Formal derivation and existence of global weak solutions of a two-dimensional bilayer model coupling shallow water and Reynolds lubrication equations

Article type: Research Article

Authors: Roamba, Brahima^a | de Dieu Zabsonré, Jean^{b; *} | Traoré, Sado^a

Affiliations: [a] UFR/ST, Université Polytechnique de Bobo-Dioulasso, 10 BP 1091 Bobo-Dioulasso, Burkina Faso. E-mails: braroamba@gmail.com, traore.sado@yahoo.fr | [b] IUT, Université Polytechnique de Bobo-Dioulasso, 10 BP 1091 Bobo-Dioulasso, Burkina Faso. E-mail: jzabsonre@gmail.com

Correspondence: [*] Corresponding author. E-mail: jzabsonre@gmail.com.

Abstract: 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 Mathematics 24(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.

Keywords: bilayer models, Reynolds equation, shallow water equations, multiscale analysis, viscosity, friction, capillary effects

DOI: 10.3233/ASY-161381

Journal: Asymptotic Analysis, vol. 99, no. 3-4, pp. 207-239, 2016