American Institute of Mathematical Sciences

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

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

Laurent Boudin 1, , Bérénice Grec 2, and  Francesco Salvarani 2,

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.
Keywords: Maxwell-Stefan's equations, Diffusion in a gaseous mixture, uphill diffusion..
Mathematics Subject Classification: Primary: 35Q35, 35B40; Secondary: 65M0.
Citation: Laurent Boudin, Bérénice Grec, Francesco Salvarani. A mathematical and numerical analysis of the Maxwell-Stefan diffusion equations. Discrete & 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.  doi: 10.4171/ZAA/1170.  Google Scholar

[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.  doi: 10.1016/j.matpur.2009.05.003.  Google Scholar

[3]

L. Boudin, D. Götz and B. Grec, Diffusion models of multicomponent mixtures in the lung,, in, 30 (2010), 90.   Google Scholar

[4]

L. Boudin, B. Grec and F. Salvarani, The Maxwell-Stefan diffusion limit for a kinetic model of mixtures,, HAL preprint, (2011).   Google Scholar

[5]

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

[6]

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

[7]

J. Crank, "The Mathematics of Diffusion,'', 2nd edition, (1975).   Google Scholar

[8]

H. Darcy, "Les Fontaines Publiques de la Ville de Dijon,'', V. Dalmont, (1856).   Google Scholar

[9]

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

[10]

A. Ern and V. Giovangigli, "Multicomponent Transport Algorithms,'', Lecture Notes in Physics, 24 (1994).   Google Scholar

[11]

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

[12]

L. C. Evans, "Partial Differential Equations,'', 2nd edition, 19 (2010).   Google Scholar

[13]

A. Fick, On liquid diffusion,, Phil. Mag., 10 (1855), 30.   Google Scholar

[14]

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

[15]

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

[16]

V. Giovangigli, "Multicomponent Flow Modeling,'', Modeling and Simulation in Science, (1999).  doi: 10.1007/978-1-4612-1580-6.  Google Scholar

[17]

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

[18]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,'', Translations of Mathematical Monographs, (1967).   Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

[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.   Google Scholar

[24]

J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,'', Oxford Mathematical Monographs, (2007).   Google Scholar

[25]

F. A. Williams, "Combustion Theory,'', 2nd edition, (1985).   Google Scholar

show all references

References:
[1]

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

[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.  doi: 10.1016/j.matpur.2009.05.003.  Google Scholar

[3]

L. Boudin, D. Götz and B. Grec, Diffusion models of multicomponent mixtures in the lung,, in, 30 (2010), 90.   Google Scholar

[4]

L. Boudin, B. Grec and F. Salvarani, The Maxwell-Stefan diffusion limit for a kinetic model of mixtures,, HAL preprint, (2011).   Google Scholar

[5]

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

[6]

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

[7]

J. Crank, "The Mathematics of Diffusion,'', 2nd edition, (1975).   Google Scholar

[8]

H. Darcy, "Les Fontaines Publiques de la Ville de Dijon,'', V. Dalmont, (1856).   Google Scholar

[9]

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

[10]

A. Ern and V. Giovangigli, "Multicomponent Transport Algorithms,'', Lecture Notes in Physics, 24 (1994).   Google Scholar

[11]

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

[12]

L. C. Evans, "Partial Differential Equations,'', 2nd edition, 19 (2010).   Google Scholar

[13]

A. Fick, On liquid diffusion,, Phil. Mag., 10 (1855), 30.   Google Scholar

[14]

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

[15]

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

[16]

V. Giovangigli, "Multicomponent Flow Modeling,'', Modeling and Simulation in Science, (1999).  doi: 10.1007/978-1-4612-1580-6.  Google Scholar

[17]

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

[18]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,'', Translations of Mathematical Monographs, (1967).   Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

[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.   Google Scholar

[24]

J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,'', Oxford Mathematical Monographs, (2007).   Google Scholar

[25]

F. A. Williams, "Combustion Theory,'', 2nd edition, (1985).   Google Scholar

[1]

