Exact solutions for two-dimensional steady flows of a power-law liquid on an incline

Under assumptions that are not too restrictive it is possible to reduce the equations that describe steady viscous gravity flows of a power-law liquid on an inclined plane to an equivalent problem consisting of an unsteady one-dimensional nonlinear diffusion process. In a paper dealing with the steady spreading flow of a Herschel–Buckley liquid, Wilson and Burgess [“The steady, spreading flow of a rivulet of mud,” J. Non-Newtonian Fluid Mech. 79, 77 (1998) ] noticed a formal analogy between the steady, two-dimensional viscous gravity flows of a power-law liquid on an incline and a one-dimensional time-dependent nonlinear diffusion phenomena; however, they did not pursue the matter further. Here we develop the analogy and show how it can be used to find a large number of exact solutions representing steady two-dimensional flows of power-law liquids, based on the available knowledge concerning nonlinear diffusion. We describe flows whose widths stay constant until a certain distance from the source, which are analogous to the well-known waiting-time solutions of nonlinear diffusion. We then introduce a phase-plane formalism that allows us to find self-similar solutions and we give as examples three different currents limited laterally by a wall that ends abruptly and currents on an inclined stripe. Finally we describe the two-dimensional currents that are analogous to the traveling wave solutions of the nonlinear diffusion equation. The approximations involved in the analogy are essentially equivalent to those of the lubrication theory, so that they do not impose restrictions more severe than those usually present in problems of this type. The present theory does not include surface tension effects, which implies that the appropriate Bond number must be large.