Lubricated axisymmetric gravity currents of power-law fluids

Abstract

The motion of glaciers over their bedrock or drops of fluid along a solid surface can become unstable when these substrates are lubricated. Previous studies modeled such systems as coupled gravity currents (GCs) consisting of one fluid that lubricates the flow of another fluid, and having two propagating fronts. When both fluid are Newtonian and discharged at constant flux, global similarity solutions were found. However, when the top fluid is strain-rate softening experiments have shown that each fluid front evolved with a different exponent. Here we explore theoretically and numerically such lubricated GCs in a model that describes the axisymmetric spreading of a power-law fluid on top of a Newtonian fluid, where the discharge of both fluids is power law in time. We find that the model admits similarity solutions only in specific cases, including the purely Newtonian case, for a certain discharge exponent, at asymptotic limits of the fluids viscosity ratio, and at the vicinity of the fluid fronts. Generally, each fluid front has a power-law time evolution with a similar exponent as a non-lubricated GC of the corresponding fluid, and intercepts that depend on both fluid properties. Consequently, we identify two mechanisms by which the inner lubricating fluid front outstrips the outer fluid front. Many aspects of our theory are found consistent with recent laboratory experiments. Discrepancies suggest that hydrofracturing or wall slip may be important near the fronts. This theory may help to understand the dynamics of various systems, including surges and ice streams.