[Paper Review] Theory of Pendular Rings Revisited

Boris Rubinstein, Leonid G. Fel|arXiv (Cornell University)|Jul 30, 2012

Advanced Topics in Algebra5 references3 citations

TL;DR

This paper revisits the theory of pendular rings—axisymmetric liquid bridges between two solids—by rigorously deriving curvature, volume, and surface area for all meniscus types (catenoid, sphere, cylinder, nodoid, unduloid) as functions of the filling angle ψ. It establishes a discrete spectrum of unduloid solutions to the Young-Laplace equation, indexed by inflection points (n) and convexity (s=±1), and identifies a bounded domain in the {ψ, H} plane where no solutions exist, introducing a topological classification of saddle points in meniscus transitions.

ABSTRACT

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 ψfor 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(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=\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(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.

Motivation & Objective

• To provide a comprehensive analytical framework for axisymmetric pendular rings between a sphere and a plane with prescribed contact angles.
• To systematically derive expressions for curvature, volume, and surface area of all meniscus types (catenoid, sphere, cylinder, nodoid, unduloid) as functions of the filling angle ψ.
• To classify unduloid meniscus solutions via a discrete spectrum indexed by the number of inflection points (n) and convexity index (s=±1).
• To identify a bounded domain in the {ψ, H} plane where no meniscus solutions exist, and to classify saddle points in meniscus transition pathways.
• To resolve the nonlinear eigenvalue problem arising from the Young-Laplace equation with boundary conditions, using elliptic integrals and asymptotic analysis.

Proposed method

• Derives exact analytical expressions for curvature, volume, and surface area of all meniscus types (Cat, Sph, Cyl, Nod, Und) using solutions to the Young-Laplace equation.
• Models meniscus shapes as surfaces of revolution with constant mean curvature, solving the YL equation via elliptic integrals and classifying solutions by the constant c in the differential equation.
• Introduces a two-index labeling (n, s) for unduloid solutions: n counts inflection points on the meridional profile, s=±1 denotes convexity or concavity of the contact segment.
• Analyzes the behavior of Hₙˢ(ψ) curves in the {ψ, H} plane, identifying a closed bounded domain where no solutions exist, and studies their tangency to the boundary curve.
• Applies asymptotic analysis and complex analysis to evaluate elliptic integrals, particularly at singular limits involving imaginary arguments and special values t*±.
• Uses conjugation identities and transformations of elliptic integrals to derive closed-form expressions for volume, surface area, and curvature of inflectional unduloids (n=1,2,3,…).

Experimental results

Research questions

• RQ1 How do curvature, volume, and surface area of pendular rings vary with the filling angle ψ for all meniscus types (catenoid, sphere, cylinder, nodoid, unduloid)?
• RQ2 What is the discrete spectrum of solutions to the Young-Laplace equation for unduloid menisci, and how are they indexed by inflection points (n) and convexity (s=±1)?
• RQ3 Why does a bounded domain exist in the {ψ, H} plane where no meniscus solutions exist, and what topological features define the boundary of this domain?
• RQ4 How do transitions between different unduloid types (e.g., Undₙ⁻ ↔ Undₙ₊₁⁺) occur, and what role do saddle points play in these transitions?
• RQ5 What are the exact analytical expressions for volume and surface area of inflectional unduloids with one, two, three, or more inflection points?

Key findings

• The curvature Hₙˢ(ψ) for unduloid menisci forms a discrete spectrum indexed by n (number of inflection points) and s=±1 (convexity), with solutions existing only within a bounded domain in the {ψ, H} plane.
• A closed, smooth boundary curve exists in the {ψ, H} plane such that Hₙˢ(ψ) curves can be tangent to it, defining the limit of existence for meniscus solutions.
• Saddle points in meniscus transitions are classified into simple, mixed, and sequence types, arising from tangency or bifurcation in the Hₙˢ(ψ) curves.
• For unduloids with n=1,2,3,… inflection points, explicit formulas for volume and surface area are derived using complete elliptic integrals K and E, with asymptotic behavior analyzed near spherical solutions.
• The paper derives exact expressions for the divergent part of the integral I₄ in the limit c→-1, showing a 1/ε singularity that governs the behavior near critical configurations.
• The analysis reveals non-monotonic behavior in nodoid meniscus characteristics (e.g., curvature), particularly for Nod⁺, and provides asymptotic expansions for all key meniscus properties.