American Institute of Mathematical Sciences

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March  2013, 6(1): 137-157. doi: 10.3934/krm.2013.6.137

Diffusion asymptotics of a kinetic model for gaseous mixtures

Laurent Boudin 1, , Bérénice Grec 2, , Milana Pavić 3, and  Francesco Salvarani 4,

1. 

UPMC Univ Paris 06, UMR 7598 LJLL, Paris, F-75005
2. 

MAP5, CNRS UMR 8145, Université Paris Descartes, Sorbonne Paris Cité, 45 Rue des Saints Pères, F-75006 Paris, France
3. 

CMLA, ENS Cachan, PRES UniverSud Paris, 61 Avenue du Président Wilson, F-94235 Cachan Cedex, France
4. 

Dipartimento di Matematica, Università degli Studi di Pavia, Via Ferrata 1 - 27100 Pavia

Received  July 2012 Revised  October 2012 Published  December 2012

In this work, we consider the non-reactive fully elastic Boltzmann equations for mixtures in the diffusive scaling. We mainly use a Hilbert expansion of the distribution functions. After briefly recalling the H-theorem, the lower-order non trivial equality obtained from the Boltzmann equations leads to a linear functional equation in the velocity variable. This equation is solved thanks to the Fredholm alternative. Since we consider multicomponent mixtures, the classical techniques introduced by Grad cannot be applied, and we propose a new method to treat the terms involving particles with different masses.
Keywords: diffusive scaling, Boltzmann equations, Fredholm's alternative, Chapman-Enskog expansion, multispecies mixture..
Mathematics Subject Classification: Primary: 35Q20, 45B05; Secondary: 35Q35, 82C4.
Citation: 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
References:
[1]

C. Bardos, F. Golse and C. D. Levermore, Sur les limites asymptotiques de la théorie cinétique conduisant à la dynamique des fluides incompressibles, C. R. Acad. Sci. Paris Sér. I Math., 309 (1989), 727-732.  Google Scholar

[2]

C. Bardos, F. Golse and C. D. Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations, J. Statist. Phys., 63 (1991), 323-344.  Google Scholar

[3]

C. Bardos, F. Golse and C. D. Levermore, Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math., 46 (1993), 667-753.  Google Scholar

[4]

S. Bastea, R. Esposito, J. L. Lebowitz and R. Marra, Binary fluids with long range segregating interaction. I. Derivation of kinetic and hydrodynamic equations, J. Statist. Phys., 101 (2000), 1087-1136.  Google Scholar

[5]

M. Bennoune, M. Lemou and L. Mieussens, An asymptotic preserving scheme for the Kac model of the Boltzmann equation in the diffusion limit, Contin. Mech. Thermodyn., 21 (2009), 401-421.  Google Scholar

[6]

M. Bisi and L. Desvillettes, Incompressible Navier-Stokes equations from kinetic models for a mixture of rarefied gases,, Work in progress., ().   Google Scholar

[7]

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

[8]

J.-F. Bourgat, L. Desvillettes, P. Le Tallec and B. Perthame, Microreversible collisions for polyatomic gases and Boltzmann's theorem, European J. Mech. B Fluids, 13 (1994), 237-254.  Google Scholar

[9]

C. Cercignani, "The Boltzmann Equation and Its Applications," volume 67 of Applied Mathematical Sciences. Springer-Verlag, New York, 1988.  Google Scholar

[10]

S. Chapman and T. G. Cowling, "The mathematical Theory of Nonuniform Gases," Cambridge Mathematical Library. Cambridge University Press, Cambridge, third edition, 1990. An account of the kinetic theory of viscosity, thermal conduction and diffusion in gases, In co-operation with D. Burnett, With a foreword by Carlo Cercignani.  Google Scholar

[11]

P. Degond, Macroscopic limits of the Boltzmann equation: A review, in "Modeling and Computational Methods for Kinetic Equations," Model. Simul. Sci. Eng. Technol., 3-57. Birkhäuser Boston, Boston, MA, (2004).  Google Scholar

[12]

L. Desvillettes, R. Monaco and F. Salvarani, A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions, Eur. J. Mech. B Fluids, 24 (2005), 219-236.  Google Scholar

[13]

C. Dogbe, Fluid dynamic limits for gas mixture. I. Formal derivations, Math. Models Methods Appl. Sci., 18 (2008), 1633-1672.  Google Scholar

