# American Institute of Mathematical Sciences

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

## 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

Received  May 2020 Revised  December 2020 Published  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.

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
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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)
,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
Dependence of $\Delta(\gamma, \alpha)$ in $\gamma$ for certain values of $\alpha$, with a particular emphasize on the behavior for $0<\gamma<2$
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
 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
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
 $\mathcal{N}$ 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
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%
 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%
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%
 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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