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 within the framework of both two-dimensional governing equations of a power-law fluid and its depth-averaged hyperbolic simplification. The conditions of the existence of roll waves for these models are formulated in terms of the Whitham criterion. A free surface evolution and the development of roll waves are numerically calculated. The amplitude of roll waves obtained by the 2D equations is slightly larger than that for the 1D model. Moreover, for certain flow parameters, the 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.

考虑在重力作用下沿倾斜平面向下的薄层泥浆层流。在幂律流体的二维控制方程及其深度平均双曲线简化的框架内研究了薄膜流动的不稳定性和有限振幅波的形成。这些模型的滚波存在条件是根据惠瑟姆准则制定的。自由表面演化和滚动波的发展进行了数值计算。二维方程得到的横摇波幅值比一维模型略大。此外，对于某些流动参数，基本解的小扰动对于 2D 方程会增加，对于深度平均模型会衰减。获得一维模型的精确分段平滑解的双参数类，并与数值解进行比较。在这些参数的区域中，构造了横滚波存在的图。