Formal derivation of a bilayer model coupling shallow water and Reynolds lubrication equations: evolution of a thin pollutant layer over water
European journal of applied mathematics
In this paper a bilayer model is derived to simulate the evolution of a thin film flow over water. This model is derived from the incompressible Navier-Stokes equations together with suitable boundary conditions including friction and capillary effects. The derivation is based on the different properties of the fluids, thus, we perform a multiscale analysis in space and time, and a different asymptotic analysis to derive a system coupling two different models: the Reynolds lubrication equation
... brication equation for the upper layer and the shallow water model for the lower one. We prove that the model is provided of a dissipative entropy inequality, up to a second order term. Moreover, we propose a correction of the model −by taking into account the second order extention for the pressure− that admits an exact dissipative entropy inequality. Two numerical tests are presented. In the first one we compare the numerical results with the viscous bilayer shallow water model proposed in [G. Narbona-Reina, J.D.D. Zabsonré, E.D. Fernández-Nieto, D. Bresch, CMES Comput. Model. Eng. Sci., 2009]. In the second test the objective is to show some of the characteristic situations that can be studied with the proposed model. We simulate a problem of pollutant dispersion near the coast. For this test the influence of the friction coefficient on the coastal area affected by the pollutant is studied.