Excess pressure inside a non spherical drop How can we derive the formula for excess pressure inside a non spherical drop which is $T(1/a_1 + 1/a_2)$ as given here .I was unable to find any derivation on the internet. Also what is $a_1$ and $a_2$ in the formula is unclear to me. It says that $a_1$ and $a_2$ are radius of curvatures of the intersection of perpendicular planes with the surface which raises the question why the intersection gives a circle. Is there any general equation of the surface of drop resting on a plane surface?
 A: There are general equations for drops on a surface but they are different equations and require other/more information. When a fluid interacts with a surface, the hydro -phobicity or -philicity also needs to be taken into account, this is usually expressed via a contact angle.
The equation talks about drops that are non-spherical in the sense that they aren't perfect spheres, as in an oval is not a circle, they are usually not talking about completely irregular shapes (which is why this come across as confusing).
a1 or a2, usually called R1 and R2, are the radii of the droplet or surface.
Where there's generally two options, the first one being the clearest example.
Think of it as a deformed spherical droplet, stretched/squeezed along 1 axis (let's say z). That way instead of looking circular from the front the droplet will look oval and you will need two radii to describe it.
As in the image below
https://www.slideserve.com/betty_james/young-laplace-equation-equation-of-capillarity
The second is a bit more complex.
You can describe any surface as a a surface made up of many (to infinity) very small (infinitesimal) surfaces. This is what's usually done when integrating over a more complex surface. Such a small tiny surface does not have to be considered flat but can also be considered(approximated) curved. This can then be used to calculate the integral of the surface tension, to approximate any irregular shape.
Where each small section could be represented as in the image below
https://upload.wikimedia.org/wikipedia/commons/f/f5/Curvature_radii.JPG
If you really want to understand the subject well, there's a derivation for the Laplace-Young equation here:
http://www.ux.uis.no/~s-skj/PetFys04/Notater/Young-Laplace/you-lap.pdf
(warning fairly complex and arduous math)
