The derivation of the Saint-Venant equations

The water motion in real river-beds is described by the mathematical models that under appropriate assumptions are various approximations of hydrodynamic equations, or the Navier-Stokes equations. The mathematical model of a water flow, based on the laws of conservation of momentum and mass of fluid, was proposed by Saint-Venant. The Saint-Venant equations (the shallow-water equations) are often used in theoretical and applied studies of the unsteady water motion in free channels. In this paper, the rigorous deduction of the shallow-water equations from the Navier-Stokes equations using the recommendations and methods proposed in [1] is presented.