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

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, 2020, 13 (6) : 1163-1174. doi: 10.3934/krm.2020041
References:
[1]

P. AndriesP. Le TallecJ.-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. BhatnagarE. 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. FunaganeS. TakataK. 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, (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. MengL. WuJ. 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. TakataH. 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

show all references

References:
[1]

P. AndriesP. Le TallecJ.-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. BhatnagarE. 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. FunaganeS. TakataK. 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, (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. MengL. WuJ. 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. TakataH. 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

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
$ 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
[1]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, 2021, 14 (1) : 115-148. doi: 10.3934/krm.2020051

[2]

Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032

[3]

François Dubois. Third order equivalent equation of lattice Boltzmann scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 221-248. doi: 10.3934/dcds.2009.23.221

[4]

Tong Yang, Seiji Ukai, Huijiang Zhao. Stationary solutions to the exterior problems for the Boltzmann equation, I. Existence. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 495-520. doi: 10.3934/dcds.2009.23.495

[5]

Sabine Hittmeir, Laura Kanzler, Angelika Manhart, Christian Schmeiser. Kinetic modelling of colonies of myxobacteria. Kinetic & Related Models, 2021, 14 (1) : 1-24. doi: 10.3934/krm.2020046

[6]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[7]

Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020284

[8]

Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374

[9]

Editorial Office. Retraction: Xiao-Qian Jiang and Lun-Chuan Zhang, A pricing option approach based on backward stochastic differential equation theory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 969-969. doi: 10.3934/dcdss.2019065

[10]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[11]

Gi-Chan Bae, Christian Klingenberg, Marlies Pirner, Seok-Bae Yun. BGK model of the multi-species Uehling-Uhlenbeck equation. Kinetic & Related Models, 2021, 14 (1) : 25-44. doi: 10.3934/krm.2020047

[12]

Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021003

[13]

Xiaoming Wang. Upper semi-continuity of stationary statistical properties of dissipative systems. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 521-540. doi: 10.3934/dcds.2009.23.521

[14]

Hongfei Yang, Xiaofeng Ding, Raymond Chan, Hui Hu, Yaxin Peng, Tieyong Zeng. A new initialization method based on normed statistical spaces in deep networks. Inverse Problems & Imaging, 2021, 15 (1) : 147-158. doi: 10.3934/ipi.2020045

[15]

Wei Ouyang, Li Li. Hölder strong metric subregularity and its applications to convergence analysis of inexact Newton methods. Journal of Industrial & Management Optimization, 2021, 17 (1) : 169-184. doi: 10.3934/jimo.2019105

[16]

Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020395

[17]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[18]

Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3171-3200. doi: 10.3934/dcds.2020054

[19]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292

[20]

Weihong Guo, Yifei Lou, Jing Qin, Ming Yan. IPI special issue on "mathematical/statistical approaches in data science" in the Inverse Problem and Imaging. Inverse Problems & Imaging, 2021, 15 (1) : I-I. doi: 10.3934/ipi.2021007

2019 Impact Factor: 1.311

Metrics

  • PDF downloads (65)
  • HTML views (139)
  • Cited by (0)

[Back to Top]