An analysis of the Sturm-Liouville eigenvalue problem of a cylinder between two spheres of equal radius

Abstract

In this paper,we examine the spectrum of the eigenvalue problem associated to a cylindrical liquid bridge connecting two spheres of equal radius. We cover all possible types of spheres: concave, convex and concave-convex spheres. We also consider the special case that a support is a plane and the other one is a sphere. We find the Morse index in all the situations, and we determine its behavior with respect to the radius of the cylinder and the separation between the two spheres as well. As a consequence, we derive results of stability and instability of Plateau-Rayleigh type. Finally, a result of bifurcation is established proving that given a cylinder of fixed radius between two concave spheres, there is a critical separation between both spheres such that new axisymmetric capillary surfaces appear with the same contact angle as that for the initial cylinder.