Interfacial viscous gravity current

A viscous current with density between those of two ambient fluids spreads along their interface. Vertical buoyancy divides its thickness above and below the undisturbed interface. When its internal shear and in-plane extension are both negligible, resistance from the two ambient Stokes flows controls its spreading.

When vertical diffusion equation controls depth, $H\sim \sqrt{Dt}$, while conserved excess solute mass gives $\Delta \rho R^{2}H\sim \Delta {\rho }_0{V}_0$. Ambient viscous resistance then leads to a $t^{1/2}$ spreading law. The enriched-fluid volume grows, so the fixed-volume $t^{1/5}$ law does not apply.

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.

Hydrostatic pressure continuity across both interfaces fixes the partition of current thickness. The modified current pressure is $\Delta \rho gh$, with $\Delta \rho =({\rho }_0-{\rho }_1)({\rho }_2-{\rho }_0)/({\rho }_2-{\rho }_1)$. The equivalent radial body force is $-\Delta \rho g{h}_r{\mathbf{e}}_r$.