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February  2018, 11(1): 71-95. doi: 10.3934/krm.2018004

Dynamical pressure in a polyatomic gas: Interplay between kinetic theory and extended thermodynamics

1. 

Dip. di Scienze Matematiche, Fisiche e Informatiche, Universitá di Parma, Parco Area delle Scienze 53/A, I-43124 Parma, Italy

2. 

Dip. di Matematica and Alma Mater Research Center, on Applied Mathematics AM$^2$, Via Saragozza, 8,40123 Bologna, Italy

* Corresponding author: T. Ruggeri

Received  November 2016 Published  August 2017

The aim of this paper is to compare different kinetic approaches to a polyatomic rarefied gas: the kinetic approach via a continuous energy parameter $I$ and the mixture-like one, based on discrete internal energy. We prove that if we consider only $6$ moments for a non-polytropic gas the two approaches give the same symmetric hyperbolic differential system previously obtained by the phenomenological Extended Thermodynamics. Both meaning and role of dynamical pressure become more clear in the present analysis.

Citation: Marzia Bisi, Tommaso Ruggeri, Giampiero Spiga. Dynamical pressure in a polyatomic gas: Interplay between kinetic theory and extended thermodynamics. Kinetic and Related Models, 2018, 11 (1) : 71-95. doi: 10.3934/krm.2018004
References:
[1]

T. ArimaA. Mentrelli and T. Ruggeri, Extended thermodynamics of rarefied polyatomic gases and characteristic velocities, Rend. Lincei Mat. Appl., 25 (2014), 275-291.  doi: 10.4171/RLM/678.

[2]

T. ArimaA. Mentrelli and T. Ruggeri, Molecular extended thermodynamics of rarefied polyatomic gases and wave velocities for increasing number of moments, Ann. Physics, 345 (2014), 111-140.  doi: 10.1016/j.aop.2014.03.011.

[3]

T. Arima, T. Ruggeri and M. Sugiyama, Duality principle from rarefied to dense gas and extended thermodynamics with $6$ fields, Phys. Rev. Fluids, 2 (2017), 013401, 22 pp.

[4]

T. ArimaT. RuggeriM. Sugiyama and S. Taniguchi, Monatomic gas as a singular limit of polyatomic gas in molecular extended thermodynamics with many moments, Ann. Physics, 372 (2016), 83-109.  doi: 10.1016/j.aop.2016.04.015.

[5]

T. ArimaT. RuggeriM. Sugiyama and S. Taniguchi, Nonlinear extended thermodynamics of real gases with 6 fields, Int. J. Non-Linear Mech., 72 (2015), 6-15. 

[6]

T. ArimaT. RuggeriS. Taniguchi and M. Sugiyama, Monatomic rarefied gas as a singular limit of polyatomic gas in extended thermodynamics, Phys. Lett. A, 377 (2013), 2136-2140.  doi: 10.1016/j.physleta.2013.06.035.

[7]

T. ArimaS. TaniguchiT. Ruggeri and M. Sugiyama, Dispersion relation for sound in rarefied polyatomic gases based on extended thermodynamics, Contin. Mech. Thermodyn., 25 (2013), 727-737.  doi: 10.1007/s00161-012-0271-8.

[8]

T. ArimaS. TaniguchiT. Ruggeri and M. Sugiyama, Extended thermodynamics of dense gases, Contin. Mech. Thermodyn., 24 (2012), 271-292.  doi: 10.1007/s00161-011-0213-x.

[9]

T. ArimaS. TaniguchiT. Ruggeri and M. Sugiyama, Extended thermodynamics of real gases with dynamic pressure: An extension of Meixner's theory, Phys. Lett. A, 376 (2012), 2799-2803.  doi: 10.1016/j.physleta.2012.08.030.

[10]

T. ArimaT. RuggeriM. Sugiyama and S. Taniguchi, Recent results on nonlinear extended thermodynamics of real gases with six fields. Part I: general theory, Ric. Mat., 65 (2016), 263-277.  doi: 10.1007/s11587-016-0283-y.

[11]

M. BisiL. Desvillettes and G. Spiga, Exponential convergence to equilibrium via Lyapunov functionals for reaction-diffusion equations arising from non reversible chemical kinetics, M2AN Math. Model. Numer. Anal., 43 (2009), 151-172.  doi: 10.1051/m2an:2008045.

