Theory of axisymmetric pendular rings

We present the theory of liquid bridges between two solids, sphere and plane, with prescribed contact angles. We give explicit expressions for curvature, volume and surface area of pendular ring as functions of the filling angle ψ for all available types of menisci: catenoid, sphere, cylinder, nodoid and unduloid (the meridional profile of the latter may have inflection points). There exists a rich set of solutions of the Young-Laplace equation for the shape of an axisymmetric meniscus of constant mean curvature. In case when the solids do not contact each other, these solutions extend Plateau's sequence of meniscus evolution observed with increase of the liquid volume to include the unduloids at small filling angle, unduloids with multiple inflection points and multiple catenoids. The Young-Laplace equation with boundary conditions can be viewed as a nonlinear eigenvalue problem. Its unduloid solutions, menisci shapes and curvatures H(n)(s)(ψ), 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=±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)(ψ) behaves in such a way that in the plane {ψ,H} there exists a bounded domain where H(n)(s)(ψ) do not exist for any distance between solids. The curves H(n)(s)(ψ) 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.