Hydrostatic rise in a cylindrical cusp ($w(z)\sim [2\gamma a/(\rho gz){]}^{1/2}$)

For vertical touching cylinders in a bath, the hydrostatic suction at height $z$ is $\rho gz$. Equating it to the meniscus suction $2\gamma a/w^2$ gives the displayed large-height width. Ideal perfect wetting permits an arbitrarily high thin cusp tail, unlike a uniform finite-gap capillary. The prediction assumes the continuum model remains valid as the gap narrows.