[12]

M. BisiG. Martaló and G. Spiga, Multi-temperature fluid-dynamic model equations from kinetic theory in a reactive gas: the steady shock problem, Comput. Math. Appl., 66 (2013), 1403-1417.  doi: 10.1016/j.camwa.2013.08.015.

[13]

M. BisiA. Rossani and G. Spiga, A conservative multi-group approach to the Boltzmann equations for reactive gas mixtures, Phys. A, 438 (2015), 603-611.  doi: 10.1016/j.physa.2015.06.021.

[14]

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

[15]

J. F. BourgatL. DesvillettesP. Le Tallec and B. Perthame, Microreversible collisions for polyatomic gases, Eur. J. Mech. B/Fluids, 13 (1994), 237-254. 

[16]

C. Cercignani, Rarefied Gas Dynamics. From Basic Concepts to Actual Calculations, , University Press, Cambridge, 2000.

[17]

S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, Cambridge, 1970.

[18]

F. ConfortoM. GroppiR. Monaco and G. Spiga, Kinetic approach to deflagration processes in a recombination reaction, Kinet. Relat. Models, 4 (2011), 259-276.  doi: 10.3934/krm.2011.4.259.

[19]

W. Dreyer, Maximization of the entropy in non-equilibrium, J. Phys. A: Math. Gen., 20 (1987), 6505-6517.  doi: 10.1088/0305-4470/20/18/047.

[20]

M. GroppiK. AokiG. Spiga and V. Tritsch, Shock structure analysis in chemically reacting gas mixtures by a relaxation-time kinetic model, Phys. Fluids, 20 (2008), 117103, 11pp.  doi: 10.1063/1.3013637.

[21]

M. GroppiP. LichtenbergerF. Schuerrer and G. Spiga, Conservative approximation schemes of kinetic equations for chemical reactions, Eur. J. Mech. B/Fluids, 27 (2008), 202-217.  doi: 10.1016/j.euromechflu.2007.05.001.

[22]

M. Groppi, S. Rjasanow and G. Spiga, A kinetic relaxation approach to fast reactive mixtures: Shock wave structure J. Stat. Mech. -Theory Exp. , (2009), P10010, 15 pp. doi: 10.1088/1742-5468/2009/10/P10010.

[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. Groppi, G. Spiga and F. Zus, Euler closure of the Boltzmann equations for resonant bimolecular reactions, Phys. Fluids, 18 (2006), 057105, 8 pp. doi: 10.1063/1.2204098.

[25]

S. Kosuge, K. Aoki and T. Goto, Shock wave structure in polyatomic gases: numerical analysis using a model Boltzmann equation, AIP Conference Proceedings of "30th International Symposium on Rarefied Gas Dynamics" (eds. A. Ketsdever and H. Struchtrup), 1786 (2016), 180004, 8 pp. doi: 10.1063/1.4967673.

[26]

J. R. Mika and J. Banasiak, Singularly Perturbed Evolution Equations with Applications to Kinetic Theory, World Scientific, Singapore, 1995. doi: 10.1142/9789812831248.

[27]

I. Müller and T. Ruggeri, Extended Thermodynamics, Springer Tracts in Natural Philosophy, 37 ($1^{\rm st}$ edition), Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4684-0447-0.

[28]

I. Müller and T. Ruggeri, Rational Extended Thermodynamics, $2^{\rm nd}$ edition, Springer, New York, 1998. doi: 10.1007/978-1-4612-2210-1.

[29]

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

[30]

A. Rossani and G. Spiga, A note on the kinetic theory of chemically reacting gases, Phys. A, 272 (1999), 563-573.  doi: 10.1016/S0378-4371(99)00336-2.

[31]

T. Ruggeri, Galilean invariance and entropy principle for systems of balance laws, Contin. Mech. Thermodyn., 1 (1989), 3-20.  doi: 10.1007/BF01125883.

[32]

T. Ruggeri, Non-linear maximum entropy principle for a polyatomic gas subject to the dynamic pressure, Bull. Inst. Math. Acad. Sin. (N.S.), 11 (2016), 1-22. 

[33]

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

[34]

Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275.  doi: 10.14492/hokmj/1381757663.

[35]

S. Taniguchi, T. Arima, T. Ruggeri and M. Sugiyama, Effect of dynamic pressure on the shock wave structure in a rarefied polyatomic gas, Phys. Fluids, 26 (2014), 016103, 15 pp. doi: 10.1063/1.4861368.

