- 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
Best Answer
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.