Unusual shapes for a catenary under the effects of surface tension and gravity: A variational treatment

The familiar catenary is the shape assumed by a chain or string as it hangs from two points. The mathematical equation of the catenary was first published more than three hundred years ago by Leibnitz and Huygen, among others. Here we consider the shapes assumed by a hanging string in the presence of gravity and surface tension. The surface tension is introduced by suspending the string from a thin horizontal rod while the area bounded by the string and the rod is covered with a soap film. The string then assumes new and wonderful shapes depending on the relative strength of the surface tension and the weight per unit length of the string. When surface tension dominates, the string is pulled inward, assuming a convex shape similar to the Greek letter {γ}. On the other hand, when gravity is dominant the string is pulled outward and assumes a concave shape best described as a distorted catenary. However, when the gravitational force normal to the string matches the surface tension, the string takes a linear configuration similar to the letter V. Under suitable conditions, the string can be made to assume any of the three configurations by adjusting the separation of its end points. The equations that describe the shape of the string are derived by minimizing the total energy of the system and are presented for the three principal configurations.