American Institute of Mathematical Sciences

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June  2021, 14(3): 483-522. doi: 10.3934/krm.2021013

Polytropic gas modelling at kinetic and macroscopic levels

Vladimir Djordjić 1,2, , Milana Pavić-Čolić 2,, and  Nikola Spasojević 3,

1. 

Applied and Computational Mathematics, RWTH Aachen University, Schinkelstr. 2, 52062 Aachen, Germany
2. 

Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia
3. 

Oden Institute for Computational Engineering and Sciences, University of Texas at Austin, 204 E 24th St, Austin TX 78712, USA

* Corresponding author: Milana Pavić-Čolić

Received  May 2020 Revised  December 2020 Published  June 2021 Early access  February 2021

In this paper, we consider the kinetic model of continuous type describing a polyatomic gas in two different settings corresponding to a different choice of the functional space used to define macroscopic quantities. Such a model introduces a single continuous variable supposed to capture all the phenomena related to the more complex structure of a polyatomic molecule. In particular, we provide a direct comparison of these two settings, and show their equivalence after the distribution function is rescaled and the cross section is reformulated. We then focus on the kinetic model for which the rigorous existence and uniqueness result in the space homogeneous case is recently proven. Using the cross section proposed in that analysis together with the maximum entropy principle, we establish macroscopic models of six and fourteen fields. In the case of six moments, we calculate the exact, nonlinear, production term and prove its total agreement with extended thermodynamics. Moreover, for the fourteen moments model, we provide new expressions for relaxation times and transport coefficients in a linearized setting, that yield both matching with the experimental data for dependence of the shear viscosity upon temperature and a satisfactory agreement with the theoretical value of the Prandtl number.

Keywords: Kinetic theory of gases, polyatomic gases, Boltzmann equation, moment equations, non-equilibrium processes.
Mathematics Subject Classification: Primary: 76P05, 82C05; Secondary: 35Q82.
Citation: Vladimir Djordjić, Milana Pavić-Čolić, Nikola Spasojević. Polytropic gas modelling at kinetic and macroscopic levels. Kinetic & Related Models, 2021, 14 (3) : 483-522. doi: 10.3934/krm.2021013
References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, 55, U.S. Government Printing Office, Washington, D.C., 1964.  Google Scholar

[2]

T. Arima, A. 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.  Google Scholar

[3]

T. Arima, T. Ruggeri, M. Sugiyama and S. Taniguchi, Non-linear extended thermodynamics of real gases with 6 fields, Int. J. Non-Lin. Mech., 72 (2015), 6-15.  doi: 10.1016/j.ijnonlinmec.2015.02.005.  Google Scholar

[4]

T. Arima, T. Ruggeri, M. 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.  Google Scholar

[5]

C. Baranger, M. Bisi, S. Brull and L. Desvillettes, On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases, Kinet. Relat. Models, 11 (2018), 821-858.  doi: 10.3934/krm.2018033.  Google Scholar

[6]

M. Bisi, T. 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

[7]

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.  Google Scholar

[8]

L. Boudin, B. Grec, M. Pavić-Čolić and F. Salvarani, A kinetic model for polytropic gases with internal energy, PAMM Proc. Appl. Math. Mech., 13 (2013), 353-354.  doi: 10.1002/pamm.201310172.  Google Scholar

[9]

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

[10]

C. Cercignani, The Boltzmann Equation and Its Applications, Applied Mathematical Sciences, 67, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[11] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, London, 1970.   Google Scholar
[12]

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

[13]

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.  doi: 10.1016/j.euromechflu.2004.07.004.  Google Scholar

[14]

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

[15]

I. M. Gamba and M. Pavić-Čolić, On the Cauchy problem for Boltzmann equation modelling a polyatomic gas, preprint, arXiv: 2005.01017. Google Scholar

[16]

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

[17]

H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407.  doi: 10.1002/cpa.3160020403.  Google Scholar

[18]

M. Groppi and G. Spiga, Kinetic approach to chemical reactions and inelastic transitions in a rarefied gas, J. Math. Chem., 26 (1999), 197-219.  doi: 10.1023/A:1019194113816.  Google Scholar

[19]

M. N. Kogan, Rarefied Gas Dynamics, Springer, Boston, MA, 1969. doi: 10.1007/978-1-4899-6381-9.  Google Scholar

[20]

