Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 329 / 3 / c

In a static meniscus, hydrostatic pressure variation $\rho g\ell$ balances capillary pressure $\gamma /\ell$, giving the capillary lengthThe small parameter satisfiesIt is reasonable to match to a nearly static meniscus because the bulk pool has negligible thin-film viscous resistance, while its local capillary-hydrostatic shape approaches the vertical wall almost tangentially.

Since $h=h_{0}H$ and $x=h_{0}X/ϵ$, dimensional curvature isMatching it to $(2\rho g/\gamma {)}^{1/2}=\sqrt{2}/\ell$ gives