On the shape of an axisymmetric meniscus rising from a static liquid pool

TL;DR

This paper develops a convergent power series solution for the shape of an axisymmetric meniscus rising from a static liquid pool, accounting for gravity and surface tension effects, validated against numerical solutions.

Contribution

It introduces a novel power series method with asymptotic matching and Euler transformation for accurately modeling meniscus shapes across a range of Bond numbers.

Findings

Power series solution matches large-distance asymptotics.

Validation against numerical solutions shows high accuracy.

Method outperforms classical asymptotic solutions for B>0.01.

Abstract

We examine the classical problem of the height of a static liquid interface that forms on the outside of a solid vertical cylinder in an unbounded stagnant pool exposed to air. Gravitational and surface tension effects compete to affect the interface shape as characterized by the Bond number, $B = \rho g R^{2}/\sigma$, where $\rho$ is fluid density, $g$ is the gravitational constant, $R$ is the radius of the cylinder, and $\sigma$ is surface tension. Here, we provide a convergent power series solution for interface shapes that rise above the horizontal pool as a function of Bond number. The power series solution is expressed in terms of a pre-factor that matches the large distance asymptotic behavior -- the modified Bessel function of zeroth order -- and an Euler transformation that moves the influence of convergence-limiting singularities out of the physical domain. The power series solution is validated through comparison with a numerical solution, and the $B\rightarrow0$ matched asymptotic solutions of Lo (1983, J. Fluid Mech., 132, p.65-78). For a benchmark static contact angle of $45$ degrees, the power series approach exceeds the accuracy of matched asymptotic solutions for $B>0.01$; this lower limit on $B$ arises from round-off error in the computation of the series coefficients in double precision arithmetic, and is reduced as the contact angle is increased.