S. Kosuge and K. Aoki, Shock-wave structure for a polyatomic gas with large bulk viscosity, Phys. Rev. Fluids, 3 (2018). doi: 10.1103/PhysRevFluids.3.023401.  Google Scholar

[21]

S. Kosuge, H.-W. Kuo and K. Aoki, A kinetic model for a polyatomic gas with temperature-dependent specific heats and its application to shock-wave structure, J. Stat. Phys., 177 (2019), 209-251.  doi: 10.1007/s10955-019-02366-5.  Google Scholar

[22]

E. W. Lemmon and R. T. Jacobsen, Viscosity and thermal conductivity equations for nitrogen, oxygen, argon and air, Int. J. Thermophys., 25 (2004), 21-69.  doi: 10.1023/B:IJOT.0000022327.04529.f3.  Google Scholar

[23]

C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys., 83 (1996), 1021-1065.  doi: 10.1007/BF02179552.  Google Scholar

[24]

T. Magin, B. Graille and M. Massot, Kinetic theory derivation of transport equations for gases with internal energy, 42nd AIAA Thermophysics Conference, Honolulu, Hawaii, USA, 2011. doi: 10.2514/6.2011-4034.  Google Scholar

[25]

G. C. Maitland and E. B. Smith, Critical reassessment of viscosities of 11 common gases, J. Chem. Eng. Data, 17 (1972), 150-156.  doi: 10.1021/je60053a015.  Google Scholar

[26]

I. Müller, T. Ruggeri, Extended Thermodynamics, Springer Tracts in Natural Philosophy, 37, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4684-0447-0.  Google Scholar

[27]

E. Nagnibeda and E. Kustova, Non-Equilibrium Reacting Gas Flows. Kinetic Theory of Transport and Relaxation Processes, Heat and Mass Transfer, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-01390-4.  Google Scholar

[28]

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

[29]

M. Pavić-Čolić, D. Madjarević and S. Simić, Polyatomic gases with dynamic pressure: Kinetic non-linear closure and the shock structure, Int. J. Non-Lin. Mech., 92 (2017), 160-175.  doi: 10.1016/j.ijnonlinmec.2017.04.008.  Google Scholar

[30]

M. Pavić-Čolić and S. Simić, Moment equations for polyatomic gases, Acta Appl. Math., 132 (2014), 469-482.  doi: 10.1007/s10440-014-9928-6.  Google Scholar

[31]

B. Rahimi and H. Struchtrup, Macroscopic and kinetic modelling of rarefied polyatomic gases, J. Fluid Mech., 806 (2016), 437-505.  doi: 10.1017/jfm.2016.604.  Google Scholar

[32]

T. Ruggeri, Maximum entropy principle closure for 14-moment system for a non-polytropic gas, Ric. Mat., (2020). doi: 10.1007/s11587-020-00510-y.  Google Scholar

[33]

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.   Google Scholar

[34]

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

[35]

S. Simić, M. Pavić-Čolić and D. Madjarević, Non-equilibrium mixtures of gases: Modelling and computation, Riv. Math Univ. Parma (N.S.), 6 (2015), 135-214.   Google Scholar

[36]

Y. Sone, Kinetic Theory and Fluid Dynamics, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2002. doi: 10.1007/978-1-4612-0061-1.  Google Scholar

[37]

Y. Sone, Molecular Gas Dynamics. Theory, Techniques, and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2007. doi: 10.1007/978-0-8176-4573-1.  Google Scholar

[38]

D. Stéphane, On the Wang Chang-Uhlenbeck equations, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 229-253.  doi: 10.3934/dcdsb.2003.3.229.  Google Scholar

[39]

H. Struchtrup, The Boltzmann equation and its properties, in Macroscopic Transport Equations for Rarefied Gas Flows, Springer, Berlin, Heidelberg, 2005, 27–51. doi: 10.1007/3-540-32386-4_3.  Google Scholar

[40]

S. Taniguchi, T. Arima, T. 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-Lin. Mech., 79 (2016), 66-75.  doi: 10.1016/j.ijnonlinmec.2015.11.003.  Google Scholar

[41]

S. Taniguchi, T. Arima, T. Ruggeri and M. Sugiyama, Shock wave structure in a rarefied polyatomic gas based on extended thermodynamics, Acta Appl. Math., 132 (2014), 583-593.  doi: 10.1007/s10440-014-9931-y.  Google Scholar

