Soap film on two noncircular frames

Abstract

The classical Plateau problem of finding minimal surfaces supported by two noncircular coaxial rings is studied theoretically and experimentally. Using a fluid dynamics analogy, we generalize the classical catenoid solution for a film on circular rings to the general cases of noncircular rings. Some examples of analytical solutions for elliptical, polygonal, and ovoidal rings are presented. The shapes of a tubular film and a film separated by a lamella at the wrist are obtained in an analytical form. The stability of these films is analyzed and compared with the classical catenoid. The data on critical parameters of all minimal surfaces are collected in the tables that can be used in practical applications. The theory is experimentally validated using soap films on elliptical identical frames. Moreover, the shapes of soap films on two different elliptical frames demonstrate a new feature: a flat separating lamella lying parallel to the rings was never observed in experiments. All lamellae appeared deformed suggesting the existence of a new family of minimal surfaces which does not exist in the case of frames of the same sizes.