Derivation of a non-hydrostatic shallow water model; Comparison with Saint-Venant and Boussinesq systems

From the free surface Navier-Stokes system, we derive the non-hydrostatic Saint-Venant system for the shallow waters including friction and viscosity. Based on an asymptotic analysis, the derivation leads to two formulations of growing complexity depending on the level of approximation chosen for the fluid pressure. The obtained models rely on formal estimates and are compared with the Boussinesq models.