Derivation of a new asymptotic viscous shallow water model with dependence on depth

A new shallow water model with viscosity and dependence on depth is presented. It is derived from the Navier–Stokes equations, with anisotropic viscosity, in a shallow domain. Asymptotic analysis has been used as in our previous works [16], [17], [18], [19], [20] and [21]: a small non-dimensional parameter ε related to the depth is introduced and it is studied what happens when ε becomes small. New diffusion terms are revealed thanks to the rigorous asymptotic analysis. The bidimensional shallow water model obtained in this way considers the possibility of a non-constant bottom and calculates the depth-averaged horizontal velocity, but also the three components of velocity for all z (when the vorticity is not zero). The authors perform some numerical computations that confirm the new model is able to approximate the solutions of Navier–Stokes equations with dependence on z (reobtaining the same velocities profile), whereas the classic model only computes the average velocity.