# American Institute of Mathematical Sciences

doi: 10.3934/krm.2020041

## On the entropic property of the Ellipsoidal Statistical model with the prandtl number below 2/3

 1 Department of Aeronautics and Astronautics & Advanced Engineering Research Center, Kyoto University, Kyoto 615-8540, Japan 2 Department of Aeronautics and Astronautics, Kyoto University, Kyoto 615-8540, Japan

* Corresponding author: Shigeru TAKATA

Received  April 2020 Revised  July 2020 Published  September 2020

Fund Project: The present work is supported in part by KAKENHI from JSPS (No. 17K18840)

Entropic property of the Ellipsoidal Statistical model with the Prandtl number Pr below 2/3 is discussed. Although 2/3 is the lower bound of Pr for the H theorem to hold unconditionally, it is shown that the theorem still holds even for $\mathrm{Pr}<2/3$, provided that anisotropy of stress tensor satisfies a certain criterion. The practical tolerance of that criterion is assessed numerically by the strong normal shock wave and the Couette flow problems. A couple of moving plate tests are also conducted.

Citation: Shigeru Takata, Masanari Hattori, Takumu Miyauchi. On the entropic property of the Ellipsoidal Statistical model with the prandtl number below 2/3. Kinetic & Related Models, doi: 10.3934/krm.2020041
##### References:

