 Hemispherical capillary imbibition

Radial capillary imbibition into a half-space has hemispherical flow area $2\pi r^2$. Its radial hydraulic resistance is proportional to $1/R_s-1/R$, so its inlet volume flux approaches $2\pi k\Delta P{R}_s/\mu$. The growing front area gives late-time $R\propto t^{1/3}$, unlike the planar $t^{1/2}$ law. Both results assume capillary-dominated geometry with negligible directional gravity effects.