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, κ^-1, and the container radius, L, is varied from a small value comparable to R to ∞. In the limit of L≪κ^-1, we analytically solve the general Young-Laplace equation governing the meniscus profile and show that the meniscus height, Δh, scales approximately with Rln(L/R). In the opposite limit where L≫κ^-1,Δh becomes independent of L and scales with Rln(κ^-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 0.4κ^-1≲L≲4κ^-1. An approximate analytical expression has been found for Δh, enabling its accurate prediction at any values of L that ranges from microscopic to macroscopic scales.