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?