[42]

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). doi: 10.1103/PhysRevE.89.013025.  Google Scholar

[43]

C. S. Wang Chang, G. E. Uhlenbeck and J. de Boer, The heat conductivity and viscosity of polyatomic gases, in Studies in Statistical Mechanics, Vol. II, North-Holland, Amsterdam; Interscience, New York, 1964,241–268.  Google Scholar

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, 55, U.S. Government Printing Office, Washington, D.C., 1964.  Google Scholar

[2]

T. Arima, A. 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.  Google Scholar

[3]

T. Arima, T. Ruggeri, M. Sugiyama and S. Taniguchi, Non-linear extended thermodynamics of real gases with 6 fields, Int. J. Non-Lin. Mech., 72 (2015), 6-15.  doi: 10.1016/j.ijnonlinmec.2015.02.005.  Google Scholar

[4]

T. Arima, T. Ruggeri, M. 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.  Google Scholar

[5]

C. Baranger, M. Bisi, S. Brull and L. Desvillettes, On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases, Kinet. Relat. Models, 11 (2018), 821-858.  doi: 10.3934/krm.2018033.  Google Scholar

[6]

M. Bisi, T. 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

[7]

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.  Google Scholar

[8]

L. Boudin, B. Grec, M. Pavić-Čolić and F. Salvarani, A kinetic model for polytropic gases with internal energy, PAMM Proc. Appl. Math. Mech., 13 (2013), 353-354.  doi: 10.1002/pamm.201310172.  Google Scholar

[9]

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

[10]

C. Cercignani, The Boltzmann Equation and Its Applications, Applied Mathematical Sciences, 67, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[11] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, London, 1970.   Google Scholar
[12]

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

[13]

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.  doi: 10.1016/j.euromechflu.2004.07.004.  Google Scholar

[14]

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

[15]

I. M. Gamba and M. Pavić-Čolić, On the Cauchy problem for Boltzmann equation modelling a polyatomic gas, preprint, arXiv: 2005.01017. Google Scholar

[16]

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

[17]

H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407.  doi: 10.1002/cpa.3160020403.  Google Scholar

[18]

M. Groppi and G. Spiga, Kinetic approach to chemical reactions and inelastic transitions in a rarefied gas, J. Math. Chem., 26 (1999), 197-219.  doi: 10.1023/A:1019194113816.  Google Scholar

[19]

M. N. Kogan, Rarefied Gas Dynamics, Springer, Boston, MA, 1969. doi: 10.1007/978-1-4899-6381-9.  Google Scholar

[20]

S. Kosuge and K. Aoki, Shock-wave structure for a polyatomic gas with large bulk viscosity, Phys. Rev. Fluids, 3 (2018). doi: 10.1103/PhysRevFluids.3.023401.  Google Scholar

[21]

S. Kosuge, H.-W. Kuo and K. Aoki, A kinetic model for a polyatomic gas with temperature-dependent specific heats and its application to shock-wave structure, J. Stat. Phys., 177 (2019), 209-251.  doi: 10.1007/s10955-019-02366-5.  Google Scholar

[22]

E. W. Lemmon and R. T. Jacobsen, Viscosity and thermal conductivity equations for nitrogen, oxygen, argon and air, Int. J. Thermophys., 25 (2004), 21-69.  doi: 10.1023/B:IJOT.0000022327.04529.f3.  Google Scholar

[23]

C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys., 83 (1996), 1021-1065.  doi: 10.1007/BF02179552.  Google Scholar

[24]

T. Magin, B. Graille and M. Massot, Kinetic theory derivation of transport equations for gases with internal energy, 42nd AIAA Thermophysics Conference, Honolulu, Hawaii, USA, 2011. doi: 10.2514/6.2011-4034.  Google Scholar

[25]

G. C. Maitland and E. B. Smith, Critical reassessment of viscosities of 11 common gases, J. Chem. Eng. Data, 17 (1972), 150-156.  doi: 10.1021/je60053a015.  Google Scholar

[26]

I. Müller, T. Ruggeri, Extended Thermodynamics, Springer Tracts in Natural Philosophy, 37, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4684-0447-0.  Google Scholar

[27]

E. Nagnibeda and E. Kustova, Non-Equilibrium Reacting Gas Flows. Kinetic Theory of Transport and Relaxation Processes, Heat and Mass Transfer, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-01390-4.  Google Scholar

