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 $\mathrm{\infty }$. 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, $\mathrm{\Delta }h$, scales approximately with $R\ln (L\mathrm{/}R)$. In the opposite limit where $L \gg \kappa^{-1}$,$\Delta h$becomesindependentof$L$andscaleswith$R\ln (\kappa^{-1}/R)$. 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 $\mathrm{\Delta }h$, enabling its accurate prediction at any values of $L$ that ranges from microscopic to macroscopic scales.