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

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

Get full access to this article

View all access options for this article.

References

Audusse, A multilayer Saint-Venant model, Disc. Cont. Dyn. Systems, B 5(2) (2005), 189–214. doi:10.3934/dcdsb.2005.5.189.

Bresch and

Desjardin, Existence of global weak solution for 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys. 238(1–3) (2003), 211–223. doi:10.1007/s00220-003-0859-8.

Bresch and

Desjardin, Some diffusive capillary model of Korteweg type, C. R. Acad. Sci. Paris, Section Mécanique 332(11) (2004), 881–886.

Bresch and

Desjardin, On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier–Stokes models, J. Math. Pures Appl. 86(4) (2006), 362–368. doi:10.1016/j.matpur.2006.06.005.

Bresch,

Desjardin and

Gérard-Varet, On compressible Navier–Stokes equations with density dependent viscosities in bounded domains, J. Math. Pures Appl. 87(9) (2007), 227–235. doi:10.1016/j.matpur.2006.10.010.

Bresch,

Desjardin and

C.-K.

Lin, On some compressible fluid models: Korteweg, lubrication and shallow water systems, Comm. Partial Diff. Equations 28(3–4) (2003), 843–868. doi:10.1081/PDE-120020499.

Bresch,

Desjardin and

Zatorska, Two-velocity hydrodynamics in fluid mechanics: Part II. Existence of global κ-entropy solutions to the compressible Navier–Stokes systems with degenerate viscosities, J. Math. Pures Appl. 104(4) (2015), 801–836. doi:10.1016/j.matpur.2015.05.004.

Fabrice and

Marche, Another proof of stability for global weak solutions of 2D degenerated shallow water models, J. Math. Fluid Mech. 1 (2009), 536–551. doi:10.1007/s00021-008-0271-4.

E.D.

Fernandez-Nieto,

Narbonareina and

J.D.

Zabsonré, Derivation of a bilayer model coupling shallow water and Reynolds lubrication equations: Evolution of a thin pollutant layer over water, European J. Applied Mathematics 24(6) (2013), 803–833. doi:10.1017/S095679251300020X.

10.

Gerbeau and

Perthame, Derivation of viscous Saint-Venant system for laminar shallow-water: Numerical validation, Disc. Syst., B 1(1) (2001), 89–102.

11.

Kitavtsev,

Laurençot and

Niethammer, Weak solutions to lubrication equations in the presence of strong slippage, Methods and Appl. of Analysis 18(2) (2011), 83–202.

12.

J.-L.

Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, 1969.

13.

Marche, Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom and capillary effects, European J. Mech. B/Fluids 26(1) (2007), 49–63. doi:10.1016/j.euromechflu.2006.04.007.

14.

Narbona-Reina,

J.D.D.

Zabsonré,

E.D.

Fernàndez-Nieto and

Bresch, Derivation of bilayer model for shalow water equations with viscosity. Numerical validation, CMES 43(1) (2009), 27–71.

15.

Oron,

S.H.

Davis and

S.G.

Bankoff, Long scale evolution of thin films, Rev. Phys. 69(3) (1997), 931–980.

16.

Reynolds, On the theory of lubrication and its application to Mr. Beauchamp tower’s experiment, Phil. Trans. R. Soc. London Part I 52 (1886), 228–310.

17.

A.F.

Vasseur and

Yu, Global weak solutions to compressible quantum Navier–Stokes equations with damping, SIAM J. Math. Anal. 48(2) (2016), 1489–1511. doi:10.1137/15M1013730.

18.

J.D.D.

Zabsonré, Modèle visqueux en sédimentation et stratification. Obtention formelle, stabilité théorique et schémes volumes finis bien équilibrés, PhD Thesis, University of Savoie, France, 2008.

19.

J.D.D.

Zabsonré,

Lucas and

E.D.

Fernandez-Nieto, An energetically consistent viscous sedimentation model, Math. Models and Methods in Appl. Sci. 19(3) (2009), 477–499. doi:10.1142/S0218202509003504.

20.

J.D.D.

Zabsonré and

Narbona-Reina, Existence of a global weak solution for 2D viscous bi-layer shalow water model, Nonlin. Analysis: Real World Appl. 10(5) (2009), 2971–2984. doi:10.1016/j.nonrwa.2008.09.004.

21.

Zatorska, Fundamental problems to equations of compressible chemically reacting flows, PhD Thesis, University of Warsaw, Poland, 2013.