A shocking viscous gravity current

We analyse propagation of a viscous gravity current over a layer of different fluid, but with the same density. Exact similarity solutions are found with two free parameters: a non-dimensional flux $Q$ and the viscosity ratio $m$. The theory predicts a jump discontinuity, or shock, at the nose of the current for $m>m_{crit}$, which develops kinematically due to the hyperbolic nature of the interfacial height equation. The jump is identified as a novel form of under-compressive shock, resulting from a non-concave relative flux function and the total height variations. Small non-zero density differences regularise this system, and a local travelling-wave solution near the nose justifies the shock conditions for the jump height as a function of $m$. Our results obtained compare well to experimental data, both in terms of the predictions of radial extent and of the overall shape. In some circumstances, the front becomes unstable.