Formal passage from kinetic theory to incompressible Navier–Stokes equations for a mixture of gases

Marzia Bisi; Laurent Desvillettes

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 4, page 1171-1197
  • ISSN: 0764-583X

Abstract

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We present in this paper the formal passage from a kinetic model to the incompressible Navier−Stokes equations for a mixture of monoatomic gases with different masses. The starting point of this derivation is the collection of coupled Boltzmann equations for the mixture of gases. The diffusion coefficients for the concentrations of the species, as well as the ones appearing in the equations for velocity and temperature, are explicitly computed under the Maxwell molecule assumption in terms of the cross sections appearing at the kinetic level.

How to cite

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Bisi, Marzia, and Desvillettes, Laurent. "Formal passage from kinetic theory to incompressible Navier–Stokes equations for a mixture of gases." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.4 (2014): 1171-1197. <http://eudml.org/doc/273132>.

@article{Bisi2014,
abstract = {We present in this paper the formal passage from a kinetic model to the incompressible Navier−Stokes equations for a mixture of monoatomic gases with different masses. The starting point of this derivation is the collection of coupled Boltzmann equations for the mixture of gases. The diffusion coefficients for the concentrations of the species, as well as the ones appearing in the equations for velocity and temperature, are explicitly computed under the Maxwell molecule assumption in terms of the cross sections appearing at the kinetic level.},
author = {Bisi, Marzia, Desvillettes, Laurent},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {kinetic theory; incompressible Navier–Stokes equations; hydrodynamic limits; incompressible Navier-Stokes equation; gas mixture; hydrodynamic limit; Boltzmann equations},
language = {eng},
number = {4},
pages = {1171-1197},
publisher = {EDP-Sciences},
title = {Formal passage from kinetic theory to incompressible Navier–Stokes equations for a mixture of gases},
url = {http://eudml.org/doc/273132},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Bisi, Marzia
AU - Desvillettes, Laurent
TI - Formal passage from kinetic theory to incompressible Navier–Stokes equations for a mixture of gases
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 4
SP - 1171
EP - 1197
AB - We present in this paper the formal passage from a kinetic model to the incompressible Navier−Stokes equations for a mixture of monoatomic gases with different masses. The starting point of this derivation is the collection of coupled Boltzmann equations for the mixture of gases. The diffusion coefficients for the concentrations of the species, as well as the ones appearing in the equations for velocity and temperature, are explicitly computed under the Maxwell molecule assumption in terms of the cross sections appearing at the kinetic level.
LA - eng
KW - kinetic theory; incompressible Navier–Stokes equations; hydrodynamic limits; incompressible Navier-Stokes equation; gas mixture; hydrodynamic limit; Boltzmann equations
UR - http://eudml.org/doc/273132
ER -