[14]

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

[15]

F. Golse and L. Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math., 155 (2004), 81-161.  Google Scholar

[16]

F. Golse and L. Saint-Raymond, The incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials, J. Math. Pures Appl. (9), 91 (2009), 508-552.  Google Scholar

[17]

H. Grad, Asymptotic theory of the Boltzmann equation, Phys. Fluids, 6 (1963), 147-181.  Google Scholar

[18]

H. Grad, Asymptotic theory of the Boltzmann equation. II, in "Rarefied Gas Dynamics (Proc. 3rd Internat. Sympos., Palais de l'UNESCO, Paris, 1962" I, 26-59. Academic Press, New York, (1963).  Google Scholar

[19]

D. E. Greene, Mathematical aspects of kinetic model equations for binary gas mixtures, J. Mathematical Phys., 16 (1975), 776-782.  Google Scholar

[20]

D. Hilbert, Mathematical problems, Bull. Amer. Math. Soc., 8 (1902), 437-479.  Google Scholar

[21]

F. Huang, Y. Wang and T. Yang, Fluid dynamic limit to the Riemann solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity, Kinet. Relat. Models, 3 (2010), 685-728.  Google Scholar

[22]

F. Huang, Y. Wang and T. Yang, Hydrodynamic limit of the Boltzmann equation with contact discontinuities, Comm. Math. Phys., 295 (2010), 293-326.  Google Scholar

[23]

S. Jin and Q. Li, A BGK-penalization asymptotic-preserving scheme for the multispecies Boltzmann equation, Technical report, 2012. Submitted. Google Scholar

[24]

R. Krishna and J. A. Wesselingh, The Maxwell-Stefan approach to mass transfer, Chem. Eng. Sci., 52 (1997), 861-911. Google Scholar

[25]

P.-L. Lions and N. Masmoudi, From the Boltzmann equations to the equations of incompressible fluid mechanics. I, II, Arch. Ration. Mech. Anal., 158 (2001), 173-193, 195-211.  Google Scholar

[26]

J. C. Maxwell, On the dynamical theory of gases, Phil. Trans. R. Soc., 157 (1866), 49-88. Google Scholar

[27]

T. F. Morse, Kinetic model equations for a gas mixture, Phys. Fluids, 7 (1964), 2012-2013.  Google Scholar

[28]

J. Ross and P. Mazur, Some deductions from a formal statistical mechanical theory of chemical kinetics, J. Chem. Phys., 35 (1961), 19-28. Google Scholar

[29]

A. Rossani and G. Spiga, A note on the kinetic theory of chemically reacting gases, Phys. A, 272 (1999), 563-573.  Google Scholar

[30]

L. Sirovich, Kinetic modeling of gas mixtures, Phys. Fluids, 5 (1962), 908-918.  Google Scholar

[31]

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

[32]

S.-H. Yu, Hydrodynamic limits with shock waves of the Boltzmann equation, Comm. Pure Appl. Math., 58 (2005), 409-443.  Google Scholar

show all references

References:
[1]

C. Bardos, F. Golse and C. D. Levermore, Sur les limites asymptotiques de la théorie cinétique conduisant à la dynamique des fluides incompressibles, C. R. Acad. Sci. Paris Sér. I Math., 309 (1989), 727-732.  Google Scholar

[2]

C. Bardos, F. Golse and C. D. Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations, J. Statist. Phys., 63 (1991), 323-344.  Google Scholar

[3]

C. Bardos, F. Golse and C. D. Levermore, Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math., 46 (1993), 667-753.  Google Scholar

[4]

S. Bastea, R. Esposito, J. L. Lebowitz and R. Marra, Binary fluids with long range segregating interaction. I. Derivation of kinetic and hydrodynamic equations, J. Statist. Phys., 101 (2000), 1087-1136.  Google Scholar

[5]

M. Bennoune, M. Lemou and L. Mieussens, An asymptotic preserving scheme for the Kac model of the Boltzmann equation in the diffusion limit, Contin. Mech. Thermodyn., 21 (2009), 401-421.  Google Scholar

[6]

M. Bisi and L. Desvillettes, Incompressible Navier-Stokes equations from kinetic models for a mixture of rarefied gases,, Work in progress., ().   Google Scholar

[7]

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

[8]

