For the purposes of modelling a fluid mechanics experiment, I'm dealing with a convex surface parametrized by the azimuth $\theta$ and an arc length $s$ along the surface. The points on the surface are described in a coordinate system of $r$ and $y$, as shown in the image below.
I would like to build an ordinary differential equation in terms of the derivatives of $y$ and $r$ which I will then solve numerically to verify the model. Since surface tension is involved, I need to find the mean curvature at any point on the surface.
Finding the curvature in the $y$-$r$ plane is straightforward: $\theta$ is fixed, and we're looking at a simple 2D parametric curve. But how can I get the curvature in the other principal direction, i.e. in the plane that lies normal to the tangent along the arc $s$? (I'm assuming those are the principal directions; maybe that's already a mistake …?)
I have looked at expressions that were worked out for cylindrical and spherical systems. Both are awfully complicated. My attempts to derive a formula from scratch haven't borne fruit so far—there are just too many terms to juggle. I'm also thankful for any suggestions that would allow me to recast the problem in a coordinate system that makes this easier to handle, or suggestions that will result in an approximate solution.
Update: if I find the curvature parallel to the $\theta$-$r$ plane, and multiply that value by the sine of the tangent angle … would that be a valid measure of the curvature in the other principal direction?