Stability of lubricated viscous gravity currents. Part 1. Internal and frontal analyses and stabilisation by horizontal shear

A novel viscous fingering instability, involving a less viscous fluid intruding underneath a current of more viscous fluid, was recently observed in the experiments of Kowal & Worster (J. Fluid Mech., vol. 766, 2015, pp. 626-655). We examine the origin of the instability by asking whether the instability is an internal instability, arising from internal dynamics, or a frontal instability, arising from viscous intrusion. We find it is the latter and characterise the instability criterion in terms of viscosity difference or, equivalently, the jump in hydrostatic pressure gradient at the intrusion front. The mechanism of this instability is similar to, but contrasts with, the Saffman-Taylor instability, which occurs as a result of a jump in dynamic pressure gradient across the intrusion front. We focus on the limit in which the two viscous fluids are of equal density, in which a frontal singularity, arising at the intrusion, or lubrication, front, becomes a jump discontinuity, and perform a local analysis in an inner region near the lubrication front, which we match asymptotically to the far field. We also investigate the large-wavenumber stabilisation by transverse shear stresses in two dynamical regimes: a regime in which the wavelength of the perturbations is much smaller than the thickness of both layers of fluid, in which case the flow of the perturbations is resisted dominantly by horizontal shear stresses; and an intermediate regime, in which both vertical and horizontal shear stresses are important.