References

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  1. [1] T. Alazard, Low Mach number limit of the full Navier–Stokes equations. Arch. Rational Mech. Anal.180 (2006) 1–73. Zbl1108.76061MR2211706
  2. [2] D. Arsenio, From Boltzmann’s equation to the incompressible Navier−Stokes–Fourier system with long-range interactions. Arch. Ration. Mech. Anal. 206 (2012) 367–488. Zbl1257.35140MR2980526
  3. [3] C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations. J. Statis. Phys. 63 (1991) 323–344. Zbl1151.35066MR1115587
  4. [4] C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation. Commun. Pure Appl. Math. 46 (1993) 667–753. Zbl0817.76002MR1213991
  5. [5] S. Bastea, R. Esposito, J.L. Lebowitz and R. Marra, Binary fluids with long range segregating interaction. I. Derivation of kinetic and hydrodynamic equations. J. Statis. Phys. 101 (2000) 1087–1136. Zbl0989.82025MR1806716
  6. [6] B.J. Bayly, D. Levermore and T. Passot, Density variations in weakly compressible flows. Phys. Fluids A4 (1992) 945–954. Zbl0756.76061MR1160287
  7. [7] A. Berti, V. Berti and D. Grandi, Well–posedness of an isothermal diffusive model for binary mixtures of incompressible fluids. Nonlinearity24 (2011) 3143–3164. Zbl1269.35029MR2844832
  8. [8] M. Bisi, M. Groppi and G. Spiga, Fluid-dynamic equations for reacting gas mixtures. Appl. Math.50 (2005) 43–62. Zbl1099.82015MR2117695
  9. [9] M. Bisi, M. Groppi and G. Spiga, Kinetic Modelling of Bimolecular Chemical Reactions, Kinetic Methods for Nonconservative and Reacting Systems. Quaderni di Matematica [Math. Ser.], vol. 16. Edited by G. Toscani. Aracne Editrice, Roma (2005) 1–143. Zbl1121.82032MR2244535
  10. [10] M. Bisi and L. Desvillettes, From reactive Boltzmann equations to reaction-diffusion systems. J. Statis. Phys.124 (2006) 881–912. Zbl1134.82323MR2264629
  11. [11] M. Bisi, G. Martalò and G. Spiga, Multi-temperature Euler hydrodynamics for a reacting gas from a kinetic approach to rarefied mixtures with resonant collisions. Europhys. Lett. 95 (2011), 55002. 
  12. [12] L. Boudin, B. Grec, M. Pavic and F. Salvarani, Diffusion asymptotics of a kinetic model for gaseous mixtures. Kinet. Relat. Models6 (2013) 137–157. Zbl1260.35100MR3005625
  13. [13] S. Brull, Habilitation thesis. Univ. Bordeaux (2012). 
  14. [14] S. Brull, V. Pavan and J. Schneider, Derivation of BGK models for mixtures. Eur. J. Mech. B-Fluids33 (2012) 74–86. Zbl1258.76122MR2896732
  15. [15] C. Cercignani, The Boltzmann Equation and its Applications. Springer, New York (1988). Zbl0646.76001MR1313028
  16. [16] V. Giovangigli, Multicomponent flow modeling, Series on Modeling and Simulation in Science, Engineering and Technology. Birkhaüser, Boston (1999). Zbl0956.76003MR1713516
  17. [17] F. Golse and L. Saint-Raymond, The Navier−Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math. 155 (2004) 81–161. Zbl1060.76101MR2025302
  18. [18] F. Golse and L. Saint-Raymond, The incompressible Navier−Stokes limit of the Boltzmann equation for hard cutoff potentials. J. Math. Pures Appl. 91 (2009) 508–552. Zbl1178.35290MR2517786
  19. [19] H. Grad, Asymptotic theory of the Boltzmann equation. Phys. Fluids6 (1963) 147–181. Zbl0115.45006MR155541
  20. [20] H. Grad, Asymptotic theory of the Boltzmann equation II, Rarefied Gas Dynamics. Proc. of 3rd Int. Sympos. Academic Press, New York I (1963) 26–59. Zbl0115.45006MR156656
  21. [21] T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag, New York (1966). Zbl0435.47001MR203473
  22. [22] D. Levermore and N. Masmoudi, From the Boltzmann equation to an incompressible Navier-Stokes-Fourier system. Arch. Rational Mech. Anal.196 (2010) 753–809. Zbl1304.35476MR2644440
  23. [23] P.L. Lions, N. Masmoudi, Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl.77 (1998) 585–627. Zbl0909.35101MR1628173
  24. [24] P.L. Lions and N. Masmoudi, From the Boltzmann equations to the equations of incompressible fluid mechanics II. Arch. Rational Mech. Anal.158 (2001) 195–211. Zbl0987.76088MR1842343
  25. [25] J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn–Hilliard fluids and topological transitions. Proc. R. Soc. A. Math. Phys. Eng. Sci.454 (1998) 2617–2654. Zbl0927.76007MR1650795
  26. [26] L. Saint-Raymond, Some recent results about the sixth problem of Hilbert. Analysis and simulation of fluid dynamics. Adv. Math. Fluid Mech. Birkhäuser, Basel (2007) 183–199. Zbl1291.35126MR2331340
  27. [27] L. Saint-Raymond, Hydrodynamic limits of the Boltzmann equation. Vol. 1971 of Lect. Notes Math. Springer-Verlag, Berlin (2009). Zbl1171.82002MR2683475
  28. [28] L. Saint-Raymond, Some recent results about the sixth problem of Hilbert: hydrodynamic limits of the Boltzmann equation, European Congress of Mathematics. Eur. Math. Soc. Zürich (2010) 419–439. Zbl1229.35163MR2648335
  29. [29] E.A. Spiegel and G. Veronis, On the Boussinesq approximation for a compressible fluid. Astrophys. J. 131 442–447. MR128767
  30. [30] A. Vorobev, Boussinesq approximation of the Cahn-Hilliard-Navier-Stokes equations. Phys. Rev. E 85 (2010) 056312. 

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