American Institute of Mathematical Sciences

Advanced
July  2011, 16(1): 361-383. doi: 10.3934/dcdsb.2011.16.361

Derivation and stability study of a rigid lid bilayer model

Timack Ngom 1, and  Mamadou Sy 2,

1. 

Laboratoire d'Analyse Numérique et Informatique, Université Gaston Berger, UFR SAT BP 234 Saint-Louis, Sénégal, LAMA, UMR 5127 CNRS, Université de Savoie, 73376 Le Bourget du lac, France
2. 

Laboratoire d'Analyse Numérique et d'Informatique (LANI), Université Gaston Berger, BP 234, Saint-Louis

Received  December 2009 Revised  September 2010 Published  April 2011

In this paper we present the derivation of a bilayer shallow water model with rigid lid hypothesis. We start from the incompressible Navier-Stokes equations, we introduce a small parameter $\varepsilon$ which is the ratio between the characteristic height and the characteristic length of the fluids domain. We use a formal asymptotic expansion then we resort to averaging to obtain the model. We also prove the stability of the model, in the following sense, up to a subsequence, every sequence of weak solutions converges to a solution of the model.
Keywords: bilayer., rigid lid, stability, shallow water, Asymptotic expansion.
Mathematics Subject Classification: Primary: 35Q35, 35Q30; Secondary: 35B3.
Citation: Timack Ngom, Mamadou Sy. Derivation and stability study of a rigid lid bilayer model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 361-383. doi: 10.3934/dcdsb.2011.16.361
References:
[1]

F. Bouchut and M. Westdickenberg, Gravity driven shallow water models for arbitrary topography,, Commun. Math. Sci., 2 (2004), 359.   Google Scholar

[2]

E. Audusse, A multilayer Saint-Venant model: derivation and numerical validation,, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 189.  doi: 10.3934/dcdsb.2005.5.189.  Google Scholar

[3]

E. Bruce Pitman and Long Le, A two-fluid model for avalanche and debris flows,, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 363 (2005), 1573.   Google Scholar

[4]

B. Di Martino, P. Orenga and M. Peybernes, On a bi-layer shallow water model with rigid-lid hypothesis,, Math. Models Methods Appl. Sci., 15 (2005), 843.  doi: 10.1142/S0218202505000583.  Google Scholar

[5]

G. Narbona-Reina, J. D. D. Zabsonré, E. D. Fernández-Nieto and D. Bresch, Derivation of a bilayer model for shallow water equations with viscosity. Numerical validation,, CMES Comput. Model. Eng. Sci., 43 (2009), 27.   Google Scholar

[6]

B. Di Martino, C. Giacomoni and P. Orenga, Analysis of some shallow water problems with rigid-lid hypothesis,, Math. Models Methods Appl. Sci., 11 (2001), 979.  doi: 10.1142/S0218202501001203.  Google Scholar

[7]

María Luz Muñoz-Ruiz, Manuel Jesú Castro-Díaz and Carlos Parés, On an one-dimensional bi-layer shallow-water problem,, Nonlinear Anal., 53 (2003), 567.  doi: 10.1016/S0362-546X(02)00137-2.  Google Scholar

[8]

Philippe Guyenne, David Lannes and Jean-Claude Saut, Well-posedness of the Cauchy problem for models of large amplitude internal waves,, Nonlinearity, 23 (2010), 237.  doi: 10.1088/0951-7715/23/2/003.  Google Scholar

[9]

D. Bresch and M. Renardy, Well-posedness of two-layer shallow water flow between two horizontal rigid plates,, To appear in nonlinearity, (2011).   Google Scholar

[10]

D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model,, Comm. Math. Phys., 238 (2003), 211.   Google Scholar

[11]

J. Simon, "Équation de Navier-Stokes,'', Cours de DEA 2002-2003 Universit茅 Blaise Pascal Clermont-Ferrand., (): 2002.   Google Scholar

[12]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,'', Dunod, (1969).   Google Scholar

[13]

