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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), 53-85. 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), 304–-334, (electronic). 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), 117103. 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), 026102. 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), 113-126. doi: 10.1080/00411458708204299.  Google Scholar

[6]

J. J. M. Beenakker, Reduced Dimensionality in Gases in Nanopores, Phys. Low-Dim. Struct., (1995), 115-124. 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), 514. 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), 379-382. 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), 4015. 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, Berlin, 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), 27-53. Google Scholar

[15]

C. Cercignani, Scattering kernels for gas-surface interaction, Proceedings of the Workshop on Hypersonic Flows for Reentry Problems, I (1990), 9-29, INRIA Antibes. Google Scholar

[16]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer: New York, 1994, 133-163.  Google Scholar

[17]

C. Cercignani and M. Lampis, Kinetic models for gas-surface interactions, Transp. Th. and Stat. Phys., 1 (1971), 101-114. 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), 1319-1336. doi: 10.1080/00411459508206026.  Google Scholar

[19]

S. Chandrasekhar, Radiative Transfer, Dover Publications, Inc., New York, 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), 124–-157 (electronic). 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), 409–-435. 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, Collége de France Seminar, XIV (1997/1998), Stud. Math. Appl. 31, North Holland, Amsterdam, (2002), 273-296. 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), 589-636. 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), 364-392. 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), 37-62. doi: 10.1081/TT-120018651.  Google Scholar

[26]

F. Golse, Knudsen layers from a computational viewpoint, Transport Theory Statist. Phys., 21 (1992), 211-236. 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), 1033–-1061. 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), 2032-2050. 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), 735. 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., Appendix (1879), 231-256. Google Scholar

[33]

F. Poupaud, Diffusion approximation of the linear semiconductor Boltzmann equation: Analysis of boundary layers, Asymptotic Anal., 4 (1991), 293-317.  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), 112001. 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), 53-85. 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), 304–-334, (electronic). 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), 117103. 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), 026102. 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), 113-126. doi: 10.1080/00411458708204299.  Google Scholar

[6]

J. J. M. Beenakker, Reduced Dimensionality in Gases in Nanopores, Phys. Low-Dim. Struct., (1995), 115-124. 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), 514. 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), 379-382. 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), 4015. 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, Berlin, 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), 27-53. Google Scholar

[15]

C. Cercignani, Scattering kernels for gas-surface interaction, Proceedings of the Workshop on Hypersonic Flows for Reentry Problems, I (1990), 9-29, INRIA Antibes. Google Scholar

[16]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer: New York, 1994, 133-163.  Google Scholar

[17]

C. Cercignani and M. Lampis, Kinetic models for gas-surface interactions, Transp. Th. and Stat. Phys., 1 (1971), 101-114. 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), 1319-1336. doi: 10.1080/00411459508206026.  Google Scholar

[19]

S. Chandrasekhar, Radiative Transfer, Dover Publications, Inc., New York, 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), 124–-157 (electronic). 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), 409–-435. 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, Collége de France Seminar, XIV (1997/1998), Stud. Math. Appl. 31, North Holland, Amsterdam, (2002), 273-296. 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), 589-636. 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), 364-392. 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), 37-62. doi: 10.1081/TT-120018651.  Google Scholar

[26]

F. Golse, Knudsen layers from a computational viewpoint, Transport Theory Statist. Phys., 21 (1992), 211-236. 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), 1033–-1061. 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), 2032-2050. 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), 735. 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., Appendix (1879), 231-256. Google Scholar

[33]

F. Poupaud, Diffusion approximation of the linear semiconductor Boltzmann equation: Analysis of boundary layers, Asymptotic Anal., 4 (1991), 293-317.  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), 112001. doi: 10.1063/1.4829907.  Google Scholar

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