[36]

S. TaniguchiT. ArimaT. Ruggeri and M. Sugiyama, Overshoot of the non-equilibrium temperature in the shock wave structure of a rarefied polyatomic gas subject to the dynamic pressure, Int. J. Non-Linear Mech., 79 (2016), 66-75. 

[37]

S. Taniguchi, T. Arima, T. Ruggeri and M. Sugiyama, Thermodynamic theory of the shock wave structure in a rarefied polyatomic gas: Beyond the Bethe-Teller theory, Phys. Rev. E, 89 (2014), 013025, 11 pp. doi: 10.1103/PhysRevE.89.013025.

show all references

References:
[1]

T. ArimaA. Mentrelli and T. Ruggeri, Extended thermodynamics of rarefied polyatomic gases and characteristic velocities, Rend. Lincei Mat. Appl., 25 (2014), 275-291.  doi: 10.4171/RLM/678.

[2]

T. ArimaA. Mentrelli and T. Ruggeri, Molecular extended thermodynamics of rarefied polyatomic gases and wave velocities for increasing number of moments, Ann. Physics, 345 (2014), 111-140.  doi: 10.1016/j.aop.2014.03.011.

[3]

T. Arima, T. Ruggeri and M. Sugiyama, Duality principle from rarefied to dense gas and extended thermodynamics with $6$ fields, Phys. Rev. Fluids, 2 (2017), 013401, 22 pp.

[4]

T. ArimaT. RuggeriM. Sugiyama and S. Taniguchi, Monatomic gas as a singular limit of polyatomic gas in molecular extended thermodynamics with many moments, Ann. Physics, 372 (2016), 83-109.  doi: 10.1016/j.aop.2016.04.015.

[5]

T. ArimaT. RuggeriM. Sugiyama and S. Taniguchi, Nonlinear extended thermodynamics of real gases with 6 fields, Int. J. Non-Linear Mech., 72 (2015), 6-15. 

[6]

T. ArimaT. RuggeriS. Taniguchi and M. Sugiyama, Monatomic rarefied gas as a singular limit of polyatomic gas in extended thermodynamics, Phys. Lett. A, 377 (2013), 2136-2140.  doi: 10.1016/j.physleta.2013.06.035.

[7]

T. ArimaS. TaniguchiT. Ruggeri and M. Sugiyama, Dispersion relation for sound in rarefied polyatomic gases based on extended thermodynamics, Contin. Mech. Thermodyn., 25 (2013), 727-737.  doi: 10.1007/s00161-012-0271-8.

[8]

T. ArimaS. TaniguchiT. Ruggeri and M. Sugiyama, Extended thermodynamics of dense gases, Contin. Mech. Thermodyn., 24 (2012), 271-292.  doi: 10.1007/s00161-011-0213-x.

[9]

T. ArimaS. TaniguchiT. Ruggeri and M. Sugiyama, Extended thermodynamics of real gases with dynamic pressure: An extension of Meixner's theory, Phys. Lett. A, 376 (2012), 2799-2803.  doi: 10.1016/j.physleta.2012.08.030.

[10]

T. ArimaT. RuggeriM. Sugiyama and S. Taniguchi, Recent results on nonlinear extended thermodynamics of real gases with six fields. Part I: general theory, Ric. Mat., 65 (2016), 263-277.  doi: 10.1007/s11587-016-0283-y.

[11]

M. BisiL. Desvillettes and G. Spiga, Exponential convergence to equilibrium via Lyapunov functionals for reaction-diffusion equations arising from non reversible chemical kinetics, M2AN Math. Model. Numer. Anal., 43 (2009), 151-172.  doi: 10.1051/m2an:2008045.

[12]

M. BisiG. Martaló and G. Spiga, Multi-temperature fluid-dynamic model equations from kinetic theory in a reactive gas: the steady shock problem, Comput. Math. Appl., 66 (2013), 1403-1417.  doi: 10.1016/j.camwa.2013.08.015.

[13]

M. BisiA. Rossani and G. Spiga, A conservative multi-group approach to the Boltzmann equations for reactive gas mixtures, Phys. A, 438 (2015), 603-611.  doi: 10.1016/j.physa.2015.06.021.

[14]

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

[15]

J. F. BourgatL. DesvillettesP. Le Tallec and B. Perthame, Microreversible collisions for polyatomic gases, Eur. J. Mech. B/Fluids, 13 (1994), 237-254. 

