Theory of Pendular Rings Revisited

We present the theory of liquid bridges between two axisymmetric solids, sphere and plane, with prescribed contact angles in a general setup, when the solids are non-touching, touching or intersecting, We give a detailed derivation of expressions for curvature, volume and surface area of pendular ring as functions of the filling angle \psi for all available types of menisci: catenoid Cat, sphere Sph, cylinder Cyl, nodoid Nod and unduloid Und (the meridional profile of the latter may have inflection points). The Young-Laplace equation with boundary conditions can be viewed as a nonlinear eigenvalue problem. Its unduloid solutions, menisci shapes z_n^s(r) and their curvatures H_n^s(\psi), exhibit a discrete spectrum and are enumerated by two indices: the number n of inflection points on the meniscus meridional profile M and the convexity index s=\pm 1 determined by the shape of a segment of M contacting the solid sphere: the shape is either convex, s=1, or concave, s=-1. For the fixed contact angles the set of the functions H_n^s(\psi) behaves in such a way that in the plane (\psi,H) there exists a bounded domain where H_n^s(\psi) do not exist for any distance between solids. The curves H_n^s(\psi) may be tangent to the boundary of domain which is a smooth closed curve. This topological representation allows to classify possible curves and introduce a saddle point notion. We observe several types of saddle points, and give their classification.