American Institute of Mathematical Sciences

Advanced
June  2014, 7(2): 219-251. doi: 10.3934/krm.2014.7.219

Gas-surface interaction and boundary conditions for the Boltzmann equation

Stéphane Brull 1, , Pierre Charrier 2, and  Luc Mieussens 1,

1. 

Institut de Mathématiques de Bordeaux, ENSEIRB-MATMECA, IPB, Université de Bordeaux, F-33405 Talence cedex, France, France
2. 

Institut de Mathméatiques de Bordeaux, ENSEIRB-MATMECA, IPB, Université de Bordeaux, F-33405 Talence cedex

Received  May 2013 Revised  February 2014 Published  March 2014

In this paper we revisit the derivation of boundary conditions for the Boltzmann Equation. The interaction between the wall atoms and the gas molecules within a thin surface layer is described by a kinetic equation introduced in [10] and used in [1]. This equation includes a Vlasov term and a linear molecule-phonon collision term and is coupled with the Boltzmann equation describing the evolution of the gas in the bulk flow. Boundary conditions are formally derived from this model by using classical tools of kinetic theory such as scaling and systematic asymptotic expansion. In a first step this method is applied to the simplified case of a flat wall. Then it is extented to walls with nanoscale roughness allowing to obtain more complex scattering patterns related to the morphology of the wall. It is proved that the obtained scattering kernels satisfy the classical imposed properties of non-negativeness, normalization and reciprocity introduced by Cercignani [13].
Keywords: Kinetic theory, surface diffusion., nanoflows.
Mathematics Subject Classification: Primary: 82C40, 76P05, 41A60, 82D05; Secondary: 74A2.
Citation: Stéphane Brull, Pierre Charrier, Luc Mieussens. Gas-surface interaction and boundary conditions for the Boltzmann equation. Kinetic & Related Models, 2014, 7 (2) : 219-251. doi: 10.3934/krm.2014.7.219
References:
[1]

K. Aoki, P. Charrier and P. Degond, A hierarchy of models related to nanoflows and surface diffusion,, Kinetic and Related Models, 4 (2011), 53.  doi: 10.3934/krm.2011.4.53.  Google Scholar

[2]

K. Aoki and P. Degond, Homogenization of a flow in a periodic channel of small section,, Multiscale Model. Simul., 1 (2003).  doi: 10.1137/S1540345902409931.  Google Scholar

[3]

K. Aoki, P. Degond, S. Takata and H. Yoshida, Diffusion models for Knudsen compressors,, Phys. Fluids, 19 (2007).  doi: 10.1063/1.2798748.  Google Scholar

[4]

G. Arya, H.-C. Chang and E. Magin, Knudsen Diffusivity of a Hard Sphere in a Rough Slit Pore,, Phys. Rev. Lett., 91 (2003).  doi: 10.1103/PhysRevLett.91.026102.  Google Scholar

[5]

H. Babovsky, Derivation of stochastic reflection laws from specular reflection,, Trans. Th. and Stat. Phys., 16 (1987), 113.  doi: 10.1080/00411458708204299.  Google Scholar

[6]

J. J. M. Beenakker, Reduced Dimensionality in Gases in Nanopores,, Phys. Low-Dim. Struct., (1995), 115.   Google Scholar

[7]

J. J. M. Beenakker, V. D. Borman and S. Yu Krylov, Molecular Transport in the Nanometer Regime,, Phys. Rev. Lett., 72 (1994).  doi: 10.1103/PhysRevLett.72.514.  Google Scholar

[8]

J. J. M. Beenakker, V. D. Borman and S. Yu. Krylov1, Molecular transport in subnanometer pores: Zero-point energy, reduced dimensionality and quantum sieving,, Chem. Phys. Letters, 232 (1995), 379.  doi: 10.1016/0009-2614(94)01372-3.  Google Scholar

[9]

J. J. M. Beenakker and S. Yu. Krylov, One-dimensional surface diffusion: Density dependence in a smooth potential,, J. Chem. Phys., 107 (1997).  doi: 10.1063/1.474757.  Google Scholar

[10]

V. D. Borman, S. Yu. Krylov and A. V. Prosyanov, Theory of nonequilibrium phenomena at a gas-solid interface,, Sov. Phys. JETP, 67 (1988).   Google Scholar

[11]

V. D. Borman, S. Yu Krylov and A. V. Prosyanov, Fundamental role of unbound surface particles in transport phenomena along a gas-solid interface,, Sov. Phys. JETP, 70 (1990).   Google Scholar