[16]

C. Cercignani, Rarefied Gas Dynamics. From Basic Concepts to Actual Calculations, , University Press, Cambridge, 2000.

[17]

S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, Cambridge, 1970.

[18]

F. ConfortoM. GroppiR. Monaco and G. Spiga, Kinetic approach to deflagration processes in a recombination reaction, Kinet. Relat. Models, 4 (2011), 259-276.  doi: 10.3934/krm.2011.4.259.

[19]

W. Dreyer, Maximization of the entropy in non-equilibrium, J. Phys. A: Math. Gen., 20 (1987), 6505-6517.  doi: 10.1088/0305-4470/20/18/047.

[20]

M. GroppiK. AokiG. Spiga and V. Tritsch, Shock structure analysis in chemically reacting gas mixtures by a relaxation-time kinetic model, Phys. Fluids, 20 (2008), 117103, 11pp.  doi: 10.1063/1.3013637.

[21]

M. GroppiP. LichtenbergerF. Schuerrer and G. Spiga, Conservative approximation schemes of kinetic equations for chemical reactions, Eur. J. Mech. B/Fluids, 27 (2008), 202-217.  doi: 10.1016/j.euromechflu.2007.05.001.

[22]

M. Groppi, S. Rjasanow and G. Spiga, A kinetic relaxation approach to fast reactive mixtures: Shock wave structure J. Stat. Mech. -Theory Exp. , (2009), P10010, 15 pp. doi: 10.1088/1742-5468/2009/10/P10010.

[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. Groppi, G. Spiga and F. Zus, Euler closure of the Boltzmann equations for resonant bimolecular reactions, Phys. Fluids, 18 (2006), 057105, 8 pp. doi: 10.1063/1.2204098.

[25]

S. Kosuge, K. Aoki and T. Goto, Shock wave structure in polyatomic gases: numerical analysis using a model Boltzmann equation, AIP Conference Proceedings of "30th International Symposium on Rarefied Gas Dynamics" (eds. A. Ketsdever and H. Struchtrup), 1786 (2016), 180004, 8 pp. doi: 10.1063/1.4967673.

[26]

J. R. Mika and J. Banasiak, Singularly Perturbed Evolution Equations with Applications to Kinetic Theory, World Scientific, Singapore, 1995. doi: 10.1142/9789812831248.

[27]

I. Müller and T. Ruggeri, Extended Thermodynamics, Springer Tracts in Natural Philosophy, 37 ($1^{\rm st}$ edition), Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4684-0447-0.

[28]

I. Müller and T. Ruggeri, Rational Extended Thermodynamics, $2^{\rm nd}$ edition, Springer, New York, 1998. doi: 10.1007/978-1-4612-2210-1.

[29]

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

[30]

A. Rossani and G. Spiga, A note on the kinetic theory of chemically reacting gases, Phys. A, 272 (1999), 563-573.  doi: 10.1016/S0378-4371(99)00336-2.

[31]

T. Ruggeri, Galilean invariance and entropy principle for systems of balance laws, Contin. Mech. Thermodyn., 1 (1989), 3-20.  doi: 10.1007/BF01125883.

[32]

T. Ruggeri, Non-linear maximum entropy principle for a polyatomic gas subject to the dynamic pressure, Bull. Inst. Math. Acad. Sin. (N.S.), 11 (2016), 1-22. 

[33]

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

[34]

Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275.  doi: 10.14492/hokmj/1381757663.

[35]

S. Taniguchi, T. Arima, T. Ruggeri and M. Sugiyama, Effect of dynamic pressure on the shock wave structure in a rarefied polyatomic gas, Phys. Fluids, 26 (2014), 016103, 15 pp. doi: 10.1063/1.4861368.

[36]

S. TaniguchiT. ArimaT. Ruggeri and M. Sugiyama, Overshoot of the non-equilibrium temperature in the shock wave structure of a rarefied polyatomic gas subject to the dynamic pressure, Int. J. Non-Linear Mech., 79 (2016), 66-75. 

[37]

S. Taniguchi, T. Arima, T. Ruggeri and M. Sugiyama, Thermodynamic theory of the shock wave structure in a rarefied polyatomic gas: Beyond the Bethe-Teller theory, Phys. Rev. E, 89 (2014), 013025, 11 pp. doi: 10.1103/PhysRevE.89.013025.

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