B. Anwasia, M. Bisi, F. Salvarani, A. J. Soares. On the Maxwell-Stefan diffusion limit for a reactive mixture of polyatomic gases in non-isothermal setting. Kinetic & Related Models, 2020, 13 (1) : 63-95. doi: 10.3934/krm.2020003
[2]

Laurent Boudin, Bérénice Grec, Milana Pavić, Francesco Salvarani. Diffusion asymptotics of a kinetic model for gaseous mixtures. Kinetic & Related Models, 2013, 6 (1) : 137-157. doi: 10.3934/krm.2013.6.137
[3]

W. Wei, H. M. Yin. Global solvability for a singular nonlinear Maxwell's equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 431-444. doi: 10.3934/cpaa.2005.4.431
[4]

Björn Birnir, Niklas Wellander. Homogenized Maxwell's equations; A model for ceramic varistors. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 257-272. doi: 10.3934/dcdsb.2006.6.257
[5]

L. Chupin. Existence result for a mixture of non Newtonian flows with stress diffusion using the Cahn-Hilliard formulation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 45-68. doi: 10.3934/dcdsb.2003.3.45
[6]

M. Eller. On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 473-481. doi: 10.3934/dcdss.2009.2.473
[7]

Oleg Yu. Imanuvilov, Masahiro Yamamoto. Calderón problem for Maxwell's equations in cylindrical domain. Inverse Problems & Imaging, 2014, 8 (4) : 1117-1137. doi: 10.3934/ipi.2014.8.1117
[8]

B. L. G. Jonsson. Wave splitting of Maxwell's equations with anisotropic heterogeneous constitutive relations. Inverse Problems & Imaging, 2009, 3 (3) : 405-452. doi: 10.3934/ipi.2009.3.405
[9]

Andreas Kirsch. An integral equation approach and the interior transmission problem for Maxwell's equations. Inverse Problems & Imaging, 2007, 1 (1) : 159-179. doi: 10.3934/ipi.2007.1.159
[10]

Cleverson R. da Luz, Gustavo Alberto Perla Menzala. Uniform stabilization of anisotropic Maxwell's equations with boundary dissipation. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 547-558. doi: 10.3934/dcdss.2009.2.547
[11]

Gang Bao, Bin Hu, Peijun Li, Jue Wang. Analysis of time-domain Maxwell's equations in biperiodic structures. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 259-286. doi: 10.3934/dcdsb.2019181
[12]

Wafa Hamrouni, Ali Abdennadher. Random walk's models for fractional diffusion equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2509-2530. doi: 10.3934/dcdsb.2016058
[13]

Massimiliano Berti. Some remarks on a variational approach to Arnold's diffusion. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 307-314. doi: 10.3934/dcds.1996.2.307
[14]

Dirk Pauly. On Maxwell's and Poincaré's constants. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 607-618. doi: 10.3934/dcdss.2015.8.607
[15]

Kim Dang Phung. Energy decay for Maxwell's equations with Ohm's law in partially cubic domains. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2229-2266. doi: 10.3934/cpaa.2013.12.2229
[16]

Angelo Morro. Nonlinear diffusion equations in fluid mixtures. Evolution Equations & Control Theory, 2016, 5 (3) : 431-448. doi: 10.3934/eect.2016012
[17]

Yuming Paul Zhang. On a class of diffusion-aggregation equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 907-932. doi: 10.3934/dcds.2020066
[18]

Mohammad Asadzadeh, Piotr Kowalczyk, Christoffer Standar. On hp-streamline diffusion and Nitsche schemes for the relativistic Vlasov-Maxwell system. Kinetic & Related Models, 2019, 12 (1) : 105-131. doi: 10.3934/krm.2019005
[19]

J. J. Morgan, Hong-Ming Yin. On Maxwell's system with a thermal effect. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 485-494. doi: 10.3934/dcdsb.2001.1.485
[20]

S. S. Krigman. Exact boundary controllability of Maxwell's equations with weak conductivity in the heterogeneous medium inside a general domain. Conference Publications, 2007, 2007 (Special) : 590-601. doi: 10.3934/proc.2007.2007.590

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (19)
  • HTML views (0)
  • Cited by (22)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences