February  2020, 13(1): 63-95. doi: 10.3934/krm.2020003

On the Maxwell-Stefan diffusion limit for a reactive mixture of polyatomic gases in non-isothermal setting

1. 

Universidade do Minho, Centro de Matemática, Campus de Gualtar, 4710-057 Braga, Portugal

2. 

Università di Parma, Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Parco Area delle Scienze 53/A, 43124 Parma, Italy

3. 

Université Paris-Dauphine, PSL Research University, Ceremade, UMR CNRS, 75775 Paris, France

4. 

Università degli Studi di Pavia, Dipartimento di Matematica, 27100 Pavia, Italy

*Corresponding author: A. J. Soares

Received  February 2019 Revised  July 2019 Published  December 2019

In this article we deduce a mathematical model of Maxwell-Stefan type for a reactive mixture of polyatomic gases with a continuous structure of internal energy. The equations of the model are derived in the diffusive limit of a kinetic system of Boltzmann equations for the considered mixture, in the general non-isothermal setting. The asymptotic analysis of the kinetic system is performed under a reactive-diffusive scaling for which mechanical collisions are dominant with respect to chemical reactions. The resulting system couples the Maxwell-Stefan equations for the diffusive fluxes with the evolution equations for the number densities of the chemical species and the evolution equation for the temperature of the mixture. The production terms due to the chemical reaction and the Maxwell-Stefan diffusion coefficients are moreover obtained in terms of general collision kernels and parameters of the kinetic model.

Citation: 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
References:
[1]

B. AnwasiaP. Gonçcalves and A. J. Soares, From the simple reacting sphere kinetic model to the reaction-diffusion system of Maxwell-Stefan type, Communications in Mathematical Sciences, 17 (2019), 507-538.  doi: 10.4310/CMS.2019.v17.n2.a9.  Google Scholar

[2]

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C. BardosF. Golse and C. D. Levermore, Fluid dynamic limits of kinetic equations. Ⅱ. Convergence proofs for the Boltzmann equation, Commun. Pure Appl. Math, 46 (1993), 667-753.  doi: 10.1002/cpa.3160460503.  Google Scholar

[4]

M. Bisi and L. Desvillettes, From reactive Boltzmann equations to reaction-diffusion systems, J. Stat. Phys., 124 (2006), 881-912.  doi: 10.1007/s10955-005-8075-x.  Google Scholar

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M. Bisi and M. J. Cáceres, A BGK relaxation model for polyatomic gas mixtures, Communications in Mathematical Sciences, 14 (2016), 297-325.  doi: 10.4310/CMS.2016.v14.n2.a1.  Google Scholar

[6]

M. Bisi, R. Monaco and A. J. Soares, A BGK model for reactive mixtures of polyatomic gases with continuous internal energy, J. Phys. A, 51 (2018), Art. ID 125501, 29 pp. doi: 10.1088/1751-8121/aaac8e.  Google Scholar

[7]

M. BisiT. Ruggeri and G. Spiga, Dynamical pressure in a polyatomic gas: Interplay between kinetic theory and extended thermodynamics, Kinet. Relat. Models, 11 (2018), 71-95.  doi: 10.3934/krm.2018004.  Google Scholar

[8]

A. BondesanL. Boudin and B. Grec, A numerical scheme for a kinetic model for mixtures in the diffusive limit using the moment methods, Numer. Methods Partial Differential Equations, 35 (2019), 1184-1205.  doi: 10.1002/num.22345.  Google Scholar

[9]

C. Borgnakke and P. S. Larsen, Statistical collision model for Monte–Carlo simulation of polyatomic gas mixtures, J. Comput. Phys., 18 (1975), 405-420.  doi: 10.1016/0021-9991(75)90094-7.  Google Scholar

[10]

D. Bothe, On the Maxwell-Stefan approach to multicomponent diffusion, in Parabolic problems, Progr. Nonlinear Differential Equations Appl., 80 (2011), 81–93, Birkhäuser, Springer Basel AG. doi: 10.1007/978-3-0348-0075-4_5.  Google Scholar

[11]

L. BoudinB. Grec and V. Pavan, The Maxwell-Stefan diffusion limit for a kinetic model of mixtures with general cross sections, Nonlinear Analysis, 159 (2017), 40-61.  doi: 10.1016/j.na.2017.01.010.  Google Scholar

[12]

L. BoudinB. GrecM. Pavić and F. Salvarani, Diffusion asymptotics of a kinetic model for gaseous mixtures, Kinet. Relat. Models, 6 (2013), 137-157.  doi: 10.3934/krm.2013.6.137.  Google Scholar

[13]

L. BoudinB. Grec and F. Salvarani, A mathematical and numerical analysis of the Maxwell-Stefan diffusion equations, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1427-1440.  doi: 10.3934/dcdsb.2012.17.1427.  Google Scholar

