July  2012, 17(5): 1427-1440. doi: 10.3934/dcdsb.2012.17.1427

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

1. 

UPMC Univ Paris 06, UMR 7598 LJLL, Paris, F-75005

2. 

INRIA Paris-Rocquencourt, REO Project team, BP 105, 78153 Le Chesnay, France, France

Received  June 2010 Revised  June 2011 Published  March 2012

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.
Citation: Laurent Boudin, Bérénice Grec, Francesco Salvarani. A mathematical and numerical analysis of the Maxwell-Stefan diffusion equations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1427-1440. doi: 10.3934/dcdsb.2012.17.1427
References:
[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.

show all references

References:
[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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