Viscous transfer of momentum across a shallow laminar flow

Abstract

In a shallow channel, the flow transfers most of its momentum vertically. Based on this observation, one often neglects the momentum that is transferred across the stream – the core assumption of the shallow-water theory. In the context of viscous flows, this approximation is referred to as the ‘lubrication theory’, in which one assumes that the shear stress exerted by the fluid on the substrate over which it flows is proportional to its velocity. Here, we revise this theory to account for the momentum that viscosity transfers across a shallow laminar flow, while keeping the problem low-dimensional. We then test the revised lubrication theory against analytical and numerical solutions of the exact problem. We find that, at a low computational cost, the present theory represents the actual flow more accurately than the classical lubrication approximation. This theoretical improvement, devised with laboratory rivers in mind, should also apply to other geophysical contexts, such as ice flows or forming lava domes.