This paper is devoted to the derivation and the study of a two-dimensional bilayer model. This model is obtained from the incompressible Navier–Stokes equations with suitable boundary conditions including friction and capillary effects as in the method used by [European J. Applied Mathematics24(6) (2013), 803–833] in the one dimensional case. We perform a multiscale analysis in space and time, as well as an asymptotic analysis to obtain a system coupling two different features: the Reynolds lubrication equations for the upper layer and the shallow water equation for the lower one. We finally prove the existence of global weak solutions in time for model containing some additional terms.
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