Gravity-driven viscous flow over partially lubricated bed

Abstract

We present an investigation into the response of a viscous fluid flowing over a sloped bed across a spatially finite patch of basal lubrication. We present a simple analytical model that captures the fundamental structure of such lubrication-induced stress and velocity perturbations in Newtonian fluids, as well as scaling arguments and numerical experiments that extend our analysis to power-law fluids. These analyses concisely reveal the underlying relationships between the system parameters (fluid thickness, $h$, slope, $\alpha$, slippery patch length, $\ell$, and sliding condition outside of the slippery patch, $\gamma$) and the magnitude and spatial extent of the resulting perturbed stresses, $\tau_{xx}$, and velocities, $u_p$. From these results, we conclude that the induced stresses are exponentially decaying functions of distance away from the patch location, and show that the amplitude of the perturbations scales linearly with surface slope and patch length while the decay length scales with thickness and patch length, and is critically dependent on the basal boundary condition outside of the slippery patch. These fundamental relationships can be incorporated into more complex models to investigate whether rapid lake drainages on ice sheets, which create a partially lubricated bed, can generate sufficient stress and velocity perturbations in the overlying ice flow to trigger lake drainage cascades.