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A mathematical and numerical analysis of the Maxwell-Stefan diffusion equations

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  • We consider the Maxwell-Stefan model of diffusion in a multicomponent gaseous mixture. After focusing on the main differences with the Fickian diffusion model, we study the equations governing a three-component gas mixture. Mostly in the case of a tridiagonal diffusion matrix, we provide a qualitative and quantitative mathematical analysis of the model. We develop moreover a standard explicit numerical scheme and investigate its main properties. We eventually include some numerical simulations underlining the uphill diffusion phenomenon.
    Mathematics Subject Classification: Primary: 35Q35, 35B40; Secondary: 65M06.

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  • [1]

    M. Bebendorf, A note on the Poincaré inequality for convex domains, Z. Anal. Anwendungen, 22 (2003), 751-756.doi: 10.4171/ZAA/1170.

    [2]

    M. Bendahmane, T. Lepoutre, A. Marrocco and B. Perthame, Conservative cross diffusions and pattern formation through relaxation, J. Math. Pures Appl. (9), 92 (2009), 651-667.doi: 10.1016/j.matpur.2009.05.003.

    [3]

    L. Boudin, D. Götz and B. Grec, Diffusion models of multicomponent mixtures in the lung, in "CEMRACS 2009: Mathematical Modelling in Medicine," ESAIM Proc., 30, EDP Sci., Les Ulis, (2010), 90-103.

    [4]

    L. Boudin, B. Grec and F. Salvarani, The Maxwell-Stefan diffusion limit for a kinetic model of mixtures, HAL preprint, submitted, 2011. Available from: http://hal.archives-ouvertes.fr/hal-00554744.

    [5]

    H. K. Chang, Multicomponent diffusion in the lung, Fed. Proc., 39 (1980), 2759-2764.

    [6]

    L. Chen and A. Jüngel, Analysis of a multidimensional parabolic population model with strong cross-diffusion, SIAM J. Math. Anal., 36 (2004), 301-322.

    [7]

    J. Crank, "The Mathematics of Diffusion,'' 2nd edition, Clarendon Press, Oxford, 1975.

    [8]

    H. Darcy, "Les Fontaines Publiques de la Ville de Dijon,'' V. Dalmont, Paris, 1856.

    [9]

    J. B. Duncan and H. L. Toor, An experimental study of three component gas diffusion, AIChE Journal, 8 (1962), 38-41.

    [10]

    A. Ern and V. Giovangigli, "Multicomponent Transport Algorithms,'' Lecture Notes in Physics, New Series m: Monographs, 24, Springer-Verlag, Berlin, 1994.

    [11]

    A. Ern and V. Giovangigli, Projected iterative algorithms with application to multicomponent transport, Linear Algebra Appl., 250 (1997), 289-315.doi: 10.1016/0024-3795(95)00502-1.

    [12]

    L. C. Evans, "Partial Differential Equations,'' 2nd edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010.

    [13]

    A. Fick, On liquid diffusion, Phil. Mag., 10 (1855), 30-39.

    [14]

    A. Fick, Über Diffusion, Poggendorff's Annalen der Physik und Chemie, 94 (1855), 59-86.doi: 10.1002/andp.18551700105.

    [15]

    V. Giovangigli, Convergent iterative methods for multicomponent diffusion, Impact Comput. Sci. Engrg., 3 (1991), 244-276.doi: 10.1016/0899-8248(91)90010-R.

    [16]

    V. Giovangigli, "Multicomponent Flow Modeling,'' Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 1999.doi: 10.1007/978-1-4612-1580-6.

    [17]

    R. Krishna and J. A. Wesselingh, The Maxwell-Stefan approach to mass transfer, Chem. Eng. Sci., 52 (1997), 861-911.doi: 10.1016/S0009-2509(96)00458-7.

    [18]

    O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,'' Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1967.

    [19]

    Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.doi: 10.1006/jdeq.1996.0157.

    [20]

    J. C. Maxwell, On the dynamical theory of gases, Phil. Trans. R. Soc., 157 (1866), 49-88.doi: 10.1098/rstl.1867.0004.

    [21]

    L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal., 5 (1960), 286-292.doi: 10.1007/BF00252910.

    [22]

    J. Stefan, Über das Gleichgewicht und die Bewegung insbesondere die Diffusion von Gasgemengen, Akad. Wiss. Wien, 63 (1871), 63-124.

    [23]

    M. Thiriet, D. Douguet, J.-C. Bonnet, C. Canonne and C. Hatzfeld, The effect on gas mixing of a He-$\mboxO_2$ mixture in chronic obstructive lung diseases, Bull. Eur. Physiopathol. Respir., 15 (1979), 1053-1068.

    [24]

    J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,'' Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, Oxford, 2007.

    [25]

    F. A. Williams, "Combustion Theory,'' 2nd edition, Benjamin Cummings, 1985.

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