Flow of a viscous fluid down a slope

A thin layer of viscous fluid flowing down a slope under gravity is considered. The stability of this flow is studied in the framework of a shallow water model. A number of equations of shallow water motion are derived with consideration of the viscous friction. The corresponding Cauchy problem is solved analytically. It is shown that the uniform flow may be unstable with respect to small initial perturbations. A criterion of instability is proposed.