[14]

L. BoudinB. Grec and F. Salvarani, The Maxwell-Stefan diffusion limit for a kinetic model of mixtures, Acta Appl. Math., 136 (2015), 79-90.  doi: 10.1007/s10440-014-9886-z.  Google Scholar

[15]

J. F. BourgatL. DesvillettesP. Le Tallec and B. Perthame, Microreversible collisions for polyatomic gases and Boltzmann's theorem, Europ. J. Mech. B/ Fluids, 13 (1994), 237-254.   Google Scholar

[16]

X. Chen and A. Jüngel, Analysis of an incompressible Navier–Stokes–Maxwell–Stefan system, Comm. Math. Phys., 340 (2015), 471-497.  doi: 10.1007/s00220-015-2472-z.  Google Scholar

[17]

L. Desvillettes, Sur un modèle de type Borgnakke-Larsen conduisant à des lois d'énergie non-linéaires en température pour les gaz parfaits polyatomiques, Ann. Fac. Sci. Toulouse Math., 6 (1997), 257-262.  doi: 10.5802/afst.864.  Google Scholar

[18]

L. DesvillettesR. Monaco and F. Salvarani, A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions, Europ. J. Mech. B/ Fluids, 24 (2005), 219-236.  doi: 10.1016/j.euromechflu.2004.07.004.  Google Scholar

[19]

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

[20]

A. Fick, Über diffusion, Poggendorff's Annel Physik, 94 (1855), 59-86.   Google Scholar

[21]

J. Geiser, Iterative solvers for the Maxwell-Stefan diffusion equations: Methods and applications in plasma and particle transport, Cogent Math., 2 (2015), Art. ID 1092913, 16 pp. doi: 10.1080/23311835.2015.1092913.  Google Scholar

[22]

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.  Google Scholar

[23]

M. Groppi and G. Spiga, Kinetic approach to chemical reactions and inelastic transitions in a rarefied gas, J. Math. Chem., 26 (1999), 197-219.   Google Scholar

[24]

M. HerbergM. MeyriesJ. Prüss and M. Wilke, Reaction-diffusion systems of Maxwell-Stefan type with reversible mass-action kinetics, Nonlinear Analysis, 159 (2017), 264-284.  doi: 10.1016/j.na.2016.07.010.  Google Scholar

[25]

H. Hutridurga and F. Salvarani, Maxwell-Stefan diffusion asymptotics for gas mixtures in non-isothermal setting, Nonlinear Analysis, 159 (2017), 285-297.  doi: 10.1016/j.na.2017.03.019.  Google Scholar

[26]

H. Hutridurga and F. Salvarani, On the Maxwell-Stefan diffusion limit for a mixture of monatomic gases, Math. Meth. Appl. Sci., 40 (2017), 803-813.  doi: 10.1002/mma.4013.  Google Scholar

[27]

H. Hutridurga and F. Salvarani, Existence and uniqueness analysis of a non-isothermal cross-diffusion system of Maxwell-Stefan type, Appl. Math. Letters, 75 (2018), 108-113.  doi: 10.1016/j.aml.2017.06.007.  Google Scholar

[28]

A. Jüngel and I. V. Stelzer, Existence analysis of Maxwell-Stefan systems for multicomponent mixtures, SIAM J. Math. Anal., 45 (2013), 2421-2440.  doi: 10.1137/120898164.  Google Scholar

[29]

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

[30]

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

[31]

M. McLeod and Y. Bourgault, Mixed finite element methods for addressing multi-species diffusion using the Maxwell-Stefan equations, Comput. Methods Appl. Mech. Engrg., 279 (2014), 515-535.  doi: 10.1016/j.cma.2014.07.010.  Google Scholar

[32]

M. Pavić, Mathematical Modelling and Analysis of polyatomic gases and mixtures in the context of kinetic theory of gases and fluid mechanics, Ph.D Thesis, École Normale Supérieure de Cachan, 2014. Google Scholar

[33]

M. PavićT. Ruggeri and S. Simić, Maximum entropy principle for polyatomic gases, Physica A, 392 (2013), 1302-1317.  doi: 10.1016/j.physa.2012.12.006.  Google Scholar

[34]

J. Polewczak and A. J. Soares, On modified simple reacting spheres kinetic model for chemically reactive gases, Kinet. Relat. Models, 10 (2017), 513-539.  doi: 10.3934/krm.2017020.  Google Scholar

[35]

T. Ruggeri and M. Sugiyama, Rational Extended Thermodynamics beyond the Monatomic Gas, Springer, Cham, 2015. doi: 10.1007/978-3-319-13341-6.  Google Scholar

[36]

