Homogenization of a flow in a periodic channel of small section

Multiscale Modeling and Simulation

Volumn 1, Issue 2, 2003, Pages 304-334

  Aoki, K ^a   Degond, P ^b  

^a KYOTO UNIVERSITY   (Japan)

^b INSTITUT DE MATHÉMATIQUES DE TOULOUSE   (France)

Author keywords

Boltzmann equation; Convection diffusion model; Diffusion approximation; Free molecular flow; Homogenization; Rarefied gas flow

Indexed keywords

BOLTZMANN EQUATION; DIFFUSION; DIFFUSION IN LIQUIDS; FLOW OF GASES; GASES; HEAT CONVECTION; HOMOGENIZATION METHOD;

ASYMPTOTIC PROCEDURES; CONVECTION-DIFFUSION EQUATIONS; CONVECTION-DIFFUSION MODELS; DIFFUSION APPROXIMATIONS; FREE MOLECULAR FLOW; HOMOGENIZATION TECHNIQUES; RAREFIED GAS FLOW; TEMPERATURE PROFILES;

DENSITY OF GASES;

EID: 84898966428     PISSN: 15403459     EISSN: 15403467     Source Type: Journal    
DOI: 10.1137/S1540345902409931     Document Type: Article

References (49)

1. K. Aoki,C. Bardos,F. Golse,M. N. Kogan,and Y. Sone, Steady flows of a rarefied gas around arbitrary obstacle distributions, Eur. J. Mech. B Fluids, 12 (1993), pp. 565-577.
2. K. Aoki,Y. Sone,S. Takata,K. Takahashi,and G. A. Bird, One-way flow of a rarefied gas induced in a circular pipe with a periodic temperature distribution, in Rarefied Gas Dynamics, T. J. Bartel and M. Gallis, eds., AIP, Melville, New York, 2001, pp. 940-947.
3. H. Babovsky, On the Knudsen flows within thin tubes, J. Statist. Phys., 44 (1986), pp. 865- 878.
4. H. Babovsky,C. Bardos,and T. Platkowski, Diffusion approximation for a Knudsen gas in a thin domain with accommodation on the boundary, Asymptot. Anal., 3 (1991), pp. 265- 289.
5. C. Bardos,R. E. Caflisch,and B. Nicolaenko, The Milne and Kramers problems for the Boltzmann equation of a hard sphere gas, Comm. Pure Appl. Math., 39 (1986), pp. 323- 352.
6. C. Bardos,F. Golse,and B. Perthame, The Rosseland approximation for the radiative transfer equations, Comm. Pure Appl. Math., 40 (1987), pp. 691-721.
7. C. Bardos,F. Golse,and B. Perthame, Erratum to The Rosseland approximation for the radiative transfer equations, Comm. Pure Appl. Math., 42 (1989), pp. 891-894.
8. C. Bardos,R. Santos,and R. Sentis, Diffusion approximation and computation of the critical size, Trans. Amer. Math. Soc., 284 (1984), pp. 617-649.
9. N. Ben Abdallah,P. Degond,A. Mellet,and F. Poupaud, Electron transport in semiconductor multiquantum well structures, Quart. Appl. Math., to appear.
10. A. Bensoussan,J. L. Lions,and G. C. Papanicolaou, Boundary layers and homogenization of transport processes, Publ. Res. Inst. Math. Sci., 15 (1979), pp. 53-157.
11. C. Börgers,C. Greengard,and E. Thomann, The diffusion limit of free molecular flow in thin plane channels, SIAM J. Appl. Math., 52 (1992), pp. 1057-1075.
12. C. Cercignani, Half-space problem in the kinetic theory of gases, in Trends in Applications of Pure Mathematics to Mechanics, E. Kröner and K. Kirchgässner, eds., Springer-Verlag, Berlin, 1986, pp. 35-50.
13. C. Cercignani, The Boltzmann Equation and Its Applications, Springer-Verlag, New York, 1988.
14. S. Chapman and T. G. Cowling, The Mathematical Theory of Non-uniform Gases, Cambridge University Press, New York, 1958.
15. P. Degond, A model of near-wall conductivity and its application to plasma thrusters, SIAM J. Appl. Math., 58 (1998), pp. 1138-1162.
16. P. Degond, Transport of Trapped Particles in a Surface Potential, Stud. Math. Appl. 31, D. Cioranescu and J. L. Lions, eds., Elsevier, Amsterdam, 2002, pp. 273-296.
17. P. Degond,T. Goudon, and F. Poupaud, Diffusion limit for non homogeneous and non micro-reversible processes, Indiana Univ. Math. J., 49 (2000), pp. 1175-1198.
18. P. Degond,V. Latocha,L. Garrigues,and J. P. Boeuf, Electron transport in stationary plasma thrusters, Transport Theory Statist. Phys., 27 (1998), pp. 203-221.
19. P. Degond,V. Latocha,S. Mancini,and A. Mellet, Diffusion dynamics of an electron gas confined between two plates, Methods Appl. Anal., 9 (2002), pp. 131-154.
20. P. Degond and M. Lemou, On the viscosity and thermal conduction of fluids with multivalued internal energy, Eur. J. Mech. B Fluids, 20 (2001), pp. 303-327.
21. P. Degond and B. Lucquin-Desreux, Transport coefficients of plasmas and disparate mass binary gases, Transport Theory Statist. Phys., 25 (1996), pp. 595-633.