A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equations,, Comm. Partial Differential Equations, 32 (2007), 431.  doi: 10.1080/03605300600857079.  Google Scholar

[14]

D. Bresch, B. Desjardins and D. Gérard-Varet, On compressible Navier-Stokes equations with density dependent viscosities in bounded domains,, J. Math. Pures Appl. (9), 87 (2007), 227.  doi: 10.1016/j.matpur.2006.10.010.  Google Scholar

[15]

F. Boyer and P. Fabrie, "Eléments D'analyse pour L'éetude de Quelques Modèles D'écoulements de Fluides Visqueux Incompressibles,'', Mathématiques & Applications (Berlin) [Mathematics & Applications]., ().   Google Scholar

[16]

F. Boyer, "Analyse Numérique des edp Elliptiques,'', Cours Master 2 2009 Université Paul Cézanne., (2009).   Google Scholar

show all references

References:
[1]

F. Bouchut and M. Westdickenberg, Gravity driven shallow water models for arbitrary topography,, Commun. Math. Sci., 2 (2004), 359.   Google Scholar

[2]

E. Audusse, A multilayer Saint-Venant model: derivation and numerical validation,, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 189.  doi: 10.3934/dcdsb.2005.5.189.  Google Scholar

[3]

E. Bruce Pitman and Long Le, A two-fluid model for avalanche and debris flows,, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 363 (2005), 1573.   Google Scholar

[4]

B. Di Martino, P. Orenga and M. Peybernes, On a bi-layer shallow water model with rigid-lid hypothesis,, Math. Models Methods Appl. Sci., 15 (2005), 843.  doi: 10.1142/S0218202505000583.  Google Scholar

[5]

G. Narbona-Reina, J. D. D. Zabsonré, E. D. Fernández-Nieto and D. Bresch, Derivation of a bilayer model for shallow water equations with viscosity. Numerical validation,, CMES Comput. Model. Eng. Sci., 43 (2009), 27.   Google Scholar

[6]

B. Di Martino, C. Giacomoni and P. Orenga, Analysis of some shallow water problems with rigid-lid hypothesis,, Math. Models Methods Appl. Sci., 11 (2001), 979.  doi: 10.1142/S0218202501001203.  Google Scholar

[7]

María Luz Muñoz-Ruiz, Manuel Jesú Castro-Díaz and Carlos Parés, On an one-dimensional bi-layer shallow-water problem,, Nonlinear Anal., 53 (2003), 567.  doi: 10.1016/S0362-546X(02)00137-2.  Google Scholar

[8]

Philippe Guyenne, David Lannes and Jean-Claude Saut, Well-posedness of the Cauchy problem for models of large amplitude internal waves,, Nonlinearity, 23 (2010), 237.  doi: 10.1088/0951-7715/23/2/003.  Google Scholar

[9]

D. Bresch and M. Renardy, Well-posedness of two-layer shallow water flow between two horizontal rigid plates,, To appear in nonlinearity, (2011).   Google Scholar

[10]

D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model,, Comm. Math. Phys., 238 (2003), 211.   Google Scholar

[11]

J. Simon, "Équation de Navier-Stokes,'', Cours de DEA 2002-2003 Universit茅 Blaise Pascal Clermont-Ferrand., (): 2002.   Google Scholar

[12]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,'', Dunod, (1969).   Google Scholar

[13]

A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equations,, Comm. Partial Differential Equations, 32 (2007), 431.  doi: 10.1080/03605300600857079.  Google Scholar

[14]

D. Bresch, B. Desjardins and D. Gérard-Varet, On compressible Navier-Stokes equations with density dependent viscosities in bounded domains,, J. Math. Pures Appl. (9), 87 (2007), 227.  doi: 10.1016/j.matpur.2006.10.010.  Google Scholar

[15]

F. Boyer and P. Fabrie, "Eléments D'analyse pour L'éetude de Quelques Modèles D'écoulements de Fluides Visqueux Incompressibles,'', Mathématiques & Applications (Berlin) [Mathematics & Applications]., ().   Google Scholar

[16]

