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When soap/shampoo bubble/film enclosing a small pressure in a small volume is formed on a (relatively hard) flat surface or inside parts of a circular cone/funnel, it forms a hemispherical bubble or respectively a segment of a sphere (as per Young-Laplace Law)

$$\kappa_1+\kappa_2 = H = p/T \tag1 $$

( Pressure differential $p$, surface tension $T$ are constants). The CMC (constant mean curvature) surface integrates to a spherical shape whose curvature increases with $ p/T.$

Similarly when a bubble forms enclosing air with pressure on an arbitrary patch of shape $f(x,y,z)=0,$ what would be the shape of the interfacial closed oval closed ring that is seen to definitely form, as a function of $(p/T, f)?$

When soap bubble moves to a new location, the ring changes shape conforming to new function $f$ of rigid shape.

We divide (1) by the following Boyle's Law to eliminate pressure $p$

$$ p \cdot V = C, \tag 2$$

getting constant pressure and volume $V$ of enclosure, since the RHS are physical and geometrical invariants:

$$ V= C/ T H \quad p = T H; \tag3 $$

Boyle's Law acts at constant temperature ..like when pressure is released volume increases in an automobile cylinder and the gas is compressed to low volume by piston moving in creating high pressure. Also a gas balloon expands due to rarified gas pressure at higher altitude would shrink back when brought back near ground.

As another example of the carrier rigid surface if a bubble is trapped inside a transparent glass torus, enclosing let us say a tenth of its inside volume $ 2 \pi^2 a^2 b $ then it forms a definite shape inside the torus ( up to arbitrary polar positions). we are attempting to find out here what shape has the film and its boundary described through differential equations.

EDIT1:

I thought in terms of Lagrangian $ H + \lambda V $ with orthogonality of film/surface as boundary in a variational problem. Assuming Monge form $ z=z(x,y) $ a functional to be extremized could be of a form

$$ \int\int \left(\frac{r}{(1+p^2)^{\frac32}}+\frac{t}{(1+q^2)^{\frac32}}-2 \lambda \, z(x,y)\right) dx\,dy$$

enter image description here

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    $\begingroup$ I don't understand what "interfacial closed oval closed ring" is supposed to mean in your question. If you believe in the Young-Laplace equation the soap film itself will be constant mean curvature. But you have a bit of a free-boundary problem at the interface. (Since changing the boundary will change the internal volume and that will change the effective pressure and then the mean curvature.) Is the solution to this free boundary problem what you are asking about? $\endgroup$ – Willie Wong Sep 3 '17 at 19:23
  • $\begingroup$ To illustrate I attached a picture. Yes, $H$ is constant, If $f$ is variable intersection is variable due to free bubble motion. However there is nothing to change bubble volume or pressure. By Boyle's Law $ p* Vol$ product is constant and by Laplace-Young Equn $ p = H \, T$. Since $T$ surface tension is constant as a property, volume and pressure remain the same by division of the two relations. The intersection line varies due to varying inside shape of shampoo bottle on which it moves. Hope it is clear. $\endgroup$ – Narasimham Sep 3 '17 at 20:18
  • $\begingroup$ The free boundary problem results in the minimal/CMC surface being orthogonal to the surface where they meet. If the fixed surface possesses a fold, as between two faces of a polyhedron, the minimal surface is orthogonal to both $\endgroup$ – Will Jagy Sep 3 '17 at 20:25
  • $\begingroup$ Yes indeed, it shows the dihedral between such two meeting faces beautifully, adjusting itself automatically that way, Cyril Isenberg's book states these facts. $\endgroup$ – Narasimham Sep 3 '17 at 20:37
  • $\begingroup$ @Narasimham: I don't get the volume/pressure constancy constraint. The mean curvature $H$ is constant along the film, but doesn't have to take any fixed value. The system of equations is the CMC equation for the film, orthogonality with the fixed surface, plus the constraint that the pressure differential between the interior and the exterior is the surface tension times the mean curvature. In the case of a flat surface the final constraint is what determines how big the the bubble is. (If the bubble is too big the pressure differential is too small to satisfy Young-Laplace etc.) $\endgroup$ – Willie Wong Sep 4 '17 at 1:17

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