Existence of a global weak solution for a 2D viscous bi-layer Shallow Water model

Zabsonré, Jean de Dieu; Narbona-Reina, Gladys

doi:10.1016/j.nonrwa.2008.09.004

Cited by 15 publications

(9 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The last step is to develop the first two terms on the left hand side of Equation (19) and to use the fact that 2ab  ✓a…”

Section: Energy Inequalitiesmentioning

confidence: 99%

“…In this paper, we are interested in another system of bilayer immiscible fluid obtained by derivation in [1]. These equations have been studied in [18,19]; the authors proved the existence of a global weak solution for viscous bilayer Shallow Water models with the BD inequality (but in two dimensions). This paper deals with strong solutions, in one dimension.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Strong solutions for a 1D viscous bilayer shallow water model

Zabsonré

Lucas

Ouédraogo

2013

Nonlinear Analysis: Real World Applications

Self Cite

View full text Add to dashboard Cite

In this paper, we consider a viscous bilayer shallow water model in one space dimension that represents two superposed immiscible fluids. For this model, we prove the existence of strong solutions in a periodic domain. The initial heights are required to be bounded above and below away from zero and we get the same bounds for every time. Our analysis is based on the construction of approximate systems which satisfy the BD entropy and on the method developed by A. Mellet and A. Vasseur to obtain the existence of global strong solutions for the one dimensional Navier-Stokes equations.

show abstract

“…The last step is to develop the first two terms on the left hand side of Equation (19) and to use the fact that 2ab  ✓a…”

Section: Energy Inequalitiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Strong solutions for a 1D viscous bilayer shallow water model

Zabsonré

Lucas

Ouédraogo

2013

Nonlinear Analysis: Real World Applications

Self Cite

View full text Add to dashboard Cite

show abstract

“…The main difficulty in these papers arises from the terms coupling the two layers. We can also find in [7] the theoretical study of the bilayer model but where the friction terms have been simplified. The friction terms considered here are more difficult to control and to pass to the limit in the weak formulation.…”

Section: Introductionmentioning

confidence: 99%

“…The proof of the Theorem 3.1 is the same as that built in [7]. We must only prove some additional strong convergences.…”

Section: Introductionmentioning

confidence: 99%

Existence of global weak solutions for a viscous 2D bilayer Shallow Water model

Narbona-Reina

Zabsonré

2011

Comptes Rendus. Mathématique

Self Cite

View full text Add to dashboard Cite

We consider a system composed by two immiscible fluids in two-dimensional space that can be modelized by a bilayer Shallow Water equations with extra friction terms and capillary effects. We give an existence theorem of global weak solutions in a periodic domain. RésuméNous considérons un système composé par deux fluides immiscibles dans un domaine bi-dimensionnel pouvantêtre représenté par un modèle bicouche visqueux de Saint-Venant avec des termes de friction additionnels et des effets de capillarité. Nous donnons un théorème d'existence de solutions faibles globales dans un domaine périodique. Version française abrégéeDans cette note, nous nous intéressonsà l'étude de l'existence de solutions faibles globales en temps d'un modèle bicouche visqueux de Saint-Venant dérivé dans [6]

show abstract

“…This derivation takes into account laminar friction at the bottom and viscous effects. Viscous and capillary effects are useful to obtain an existence result of global weak solutions in [41]. In [24], a viscous two dimensional one-layer shallow water system, taking into account surface tension, capillary effects and quadratic friction terms, has been derived.…”

Section: Introductionmentioning

confidence: 99%

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

2013

View full text Add to dashboard Cite

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 is provided of 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 extention for the pressure− that admits an exact dissipative entropy inequality. Two numerical tests are presented. In the first one we compare the numerical results with the viscous bilayer shallow water model proposed in [G. Narbona-Reina, J.D.D. Zabsonré, E.D. Fernández-Nieto, D. Bresch, CMES Comput. Model. Eng. Sci., 2009]. 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.

show abstract

Existence of a global weak solution for a 2D viscous bi-layer Shallow Water model

Cited by 15 publications

References 15 publications

Strong solutions for a 1D viscous bilayer shallow water model

Strong solutions for a 1D viscous bilayer shallow water model

Existence of global weak solutions for a viscous 2D bilayer Shallow Water model

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

Contact Info

Product

Resources

About