In this paper, we obtain new models for gravity driven shallow water laminar flows in several space dimensions over a general topography. These models are derived from the incompressible Navier-Stokes equations with no-slip condition at the bottom and include capillary effects. No particular assumption is made on the size of the viscosity and on the variations of the slope. The equations are written for an arbitrary parametrization of the bottom, and an explicit formulation is given in the orthogonal courvilinear coordinates setting and for a particular parametrization so-called “steepest descent” curvilinear coordinates.
"Shallow water viscous flows for arbitrary topopgraphy." Commun. Math. Sci. 6 (1) 29 - 55, March 2008.