[12]

F. Celestini and F. Mortessagne, The cosine law at the atomic scale: Toward realistic simulations of Knudsen diffusion,, Phys.Rev. E, 77 (2008).  doi: 10.1103/PhysRevE.77.021202.  Google Scholar

[13]

C. Cercignani, The Boltzman Equation and Its Applications,, Springer, (1988).  doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[14]

C. Cercignani, Scattering kernels for gas-surface interactions,, Transp. Th. and Stat. Phys., 2 (1972), 27.   Google Scholar

[15]

C. Cercignani, Scattering kernels for gas-surface interaction,, Proceedings of the Workshop on Hypersonic Flows for Reentry Problems, I (1990), 9.   Google Scholar

[16]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases,, Springer: New York, (1994), 133.   Google Scholar

[17]

C. Cercignani and M. Lampis, Kinetic models for gas-surface interactions,, Transp. Th. and Stat. Phys., 1 (1971), 101.  doi: 10.1080/00411457108231440.  Google Scholar

[18]

C. Cercignani, M. Lampis and A. Lentati, A new scattering kernel in kinetic theory of gases,, Trans. Theory Statist. Phys., 24 (1995), 1319.  doi: 10.1080/00411459508206026.  Google Scholar

[19]

S. Chandrasekhar, Radiative Transfer,, Dover Publications, (1960).   Google Scholar

[20]

P. Charrier and B. Dubroca, Asymptotic transport models for heat and mass transfer in reactive porous media,, Multiscale Model. Simul., 2 (2003).  doi: 10.1137/S1540345902411736.  Google Scholar

[21]

F. Coron, F. Golse and C. Sulem, A classification of well-posed kinetic layer problems,, Commun. Pure Appl. Math., 41 (1988).  doi: 10.1002/cpa.3160410403.  Google Scholar

[22]

P. Degond, Transport of trapped particles in a surface potential,, in Nonlinear Partial Differential Equations and Their Applications, XIV (2002), 273.  doi: 10.1016/S0168-2024(02)80014-5.  Google Scholar

[23]

P. Degond and S. Mas-Gallic, Existence of solutions and diffusion approximation for a model Fokker-Planck equation,, Transport Theory Statist. Phys., 16 (1987), 589.  doi: 10.1080/00411458708204307.  Google Scholar

[24]

P. Degond, C. Parzani and M.-H. Vignal, A Boltzmann Model for Trapped Particles in a Surface Potential,, SIAM J. Multiscale model. Simul., 5 (2006), 364.  doi: 10.1137/050642897.  Google Scholar

[25]

L. Falk, Existence of solutions to the stationary linear boltzmann equation,, Transport Theory and Statistical Physics, 32 (2003), 37.  doi: 10.1081/TT-120018651.  Google Scholar

[26]

F. Golse, Knudsen layers from a computational viewpoint,, Transport Theory Statist. Phys., 21 (1992), 211.  doi: 10.1080/00411459208203921.  Google Scholar

[27]

F. Golse and A. Klar, A numerical method for computing asymptotic states and outgoing distributions for kinetic linear Half-Space problems,, J. of Sta. Physics, 80 (1995).  doi: 10.1007/BF02179863.  Google Scholar

[28]

G. Karniadakis, A. Beskok and N. Aluru, Microflows and Nanoflows,, Springer, (2005).   Google Scholar

[29]

A. Klar, Asymptotic-induced domain decomposition methods for kinetic and drift-diffusion equations,, SIAM J. Sci. Comput., 19 (1998), 2032.  doi: 10.1137/S1064827595286177.  Google Scholar

[30]

S. Yu. Krylov, A. V. Prosyanov and J. J. M. Beenakker, One dimensional surface diffusion. II. Density dependence in a corrugated potential,, J. Chem. Phys., 107 (1997).  doi: 10.1063/1.474937.  Google Scholar

[31]

S. Yu Krylov, Molecular transport in Sub-Nano-Scale systems,, RGD, 663 (2003).  doi: 10.1063/1.1581616.  Google Scholar

[32]

J. C. Maxwell, On stresses in rarefied gases arising from inequalities of temperature,, Phil. Trans. Royal Soc., (1879), 231.   Google Scholar

[33]

F. Poupaud, Diffusion approximation of the linear semiconductor Boltzmann equation: Analysis of boundary layers,, Asymptotic Anal., 4 (1991), 293.   Google Scholar

[34]

Y. Sone, Kinetic Theory and Fluid Dynamics,, Birkäuser, (2002).  doi: 10.1007/978-1-4612-0061-1.  Google Scholar

[35]

Y. Sone, Molecular Gas Dynamics,, Birkäuser, (2007).  doi: 10.1007/978-0-8176-4573-1.  Google Scholar

[36]

H. Struchtrup, Maxwell boundary condition and velocity dependent accommodation coefficients,, Phys. Fluids, 25 (2013).  doi: 10.1063/1.4829907.  Google Scholar

show all references

References:
[1]

K. Aoki, P. Charrier and P. Degond, A hierarchy of models related to nanoflows and surface diffusion,, Kinetic and Related Models, 4 (2011), 53.  doi: 10.3934/krm.2011.4.53.  Google Scholar

[2]

K. Aoki and P. Degond, Homogenization of a flow in a periodic channel of small section,, Multiscale Model. Simul., 1 (2003).  doi: 10.1137/S1540345902409931.  Google Scholar

[3]

K. Aoki, P. Degond, S. Takata and H. Yoshida, Diffusion models for Knudsen compressors,, Phys. Fluids, 19 (2007).  doi: 10.1063/1.2798748.  Google Scholar

[4]

G. Arya, H.-C. Chang and E. Magin, Knudsen Diffusivity of a Hard Sphere in a Rough Slit Pore,, Phys. Rev. Lett., 91 (2003).  doi: 10.1103/PhysRevLett.91.026102.  Google Scholar

[5]

H. Babovsky, Derivation of stochastic reflection laws from specular reflection,, Trans. Th. and Stat. Phys., 16 (1987), 113.  doi: 10.1080/00411458708204299.  Google Scholar

[6]

J. J. M. Beenakker, Reduced Dimensionality in Gases in Nanopores,, Phys. Low-Dim. Struct., (1995), 115.   Google Scholar

[7]

J. J. M. Beenakker, V. D. Borman and S. Yu Krylov, Molecular Transport in the Nanometer Regime,, Phys. Rev. Lett., 72 (1994).  doi: 10.1103/PhysRevLett.72.514.  Google Scholar

[8]

J. J. M. Beenakker, V. D. Borman and S. Yu. Krylov1, Molecular transport in subnanometer pores: Zero-point energy, reduced dimensionality and quantum sieving,, Chem. Phys. Letters, 232 (1995), 379.  doi: 10.1016/0009-2614(94)01372-3.  Google Scholar

[9]

J. J. M. Beenakker and S. Yu. Krylov, One-dimensional surface diffusion: Density dependence in a smooth potential,, J. Chem. Phys., 107 (1997).  doi: 10.1063/1.474757.  Google Scholar

[10]

V. D. Borman, S. Yu. Krylov and A. V. Prosyanov, Theory of nonequilibrium phenomena at a gas-solid interface,, Sov. Phys. JETP, 67 (1988).   Google Scholar

[11]

V. D. Borman, S. Yu Krylov and A. V. Prosyanov, Fundamental role of unbound surface particles in transport phenomena along a gas-solid interface,, Sov. Phys. JETP, 70 (1990).   Google Scholar

[12]

F. Celestini and F. Mortessagne, The cosine law at the atomic scale: Toward realistic simulations of Knudsen diffusion,, Phys.Rev. E, 77 (2008).  doi: 10.1103/PhysRevE.77.021202.  Google Scholar

[13]

C. Cercignani, The Boltzman Equation and Its Applications,, Springer, (1988).  doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[14]

C. Cercignani, Scattering kernels for gas-surface interactions,, Transp. Th. and Stat. Phys., 2 (1972), 27.   Google Scholar

[15]

C. Cercignani, Scattering kernels for gas-surface interaction,, Proceedings of the Workshop on Hypersonic Flows for Reentry Problems, I (1990), 9.   Google Scholar

[16]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases,, Springer: New York, (1994), 133.   Google Scholar

[17]

C. Cercignani and M. Lampis, Kinetic models for gas-surface interactions,, Transp. Th. and Stat. Phys., 1 (1971), 101.  doi: 10.1080/00411457108231440.  Google Scholar

[18]

C. Cercignani, M. Lampis and A. Lentati, A new scattering kernel in kinetic theory of gases,, Trans. Theory Statist. Phys., 24 (1995), 1319.  doi: 10.1080/00411459508206026.  Google Scholar

[19]

S. Chandrasekhar, Radiative Transfer,, Dover Publications, (1960).   Google Scholar

[20]

P. Charrier and B. Dubroca, Asymptotic transport models for heat and mass transfer in reactive porous media,, Multiscale Model. Simul., 2 (2003).  doi: 10.1137/S1540345902411736.  Google Scholar

[21]

F. Coron, F. Golse and C. Sulem, A classification of well-posed kinetic layer problems,, Commun. Pure Appl. Math., 41 (1988).  doi: 10.1002/cpa.3160410403.  Google Scholar

[22]

P. Degond, Transport of trapped particles in a surface potential,, in Nonlinear Partial Differential Equations and Their Applications, XIV (2002), 273.  doi: 10.1016/S0168-2024(02)80014-5.  Google Scholar

[23]

P. Degond and S. Mas-Gallic, Existence of solutions and diffusion approximation for a model Fokker-Planck equation,, Transport Theory Statist. Phys., 16 (1987), 589.  doi: 10.1080/00411458708204307.  Google Scholar

[24]

P. Degond, C. Parzani and M.-H. Vignal, A Boltzmann Model for Trapped Particles in a Surface Potential,, SIAM J. Multiscale model. Simul., 5 (2006), 364.  doi: 10.1137/050642897.  Google Scholar

[25]

L. Falk, Existence of solutions to the stationary linear boltzmann equation,, Transport Theory and Statistical Physics, 32 (2003), 37.  doi: 10.1081/TT-120018651.  Google Scholar

[26]

F. Golse, Knudsen layers from a computational viewpoint,, Transport Theory Statist. Phys., 21 (1992), 211.  doi: 10.1080/00411459208203921.  Google Scholar

[27]

F. Golse and A. Klar, A numerical method for computing asymptotic states and outgoing distributions for kinetic linear Half-Space problems,, J. of Sta. Physics, 80 (1995).  doi: 10.1007/BF02179863.  Google Scholar

[28]

G. Karniadakis, A. Beskok and N. Aluru, Microflows and Nanoflows,, Springer, (2005).   Google Scholar

[29]

A. Klar, Asymptotic-induced domain decomposition methods for kinetic and drift-diffusion equations,, SIAM J. Sci. Comput., 19 (1998), 2032.  doi: 10.1137/S1064827595286177.  Google Scholar

[30]

S. Yu. Krylov, A. V. Prosyanov and J. J. M. Beenakker, One dimensional surface diffusion. II. Density dependence in a corrugated potential,, J. Chem. Phys., 107 (1997).  doi: 10.1063/1.474937.  Google Scholar

[31]

S. Yu Krylov, Molecular transport in Sub-Nano-Scale systems,, RGD, 663 (2003).  doi: 10.1063/1.1581616.  Google Scholar

[32]

J. C. Maxwell, On stresses in rarefied gases arising from inequalities of temperature,, Phil. Trans. Royal Soc., (1879), 231.   Google Scholar

[33]

F. Poupaud, Diffusion approximation of the linear semiconductor Boltzmann equation: Analysis of boundary layers,, Asymptotic Anal., 4 (1991), 293.   Google Scholar

[34]

Y. Sone, Kinetic Theory and Fluid Dynamics,, Birkäuser, (2002).  doi: 10.1007/978-1-4612-0061-1.  Google Scholar

[35]

Y. Sone, Molecular Gas Dynamics,, Birkäuser, (2007).  doi: 10.1007/978-0-8176-4573-1.  Google Scholar

[36]

H. Struchtrup, Maxwell boundary condition and velocity dependent accommodation coefficients,, Phys. Fluids, 25 (2013).  doi: 10.1063/1.4829907.  Google Scholar

[1]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051
[2]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458
[3]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326
[4]

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
[5]

Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020350
[6]

Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336
[7]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024
[8]

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
[9]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316
[10]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321
[11]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354
[12]

Puneet Pasricha, Anubha Goel. Pricing power exchange options with hawkes jump diffusion processes. Journal of Industrial & Management Optimization, 2021, 17 (1) : 133-149. doi: 10.3934/jimo.2019103
[13]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319
[14]

S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435
[15]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433
[16]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242
[17]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118
[18]

Thomas Frenzel, Matthias Liero. Effective diffusion in thin structures via generalized gradient systems and EDP-convergence. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 395-425. doi: 10.3934/dcdss.2020345

2019 Impact Factor: 1.311

Tools

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences