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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[19]

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

[20]

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

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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.

[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.

[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.

[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.

[35]

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

[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.

[37]

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

[38]

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

[39]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[19]

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

[20]

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

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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.

[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.

[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.

[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.

[35]

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

[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.

[37]

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

[38]

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

[39]

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

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