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

Shigeru Takata; Masanari Hattori; Takumu Miyauchi

doi:10.3934/krm.2020041

Article Contents

2020, Volume 13, Issue 6: 1163-1174. Doi: 10.3934/krm.2020041

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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
^* Corresponding author: Shigeru TAKATA

Received: March 31, 2020

Revised: June 30, 2020

Early access: September 2020

Published: November 2020

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

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Abstract

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.

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Mathematics Subject Classification: Primary:76P05;Secondary:82C40.

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Figure 1. 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 $

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Figure 2. 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

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Figure 3. 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 $

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Figure 4. 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

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Table 1. 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	0.5590	0.5593
$ (3/2)\mathrm{Pr}_* $	0.7776	0.8181	0.8309	0.8364	0.8385	0.8390

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