[28]

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

[29]

M. Pavić-Čolić, D. Madjarević and S. Simić, Polyatomic gases with dynamic pressure: Kinetic non-linear closure and the shock structure, Int. J. Non-Lin. Mech., 92 (2017), 160-175.  doi: 10.1016/j.ijnonlinmec.2017.04.008.  Google Scholar

[30]

M. Pavić-Čolić and S. Simić, Moment equations for polyatomic gases, Acta Appl. Math., 132 (2014), 469-482.  doi: 10.1007/s10440-014-9928-6.  Google Scholar

[31]

B. Rahimi and H. Struchtrup, Macroscopic and kinetic modelling of rarefied polyatomic gases, J. Fluid Mech., 806 (2016), 437-505.  doi: 10.1017/jfm.2016.604.  Google Scholar

[32]

T. Ruggeri, Maximum entropy principle closure for 14-moment system for a non-polytropic gas, Ric. Mat., (2020). doi: 10.1007/s11587-020-00510-y.  Google Scholar

[33]

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.   Google Scholar

[34]

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

[35]

S. Simić, M. Pavić-Čolić and D. Madjarević, Non-equilibrium mixtures of gases: Modelling and computation, Riv. Math Univ. Parma (N.S.), 6 (2015), 135-214.   Google Scholar

[36]

Y. Sone, Kinetic Theory and Fluid Dynamics, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2002. doi: 10.1007/978-1-4612-0061-1.  Google Scholar

[37]

Y. Sone, Molecular Gas Dynamics. Theory, Techniques, and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2007. doi: 10.1007/978-0-8176-4573-1.  Google Scholar

[38]

D. Stéphane, On the Wang Chang-Uhlenbeck equations, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 229-253.  doi: 10.3934/dcdsb.2003.3.229.  Google Scholar

[39]

H. Struchtrup, The Boltzmann equation and its properties, in Macroscopic Transport Equations for Rarefied Gas Flows, Springer, Berlin, Heidelberg, 2005, 27–51. doi: 10.1007/3-540-32386-4_3.  Google Scholar

[40]

S. Taniguchi, T. Arima, T. 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-Lin. Mech., 79 (2016), 66-75.  doi: 10.1016/j.ijnonlinmec.2015.11.003.  Google Scholar

[41]

S. Taniguchi, T. Arima, T. Ruggeri and M. Sugiyama, Shock wave structure in a rarefied polyatomic gas based on extended thermodynamics, Acta Appl. Math., 132 (2014), 583-593.  doi: 10.1007/s10440-014-9931-y.  Google Scholar

[42]

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). doi: 10.1103/PhysRevE.89.013025.  Google Scholar

[43]

C. S. Wang Chang, G. E. Uhlenbeck and J. de Boer, The heat conductivity and viscosity of polyatomic gases, in Studies in Statistical Mechanics, Vol. II, North-Holland, Amsterdam; Interscience, New York, 1964,241–268.  Google Scholar

