The Meniscus on the Outside of a Circular Cylinder: from Microscopic to Macroscopic Scales

We systematically study the meniscus on the outside of a small circular cylinder vertically immersed in a liquid bath in a cylindrical container that is coaxial with the cylinder. The cylinder has a radius $R$ much smaller than the capillary length, $\kappa^{-1}$, and the container radius, $L$, is varied from a small value comparable to $R$ to $\infty$. In the limit of $L \ll \kappa^{-1}$, we analytically solve the general Young-Laplace equation governing the meniscus profile and show that the meniscus height, $\Delta h$, scales approximately with $R\ln (L/R)$. In the opposite limit where $L \gg \kappa^{-1}$, $\Delta h$ becomes independent of $L$ and scales with $R\ln (\kappa^{-1}/R)$. We implement a numerical scheme to solve the general Young-Laplace equation for an arbitrary $L$ and demonstrate the crossover of the meniscus profile between these two limits. The crossover region has been determined to be roughly $0.4\kappa^{-1} \lesssim L \lesssim 4\kappa^{-1}$. An approximate analytical expression has been found for $\Delta h$, enabling its accurate prediction at any values of $L$ that ranges from microscopic to macroscopic scales.