shape of membrane on circular frame with pressure difference This is a problem I thought about recently, but I have no idea how to go about it: Consider a membrane evenly stretched across a round frame. What shape does this membrane take, when you have different air pressures on each side of the membrane? 
Basically we have a constant force in normal direction for every small piece of the same area. 

This is classic  minimal area soap film surface , it is assumed that the property of surface tension does not change with pressure, or that that the membrane is very thin.
Normal curvature $k_n$ is constant everywhere and is given by mathematical treatment as Constant Mean Curvature  (CMC) surfaces.
For a circular boundary film a spherical bubble grows in proportion to the  pressure ( force per unit area). It is given in physics also under surface tension.
$$ p = 2 T k_n  = 2 T /R $$ 
A book by Cyril Isenberg ( Science of Soap films) is insightful.
