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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 |
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.
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. |




2 | 5 | 10 | 20 | 40 | 60 | |
0.2786 | 0.5506 | 0.6137 | 0.6387 | 0.6478 | 0.6499 | |
0.5184 | 0.5454 | 0.5539 | 0.5576 | 0.5590 | 0.5593 | |
0.7776 | 0.8181 | 0.8309 | 0.8364 | 0.8385 | 0.8390 |
2 | 5 | 10 | 20 | 40 | 60 | |
0.2786 | 0.5506 | 0.6137 | 0.6387 | 0.6478 | 0.6499 | |
0.5184 | 0.5454 | 0.5539 | 0.5576 | 0.5590 | 0.5593 | |
0.7776 | 0.8181 | 0.8309 | 0.8364 | 0.8385 | 0.8390 |
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