  A laminar flow of a thin layer of mud down an inclined plane under the action
of gravity is considered. The instability of a film flow and the formation of
finite amplitude waves are studied in the framework of both two-dimensional
governing equations of a power-law fluid and its depth-averaged hyperbolic
simplification. Conditions of roll waves existence for these models are
formulated in terms of Whitham criterion. Numerical calculations of the free
surface evolution and the roll waves development are performed. It is shown
that the roll waves amplitude obtained by the 2D equations is slightly larger
than for the 1D model. Moreover, for certain flow parameters, small
perturbations of the basic solution grow for the 2D equations and decay for the
depth-averaged model. A two-parameter class of exact piecewise-smooth solutions
of 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 are
constructed.
