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.
Mots-clés : 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 }, pages = {1171--1197}, publisher = {EDP-Sciences}, volume = {48}, number = {4}, year = {2014}, doi = {10.1051/m2an/2013135}, mrnumber = {3264350}, zbl = {1301.82046}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2013135/} }
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 PY - 2014 SP - 1171 EP - 1197 VL - 48 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2013135/ DO - 10.1051/m2an/2013135 LA - en ID - M2AN_2014__48_4_1171_0 ER -
%0 Journal Article %A Bisi, Marzia %A Desvillettes, Laurent %T Formal passage from kinetic theory to incompressible Navier-Stokes equations for a mixture of gases %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 1171-1197 %V 48 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2013135/ %R 10.1051/m2an/2013135 %G en %F 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 , Tome 48 (2014) no. 4, pp. 1171-1197. doi : 10.1051/m2an/2013135. http://www.numdam.org/articles/10.1051/m2an/2013135/
[1] Low Mach number limit of the full Navier-Stokes equations. Arch. Rational Mech. Anal. 180 (2006) 1-73. | MR | Zbl
,[2] 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] Fluid dynamic limits of kinetic equations. I. Formal derivations. J. Statis. Phys. 63 (1991) 323-344. | MR | Zbl
, and ,[4] Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation. Commun. Pure Appl. Math. 46 (1993) 667-753. | MR | Zbl
, and ,[5] Binary fluids with long range segregating interaction. I. Derivation of kinetic and hydrodynamic equations. J. Statis. Phys. 101 (2000) 1087-1136. | MR | Zbl
, , and ,[6] Density variations in weakly compressible flows. Phys. Fluids A 4 (1992) 945-954. | MR | Zbl
, and ,[7] Well-posedness of an isothermal diffusive model for binary mixtures of incompressible fluids. Nonlinearity 24 (2011) 3143-3164. | MR | Zbl
, and ,[8] Fluid-dynamic equations for reacting gas mixtures. Appl. Math. 50 (2005) 43-62. | MR | Zbl
, and ,[9] 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
, and ,[10] From reactive Boltzmann equations to reaction-diffusion systems. J. Statis. Phys. 124 (2006) 881-912. | MR | Zbl
and ,[11] Multi-temperature Euler hydrodynamics for a reacting gas from a kinetic approach to rarefied mixtures with resonant collisions. Europhys. Lett. 95 (2011), 55002.
, and ,[12] Diffusion asymptotics of a kinetic model for gaseous mixtures. Kinet. Relat. Models 6 (2013) 137-157. | MR | Zbl
, , and ,[13] Habilitation thesis. Univ. Bordeaux (2012).
,[14] Derivation of BGK models for mixtures. Eur. J. Mech. B-Fluids 33 (2012) 74-86. | MR | Zbl
, and ,[15] The Boltzmann Equation and its Applications. Springer, New York (1988). | MR | Zbl
,[16] Multicomponent flow modeling, Series on Modeling and Simulation in Science, Engineering and Technology. Birkhaüser, Boston (1999). | MR | Zbl
,[17] The Navier−Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math. 155 (2004) 81-161. | MR | Zbl
and ,[18] The incompressible Navier−Stokes limit of the Boltzmann equation for hard cutoff potentials. J. Math. Pures Appl. 91 (2009) 508-552. | MR | Zbl
and ,[19] Asymptotic theory of the Boltzmann equation. Phys. Fluids 6 (1963) 147-181. | MR | Zbl
,[20] 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] Perturbation Theory for Linear Operators. Springer-Verlag, New York (1966). | MR | Zbl
,[22] From the Boltzmann equation to an incompressible Navier-Stokes-Fourier system. Arch. Rational Mech. Anal. 196 (2010) 753-809. | MR
and ,[23] Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. 77 (1998) 585-627. | MR | Zbl
, ,[24] From the Boltzmann equations to the equations of incompressible fluid mechanics II. Arch. Rational Mech. Anal. 158 (2001) 195-211. | MR | Zbl
and ,[25] Quasi-incompressible Cahn-Hilliard fluids and topological transitions. Proc. R. Soc. A. Math. Phys. Eng. Sci. 454 (1998) 2617-2654. | MR | Zbl
and ,[26] 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] Hydrodynamic limits of the Boltzmann equation. Vol. 1971 of Lect. Notes Math. Springer-Verlag, Berlin (2009). | MR | Zbl
,[28] 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] On the Boussinesq approximation for a compressible fluid. Astrophys. J. 131 442-447. | MR
and ,[30] Boussinesq approximation of the Cahn-Hilliard-Navier-Stokes equations. Phys. Rev. E 85 (2010) 056312.
,Cité par Sources :