Here we consider the problem of finding the equilibrium configurations of a rotating liquid drop, paying special attention to the cases when the droplet takes the shape of a Delaunay surface. By making use of the canonical forms of the elliptic integrals and the Jacobian elliptic functions we have derived several explicit parameterizations of the Delaunay surfaces. They are expressed relying on two independent real parameters accounting respectively the size and the shape so that all possible Delaunay surfaces are represented in a unified way.
"Rotating Liquid Drops and Delaunay Surfaces." J. Geom. Symmetry Phys. 54 55 - 78, 2019. https://doi.org/10.7546/jgsp-54-2019-55-78