Waves of finite amplitude on a thin layer of non-Newtonian fluid modelled as a power-law fluid are considered. In the long wave approximation, the system of equations taking into account the viscous and nonlinear effects has the hyper- bolic type. For the two-parameter family of periodic waves in the film flow on a vertical wall the modulation equations for nonlinear wave trains are derived and investigated. The stability criterium for roll waves based on the hyperbolicity of the modulation equations is suggested. It is shown that the evolution of stable roll waves can be described by self-similar solutions of the modulation equations.
F. Dressler, “Mathematical Solution of the Problem of Roll Waves in Inclined Open Channels,” Communications on Pure and Applied Mathematics, Vol. 2, No. 2-3, 1949, pp. 149-194. doi:10.1002/cpa.3160020203
C. E. Mesa and V. Balakotaiah, “Modelling and Experimental Studies of Large Amplitude Waves on Vertically Falling Films,” Chemical Engineering Science, Vol. 63, No. 19, 2008, pp. 4704-4734.