= Hemispherical capillary imbibition
{title2=$t=\frac{\mu\phi}{k\Delta P R_s}(R^3/3-R_sR^2/2+R_s^3/6)$}
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.
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