  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.

Soft Condensed Matter;Mesoscale and Nanoscale Physics;Fluid Dynamics