F. Boyer, "Analyse Numérique des edp Elliptiques,'', Cours Master 2 2009 Université Paul Cézanne., (2009).   Google Scholar

[1]

Bilal Al Taki, Khawla Msheik, Jacques Sainte-Marie. On the rigid-lid approximation of shallow water Bingham. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020146
[2]

Vincent Duchêne, Samer Israwi, Raafat Talhouk. Shallow water asymptotic models for the propagation of internal waves. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 239-269. doi: 10.3934/dcdss.2014.7.239
[3]

Robert McOwen, Peter Topalov. Asymptotics in shallow water waves. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3103-3131. doi: 10.3934/dcds.2015.35.3103
[4]

Andreas Hiltebrand, Siddhartha Mishra. Entropy stability and well-balancedness of space-time DG for the shallow water equations with bottom topography. Networks & Heterogeneous Media, 2016, 11 (1) : 145-162. doi: 10.3934/nhm.2016.11.145
[5]

Julien Chambarel, Christian Kharif, Olivier Kimmoun. Focusing wave group in shallow water in the presence of wind. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 773-782. doi: 10.3934/dcdsb.2010.13.773
[6]

Anna Geyer, Ronald Quirchmayr. Shallow water models for stratified equatorial flows. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4533-4545. doi: 10.3934/dcds.2019186
[7]

Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209
[8]

Zhigang Wang. Vanishing viscosity limit of the rotating shallow water equations with far field vacuum. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 311-328. doi: 10.3934/dcds.2018015
[9]

Chengchun Hao. Cauchy problem for viscous shallow water equations with surface tension. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 593-608. doi: 10.3934/dcdsb.2010.13.593
[10]

Issam S. Strub, Julie Percelay, Olli-Pekka Tossavainen, Alexandre M. Bayen. Comparison of two data assimilation algorithms for shallow water flows. Networks & Heterogeneous Media, 2009, 4 (2) : 409-430. doi: 10.3934/nhm.2009.4.409
[11]

Denys Dutykh, Dimitrios Mitsotakis. On the relevance of the dam break problem in the context of nonlinear shallow water equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 799-818. doi: 10.3934/dcdsb.2010.13.799
[12]

Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Jacques Sainte-Marie. A combined finite volume - finite element scheme for a dispersive shallow water system. Networks & Heterogeneous Media, 2016, 11 (1) : 1-27. doi: 10.3934/nhm.2016.11.1
[13]

Olivier Delestre, Arthur R. Ghigo, José-Maria Fullana, Pierre-Yves Lagrée. A shallow water with variable pressure model for blood flow simulation. Networks & Heterogeneous Media, 2016, 11 (1) : 69-87. doi: 10.3934/nhm.2016.11.69
[14]

Roberto Camassa. Characteristics and the initial value problem of a completely integrable shallow water equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 115-139. doi: 10.3934/dcdsb.2003.3.115
[15]

Werner Bauer, François Gay-Balmaz. Towards a geometric variational discretization of compressible fluids: The rotating shallow water equations. Journal of Computational Dynamics, 2019, 6 (1) : 1-37. doi: 10.3934/jcd.2019001
[16]

Xiaoping Zhai, Hailong Ye. On global large energy solutions to the viscous shallow water equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020097
[17]

Marcel Oliver, Sergiy Vasylkevych. Hamiltonian formalism for models of rotating shallow water in semigeostrophic scaling. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 827-846. doi: 10.3934/dcds.2011.31.827
[18]

Madalina Petcu, Roger Temam. An interface problem: The two-layer shallow water equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5327-5345. doi: 10.3934/dcds.2013.33.5327
[19]

Mouhamadou Aliou M. T. Baldé, Diaraf Seck. Coupling the shallow water equation with a long term dynamics of sand dunes. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1521-1551. doi: 10.3934/dcdss.2016061
[20]

Xue Yang, Xinglong Wu. Wave breaking and persistent decay of solution to a shallow water wave equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2149-2165. doi: 10.3934/dcdss.2016089

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (40)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences