March  2014, 7(1): 133-168. doi: 10.3934/krm.2014.7.133

Thermodynamical considerations implying wall/particles scattering kernels

1. 

IUSTI, 5 rue Enrico Fermi, 13453 Marseille CEDEX 13, France

Received  September 2011 Revised  March 2013 Published  December 2013

In this paper we study the thermodynamics of a rarefied gas contained in a closed vessel at constant volume. By adding axiomatic rules to the usual ones derived by Cercignani, we obtain a new symmetry property in the wall/particle scattering kernel. This new symmetry property enables us to show the first and second law of macroscopic thermodynamics for a rarefied gas having collisions with walls. Then we study the behavior of the rarefied gas when it is in contact with several (moving) thermostats at the same time. We show the existence, uniqueness and long time behavior of the solution to the homogeneous (linear) evolution equation describing the system. Finally we apply our thermodynamical model of rarefied gas to the measurement of heat flux in very low density systems and compare it to the experimental results, shading into light new interpretation of observed behaviors.
Citation: Vincent Pavan. Thermodynamical considerations implying wall/particles scattering kernels. Kinetic & Related Models, 2014, 7 (1) : 133-168. doi: 10.3934/krm.2014.7.133
References:
[1]

K. Aoki and F. Golse, On the speed of approach to equilibrium for collisionless gas, Kinet. Relat. Models, 4 (2011), 87-107. doi: 10.3934/krm.2011.4.87.  Google Scholar

[2]

W. Arendt, Positive semigroups of Kernel operators, Positivity, 12 (2008), 25-44. doi: 10.1007/s11117-007-2137-z.  Google Scholar

[3]

L. Arkeryd and A. Nouri, Boltzmann asymptotics with diffuse reflection boundary conditions, Monatsh. Math., 123 (1997), 285-298. doi: 10.1007/BF01326764.  Google Scholar

[4]

V. Bagland, P. Degond and M. Lemou, Moment systems derived from relativistic kinetic equations, J. Stat. Phys., 125 (2006), 621-659. doi: 10.1007/s10955-006-9173-0.  Google Scholar

[5]

J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups With Applications, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2006.  Google Scholar

[6]

C. Cercignani, Slow Rarefied Flows. Theory and Application to Micro-Electro-Mechanical Systems, Progress in Mathematical Physics, 41, Birkhäuser Verlag, Basel, 2006.  Google Scholar

[7]

C. Cercignani, The Boltzmann Equation and its Application, Applied Mathematical Sciences, 67, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[8]

S. K. Diadze and J. G. Méolans, Temperature jump and slip velocity calculations from an anisotropic scattering kernel, Physica A, 358 (2005), 328-346. doi: 10.1016/j.physa.2005.04.013.  Google Scholar

[9]

V. Keicher, Convergence of Positive $C_0$-Semigroups to Rotation Groups, Ph.D thesis, Eberhard Karls Universität Tübingen, 2008. Google Scholar

[10]

I. Kuscer, Reciprocity in scattering of gas molecules by surfaces, Surface Science, 25 (1971), 225-237. Google Scholar

[11]

J. Lebowitz and P. G. Bergmann, Irreversible Gibsian ensembles, Ann. Physics, 1 (1957), 1-23. doi: 10.1016/0003-4916(57)90002-7.  Google Scholar

[12]

C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552.  Google Scholar

[13]

C. Mouhot, Quantitative linearized study of the Boltzmann collision operator and applications,, Commun. Math. Sci., 2007 (): 73.  doi: 10.4310/CMS.2007.v5.n5.a6.  Google Scholar

[14]

B.-S. Tam, A cone theoretic approach to the spectral theory of positive linear operators: The finite dimensional case, Tawainese J. Math., 5 (2001), 207-277.  Google Scholar

[15]

H. Yamaguchi, K. Anazawa, Y. Matsuda, T. Niimi, A. Polikarpov and I. Graur, Investigation of heat transfer between two coaxial cylinders for measurement of thermal accomodation coefficient, Physics of Fluids, 24 (2012), 062002. doi: 10.1063/1.4726059.  Google Scholar

show all references

References:
[1]

K. Aoki and F. Golse, On the speed of approach to equilibrium for collisionless gas, Kinet. Relat. Models, 4 (2011), 87-107. doi: 10.3934/krm.2011.4.87.  Google Scholar

[2]

W. Arendt, Positive semigroups of Kernel operators, Positivity, 12 (2008), 25-44. doi: 10.1007/s11117-007-2137-z.  Google Scholar

[3]

L. Arkeryd and A. Nouri, Boltzmann asymptotics with diffuse reflection boundary conditions, Monatsh. Math., 123 (1997), 285-298. doi: 10.1007/BF01326764.  Google Scholar

[4]

V. Bagland, P. Degond and M. Lemou, Moment systems derived from relativistic kinetic equations, J. Stat. Phys., 125 (2006), 621-659. doi: 10.1007/s10955-006-9173-0.  Google Scholar

[5]

J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups With Applications, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2006.  Google Scholar

[6]

C. Cercignani, Slow Rarefied Flows. Theory and Application to Micro-Electro-Mechanical Systems, Progress in Mathematical Physics, 41, Birkhäuser Verlag, Basel, 2006.  Google Scholar

[7]

C. Cercignani, The Boltzmann Equation and its Application, Applied Mathematical Sciences, 67, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[8]

S. K. Diadze and J. G. Méolans, Temperature jump and slip velocity calculations from an anisotropic scattering kernel, Physica A, 358 (2005), 328-346. doi: 10.1016/j.physa.2005.04.013.  Google Scholar

[9]

V. Keicher, Convergence of Positive $C_0$-Semigroups to Rotation Groups, Ph.D thesis, Eberhard Karls Universität Tübingen, 2008. Google Scholar

[10]

I. Kuscer, Reciprocity in scattering of gas molecules by surfaces, Surface Science, 25 (1971), 225-237. Google Scholar

[11]

J. Lebowitz and P. G. Bergmann, Irreversible Gibsian ensembles, Ann. Physics, 1 (1957), 1-23. doi: 10.1016/0003-4916(57)90002-7.  Google Scholar

[12]

C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552.  Google Scholar

[13]

C. Mouhot, Quantitative linearized study of the Boltzmann collision operator and applications,, Commun. Math. Sci., 2007 (): 73.  doi: 10.4310/CMS.2007.v5.n5.a6.  Google Scholar

[14]

B.-S. Tam, A cone theoretic approach to the spectral theory of positive linear operators: The finite dimensional case, Tawainese J. Math., 5 (2001), 207-277.  Google Scholar

[15]

H. Yamaguchi, K. Anazawa, Y. Matsuda, T. Niimi, A. Polikarpov and I. Graur, Investigation of heat transfer between two coaxial cylinders for measurement of thermal accomodation coefficient, Physics of Fluids, 24 (2012), 062002. doi: 10.1063/1.4726059.  Google Scholar

[1]

Alexander V. Bobylev, Sergey V. Meleshko. On group symmetries of the hydrodynamic equations for rarefied gas. Kinetic & Related Models, 2021, 14 (3) : 469-482. doi: 10.3934/krm.2021012

[2]

Kazuo Aoki, Yoshiaki Abe. Stagnation-point flow of a rarefied gas impinging obliquely on a plane wall. Kinetic & Related Models, 2011, 4 (4) : 935-954. doi: 10.3934/krm.2011.4.935

[3]

Raffaele Esposito, Mario Pulvirenti. Rigorous validity of the Boltzmann equation for a thin layer of a rarefied gas. Kinetic & Related Models, 2010, 3 (2) : 281-297. doi: 10.3934/krm.2010.3.281

[4]

Nicolas Crouseilles, Giacomo Dimarco, Mohammed Lemou. Asymptotic preserving and time diminishing schemes for rarefied gas dynamic. Kinetic & Related Models, 2017, 10 (3) : 643-668. doi: 10.3934/krm.2017026

[5]

Sébastien Guisset. Angular moments models for rarefied gas dynamics. Numerical comparisons with kinetic and Navier-Stokes equations. Kinetic & Related Models, 2020, 13 (4) : 739-758. doi: 10.3934/krm.2020025

[6]

Giovanni Russo, Francis Filbet. Semilagrangian schemes applied to moving boundary problems for the BGK model of rarefied gas dynamics. Kinetic & Related Models, 2009, 2 (1) : 231-250. doi: 10.3934/krm.2009.2.231

[7]

Ali Akgül. Analysis and new applications of fractal fractional differential equations with power law kernel. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3401-3417. doi: 10.3934/dcdss.2020423

[8]

Elena Celledoni, Brynjulf Owren. Preserving first integrals with symmetric Lie group methods. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 977-990. doi: 10.3934/dcds.2014.34.977

[9]

Armand Bernou. A semigroup approach to the convergence rate of a collisionless gas. Kinetic & Related Models, 2020, 13 (6) : 1071-1106. doi: 10.3934/krm.2020038

[10]

Gerardo Hernández, Ernesto A. Lacomba. Are the geometries of the first and second laws of thermodynamics compatible?. Discrete & Continuous Dynamical Systems, 2013, 33 (3) : 1113-1116. doi: 10.3934/dcds.2013.33.1113

[11]

Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1285-1301. doi: 10.3934/cpaa.2012.11.1285

[12]

Jun Fan, Dao-Hong Xiang. Quantitative convergence analysis of kernel based large-margin unified machines. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4069-4083. doi: 10.3934/cpaa.2020180

[13]

Mingshang Hu, Xiaojuan Li, Xinpeng Li. Convergence rate of Peng’s law of large numbers under sublinear expectations. Probability, Uncertainty and Quantitative Risk, 2021, 6 (3) : 261-266. doi: 10.3934/puqr.2021013

[14]

Masashi Ohnawa. Convergence rates towards the traveling waves for a model system of radiating gas with discontinuities. Kinetic & Related Models, 2012, 5 (4) : 857-872. doi: 10.3934/krm.2012.5.857

[15]

Steinar Evje, Kenneth Hvistendahl Karlsen. Global weak solutions for a viscous liquid-gas model with singular pressure law. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1867-1894. doi: 10.3934/cpaa.2009.8.1867

[16]

Steinar Evje, Huanyao Wen. Weak solutions of a gas-liquid drift-flux model with general slip law for wellbore operations. Discrete & Continuous Dynamical Systems, 2013, 33 (10) : 4497-4530. doi: 10.3934/dcds.2013.33.4497

[17]

Fabio Camilli, Francisco Silva. A semi-discrete approximation for a first order mean field game problem. Networks & Heterogeneous Media, 2012, 7 (2) : 263-277. doi: 10.3934/nhm.2012.7.263

[18]

Nicolás Borda, Javier Fernández, Sergio Grillo. Discrete second order constrained Lagrangian systems: First results. Journal of Geometric Mechanics, 2013, 5 (4) : 381-397. doi: 10.3934/jgm.2013.5.381

[19]

Adolfo Damiano Cafaro, Simone Fiori. Optimization of a control law to synchronize first-order dynamical systems on Riemannian manifolds by a transverse component. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021213

[20]

Joachim Escher, Rossen Ivanov, Boris Kolev. Euler equations on a semi-direct product of the diffeomorphisms group by itself. Journal of Geometric Mechanics, 2011, 3 (3) : 313-322. doi: 10.3934/jgm.2011.3.313

2020 Impact Factor: 1.432

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]