Abstract
We consider the non-reactive elastic Boltzmann equation for multicomponent gaseous mixtures. We deduce, under the standard diffusive scaling, that well prepared initial conditions lead to solutions satisfying the Maxwell-Stefan diffusion equations in the vanishing Mach and Knudsen numbers limit.
Similar content being viewed by others
References
Andries, P., Aoki, K., Perthame, B.: A consistent BGK-type model for gas mixtures. J. Stat. Phys. 106(5–6), 993–1018 (2002)
Bardos, C., Golse, F., Levermore, C.D.: Sur les limites asymptotiques de la théorie cinétique conduisant à la dynamique des fluides incompressibles. C. R. Acad. Sci., Sér. 1 Math. 309(11), 727–732 (1989)
Bardos, C., Golse, F., Levermore, C.D.: Fluid dynamic limits of kinetic equations. I. Formal derivations. J. Stat. Phys. 63(1-2), 323–344 (1991)
Bardos, C., Golse, F., Levermore, C.D.: Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation. Commun. Pure Appl. Math. 46(5), 667–753 (1993)
Bisi, M., Desvillettes, L.: Formal passage from kinetic theory to incompressible Navier-Stokes equations for a mixture of gases. Modél. Math. Anal. Numér. (2014, to appear). doi:10.1051/m2an/2013135
Bothe, D.: On the Maxwell-Stefan approach to multicomponent diffusion. In: Parabolic Problems. Progr. Nonlinear Differential Equations Appl., vol. 80, pp. 81–93. Birkhäuser/Springer, Basel (2011)
Boudin, L., Götz, D., Grec, B.: Diffusion models of multicomponent mixtures in the lung. In: CEMRACS 2009: Mathematical modelling in medicine. ESAIM Proc., vol. 30, pp. 90–103. EDP Sciences, Les Ulis (2010)
Boudin, L., Grec, B., Pavić, M., Salvarani, F.: Diffusion asymptotics of a kinetic model for gaseous mixtures. Kinet. Relat. Models 6(1), 137–157 (2013)
Boudin, L., Grec, B., Salvarani, F.: A mathematical and numerical analysis of the Maxwell-Stefan diffusion equations. Discrete Contin. Dyn. Syst., Ser. B 17(5), 1427–1440 (2012)
Brull, S., Pavan, V., Schneider, J.: Derivation of a BGK model for mixtures. Eur. J. Mech. B, Fluids 33, 74–86 (2012)
Chang, H.K.: Multicomponent diffusion in the lung. Fed. Proc. 39(10), 2759–2764 (1980)
Desvillettes, L., Monaco, R., Salvarani, F.: A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions. Eur. J. Mech. B, Fluids 24(2), 219–236 (2005)
Duncan, J.B., Toor, H.L.: An experimental study of three component gas diffusion. AIChE J. 8(1), 38–41 (1962)
Garzó, V., Santos, A., Brey, J.J.: A kinetic model for a multicomponent gas. Phys. Fluids A 1(2), 380–383 (1989)
Giovangigli, V.: Multicomponent flow modeling. In: Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston, Cambridge (1999)
Golse, F., Saint-Raymond, L.: The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math. 155(1), 81–161 (2004)
Golse, F., Saint-Raymond, L.: The incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials. J. Math. Pures Appl. (9) 91(5), 508–552 (2009)
Hilbert, D.: Mathematical problems. Bull. Am. Math. Soc. 8(10), 437–479 (1902)
Jüngel, A., Stelzer, I.V.: Existence analysis of Maxwell-Stefan systems for multicomponent mixtures. SIAM J. Math. Anal. 45(4), 2421–2440 (2013)
Krishna, R., Wesselingh, J.A.: The Maxwell-Stefan approach to mass transfer. Chem. Eng. Sci. 52(6), 861–911 (1997)
Maxwell, J.C.: On the dynamical theory of gases. Philos. Trans. R. Soc. Lond. 157, 49–88 (1866)
Morse, T.F.: Kinetic model equations for a gas mixture. Phys. Fluids 7, 2012–2013 (1964)
Shigesada, N., Kawasaki, K., Teramoto, E.: Spatial segregation of interacting species. J. Theor. Biol. 79(1), 83–99 (1979)
Sirovich, L.: Kinetic modeling of gas mixtures. Phys. Fluids 5, 908–918 (1962)
Stefan, J.: Ueber das Gleichgewicht und die Bewegung insbesondere die Diffusion von Gasgemengen. Akad. Wiss. Wien 63, 63–124 (1871)
Thiriet, M., Douguet, D., Bonnet, J.-C., Canonne, C., Hatzfeld, C.: The effect on gas mixing of a He-O2 mixture in chronic obstructive lung diseases. Bull. Eur. Physiopathol. Respir. 15(5), 1053–1068 (1979). In French
Williams, F.A.: Combustion Theory, 2nd edn. Benjamin-Cummings, Redwood City (1985)
Acknowledgements
This work was partially funded by the ANR-08-JCJC-013-01 project M3RS, headed by Céline Grandmont, and by the ANR-11-TECS-0006 project OxHelease, coordinated by Caroline Majoral. B. Grec and F. Salvarani also want to acknowledge the Reo project-team from Inria Paris-Rocquencourt, for its hospitality which allowed to carry out this article.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Boudin, L., Grec, B. & Salvarani, F. The Maxwell-Stefan Diffusion Limit for a Kinetic Model of Mixtures. Acta Appl Math 136, 79–90 (2015). https://doi.org/10.1007/s10440-014-9886-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10440-014-9886-z