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

You will always have a local curvature for your 2D curve. From the coordinates of adjacent points of the discretized domain, you can calculate this curvature.

The second radius of curvature can be obtained by using the distance to the axis in some way. (The link that you provided gives some hints about that)

But you have to be careful in determining the pressure inside the bubble. The problem you're studying is not steady state. You will see that there are pressure differences inside the bubble, that will force the bubble into spherical state eventually.

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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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