F. Salvarani and A. J. Soares, On the relaxation of the Maxwell-Stefan system to linear diffusion, Applied Mathematics Letters, 85 (2018), 15-21.  doi: 10.1016/j.aml.2018.05.012.  Google Scholar

[37]

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

[38]

R. Taylor and R. Krishna, Multicomponent Mass Transfer, John Wiley, New York, USA, 1992. Google Scholar

[39]

C. Villani, A review of mathematical topics in collisional kinetic theory, North-Holland, Amsterdam, Handbook of Mathematical Fluid Dynamics, 1 (2002), 71-305.   Google Scholar

show all references

References:
[1]

B. AnwasiaP. Gonçcalves and A. J. Soares, From the simple reacting sphere kinetic model to the reaction-diffusion system of Maxwell-Stefan type, Communications in Mathematical Sciences, 17 (2019), 507-538.  doi: 10.4310/CMS.2019.v17.n2.a9.  Google Scholar

[2]

C. BardosF. Golse and C. D. Levermore, Fluid dynamic limits of kinetic equations. Ⅰ. Formal derivations, J. Stat. Phys., 63 (1991), 323-344.  doi: 10.1007/BF01026608.  Google Scholar

[3]

C. BardosF. Golse and C. D. Levermore, Fluid dynamic limits of kinetic equations. Ⅱ. Convergence proofs for the Boltzmann equation, Commun. Pure Appl. Math, 46 (1993), 667-753.  doi: 10.1002/cpa.3160460503.  Google Scholar

[4]

M. Bisi and L. Desvillettes, From reactive Boltzmann equations to reaction-diffusion systems, J. Stat. Phys., 124 (2006), 881-912.  doi: 10.1007/s10955-005-8075-x.  Google Scholar

[5]

M. Bisi and M. J. Cáceres, A BGK relaxation model for polyatomic gas mixtures, Communications in Mathematical Sciences, 14 (2016), 297-325.  doi: 10.4310/CMS.2016.v14.n2.a1.  Google Scholar

[6]

M. Bisi, R. Monaco and A. J. Soares, A BGK model for reactive mixtures of polyatomic gases with continuous internal energy, J. Phys. A, 51 (2018), Art. ID 125501, 29 pp. doi: 10.1088/1751-8121/aaac8e.  Google Scholar

[7]

M. BisiT. Ruggeri and G. Spiga, Dynamical pressure in a polyatomic gas: Interplay between kinetic theory and extended thermodynamics, Kinet. Relat. Models, 11 (2018), 71-95.  doi: 10.3934/krm.2018004.  Google Scholar

[8]

A. BondesanL. Boudin and B. Grec, A numerical scheme for a kinetic model for mixtures in the diffusive limit using the moment methods, Numer. Methods Partial Differential Equations, 35 (2019), 1184-1205.  doi: 10.1002/num.22345.  Google Scholar

[9]

C. Borgnakke and P. S. Larsen, Statistical collision model for Monte–Carlo simulation of polyatomic gas mixtures, J. Comput. Phys., 18 (1975), 405-420.  doi: 10.1016/0021-9991(75)90094-7.  Google Scholar

[10]

D. Bothe, On the Maxwell-Stefan approach to multicomponent diffusion, in Parabolic problems, Progr. Nonlinear Differential Equations Appl., 80 (2011), 81–93, Birkhäuser, Springer Basel AG. doi: 10.1007/978-3-0348-0075-4_5.  Google Scholar

[11]

L. BoudinB. Grec and V. Pavan, The Maxwell-Stefan diffusion limit for a kinetic model of mixtures with general cross sections, Nonlinear Analysis, 159 (2017), 40-61.  doi: 10.1016/j.na.2017.01.010.  Google Scholar

[12]

L. BoudinB. GrecM. Pavić and F. Salvarani, Diffusion asymptotics of a kinetic model for gaseous mixtures, Kinet. Relat. Models, 6 (2013), 137-157.  doi: 10.3934/krm.2013.6.137.  Google Scholar

[13]

L. BoudinB. Grec and F. Salvarani, A mathematical and numerical analysis of the Maxwell-Stefan diffusion equations, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1427-1440.  doi: 10.3934/dcdsb.2012.17.1427.  Google Scholar

[14]

L. BoudinB. Grec and F. Salvarani, The Maxwell-Stefan diffusion limit for a kinetic model of mixtures, Acta Appl. Math., 136 (2015), 79-90.  doi: 10.1007/s10440-014-9886-z.  Google Scholar

[15]

J. F. BourgatL. DesvillettesP. Le Tallec and B. Perthame, Microreversible collisions for polyatomic gases and Boltzmann's theorem, Europ. J. Mech. B/ Fluids, 13 (1994), 237-254.   Google Scholar