22. P. Degond and S. Mancini, Diffusion driven by collisions with the boundary, Asymptot. Anal., 27 (2001), pp. 47-73.
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), pp. 589-636.
24. P. Degond and C. Schmeiser, Macroscopic models for semiconductor heterostructures, J. Math. Phys., 39 (1998), pp. 1-30.
25. W. E, Homogenization of linear and non linear transport equations, Comm. Pure Appl. Math., 45 (1992), pp. 301-326.
26. F. Golse, Anomalous diffusion limit for the Knudsen gas, Asymptot. Anal., 17 (1998), pp. 1-12.
27. F. Golse and F. Poupaud, Stationary solutions of the linearized Boltzmann equation in a half-space, Math. Methods Appl. Sci., 11 (1989), pp. 483-502.
28. F. Golse and F. Poupaud, Limite fluide deséquations de Boltzmann des semiconducteurs pour une statistique de Fermi-Dirac, Asymptot. Anal., 6 (1992), pp. 135-160.
29. T. Goudon and F. Poupaud, Approximation by homogenization and diffusion of kinetic equations, Comm. Partial Differential Equations, 26 (2001), pp. 537-569.
30. H. Grad, Singular and nonuniform limits of solutions of the Boltzmann equation, in Transport Theory, R. Bellmann, G. Birkhoff, and I. Abu-Shumays, eds., AMS, Providence, RI, 1969, pp. 269-308.
31. E. H. Kennard, Kinetic Theory of Gases, Section 184, McGraw-Hill, New York, 1938.
32. M. Knudsen, Eine Revision der Gleichgewichtsbedingung der Gase. Thermische Molekularstr ömung, Ann. Physik, 31 (1910), pp. 205-229.
33. E. W. Larsen, Neutron transport and diffusion in inhomogeneous media I, J. Math. Phys., 16 (1975), pp. 1421-1427.
34. E. W. Larsen, Neutron transport and diffusion in inhomogeneous media II, Nuclear Sci. Eng., 60 (1976), pp. 357-368.
35. E. W. Larsen and J. B. Keller, Asymptotic solution of neutron transport problems for small mean free paths, J. Math. Phys., 15 (1974), pp. 75-81.
36. F. Poupaud, Diffusion approximation of the linear semiconductor equation: Analysis of boundary layers, Asymptot. Anal., 4 (1991), pp. 293-317.
37. Y. Sone, Some remarks on Knudsen layer, J. Phys. Soc. Japan, 21 (1966), pp. 1620-1621.
38. Y. Sone, Thermal creep in rarefied gas, J. Phys. Soc. Japan, 21 (1966), pp. 1836-1837.
39. Y. Sone, Asymptotic theory of flow of rarefied gas over a smooth boundary I, in Rarefied Gas Dynamics, Vol. 1, L. Trilling and H. Y. Wachman, eds., Academic Press, New York, 1969, pp. 243-253.
40. Y. Sone, Asymptotic theory of flow of rarefied gas over a smooth boundary II, in Rarefied Gas Dynamics, Vol. II, D. Dini, ed., Editrice Tecnico Scientifica, Pisa, Italy, 1971, pp. 737-749.
41. Y. Sone, Highly rarefied gas around a group of bodies with various temperature distributions. I. Small temperature variation, J. Méc. Théor. Appl., 3 (1984), pp. 315-328.
42. Y. Sone, Highly rarefied gas around a group of bodies with various temperature distributions. II. Arbitrary temperature variation, J. Méc. Théor. Appl., 3 (1985), pp. 1-14.
43. Y. Sone, Theoretical and Numerical Analyses of the Boltzmann Equation-Theory and Analysis of Rarefied Gas Flows-Part I, Lecture Notes, Department of Aeronautics and Astronautics, Graduate School of Engineering, Kyoto University, Kyoto, Japan, 1998; also available online from http://www.users.kudpc.kyoto-u.ac.jp/~a50077/.
44. Y. Sone, Kinetic Theory and Fluid Dynamics, Birkhäuser, Boston, 2002.
45. Y. Sone, Flow induced by temperature fields in a rarefied gas and their ghost effect on the behavior of a gas in the continuum limit, in Annual Review of Fluid Mechanics, Vol. 32, Annual Reviews, Palo Alto, CA, 2000, pp. 779-811.
46. Y. Sone,T. Fukuda,T. Hokazono,and H. Sugimoto, Experimemt on a one-way flow of a rarefied gas through a straight circular pipe without average temperature and pressure gradients, in Rarefied Gas Dynamics, T. J. Bartel and M. Gallis, eds., AIP, Melville, New York, 2001, pp. 948-955.
47. Y. Sone and K. Sato, Demonstration of a one-way flow of a rarefied gas induced through a pipe without average pressure and temperature gradients, Phys. Fluids, 12 (2000), pp. 1864- 1868.
48. Y. Sone,Y. Waniguchi,and K. Aoki, One-way flow of a rarefied gas induced in a channel with a periodic temperature distribution, Phys. Fluids, 8 (1996), pp. 2227-2235.
49. Y. Sone and K. Yamamoto, Flow of rarefied gas through a circular pipe, Phys. Fluids, 11 (1968), pp. 1672-1678.