Interfacial plug-current similarity profile ($h=H(1-r^2/R^2{)}^{1/4}$)

For fixed volume and equal ambient viscosities, radial volume conservation gives $u=r/(5t)$. Summing both ambient-face tractions gives $H^2=32\mu R/(5\pi \Delta \rho gt)$ and $R^5=125\Delta \rho g{V}^{2}t/(512\pi \mu )$. The edge slope is singular, so the profile is a leading bulk thin-current approximation.