The shape of an axisymmetric meniscus in a static liquid pool: effective implementation of the Euler transformation.

This article addresses the classical fluid mechanics problem of determining the static shape of a liquid meniscus formed on the exterior of a vertical cylinder submerged in an infinite stagnant liquid pool exposed to air. The interface shape is governed by a nonlinear ordinary differential equation involving the Bond number, which quantifies the relative effects of gravity and surface tension, and the contact angle between the liquid and the cylinder. The authors develop, for the first time, an exact analytical solution expressed as a convergent power series over the entire semi-infinite domain by applying an Euler transformation to a divergent standard power series and incorporating the known asymptotic behavior via a modified Bessel function prefactor. They also present a robust algorithm using Newton's method to predict the meniscus height at the cylinder wall without relying on numerical inputs. The convergent power series solution is validated against numerical simulations and existing matched asymptotic expansions, demonstrating superior accuracy and applicability across a broad range of Bond numbers and contact angles, while avoiding the complexity and limitations of asymptotic methods.