[16]

X. Chen and A. Jüngel, Analysis of an incompressible Navier–Stokes–Maxwell–Stefan system, Comm. Math. Phys., 340 (2015), 471-497.  doi: 10.1007/s00220-015-2472-z.  Google Scholar

[17]

L. Desvillettes, Sur un modèle de type Borgnakke-Larsen conduisant à des lois d'énergie non-linéaires en température pour les gaz parfaits polyatomiques, Ann. Fac. Sci. Toulouse Math., 6 (1997), 257-262.  doi: 10.5802/afst.864.  Google Scholar

[18]

L. DesvillettesR. Monaco and F. Salvarani, A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions, Europ. J. Mech. B/ Fluids, 24 (2005), 219-236.  doi: 10.1016/j.euromechflu.2004.07.004.  Google Scholar

[19]

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

[20]

A. Fick, Über diffusion, Poggendorff's Annel Physik, 94 (1855), 59-86.   Google Scholar

[21]

J. Geiser, Iterative solvers for the Maxwell-Stefan diffusion equations: Methods and applications in plasma and particle transport, Cogent Math., 2 (2015), Art. ID 1092913, 16 pp. doi: 10.1080/23311835.2015.1092913.  Google Scholar

[22]

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.  Google Scholar

[23]

M. Groppi and G. Spiga, Kinetic approach to chemical reactions and inelastic transitions in a rarefied gas, J. Math. Chem., 26 (1999), 197-219.   Google Scholar

[24]

M. HerbergM. MeyriesJ. Prüss and M. Wilke, Reaction-diffusion systems of Maxwell-Stefan type with reversible mass-action kinetics, Nonlinear Analysis, 159 (2017), 264-284.  doi: 10.1016/j.na.2016.07.010.  Google Scholar

[25]

H. Hutridurga and F. Salvarani, Maxwell-Stefan diffusion asymptotics for gas mixtures in non-isothermal setting, Nonlinear Analysis, 159 (2017), 285-297.  doi: 10.1016/j.na.2017.03.019.  Google Scholar

[26]

H. Hutridurga and F. Salvarani, On the Maxwell-Stefan diffusion limit for a mixture of monatomic gases, Math. Meth. Appl. Sci., 40 (2017), 803-813.  doi: 10.1002/mma.4013.  Google Scholar

[27]

H. Hutridurga and F. Salvarani, Existence and uniqueness analysis of a non-isothermal cross-diffusion system of Maxwell-Stefan type, Appl. Math. Letters, 75 (2018), 108-113.  doi: 10.1016/j.aml.2017.06.007.  Google Scholar

[28]

A. Jüngel and I. V. Stelzer, Existence analysis of Maxwell-Stefan systems for multicomponent mixtures, SIAM J. Math. Anal., 45 (2013), 2421-2440.  doi: 10.1137/120898164.  Google Scholar

[29]

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

[30]

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

[31]

M. McLeod and Y. Bourgault, Mixed finite element methods for addressing multi-species diffusion using the Maxwell-Stefan equations, Comput. Methods Appl. Mech. Engrg., 279 (2014), 515-535.  doi: 10.1016/j.cma.2014.07.010.  Google Scholar

[32]

M. Pavić, Mathematical Modelling and Analysis of polyatomic gases and mixtures in the context of kinetic theory of gases and fluid mechanics, Ph.D Thesis, École Normale Supérieure de Cachan, 2014. Google Scholar

[33]

M. PavićT. Ruggeri and S. Simić, Maximum entropy principle for polyatomic gases, Physica A, 392 (2013), 1302-1317.  doi: 10.1016/j.physa.2012.12.006.  Google Scholar

[34]

J. Polewczak and A. J. Soares, On modified simple reacting spheres kinetic model for chemically reactive gases, Kinet. Relat. Models, 10 (2017), 513-539.  doi: 10.3934/krm.2017020.  Google Scholar

[35]

T. Ruggeri and M. Sugiyama, Rational Extended Thermodynamics beyond the Monatomic Gas, Springer, Cham, 2015. doi: 10.1007/978-3-319-13341-6.  Google Scholar

[36]

F. Salvarani and A. J. Soares, On the relaxation of the Maxwell-Stefan system to linear diffusion, Applied Mathematics Letters, 85 (2018), 15-21.  doi: 10.1016/j.aml.2018.05.012.  Google Scholar

[37]

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

[38]

R. Taylor and R. Krishna, Multicomponent Mass Transfer, John Wiley, New York, USA, 1992. Google Scholar

[39]

C. Villani, A review of mathematical topics in collisional kinetic theory, North-Holland, Amsterdam, Handbook of Mathematical Fluid Dynamics, 1 (2002), 71-305.   Google Scholar

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