Shallow flows of generalised Newtonian fluids on an inclined plane

We derive a general evolution equation for a shallow layer of a generalised Newtonian fluid undergoing two-dimensional gravity-driven flow on an inclined plane. The flux term appearing in this equation is expressed in terms of an integral involving the prescribed constitutive relation and, crucially, does not require explicit knowledge of the velocity profile of the flow; this allows the equation to be formulated for any generalised Newtonian fluid. In particular, we develop general solutions for travelling waves on a mild slope and for kinematic waves on a moderately steep slope; these results provide simple and accessible models of, for example, the propagation of non-Newtonian mud and debris flows.