Solving the Young-Laplace equation for arbitrary axisymmetric geometry

Say I have a non-ellipsoidal soap bubble and I want to numerically analyse the pressure in the inner lobe of this bubble here:

The Young Laplace equation gives the pressure difference across a fluid interface as a function of the curvatures. I have a set of points in 2D space (axisymmetry is assumed) for the inner lobe.

How can I obtain the net force that acts over the entire inner lobe surface due to the Young-Laplace pressure gradient? This would be easy if the lobe itself was approximately ellipsoidal - then there are only two principal radii of curvature, and the pressure gradient follows from there.

But what if I had a more complex shape for the inner lobe that wasn't ellipsoidal? Do I try to break the shape into many ellipses, however improbable that sounds?

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You can calculate the curvature of such a surface if you know a good description of it. What's wrong with that? And there are always two principal curvatures... –  Ryan Thorngren Oct 14 '12 at 23:08

To calculate the Laplace pressure for a given surface involves a bit of math, but is not particularly difficult. For the curvature in Cartesian coordinates you will get the following monster of a non-linear second order partial differential equation:

$$\frac{\Delta P}{\gamma}=\left(\frac{1}{R_1}+\frac{1}{R_2}\right)=\frac{\partial_{xx}z\left[1+\left(\partial_yz\right)^2\right]-2\left(\partial_xz\right)\left(\partial_xz\right)\left(\partial_{xy}z\right)+\partial_{yy}z\left[1+\left(\partial_xz\right)^2\right]}{\left[1+\left(\partial_xz\right)^2+\left(\partial_yz\right)^2\right]^{3/2}}$$

As a PDE with boundary conditions this thing is very hard to solve, but if you have a given surface, i.e. $z(x,y)$ it should be straightforward to calculate the Laplace pressure for any given position $(x,y)$ on that surface.

If you are interested in some simplifications of this equation (e.g. 2D), check out pages 27 and onwards of this document on capillarity and wetting. It comes from a graduate level course on the topic.

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