Formal passage from kinetic theory to incompressible Navier-Stokes equations for a mixture of gases
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 4, pp. 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 : 10.1051/m2an/2013135
Classification : 82C40, 76P05, 76D05
Mots clés : kinetic theory, incompressible Navier-Stokes equations, hydrodynamic limits
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     title = {Formal passage from kinetic theory to incompressible {Navier-Stokes} equations for a mixture of gases},
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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 , Tome 48 (2014) no. 4, pp. 1171-1197. doi : 10.1051/m2an/2013135. http://archive.numdam.org/articles/10.1051/m2an/2013135/

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

[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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

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