# American Institute of Mathematical Sciences

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

## Gas-surface interaction and boundary conditions for the Boltzmann equation

 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].
Citation: Stéphane Brull, Pierre Charrier, Luc Mieussens. Gas-surface interaction and boundary conditions for the Boltzmann equation. Kinetic and 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. [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. [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. [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. [5] H. Babovsky, Derivation of stochastic reflection laws from specular reflection, Trans. Th. and Stat. Phys., 16 (1987), 113-126. doi: 10.1080/00411458708204299. [6] J. J. M. Beenakker, Reduced Dimensionality in Gases in Nanopores, Phys. Low-Dim. Struct., (1995), 115-124. [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. [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. [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. [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). [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). [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. [13] C. Cercignani, The Boltzman Equation and Its Applications, Springer, Berlin, 1988. doi: 10.1007/978-1-4612-1039-9. [14] C. Cercignani, Scattering kernels for gas-surface interactions, Transp. Th. and Stat. Phys., 2 (1972), 27-53. [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. [16] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer: New York, 1994, 133-163. [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. [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. [19] S. Chandrasekhar, Radiative Transfer, Dover Publications, Inc., New York, 1960 [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. [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. [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. [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. [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. [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. [26] F. Golse, Knudsen layers from a computational viewpoint, Transport Theory Statist. Phys., 21 (1992), 211-236. doi: 10.1080/00411459208203921. [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. [28] G. Karniadakis, A. Beskok and N. Aluru, Microflows and Nanoflows, Springer, 2005. [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. [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. [31] S. Yu Krylov, Molecular transport in Sub-Nano-Scale systems, RGD, 663 (2003), 735. doi: 10.1063/1.1581616. [32] J. C. Maxwell, On stresses in rarefied gases arising from inequalities of temperature, Phil. Trans. Royal Soc., Appendix (1879), 231-256. [33] F. Poupaud, Diffusion approximation of the linear semiconductor Boltzmann equation: Analysis of boundary layers, Asymptotic Anal., 4 (1991), 293-317. [34] Y. Sone, Kinetic Theory and Fluid Dynamics, Birkäuser, 2002. doi: 10.1007/978-1-4612-0061-1. [35] Y. Sone, Molecular Gas Dynamics, Birkäuser, 2007. doi: 10.1007/978-0-8176-4573-1. [36] H. Struchtrup, Maxwell boundary condition and velocity dependent accommodation coefficients, Phys. Fluids, 25 (2013), 112001. doi: 10.1063/1.4829907.

