Axisymmetric free-surface curvature ($\kappa =-r^{-1}{\partial }_r[r{h}^{\prime }/\sqrt{1+h^{\prime 2}}]$)

For an axisymmetric graph $z=h(r)$ with the normal pointing out of the liquid, the sum of principal curvatures is the displayed expression. In the small-slope limit it becomes $\kappa =-(h^{\prime \prime }+h^{\prime }/r)$, giving the capillary pressure $p_{\mathrm{liquid}}-p_{\mathrm{air}}=\gamma \kappa$. Regularity at the axis gives $h^{\prime }(0)=0$ and $h^{\prime }/r\to h^{\prime \prime }(0)$: the azimuthal contribution does not disappear at the axis. For example, the paraboloid $h=H(1-r^2/R^2)$ has $\kappa \approx 4H/R^2$, twice the meridional contribution.