show all references

##### References:
Structure of the shock wave for typical Mach numbers $M_-$ in the case of $\mathrm{Pr} = 2/3$. (a) $M_- = 2$, (b) $M_- = 5$, (c) $M_- = 20$, and (d) $M_- = 40$. Here, $\ell_-$ is the mean free path of molecules in the equilibrium state at rest with density $\rho_-$ and temperature $T_-$. In each panel, solid lines indicate $\rho_* = (\rho-\rho_-)/(\rho_+-\rho_-)$, $u_* = (v_1-u_+)/(u_–u_+)$, and $T_* = (T-T_-)/(T_+-T_-)$ respectively, while a dash-dotted line indicates $\epsilon$
The maximum value of $\epsilon$ vs. $(3/2)\mathrm{Pr}$ for various upstream Mach numbers $M_-$. Symbols indicate the results of computations. The results of common $M_-$ are connected by solid lines in the acceptable range of $\mathrm{Pr}$ and by dashed lines in the range where condition (8) is violated
The functions $\mathcal{F}_D$ and $\epsilon_S$ in the criterion (16). The solid lines indicate $\mathcal{F}_D$ for $(3/2)\mathrm{Pr} = 0.76, 0.78, 0.8,\dots,0.96$, while the dashed line indicates $\epsilon_S$
The function $\epsilon_P$ and the dimensionless density $\hat{\rho}$ in the range $-5<\hat{U}<5$. (a) $\epsilon_P$, (b) $\hat{\rho}$. In (a), the values of $\mathcal{S}(\mathrm{Pr})$ for $(3/2)\mathrm{Pr} = 0.76,0.8,0.84,\dots,0.96$ are also indicated by dash-dotted lines for reference
The maximum $\epsilon$ for various Mach numbers $M_-$ in the case of $\mathrm{Pr} = 2/3$. $\mathrm{Pr}_*$ is the lower bound of acceptable Prandtl number suggested by $\max\epsilon = \mathcal{S}(\mathrm{Pr}_*)$; see (12)
 $M_-$ 2 5 10 20 40 60 $\max\epsilon$ 0.2786 0.5506 0.6137 0.6387 0.6478 0.6499 $\mathrm{Pr}_*$ 0.5184 0.5454 0.5539 0.5576 0.559 0.5593 $(3/2)\mathrm{Pr}_*$ 0.7776 0.8181 0.8309 0.8364 0.8385 0.839
 $M_-$ 2 5 10 20 40 60 $\max\epsilon$ 0.2786 0.5506 0.6137 0.6387 0.6478 0.6499 $\mathrm{Pr}_*$ 0.5184 0.5454 0.5539 0.5576 0.559 0.5593 $(3/2)\mathrm{Pr}_*$ 0.7776 0.8181 0.8309 0.8364 0.8385 0.839
 [1] Lei Jing, Jiawei Sun. Global existence and long time behavior of the Ellipsoidal-Statistical-Fokker-Planck model for diatomic gases. Kinetic & Related Models, 2020, 13 (2) : 373-400. doi: 10.3934/krm.2020013 [2] Gilberto M. Kremer, Filipe Oliveira, Ana Jacinta Soares. $\mathcal H$-Theorem and trend to equilibrium of chemically reacting mixtures of gases. Kinetic & Related Models, 2009, 2 (2) : 333-343. doi: 10.3934/krm.2009.2.333 [3] Jacek Polewczak, Ana Jacinta Soares. On modified simple reacting spheres kinetic model for chemically reactive gases. Kinetic & Related Models, 2017, 10 (2) : 513-539. doi: 10.3934/krm.2017020 [4] Gilberto M. Kremer, Wilson Marques Jr.. Fourteen moment theory for granular gases. Kinetic & Related Models, 2011, 4 (1) : 317-331. doi: 10.3934/krm.2011.4.317 [5] Seok-Bae Yun. Entropy production for ellipsoidal BGK model of the Boltzmann equation. Kinetic & Related Models, 2016, 9 (3) : 605-619. doi: 10.3934/krm.2016009 [6] Céline Baranger, Marzia Bisi, Stéphane Brull, Laurent Desvillettes. On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases. Kinetic & Related Models, 2018, 11 (4) : 821-858. doi: 10.3934/krm.2018033 [7] Manuel Torrilhon. H-Theorem for nonlinear regularized 13-moment equations in kinetic gas theory. Kinetic & Related Models, 2012, 5 (1) : 185-201. doi: 10.3934/krm.2012.5.185 [8] Raffaele Esposito, Mario Pulvirenti. Rigorous validity of the Boltzmann equation for a thin layer of a rarefied gas. Kinetic & Related Models, 2010, 3 (2) : 281-297. doi: 10.3934/krm.2010.3.281 [9] Shigeru Takata, Hitoshi Funagane, Kazuo Aoki. Fluid modeling for the Knudsen compressor: Case of polyatomic gases. Kinetic & Related Models, 2010, 3 (2) : 353-372. doi: 10.3934/krm.2010.3.353 [10] Eugenio Aulisa, Lidia Bloshanskaya, Akif Ibragimov. Well productivity index for compressible fluids and gases. Evolution Equations & Control Theory, 2016, 5 (1) : 1-36. doi: 10.3934/eect.2016.5.1 [11] Zhongyi Huang, Peter A. Markowich, Christof Sparber. Numerical simulation of trapped dipolar quantum gases: Collapse studies and vortex dynamics. Kinetic & Related Models, 2010, 3 (1) : 181-194. doi: 10.3934/krm.2010.3.181 [12] Pedro M. Jordan. Second-sound phenomena in inviscid, thermally relaxing gases. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2189-2205. doi: 10.3934/dcdsb.2014.19.2189 [13] José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401 [14] Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361 [15] Marc Briant. Perturbative theory for the Boltzmann equation in bounded domains with different boundary conditions. Kinetic & Related Models, 2017, 10 (2) : 329-371. doi: 10.3934/krm.2017014 [16] Daewa Kim, Annalisa Quaini. A kinetic theory approach to model pedestrian dynamics in bounded domains with obstacles. Kinetic & Related Models, 2019, 12 (6) : 1273-1296. doi: 10.3934/krm.2019049 [17] 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 & Related Models, 2020, 13 (1) : 63-95. doi: 10.3934/krm.2020003 [18] P. M. Jordan. The effects of coupling on finite-amplitude acoustic traveling waves in thermoviscous gases: Blackstock's models. Evolution Equations & Control Theory, 2016, 5 (3) : 383-397. doi: 10.3934/eect.2016010 [19] Yulan Xu, Yanping Dou. Large BV solutions to Euler equations in the isothermal self-gravitating gases with damping. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1451-1467. doi: 10.3934/cpaa.2009.8.1451 [20] Fei Hou, Huicheng Yin. On global axisymmetric solutions to 2D compressible full Euler equations of Chaplygin gases. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1435-1492. doi: 10.3934/dcds.2020083

2019 Impact Factor: 1.311

## Tools

Article outline

Figures and Tables