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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

Shigeru Takata 1,, , Masanari Hattori 1, and  Takumu Miyauchi 2,

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  December 2020 Early access  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.

Keywords: kinetic theory of gases, H theorem, ellipsoidal statistical model, rarefied gases, Boltzmann equation.
Mathematics Subject Classification: Primary:76P05;Secondary:82C40.
Citation: Shigeru Takata, Masanari Hattori, Takumu Miyauchi. On the entropic property of the Ellipsoidal Statistical model with the prandtl number below 2/3. Kinetic and Related Models, 2020, 13 (6) : 1163-1174. doi: 10.3934/krm.2020041
References:
[1]

P. Andries, P. Le Tallec, J.-P. Perlat and B. Perthame, The Gaussian-BGK model of Boltzmann equation with small Prandtl number, Eur. J. Mech. B-Fluids, 19 (2000), 813-830.  doi: 10.1016/S0997-7546(00)01103-1.

[2] G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1967. 
[3]

P. L. Bhatnagar, E. P. Gross and M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511-525.  doi: 10.1103/PhysRev.94.511.

[4]

S. Brull and J. Schneider, A new approach for the ellipsoidal statistical model, Contin. Mech. Thermodyn., 20 (2008), 63-74.  doi: 10.1007/s00161-008-0068-y.

[5]

C. K. Chu, Kinetic-theoretic description of the formation of a shock wave, Phys. Fluids, 8 (1965), 12-22.  doi: 10.1063/1.1761077.

[6]

H. Funagane, S. Takata, K. Aoki and K. Kugimoto, Poiseuille flow and thermal transpiration of a rarefied polyatomic gas through a circular tube with applications to microflows, Bollettino U. M. I. Ser. IX, Ⅳ (2011), 19-46. 

[7]

M. A. Gallis and J. R. Torczynski, Investigation of the ellipsoidal-statistical Bhatnagar–Gross–Krook kinetic model applied to gas-phase transport of heat and tangential momentum between parallel walls, Phys. Fluids, 23 (2011), 030601. doi: 10.1063/1.3558869.

[8]

M. Groppi, S. Monica and G. Spiga, A kinetic ellipsoidal BGK model for a binary gas mixture, Europhys. Lett., 96 (2011), 64002. doi: 10.1209/0295-5075/96/64002.

[9]

L. H. Holway, New statistical models for kinetic theory: Methods of construction, Phys. Fluids, 9 (1966), 1658-1673.  doi: 10.1063/1.1761920.

[10]

C. Klingenberg, M. Pirner and G. Puppo, Kinetic ES-BGK models for a multi-component gas mixture, in XVI International Conference on Hyperbolic Problems: Theory, Numerics, Applications, Springer, (2016), 195–208.

[11]

H. W. Liepmann and A. Roshko, Elements of Gasdynamics, Dover, New York, 2001, Sec. 2.13.

[12]

J. Meng, L. Wu, J. M. Reese and Y. Zhang, Assessment of the ellipsoidal-statistical Bhatnagar–Gross–Krook model for force-driven Poiseuille flows, J. Comp. Phys., 251 (2013), 383-395.  doi: 10.1016/j.jcp.2013.05.045.

[13]

Y. Sone, Kinetic theory analysis of linearized Rayleigh problem, J. Phys. Soc. Jpn, 19 (1964), 1463-1473. 

[14]

Y. Sone, Molecular Gas Dynamics, Birkhäuser, Boston, 2007, Sec. 3.1.9; Supplementary Notes and Errata is available from Kyoto University Research Information Repository (http://hdl.handle.net/2433/66098). doi: 10.1007/978-0-8176-4573-1.

[15]

S. Takata, H. Funagane and K. Aoki, Fluid modeling for the Knudsen compressor: Case of polyatomic gases, Kinetic and Related Models, 3 (2010), 353-372.  doi: 10.3934/krm.2010.3.353.

show all references

References:
[1]

P. Andries, P. Le Tallec, J.-P. Perlat and B. Perthame, The Gaussian-BGK model of Boltzmann equation with small Prandtl number, Eur. J. Mech. B-Fluids, 19 (2000), 813-830.  doi: 10.1016/S0997-7546(00)01103-1.

[2] G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1967. 
[3]

P. L. Bhatnagar, E. P. Gross and M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511-525.  doi: 10.1103/PhysRev.94.511.

[4]

S. Brull and J. Schneider, A new approach for the ellipsoidal statistical model, Contin. Mech. Thermodyn., 20 (2008), 63-74.  doi: 10.1007/s00161-008-0068-y.

[5]

C. K. Chu, Kinetic-theoretic description of the formation of a shock wave, Phys. Fluids, 8 (1965), 12-22.  doi: 10.1063/1.1761077.

[6]

H. Funagane, S. Takata, K. Aoki and K. Kugimoto, Poiseuille flow and thermal transpiration of a rarefied polyatomic gas through a circular tube with applications to microflows, Bollettino U. M. I. Ser. IX, Ⅳ (2011), 19-46. 

[7]

M. A. Gallis and J. R. Torczynski, Investigation of the ellipsoidal-statistical Bhatnagar–Gross–Krook kinetic model applied to gas-phase transport of heat and tangential momentum between parallel walls, Phys. Fluids, 23 (2011), 030601. doi: 10.1063/1.3558869.

[8]

M. Groppi, S. Monica and G. Spiga, A kinetic ellipsoidal BGK model for a binary gas mixture, Europhys. Lett., 96 (2011), 64002. doi: 10.1209/0295-5075/96/64002.

[9]

L. H. Holway, New statistical models for kinetic theory: Methods of construction, Phys. Fluids, 9 (1966), 1658-1673.  doi: 10.1063/1.1761920.

[10]

C. Klingenberg, M. Pirner and G. Puppo, Kinetic ES-BGK models for a multi-component gas mixture, in XVI International Conference on Hyperbolic Problems: Theory, Numerics, Applications, Springer, (2016), 195–208.

[11]

H. W. Liepmann and A. Roshko, Elements of Gasdynamics, Dover, New York, 2001, Sec. 2.13.

[12]

J. Meng, L. Wu, J. M. Reese and Y. Zhang, Assessment of the ellipsoidal-statistical Bhatnagar–Gross–Krook model for force-driven Poiseuille flows, J. Comp. Phys., 251 (2013), 383-395.  doi: 10.1016/j.jcp.2013.05.045.

[13]

Y. Sone, Kinetic theory analysis of linearized Rayleigh problem, J. Phys. Soc. Jpn, 19 (1964), 1463-1473. 

[14]

Y. Sone, Molecular Gas Dynamics, Birkhäuser, Boston, 2007, Sec. 3.1.9; Supplementary Notes and Errata is available from Kyoto University Research Information Repository (http://hdl.handle.net/2433/66098). doi: 10.1007/978-0-8176-4573-1.

[15]

S. Takata, H. Funagane and K. Aoki, Fluid modeling for the Knudsen compressor: Case of polyatomic gases, Kinetic and Related Models, 3 (2010), 353-372.  doi: 10.3934/krm.2010.3.353.

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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$ 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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