Polytropic gas modelling at kinetic and macroscopic levels

Vladimir Djordjić; Milana Pavić-Čolić; Nikola Spasojević

doi:10.3934/krm.2021013

Article Contents

2021, Volume 14, Issue 3: 483-522. Doi: 10.3934/krm.2021013

This issue Previous Article On group symmetries of the hydrodynamic equations for rarefied gas Next Article A model of cultural evolution in the context of strategic conflict

Polytropic gas modelling at kinetic and macroscopic levels

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
^* Corresponding author: Milana Pavić-Čolić

Received: May 2020

Revised: December 2020

Early access: February 2021

Published: June 2021

Abstract / Introduction Full Text(HTML) Figure(3) / Table(4) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 76P05, 82C05; Secondary: 35Q82.

Citation:

Full Text(HTML)

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)

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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 $

Download: Full-size image PowerPoint slide

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	Translation, rotation and vibration
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

| Show Table

DownLoad: CSV

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

| Show Table

DownLoad: CSV

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%

| Show Table

DownLoad: CSV

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%

| Show Table

DownLoad: CSV

Related Papers

Cited by

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[11]	S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, London, 1970.
[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.
[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.
[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.
[15]	I. M. Gamba and M. Pavić-Čolić, On the Cauchy problem for Boltzmann equation modelling a polyatomic gas, preprint, arXiv: 2005.01017.
[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.
[17]	H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407. doi: 10.1002/cpa.3160020403.
[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.
[19]	M. N. Kogan, Rarefied Gas Dynamics, Springer, Boston, MA, 1969. doi: 10.1007/978-1-4899-6381-9.
[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.
[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.
[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.
[23]	C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[34]	T. Ruggeri and M. Sugiyama, Rational Extended Thermodynamics Beyond the Monatomic Gas, Springer, Cham, 2015. doi: 10.1007/978-3-319-13341-6.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.