Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-03T22:11:24.995Z Has data issue: false hasContentIssue false
  • English
  • Français

An example of low Mach (Froude) number effects for compressible flows with nonconstant density (height) limit

Published online by Cambridge University Press:  15 June 2005

Didier Bresch ,
Marguerite Gisclon  and
Chi-Kun Lin
Didier Bresch
Affiliation:
LMC-IMAG, (CNRS, INPG, UJF, INRIA) 38051 Grenoble Cedex, France. didier.bresch@imag.fr
Marguerite Gisclon
Affiliation:
Université de Savoie, LAMA, UMR CNRS 5127, 73376 Le Bourget-du-lac, France. marguerite.gisclon@univ-savoie.fr
Chi-Kun Lin
Affiliation:
Department of Mathematics, National Cheng Kung University, Tianan 701 Taiwan. cklin@mail.ncku.edu.tw
Get access

Abstract

The purpose of this work is to study an example of low Mach (Froude) number limit of compressible flows when the initial density (height) is almost equal to a function depending on x. This allows us to connect the viscous shallow water equation and the viscous lake equations. More precisely, we study this asymptotic with well prepared data in a periodic domain looking at the influence of the variability of the depth. The result concerns weak solutions. In a second part, we discuss the general low Mach number limit for standard compressible flows given in P.–L. Lions' book that means with constant viscosity coefficients.

Keywords

Compressible flowsNavier-Stokes equationslow Mach (Froude) Number limit shallow-water equationslake equationsnonconstant density.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 3: Special issue on Low Mach Number Flows Conference , May 2005 , pp. 477 - 486
Copyright
© EDP Sciences, SMAI, 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

T. Alazard, Incompressible limit of the non-isentropic Euler equations with solid wall boundary conditions. Submitted (2004).
Bresch, D. and Desjardins, B., Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model. Comm. Math. Phys. 238 (2003) 211–223. CrossRef
Bresch, D., Desjardins, B. and Gérard-Varet, D., Rotating fluids in a cylinder. Discrete Contin. Dynam. Systems Ser. A 11 (2004) 47–82.
Bresch, D., Desjardins, B. and Lin, C.-K., On some compressible fluid models: Korteweg, lubrication and shallow water systems. Comm. Partial Differential Equations 28 (2003) 1009–1037. CrossRef
Bresch, D., Desjardins, B., Grenier, E. and Lin, C.-K., Low Mach number limit of viscous polytropic flows: formal asymptotics in the periodic case. Stud. Appl. Math. 109 (2002) 125–148. CrossRef
R. Danchin, Fluides légèrement compressibles et limite incompressible. Séminaire École Polytechnique (France), Exposé No. III (2000).
Desjardins, B., Grenier, E., P.–L. Lions and N. Masmoudi, Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions. J. Math. Pures Appl. 78 (1999) 461–471. CrossRef
I. Gallagher, Résultats récents sur la limite incompressible. Séminaire Bourbaki (France), No. 926 (2003).
Gerbeau, J.F. and Perthame, B., Derivation of viscous Saint-Venant system for laminar Shallow water; Numerical results. Discrete Contin. Dynam. Systems Ser. B 1 (2001) 89–102.
Grenier, E., Oscillatory perturbations of the Navier-Stokes equations. J. Math. Pures Appl. 76 (1997) 477–498. CrossRef
Levermore, C.D. and Sammartino, M., A shallow water model with eddy viscosity for basins with varying bottom topography. Nonlinearity 14 (2001) 1493–1515. CrossRef
Levermore, C.D., Oliver, M. and Titi, E.S., Global well-posedness for a models of shallow water in a basin with a varying bottom. Indiana Univ. Math. J. 45 (1996) 479–510. CrossRef
P.-L. Lions, Mathematical topics in fluid dynamics, Vol. 2, Compressible models. Oxford Science Publication, Oxford (1998).
Lions, P.-L. and Masmoudi, N., Incompressible limit for a viscous compressible fluids. J. Math. Pures Appl. 77 (1998) 585–627. CrossRef
Métivier, G. and Schochet, S., The incompressible limit of the non-isentropic Euler equations. Arch. Rational Mech. Anal. 158 (2001) 61–90. CrossRef
G. Métivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, in Séminaire Équations aux Dérivées Partielles, École Polytechnique (2001).
Oliver, M., Justification of the shallow water limit for a rigid lid with bottom topography. Theor. Comp. Fluid Dyn. 9 (1997) 311–324. CrossRef
J. Pedlosky, Geophysical fluid dynamics. Berlin Heidelberg-New York, Springer-Verlag (1987).