Flow of a viscous fluid down a slope

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