Lubricated viscous gravity currents

Abstract

We present a theoretical and experimental study of viscous gravity currents lubricated by another viscous fluid from below. We use lubrication theory to model both layers as Newtonian fluids spreading under their own weight in two-dimensional and axisymmetric settings over a smooth rigid horizontal surface and consider the limit in which vertical shear provides the dominant resistance to the flow in both layers. There are contributions from Poiseuille-like flow driven by buoyancy and Couette-like flow driven by viscous coupling between the layers. The flow is self-similar if both fluids are released simultaneously, and exhibits initial transient behaviour when there is a delay between the initiation of flow in the two layers. We solve for both situations and show that the latter converges towards self-similarity at late times. The flow depends on three key dimensionless parameters relating the relative dynamic viscosities, input fluxes and density differences between the two layers. Provided the density difference between the two layers is bounded away from zero, we find an asymptotic solution in which the front of the lubricant is driven by its own gravitational spreading. There is a singular limit of equal densities in which the lubricant no longer spreads under its own weight in the vicinity of its nose and ends abruptly with a non-zero thickness there. We explore various regimes, from thin lubricating layers underneath a more viscous current to thin surface films coating an underlying more viscous current and find that although a thin film does not greatly influence the more viscous current if it forms a surface coating, it begins to cause interesting dynamics if it lubricates the more viscous current from below. We find experimentally that a lubricated gravity current is prone to a fingering instability.