Viscous gravity currents with linear basal slip in planar and axisymmetric geometries

We investigate the impact of basal sliding on the spreading of a viscous Newtonian gravity current (GC) propagating over a slippery substrate, under the lubrication approximation and assuming laminar flow. The current volume is assumed to vary in time according to a power-law injection protocol. The basal slip is modeled through a Navier slip condition, introducing a nonzero slip velocity at the base of the current. This results in an additional contribution to the governing partial differential equation, which in the dimensionless form depends on a slip number Ns⁠. This parameter encapsulates the injection protocol, fluid properties, and slip length and quantifies the relative importance of basal sliding. A unified theoretical formulation valid for both planar and axisymmetric geometries is derived. Within this framework, similarity solutions exist only in two asymptotic regimes: a no-slip limit, which recovers classical results from the literature, and a very-slippery limit, for which new similarity solutions are obtained. The transient regime connecting these limits is resolved using a fully numerical integration scheme. Comparisons between numerical and asymptotic solutions show that their range of validity depends on time, geometry, and the values of the injection exponent and slip number. A dimensional case study describing the spreading of a fluid with a macroscopic slip length illustrates that the presence of a highly slippery substrate strongly influences the current propagation, while for microscopic slip lengths, basal sliding becomes dynamically relevant primarily for micro-scale GCs. Finally, the role of alternative nondimensionalizations and typical ranges of slip numbers inferred from experimental data are discussed in dedicated appendices.