Figure 1.  The collision frequency $ \hat{\nu}_{\gamma, \alpha}(\hat{c}, \hat{I}) $ defined in (45) as a function of the dimensionless peculiar speed $ \hat{c} $ and internal energy $ \hat{I} $ introduced in (44) for certain values of $ \gamma $ ranging in the interval $ (0, 2] $, and for $ \alpha = 0 $ corresponding to linear molecules (solid line) and $ \alpha = \frac{1}{2} $ corresponding to non-linear molecules with translational and rotational degrees of freedom (dashed line)
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22,25], while solid lines represent fitted curves">Figure 2.  Shear viscosity as a function of the temperature in the form $ \mu(T) = A T^s $. Points on the plot are experimentally observed values [22,25], while solid lines represent fitted curves
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Figure 3.  Dependence of $ \Delta(\gamma, \alpha) $ in $ \gamma $ for certain values of $ \alpha $, with a particular emphasize on the behavior for $ 0<\gamma<2 $
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Table 1.  Number of degrees of freedom $D$ for different modes (combinations of translation/rotation/vibration) where $\mathcal{N} \geq 2$ is the number of atoms in a polyatomic molecule, with the corresponding value of $\alpha = \frac{D-5}{2}$, theoretical value of the Prandtl number from (72) and the value of $\gamma$ enabling that this theoretical value of the Prandtl number coincides with the one given in (71), i.e. enabling that the two expressions (71) and (72) are equal
Translation and rotation Translation, rotation and vibration
Linear molecule Non-linear molecule
Degrees of freedom 5 6 $3\mathcal{N}$
$\alpha$ 0 $\frac{1}{2}$ $\frac{1}{2}(3 \mathcal{N}-5) $
$\text{Pr}$ from (72) $\frac{14}{19}$ $\frac{16}{21}$ $\frac{6\mathcal{N}+4}{6\mathcal{N}+9}$
$\gamma$ $2.153$ $2.368$ Table 2
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Translation and rotation Translation, rotation and vibration
Linear molecule Non-linear molecule
Degrees of freedom 5 6 $3\mathcal{N}$
$\alpha$ 0 $\frac{1}{2}$ $\frac{1}{2}(3 \mathcal{N}-5) $
$\text{Pr}$ from (72) $\frac{14}{19}$ $\frac{16}{21}$ $\frac{6\mathcal{N}+4}{6\mathcal{N}+9}$
$\gamma$ $2.153$ $2.368$ Table 2
Table 2.  The number $\mathcal{N}$ of atoms in a polyatomic molecule and the corresponding value of potential $\gamma$ such that the theoretical value of the Prandtl number from (72) is equal to the one in (71)
$\mathcal{N}$ 3 4 5 6 7 8 9 10
$\gamma$ 4.063 9.469 17.262 25.801 34.705 43.835 53.123 62.526
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$\mathcal{N}$ 3 4 5 6 7 8 9 10
$\gamma$ 4.063 9.469 17.262 25.801 34.705 43.835 53.123 62.526
Table 3.  Experimental values of $s$ [11] for different molecules revealing the dependence of the shear viscosity upon temperature $\mu \sim T^s$ given in (74), the corresponding value of $\gamma$ by virtue of (75), and the Prandtl number from (71). This value of the Prandtl number is further compared to the theoretical one (72) and the relative error is provided
Gas $s$ $\gamma$ Pr from (71) Pr from (72) Relative error
H$_2$ 0.668 0.664 0.816 0.737 10.7%
CO 0.734 0.532 0.819 0.737 11.1%
N$_2$ 0.738 0.524 0.819 0.737 11.1%
NO 0.788 0.424 0.82 0.737 11.3%
O$_2$ 0.773 0.454 0.82 0.737 11.3%
CO$_2$ 0.933 0.134 0.819 0.737 11.1%
N$_2$O 0.943 0.114 0.819 0.737 11.1%
CH$_4$ 0.836 0.328 0.849 0.762 10.3%
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Gas $s$ $\gamma$ Pr from (71) Pr from (72) Relative error
H$_2$ 0.668 0.664 0.816 0.737 10.7%
CO 0.734 0.532 0.819 0.737 11.1%
N$_2$ 0.738 0.524 0.819 0.737 11.1%
NO 0.788 0.424 0.82 0.737 11.3%
O$_2$ 0.773 0.454 0.82 0.737 11.3%
CO$_2$ 0.933 0.134 0.819 0.737 11.1%
N$_2$O 0.943 0.114 0.819 0.737 11.1%
CH$_4$ 0.836 0.328 0.849 0.762 10.3%
Table 4.  Value of experimental observed parameter $s$ [22,25] and the corresponding value of $\gamma$ by means of (75), expressing the dependence of shear viscosity $\mu$ of the shape (73) upon high temperature for different molecules and the Prandtl number from (71). This value of the Prandtl number is further compared to the theoretical one (72) and the relative error is provided
Gas $s$ $\gamma$ Pr from (71) Pr from (72) Relative error
H$_2$ 0.688 0.624 0.847 0.762 11.2%
N$_2$ 0.684 0.704 0.846 0.762 11.0%
CO$_2$ 0.7 0.599 0.894 0.815 9.7%
CH$_4$ 0.689 0.419 0.930 0.872 6.8%
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Gas $s$ $\gamma$ Pr from (71) Pr from (72) Relative error
H$_2$ 0.688 0.624 0.847 0.762 11.2%
N$_2$ 0.684 0.704 0.846 0.762 11.0%
CO$_2$ 0.7 0.599 0.894 0.815 9.7%
CH$_4$ 0.689 0.419 0.930 0.872 6.8%
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