Thin-Film Flows with Wall Slip: An Asymptotic Analysis of Higher Order Glacier Flow Models

Free-surface thin-film flows can principally be described by two types of models. Lubrication models assume that shear stresses are dominant in the force balance of the flow and are appropriate where there is little or no slip at the base of the flow. Conversely, membrane or ‘free-film’ models are appropriate in situations where there is rapid slip and normal (or extensional) stresses play a significant role in force balance. In some physical applications, notably in glaciology, both rapid and slow slip can occur within the same fluid film. In order to capture the dynamics of rapid and slow slip in a single model that describes the entire fluid film, a hybrid of membrane and lubrication models is therefore required. Several of these hybrid models have been constructed on an ad hoc basis in glaciology, where they are usually termed ‘higher order models’. Here, we present a self-consistent asymptotic analysis of the most common of these models due originally to Blatter. We show that Blatter's model reproduces the solution to the underlying Stokes equations to second order in the film's aspect ratio, regardless of the amount of slip at the base of the fluid. In doing so, we also construct asymptotic expansions for the Stokes equations to this order for shear-thinning power-law fluids, paying particular attention to a high-viscosity boundary layer that develops at the free surface when there is little or no slip at the base. Lastly, we demonstrate that a depth-integrated hybrid model of comparable accuracy to Blatter's model—which cannot be depth integrated—can also be constructed, which we suggest as a viable tool for numerical simulations of thin films that contain both slowly and rapidly sliding parts.