# 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:
 [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.  Google Scholar [2] G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1967.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [9] L. H. Holway, New statistical models for kinetic theory: Methods of construction, Phys. Fluids, 9 (1966), 1658-1673.  doi: 10.1063/1.1761920.  Google Scholar [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. Google Scholar [11] H. W. Liepmann and A. Roshko, Elements of Gasdynamics, Dover, New York, 2001, Sec. 2.13.  Google Scholar [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.  Google Scholar [13] Y. Sone, Kinetic theory analysis of linearized Rayleigh problem, J. Phys. Soc. Jpn, 19 (1964), 1463-1473.   Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [2] G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1967.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [9] L. H. Holway, New statistical models for kinetic theory: Methods of construction, Phys. Fluids, 9 (1966), 1658-1673.  doi: 10.1063/1.1761920.  Google Scholar [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. Google Scholar [11] H. W. Liepmann and A. Roshko, Elements of Gasdynamics, Dover, New York, 2001, Sec. 2.13.  Google Scholar [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.  Google Scholar [13] Y. Sone, Kinetic theory analysis of linearized Rayleigh problem, J. Phys. Soc. Jpn, 19 (1964), 1463-1473.   Google Scholar [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.  Google Scholar [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.  Google Scholar
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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