Theory of Pendular Rings Revisited
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چکیده
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 ψ 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 zs n (r) and their curvatures Hs n (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 Hs n (ψ) behaves in such a way that in the plane {ψ,H} there exists a bounded domain where Hs n (ψ) do not exist for any distance between solids. The curves Hs n (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.
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تاریخ انتشار 2013