Ambient-controlled interfacial plug current ($u\sim \Delta \rho g{H}^2/\mu$)

The gravity force per interfacial area is $\Delta \rho g{H}^2/R$, while ambient viscous traction is $\mu u/R$. Their balance gives the radial velocity scale independently of current viscosity within the appropriate plug-flow window.

Linearity of Stokes flow and Uniqueness of Stokes flow identify the ambient resistance to an expanding thin plug with the reversed perturbation around a stationary disc in an axisymmetric straining flow. The imposed straining flow has zero tangential stress on the disc plane. Each face then resists radial velocity $Er$ with traction $-8\mu Er/[\pi \sqrt{R^2-r^2}]$.

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.

The lower bound makes internal vertical velocity variation negligible; the upper bound makes internal in-plane extensional stress negligible compared with ambient traction. Both are needed for an ambient-controlled interfacial plug current.