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A laminar flow of a thin layer of mud down an inclined plane under the actionof gravity is considered. The instability of a film flow and the formation offinite amplitude waves are studied in the framework of both two-dimensionalgoverning equations of a power-law fluid and its depth-averaged hyperbolicsimplification. Conditions of roll waves existence for these models areformulated in terms of Whitham criterion. Numerical calculations of the freesurface evolution and the roll waves development are performed. It is shownthat the roll waves amplitude obtained by the 2D equations is slightly largerthan for the 1D model. Moreover, for certain flow parameters, smallperturbations of the basic solution grow for the 2D equations and decay for thedepth-averaged model. A two-parameter class of exact piecewise-smooth solutionsof the 1D model is obtained and a comparison with a numerical solution is made.In the region of these parameters, diagrams of the roll waves existence areconstructed.

Formation and evolution of roll waves in a shallow free surface flow of a power-law fluid down an inclined plane