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 modelled 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 fluids 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 consisting of axisymmetric spreading of a power-law fluid on top of a Newtonian fluid, where each fluid volume grows in time like$t^{\alpha }$. We find that the structure imposed by the non-Newtonian flow precludes general self-similarity, unlike purely Newtonian GCs. Consequently, we identify outstripping solutions in which the inner fluid front outstrips the outer fluid front. Despite the absence of a general global similarity solution, we find similarity solutions in several asymptotic limits. These include the purely Newtonian limit for any$\alpha$, the case of$\alpha =5$for a general power-law fluid, asymptotic limits in the viscosity ratio, and in the vicinity of the fluid fronts. Many of our theoretical predictions are found to be consistent with recent laboratory experiments. Discrepancies suggest the presence of hydrofracturing or wall slip near the fronts, and potentially, a progressive significance of extensional stresses as front outstripping is approached.