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Formal derivation of a bilayer model coupling shallow water and Reynolds lubrication equations: evolution of a thin pollutant layer over water

Published online by Cambridge University Press:  27 June 2013

E. D. FERNÁNDEZ-NIETO ,
G. NARBONA-REINA  and
J. D. ZABSONRÉ
E. D. FERNÁNDEZ-NIETO
Affiliation:
Departamento Matemática Aplicada I, Universidad de Sevilla, Seville, Spain emails: edofer@us.es, gnarbona@us.es
G. NARBONA-REINA
Affiliation:
Departamento Matemática Aplicada I, Universidad de Sevilla, Seville, Spain emails: edofer@us.es, gnarbona@us.es
J. D. ZABSONRÉ
Affiliation:
Unité de Formation et de Recherche en Sciences et Techniques, Département de Mathématiques, Université Polytechnique de Bobo-Dioulasso, Bobo-Dioulasso, Burkina Faso email: jzabsonre@gmail.com
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Abstract

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 for the upper layer and the shallow water model for the lower one. We prove that the model verifies 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 extension for the pressure – that admits an exact dissipative entropy inequality. Two numerical tests are presented. In the first test, we compare the numerical results with the viscous bilayer shallow water model proposed in Narbona-Reina et al. (Comput. Model. Eng. Sci., 2009, Vol. 43, pp. 27–71). 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.

Keywords

Bilayer modelReynolds equationshallow watermultiscale analysis
Type
Papers
Information
European Journal of Applied Mathematics , Volume 24 , Issue 6 , December 2013 , pp. 803 - 833
Copyright
Copyright © Cambridge University Press 2013 

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