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. [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. [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. [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. [5] H. Babovsky, Derivation of stochastic reflection laws from specular reflection, Trans. Th. and Stat. Phys., 16 (1987), 113-126. doi: 10.1080/00411458708204299. [6] J. J. M. Beenakker, Reduced Dimensionality in Gases in Nanopores, Phys. Low-Dim. Struct., (1995), 115-124. [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. [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. [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. [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). [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). [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. [13] C. Cercignani, The Boltzman Equation and Its Applications, Springer, Berlin, 1988. doi: 10.1007/978-1-4612-1039-9. [14] C. Cercignani, Scattering kernels for gas-surface interactions, Transp. Th. and Stat. Phys., 2 (1972), 27-53. [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. [16] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer: New York, 1994, 133-163. [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. [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. [19] S. Chandrasekhar, Radiative Transfer, Dover Publications, Inc., New York, 1960 [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. [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. [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. [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. [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. [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. [26] F. Golse, Knudsen layers from a computational viewpoint, Transport Theory Statist. Phys., 21 (1992), 211-236. doi: 10.1080/00411459208203921. [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. [28] G. Karniadakis, A. Beskok and N. Aluru, Microflows and Nanoflows, Springer, 2005. [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. [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. [31] S. Yu Krylov, Molecular transport in Sub-Nano-Scale systems, RGD, 663 (2003), 735. doi: 10.1063/1.1581616. [32] J. C. Maxwell, On stresses in rarefied gases arising from inequalities of temperature, Phil. Trans. Royal Soc., Appendix (1879), 231-256. [33] F. Poupaud, Diffusion approximation of the linear semiconductor Boltzmann equation: Analysis of boundary layers, Asymptotic Anal., 4 (1991), 293-317. [34] Y. Sone, Kinetic Theory and Fluid Dynamics, Birkäuser, 2002. doi: 10.1007/978-1-4612-0061-1. [35] Y. Sone, Molecular Gas Dynamics, Birkäuser, 2007. doi: 10.1007/978-0-8176-4573-1. [36] H. Struchtrup, Maxwell boundary condition and velocity dependent accommodation coefficients, Phys. Fluids, 25 (2013), 112001. doi: 10.1063/1.4829907.
 [1] Kazuo Aoki, Pierre Charrier, Pierre Degond. A hierarchy of models related to nanoflows and surface diffusion. Kinetic and Related Models, 2011, 4 (1) : 53-85. doi: 10.3934/krm.2011.4.53 [2] Darryl D. Holm, Vakhtang Putkaradze, Cesare Tronci. Collisionless kinetic theory of rolling molecules. Kinetic and Related Models, 2013, 6 (2) : 429-458. doi: 10.3934/krm.2013.6.429 [3] Emmanuel Frénod, Mathieu Lutz. On the Geometrical Gyro-Kinetic theory. Kinetic and Related Models, 2014, 7 (4) : 621-659. doi: 10.3934/krm.2014.7.621 [4] Paolo Barbante, Aldo Frezzotti, Livio Gibelli. A kinetic theory description of liquid menisci at the microscale. Kinetic and Related Models, 2015, 8 (2) : 235-254. doi: 10.3934/krm.2015.8.235 [5] Hung-Wen Kuo. Effect of abrupt change of the wall temperature in the kinetic theory. Kinetic and Related Models, 2019, 12 (4) : 765-789. doi: 10.3934/krm.2019030 [6] José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic and Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401 [7] Arnaud Debussche, Julien Vovelle. Diffusion limit for a stochastic kinetic problem. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2305-2326. doi: 10.3934/cpaa.2012.11.2305 [8] Joachim Escher, Piotr B. Mucha. The surface diffusion flow on rough phase spaces. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 431-453. doi: 10.3934/dcds.2010.26.431 [9] Jeremy LeCrone, Yuanzhen Shao, Gieri Simonett. The surface diffusion and the Willmore flow for uniformly regular hypersurfaces. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3503-3524. doi: 10.3934/dcdss.2020242 [10] Emeric Bouin, Jean Dolbeault, Christian Schmeiser. Diffusion and kinetic transport with very weak confinement. Kinetic and Related Models, 2020, 13 (2) : 345-371. doi: 10.3934/krm.2020012 [11] Giada Basile, Tomasz Komorowski, Stefano Olla. Diffusion limit for a kinetic equation with a thermostatted interface. Kinetic and Related Models, 2019, 12 (5) : 1185-1196. doi: 10.3934/krm.2019045 [12] Laurent Boudin, Bérénice Grec, Milana Pavić, Francesco Salvarani. Diffusion asymptotics of a kinetic model for gaseous mixtures. Kinetic and Related Models, 2013, 6 (1) : 137-157. doi: 10.3934/krm.2013.6.137 [13] Ho-Youn Kim, Yong-Jung Kim, Hyun-Jin Lim. Heterogeneous discrete kinetic model and its diffusion limit. Kinetic and Related Models, 2021, 14 (5) : 749-765. doi: 10.3934/krm.2021023 [14] Daewa Kim, Annalisa Quaini. A kinetic theory approach to model pedestrian dynamics in bounded domains with obstacles. Kinetic and Related Models, 2019, 12 (6) : 1273-1296. doi: 10.3934/krm.2019049 [15] José A. Carrillo, M. R. D’Orsogna, V. Panferov. Double milling in self-propelled swarms from kinetic theory. Kinetic and Related Models, 2009, 2 (2) : 363-378. doi: 10.3934/krm.2009.2.363 [16] Marzia Bisi, Tommaso Ruggeri, Giampiero Spiga. Dynamical pressure in a polyatomic gas: Interplay between kinetic theory and extended thermodynamics. Kinetic and Related Models, 2018, 11 (1) : 71-95. doi: 10.3934/krm.2018004 [17] Carlos Escudero, Fabricio Macià, Raúl Toral, Juan J. L. Velázquez. Kinetic theory and numerical simulations of two-species coagulation. Kinetic and Related Models, 2014, 7 (2) : 253-290. doi: 10.3934/krm.2014.7.253 [18] Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic and Related Models, 2021, 14 (1) : 115-148. doi: 10.3934/krm.2020051 [19] Pedro Aceves-Sánchez, Christian Schmeiser. Fractional diffusion limit of a linear kinetic equation in a bounded domain. Kinetic and Related Models, 2017, 10 (3) : 541-551. doi: 10.3934/krm.2017021 [20] H.J. Hwang, K. Kang, A. Stevens. Drift-diffusion limits of kinetic models for chemotaxis: A generalization. Discrete and Continuous Dynamical Systems - B, 2005, 5 (2) : 319-334. doi: 10.3934/dcdsb.2005.5.319

2021 Impact Factor: 1.398