Gas-surface interaction and boundary conditions for the Boltzmann equation

Stéphane Brull; Pierre Charrier; Luc Mieussens

doi:10.3934/krm.2014.7.219

Previous Article
Kinetic theory and numerical simulations of two-species coagulation
KRM Home
This Issue
Next Article
Non-existence and non-uniqueness for multidimensional sticky particle systems

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

Abstract
Related Papers

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]	Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311
[2]	Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1717-1746. doi: 10.3934/dcdss.2020451
[3]	Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243
[4]	Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009
[5]	Andrea Tosin, Mattia Zanella. Uncertainty damping in kinetic traffic models by driver-assist controls. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021018
[6]	W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349
[7]	Dayalal Suthar, Sunil Dutt Purohit, Haile Habenom, Jagdev Singh. Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021019
[8]	Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109
[9]	Guangying Lv, Jinlong Wei, Guang-an Zou. Noise and stability in reaction-diffusion equations. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021005
[10]	John Leventides, Costas Poulios, Georgios Alkis Tsiatsios, Maria Livada, Stavros Tsipras, Konstantinos Lefcaditis, Panagiota Sargenti, Aleka Sargenti. Systems theory and analysis of the implementation of non pharmaceutical policies for the mitigation of the COVID-19 pandemic. Journal of Dynamics & Games, 2021 doi: 10.3934/jdg.2021004
[11]	Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367
[12]	Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
[13]	Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258
[14]	Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223
[15]	Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2599-2623. doi: 10.3934/dcdsb.2020197
[16]	Nabahats Dib-Baghdadli, Rabah Labbas, Tewfik Mahdjoub, Ahmed Medeghri. On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021004
[17]	Meiqiao Ai, Zhimin Zhang, Wenguang Yu. First passage problems of refracted jump diffusion processes and their applications in valuing equity-linked death benefits. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021039
[18]	Vo Anh Khoa, Thi Kim Thoa Thieu, Ekeoma Rowland Ijioma. On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2451-2477. doi: 10.3934/dcdsb.2020190
[19]	Bo Duan, Zhengce Zhang. A reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021067

2019 Impact Factor: 1.311

American Institute of Mathematical Sciences

Gas-surface interaction and boundary conditions for the Boltzmann equation

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Gas-surface interaction and boundary conditions for the Boltzmann equation

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors