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

  • * Corresponding author: Shigeru TAKATA

    * Corresponding author: Shigeru TAKATA 

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

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  • 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.

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

    Citation:

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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 $

    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

    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 $

    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

    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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