J.-F. Bourgat, L. Desvillettes, P. Le Tallec and B. Perthame, Microreversible collisions for polyatomic gases and Boltzmann's theorem, European J. Mech. B Fluids, 13 (1994), 237-254.  Google Scholar

[9]

C. Cercignani, "The Boltzmann Equation and Its Applications," volume 67 of Applied Mathematical Sciences. Springer-Verlag, New York, 1988.  Google Scholar

[10]

S. Chapman and T. G. Cowling, "The mathematical Theory of Nonuniform Gases," Cambridge Mathematical Library. Cambridge University Press, Cambridge, third edition, 1990. An account of the kinetic theory of viscosity, thermal conduction and diffusion in gases, In co-operation with D. Burnett, With a foreword by Carlo Cercignani.  Google Scholar

[11]

P. Degond, Macroscopic limits of the Boltzmann equation: A review, in "Modeling and Computational Methods for Kinetic Equations," Model. Simul. Sci. Eng. Technol., 3-57. Birkhäuser Boston, Boston, MA, (2004).  Google Scholar

[12]

L. Desvillettes, R. Monaco and F. Salvarani, A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions, Eur. J. Mech. B Fluids, 24 (2005), 219-236.  Google Scholar

[13]

C. Dogbe, Fluid dynamic limits for gas mixture. I. Formal derivations, Math. Models Methods Appl. Sci., 18 (2008), 1633-1672.  Google Scholar

[14]

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

[15]

F. Golse and L. Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math., 155 (2004), 81-161.  Google Scholar

[16]

F. Golse and L. Saint-Raymond, The incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials, J. Math. Pures Appl. (9), 91 (2009), 508-552.  Google Scholar

[17]

H. Grad, Asymptotic theory of the Boltzmann equation, Phys. Fluids, 6 (1963), 147-181.  Google Scholar

[18]

H. Grad, Asymptotic theory of the Boltzmann equation. II, in "Rarefied Gas Dynamics (Proc. 3rd Internat. Sympos., Palais de l'UNESCO, Paris, 1962" I, 26-59. Academic Press, New York, (1963).  Google Scholar

[19]

D. E. Greene, Mathematical aspects of kinetic model equations for binary gas mixtures, J. Mathematical Phys., 16 (1975), 776-782.  Google Scholar

[20]

D. Hilbert, Mathematical problems, Bull. Amer. Math. Soc., 8 (1902), 437-479.  Google Scholar

[21]

F. Huang, Y. Wang and T. Yang, Fluid dynamic limit to the Riemann solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity, Kinet. Relat. Models, 3 (2010), 685-728.  Google Scholar

[22]

F. Huang, Y. Wang and T. Yang, Hydrodynamic limit of the Boltzmann equation with contact discontinuities, Comm. Math. Phys., 295 (2010), 293-326.  Google Scholar

[23]

S. Jin and Q. Li, A BGK-penalization asymptotic-preserving scheme for the multispecies Boltzmann equation, Technical report, 2012. Submitted. Google Scholar

[24]

R. Krishna and J. A. Wesselingh, The Maxwell-Stefan approach to mass transfer, Chem. Eng. Sci., 52 (1997), 861-911. Google Scholar

[25]

P.-L. Lions and N. Masmoudi, From the Boltzmann equations to the equations of incompressible fluid mechanics. I, II, Arch. Ration. Mech. Anal., 158 (2001), 173-193, 195-211.  Google Scholar

[26]

J. C. Maxwell, On the dynamical theory of gases, Phil. Trans. R. Soc., 157 (1866), 49-88. Google Scholar

[27]

T. F. Morse, Kinetic model equations for a gas mixture, Phys. Fluids, 7 (1964), 2012-2013.  Google Scholar

[28]

J. Ross and P. Mazur, Some deductions from a formal statistical mechanical theory of chemical kinetics, J. Chem. Phys., 35 (1961), 19-28. Google Scholar

[29]

A. Rossani and G. Spiga, A note on the kinetic theory of chemically reacting gases, Phys. A, 272 (1999), 563-573.  Google Scholar

[30]

L. Sirovich, Kinetic modeling of gas mixtures, Phys. Fluids, 5 (1962), 908-918.  Google Scholar

[31]

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

[32]

S.-H. Yu, Hydrodynamic limits with shock waves of the Boltzmann equation, Comm. Pure Appl. Math., 58 (2005), 409-443.  Google Scholar

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