A flat plane . This is best evaluated using the axisymmetric form of the equations. R , and using the formula for the mean curvature in cylindrical coordinates implies that Thus, the pressure inside the cylinder must be for this to work. It is an easy exercise to check that this is satisfied, and leads to a pressure . These can't exist in a liquid, but can exist in a soap film .
Soap films are extensible, but there is no internal volume constraint to satisfy.
For example , consider a soap film which extends cylindrically from one ring of radius to another ring of radius . We would expect the solutions to be axisymmetric, and so should just solve the axisymmetric equation. The solutions to these equations in this geometry are called catenoids . In the special case where it is even possible to solve the equations analytically. Show that a solution to the equation in cylindrical coordinates is where is arbitrary parameter, and c is fixed uniquely in terms of the radius and the separation between the two rings.