Shallow flow of an inhomogeneous incompressible fluid down an inclined plane

In this paper, we study a model for the shallow flow of an inhomogeneous incompressible fluid down an inclined plane. The constitutive response of this class of fluids has been discussed in Malek and Rajagopal (Mech Mater 38(3):233-242, 2006). We consider a planar geometry in which the length scale ratio is small so that the lubrication approximation can be applied. We analyze two different flow regimes: in the first one, the viscous and the gravitational forces are balanced and the system is governed by a hyperbolic equation for the fluid film thickness. In the second regime, the frictional forces are balanced by the pressure gradient and the system is governed by a parabolic equation for the film thickness. For both regimes, we perform numerical simulations that show the evolution of the film thickness using some general data that are well suited for highlighting the main features of the flow.