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

Published online by Cambridge University Press:  15 July 2014

Marzia Bisi  and
Laurent Desvillettes
Marzia Bisi
Affiliation:
Dip. di Matematica e Informatica, Università di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy. marzia.bisi@unipr.it
Laurent Desvillettes
Affiliation:
CMLA, ENS Cachan, CNRS, 61, Av. du Pdt Wilson, 94235 Cachan cedex, France; desville@cmla.ens-cachan.fr
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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.

Keywords

Kinetic theoryincompressible Navier–Stokes equationshydrodynamic limits

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Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 4 , July 2014 , pp. 1171 - 1197
Copyright
© EDP Sciences, SMAI, 2014

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References

Alazard, T., Low Mach number limit of the full Navier–Stokes equations. Arch. Rational Mech. Anal. 180 (2006) 1–73. Google Scholar
Arsenio, D., From Boltzmann’s equation to the incompressible Navier−Stokes–Fourier system with long-range interactions. Arch. Ration. Mech. Anal. 206 (2012) 367–488. Google Scholar
Bardos, C., Golse, F. and Levermore, D., Fluid dynamic limits of kinetic equations. I. Formal derivations. J. Statis. Phys. 63 (1991) 323–344. Google Scholar
Bardos, C., Golse, F. and Levermore, D., Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation. Commun. Pure Appl. Math. 46 (1993) 667–753. Google Scholar
Bastea, S., Esposito, R., Lebowitz, J.L. and Marra, R., Binary fluids with long range segregating interaction. I. Derivation of kinetic and hydrodynamic equations. J. Statis. Phys. 101 (2000) 1087–1136. Google Scholar
Bayly, B.J., Levermore, D. and Passot, T., Density variations in weakly compressible flows. Phys. Fluids A 4 (1992) 945–954. Google Scholar
Berti, A., Berti, V. and Grandi, D., Well–posedness of an isothermal diffusive model for binary mixtures of incompressible fluids. Nonlinearity 24 (2011) 3143–3164. Google Scholar
Bisi, M., Groppi, M. and Spiga, G., Fluid-dynamic equations for reacting gas mixtures. Appl. Math. 50 (2005) 43–62. Google Scholar
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.
Bisi, M. and Desvillettes, L., From reactive Boltzmann equations to reaction-diffusion systems. J. Statis. Phys. 124 (2006) 881–912. Google Scholar
Bisi, M., Martalò, G. and Spiga, G., Multi-temperature Euler hydrodynamics for a reacting gas from a kinetic approach to rarefied mixtures with resonant collisions. Europhys. Lett. 95 (2011), 55002. Google Scholar
Boudin, L., Grec, B., Pavic, M. and Salvarani, F., Diffusion asymptotics of a kinetic model for gaseous mixtures. Kinet. Relat. Models 6 (2013) 137–157. Google Scholar
S. Brull, Habilitation thesis. Univ. Bordeaux (2012).
Brull, S., Pavan, V. and Schneider, J., Derivation of BGK models for mixtures. Eur. J. Mech. B-Fluids 33 (2012) 74–86. Google Scholar
C. Cercignani, The Boltzmann Equation and its Applications. Springer, New York (1988).
V. Giovangigli, Multicomponent flow modeling, Series on Modeling and Simulation in Science, Engineering and Technology. Birkhaüser, Boston (1999).
Golse, F. and Saint-Raymond, L., The Navier−Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math. 155 (2004) 81–161. Google Scholar
Golse, F. and Saint-Raymond, L., The incompressible Navier−Stokes limit of the Boltzmann equation for hard cutoff potentials. J. Math. Pures Appl. 91 (2009) 508–552. Google Scholar
Grad, H., Asymptotic theory of the Boltzmann equation. Phys. Fluids 6 (1963) 147–181. Google Scholar
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.
T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag, New York (1966).
Levermore, D. and Masmoudi, N., From the Boltzmann equation to an incompressible Navier-Stokes-Fourier system. Arch. Rational Mech. Anal. 196 (2010) 753–809. Google Scholar
Lions, P.L., Masmoudi, N., Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. 77 (1998) 585–627. Google Scholar
Lions, P.L. and Masmoudi, N., From the Boltzmann equations to the equations of incompressible fluid mechanics II. Arch. Rational Mech. Anal. 158 (2001) 195–211. Google Scholar
Lowengrub, J. and Truskinovsky, L., Quasi-incompressible Cahn–Hilliard fluids and topological transitions. Proc. R. Soc. A. Math. Phys. Eng. Sci. 454 (1998) 2617–2654. Google Scholar
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.
L. Saint-Raymond, Hydrodynamic limits of the Boltzmann equation. Vol. 1971 of Lect. Notes Math. Springer-Verlag, Berlin (2009).
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.
Spiegel, E.A. and Veronis, G., On the Boussinesq approximation for a compressible fluid. Astrophys. J. 131 442–447.
Vorobev, A., Boussinesq approximation of the Cahn-Hilliard-Navier-Stokes equations. Phys. Rev. E 85 (2010) 056312. Google Scholar