A study of pendular liquid bridge between two equal solid spheres

TL;DR

This paper mathematically analyzes pendular liquid bridges between equal spheres, comparing approximations and formulas to understand capillary forces and bridge formation, highlighting the accuracy and limitations of various models.

Contribution

It provides a detailed comparison of analytical approximations and numerical solutions for pendular liquid bridges, emphasizing the role of separation distance and the accuracy of different formulas.

Findings

Toroidal approximation is reasonably accurate at small separations.

Elliptic meridional profile offers more precise results but is complex.

Curve-fitting formulas have inherent limitations and potential errors.

Abstract

Pendular liquid bridges with concave meridians between two equal rigid spheres are mathematically studied emphasizing some less analyzed facts in the literature. Discrepancies from the numerical solution of the Young-Laplace equation are examined among typical simplifying approximations and a few curve-fitting formulas. An in-depth analysis is provided about the important role of separation distance between spheres played in pendular ring formation via capillary condensation at a given relative humidity and the strength of subsequent capillary forces. For most practical situations, the toroidal approximation could be reasonably accurate (especially with diminishing separation distance) and provide valuable mathematical insights at least in a qualitative sense with its relatively simple analytical formulas. Using the elliptic meridional profile generally offers more accurate approximations, but with such complicated analytical formulas that it would limit its convenience for practical applications. With a few examples, the present study shows that curve-fitting formulas cannot be perfect and, by their approximative nature, would always leave room for improvements; therefore, care should be taken when applying curve-fitting formulas, to avoid undesirable errors