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A variational principle for boundary value problems in kinetic theory

Abstract

A variational principle which applies directly to the integrodifferential form of the linearized Boltzmann equation is introduced. Extremely general boundary conditions and collision terms are allowed. For a class of interesting problems, the value of the functional to be varied is shown to be closely related to quantities of great physical interest. The formalism is applied to the treatment of plane Couette flow for different forms of the collision term (BGK model, rigid spheres, Maxwell's molecules).

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References

  1. C. Cercignani and C. D. Pagani,Phys. Fluids 9:1167 (1966).

    Google Scholar 

  2. C. Cercignani and C. D. Pagani, in:Rarefied Gas Dynamics, C. L. Brundin, ed. (Academic Press, New York, 1967), Vol. I, p. 555.

    Google Scholar 

  3. P. Bassanini, C. Cercignani, and C. D. Pagani,Intern. J. Heat Mass Transfer 10:447 (1967).

    Google Scholar 

  4. P. Bassanini, C. Cercignani, and C. D. Pagani,Intern. J. Heat Mass Transfer 11:1359 (1968).

    Google Scholar 

  5. C. Cercignani and C. D. Pagani,Phys. Fluids 11:1395 (1968).

    Google Scholar 

  6. C. Cercignani, C. D. Pagani, and P. Bassanini,Phys. Fluids 11:1399 (1968).

    Google Scholar 

  7. J. H. Ferziger,Phys. Fluids 10:1448 (1967).

    Google Scholar 

  8. S. K. Loyalka and J. H. Ferziger,Phys. Fluids 10:1833 (1967).

    Google Scholar 

  9. S. K. Loyalka and J. H. Ferziger,Phys. Fluids 11:1668 (1968).

    Google Scholar 

  10. C. Cercignani, P. Foresti, and F. Sernagiotto,Nuovo Cimenta, X,57B:297 (1968).

    Google Scholar 

  11. G. C. Pomraning and M. Clark, Jr.,Nucl. Sci. Eng. 16:147 (1963).

    Google Scholar 

  12. G. C. Pomraning and M. Clark, Jr.,Nucl. Sci. Eng. 16:144 (1963).

    Google Scholar 

  13. C. Cercignani, in:Fisica del Reattore (Consiglio Nazionale delie Ricerche, Roma, 1966), p. 633.

    Google Scholar 

  14. C. Cercignani,Mathematical Methods in Kinetic Theory (Plenum Press, New York, 1969).

    Google Scholar 

  15. C. Cercignani, in:Transport Theory, R. Bellmann, G. Birkhoff, and I. Abu-Shumays, eds. (American Mathematical Society, Providence, Rhode Island, 1969), p. 249.

    Google Scholar 

  16. D. R. Willis,Phys. Fluids 5:127 (1962).

    Google Scholar 

  17. C. Cercignani,J. Math. Anal. Appl. 12:254 (1965).

    Google Scholar 

  18. S. Ziering,Phys. Fluids 3:503 (1960).

    Google Scholar 

  19. E. P. Gross and S. Ziering,Phys. Fluids 1:215 (1958).

    Google Scholar 

  20. C. Cercignani,Nuovo Cimento, X,27:1240 (1963).

    Google Scholar 

  21. S. Chapman and T. G. Cowling,The Mathematical Theory of Nonuniform Gases (Cambridge Univ. Press, Cambridge, 1952).

    Google Scholar 

  22. L. Lees, Guggenheim Aeronautical Laboratory, Calif. Inst. Technology, Hypersonic Research Project, Mem. N. 51 (1959).

  23. S. Albertoni, C. Cercignani, and L. Gotusso,Phys. Fluids 6:993 (1963).

    Google Scholar 

  24. C. Cercignani and G. Tironi,Nuovo Cimento, X,43:64 (1966).

    Google Scholar 

  25. C. Cercignani and C. D. Pagani, in:Rarefied Gas Dynamics, L. Trilling and H. Wachman, eds. (Academic Press, New York, 1969), Vol. I, p. 269.

    Google Scholar 

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Research sponsored by the Air Force Office of Scientific Research under contract F 61(052)-68-C-0020, through the European Office of Aerospace Research, OAR, United States Air Force.

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Cercignani, C. A variational principle for boundary value problems in kinetic theory. J Stat Phys 1, 297–311 (1969). https://doi.org/10.1007/BF01007482

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  • DOI: https://doi.org/10.1007/BF01007482

Key words

  • kinetic theory
  • boundary value problems
  • variational principles
  • Couette flow
  • transition regime
  • Boltzmann equation
  • rarefied gas dynamics