# American Institute of Mathematical Sciences

December  2020, 13(6): 1163-1174. 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, 2020, 13 (6) : 1163-1174. doi: 10.3934/krm.2020041
##### References:

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##### 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
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