We examine the stability of radially spreading, gravity-driven thin films of power-law fluids, lubricated from below by another power-law viscous fluid. Such flows are susceptible to a viscous fingering instability, also known as a non-porous viscous fingering instability, when a less viscous fluid intrudes beneath a more viscous fluid. In contrast to the Saffman–Taylor instability, such instabilities originate from a jump in hydrostatic pressure gradient across the intrusion front, associated with gradients in the upper surface. These are stabilised by buoyancy forces associated with the lower layer near its nose, and all instabilities are suppressed above a critical density difference. We find that shear-thinning flows are more prone to instability than Newtonian and shear-thickening flows. Lower consistency ratios are sufficient for the onset of instability of shear-thinning flows, and the stabilising influences of buoyancy forces are suppressed. As such, higher density differences are required to suppress the instability completely.

We examine the gravity-driven flow of thin films of viscous fluid of power-law rheology, lubricated by another power-law viscous fluid from below. Such flows are relevant to a range of geophysical and industrial settings, including the flow of ice sheets and fast-flowing ice streams lubricated from below by a layer of subglacial till. We model both layers using lubrication theory in two-dimensional and axisymmetric settings, in the limit in which vertical shear provides the dominant resistance to the flow. The flow is self-similar if the power-law exponents, describing the rheology of the two layers of fluid, are equal. We examine the similarity solutions in both geometries and describe the flow in terms of four distinct flow regimes ranging from thin films of viscous fluid coating a more viscous fluid from above, to thin layers of fluid lubricating a more viscous fluid from below. In contrast to the former scenario, a thin film of a low-viscosity fluid strikingly alters the dynamics of a more viscous fluid when it lubricates it from below: the overlying layer thins, the upper surface gradients lessen and most of the shear is confined to the lower layer. Such features amplify, and this flow regime becomes increasingly dominant when the viscous fluids are shear thinning, like the deformation of glacial ice on the large scale. This flow regime is most relevant to the flow of lubricated ice sheets, which thin and accelerate, forming fast-flowing ice streams, when they are well lubricated from below.