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, Volume 48 (2014) no. 4, p. 1171-1197

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.

DOI : https://doi.org/10.1051/m2an/2013135
Classification:  82C40,  76P05,  76D05
Keywords: kinetic theory, incompressible Navier-Stokes equations, hydrodynamic limits
@article{M2AN_2014__48_4_1171_0,
author = {Bisi, Marzia and Desvillettes, Laurent},
title = {Formal passage from kinetic theory to incompressible Navier-Stokes equations for a mixture of gases},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {48},
number = {4},
year = {2014},
pages = {1171-1197},
doi = {10.1051/m2an/2013135},
zbl = {1301.82046},
mrnumber = {3264350},
language = {en},
url = {http://www.numdam.org/item/M2AN_2014__48_4_1171_0}
}

Bisi, Marzia; 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, Volume 48 (2014) no. 4, pp. 1171-1197. doi : 10.1051/m2an/2013135. http://www.numdam.org/item/M2AN_2014__48_4_1171_0/

[1] T. Alazard, Low Mach number limit of the full Navier-Stokes equations. Arch. Rational Mech. Anal. 180 (2006) 1-73. | MR 2211706 | Zbl 1108.76061

[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. | MR 2980526 | Zbl 1257.35140

[3] C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations. J. Statis. Phys. 63 (1991) 323-344. | MR 1115587 | Zbl 1151.35066

[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. | MR 1213991 | Zbl 0817.76002

[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. | MR 1806716 | Zbl 0989.82025

[6] B.J. Bayly, D. Levermore and T. Passot, Density variations in weakly compressible flows. Phys. Fluids A 4 (1992) 945-954. | MR 1160287 | Zbl 0756.76061

[7] A. Berti, V. Berti and D. Grandi, Well-posedness of an isothermal diffusive model for binary mixtures of incompressible fluids. Nonlinearity 24 (2011) 3143-3164. | MR 2844832 | Zbl 1269.35029

[8] M. Bisi, M. Groppi and G. Spiga, Fluid-dynamic equations for reacting gas mixtures. Appl. Math. 50 (2005) 43-62. | MR 2117695 | Zbl 1099.82015

[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. | MR 2244535 | Zbl 1121.82032

[10] M. Bisi and L. Desvillettes, From reactive Boltzmann equations to reaction-diffusion systems. J. Statis. Phys. 124 (2006) 881-912. | MR 2264629 | Zbl 1134.82323

[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] L. Boudin, B. Grec, M. Pavic and F. Salvarani, Diffusion asymptotics of a kinetic model for gaseous mixtures. Kinet. Relat. Models 6 (2013) 137-157. | MR 3005625 | Zbl 1260.35100

[13] S. Brull, Habilitation thesis. Univ. Bordeaux (2012).

[14] S. Brull, V. Pavan and J. Schneider, Derivation of BGK models for mixtures. Eur. J. Mech. B-Fluids 33 (2012) 74-86. | MR 2896732 | Zbl 1258.76122

[15] C. Cercignani, The Boltzmann Equation and its Applications. Springer, New York (1988). | MR 1313028 | Zbl 0646.76001

[16] V. Giovangigli, Multicomponent flow modeling, Series on Modeling and Simulation in Science, Engineering and Technology. Birkhaüser, Boston (1999). | MR 1713516 | Zbl 0956.76003

[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. | MR 2025302 | Zbl 1060.76101

[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. | MR 2517786 | Zbl 1178.35290

[19] H. Grad, Asymptotic theory of the Boltzmann equation. Phys. Fluids 6 (1963) 147-181. | MR 155541 | Zbl 0115.45006

[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. | MR 156656 | Zbl 0115.45006

[21] T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag, New York (1966). | MR 203473 | Zbl 0435.47001

[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. | MR 2644440 | Zbl pre05731026

[23] P.L. Lions, N. Masmoudi, Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. 77 (1998) 585-627. | MR 1628173 | Zbl 0909.35101

[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. | MR 1842343 | Zbl 0987.76088

[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. | MR 1650795 | Zbl 0927.76007

[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. | MR 2331340 | Zbl 1291.35126

[27] L. Saint-Raymond, Hydrodynamic limits of the Boltzmann equation. Vol. 1971 of Lect. Notes Math. Springer-Verlag, Berlin (2009). | MR 2683475 | Zbl 1171.82002

[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. | MR 2648335 | Zbl 1229.35163

[29] E.A. Spiegel and G. Veronis, On the Boussinesq approximation for a compressible fluid. Astrophys. J. 131 442-447. | MR 128767

[30] A. Vorobev, Boussinesq approximation of the Cahn-Hilliard-Navier-Stokes equations. Phys. Rev. E 85 (2010) 056312.