Methods of Solution of the Boltzmann Equation for Rarefied Gases

  • Carlo Cercignani
Part of the International Centre for Mechanical Sciences book series (CISM, volume 224)


According to the molecular theory of matter, a macroscopic volume of gas (say, 1 cm3) is a system of a very large number (say, 1020) of molecules moving in a rather irregular way. In principle, we may assume, ignoring quantum effects, that the molecules are particles (mass points or other systems with a small number of degrees of freedom) obeying the laws of classical mechanics. We may also assume that the laws of interaction between the molecules are perfectly known so that, in principle, the evolution of the system is computable, provided suitable initial data are given.


Boltzmann Equation Knudsen Number Couette Flow Poiseuille Flow Mass Velocity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    BRUSH, S.G., “Kinetic Theory”, vol. 2 — Pergamon Press, Oxford (1966).Google Scholar
  2. [2]
    CERCIGNANI, C., “Mathematical Methods in Kinetic Theory”, Plenum Press McMillan, New York (1969).CrossRefMATHGoogle Scholar
  3. [3]
    GRAD, H. — “Principles of Kinetic Theory in ”Handbuch der Physik“, vol. XXII, Springer (1958).Google Scholar
  4. [4]
    CERCIGNANI, C., “Transport Theory and Statistical Physics”, 2, 211 (1971)CrossRefMathSciNetGoogle Scholar
  5. [5]
    CERCIGNANI, C., “Theory and Application of the Boltzmann Equation”, Scottish Academic Press (1975).Google Scholar
  6. [6]
    BOLTZMANN, L., Sitzungsberichte Akad. Wiss., Vienna p. II, 66, 275 (1872).MATHGoogle Scholar
  7. [7]
    CHAPMAN, S. and COWLING, T.G., “The Mathematical Theory of Nonuniform Gases”, Cambridge University Press, Cambridge (1952).MATHGoogle Scholar
  8. [8]
    LANFORD, O.H., III: Private Communication.Google Scholar
  9. [9]
    GRAD, H., “On the Kinetic Theory of Rarefied Gases”, Comm. Pure and Appl. Mathematics 2, 331–407, 1949.CrossRefMATHMathSciNetGoogle Scholar
  10. [10]
    CARLEMAN, T., “Problèmes Mathematiques dans la Theorie Mathematique des gaz” Almqvist and Wicksells, Uppsala (1953).Google Scholar
  11. [11]
    HIRSCHFELDER, J.O., CURTISS, C.F., and BIRD, R.B., “Molecular Theory of Gases and Liquids”, Wiley, New York (1954).MATHGoogle Scholar
  12. [12]
    DARROZES, J.S., and GUIRAUD, J.P., Compt., Rend. Ac. Sci. (Paris), A262, 1368 (1966).Google Scholar
  13. [13]
    CERCIGNANI, C., “Boundary Value Problems in Linearized Kinetic Theory” In: “Transport Theory”, G. Birkhoff et al., eds. SIAM-AMS Proceedings, vol. I, p. 240, AMS, Providence (1968).Google Scholar
  14. [14]
    KUSCER L, in “Transport Theory Conference” AEC Report ORO-3858–1, Blacksburgh, Virginia (1963).Google Scholar
  15. [15]
    SHEN, S.F., “Entropie”, 18, 138 (1967).Google Scholar
  16. [16]
    CERCIGNANI, C. and LAMPIS, M., “Transport Theory and Statistical Physics”, 6, 101 (1971).CrossRefMathSciNetGoogle Scholar
  17. [17]
    CERCIGNANI, C., “Transport Theory and Statistical Physics”, 2, 27 (1972).CrossRefMathSciNetGoogle Scholar
  18. [18]
    BHATNAGAR, P.L., GROSS, E.P., KROOK, M., “A Model for Collision Processes in Gases”. Physical Review, 94, 511–525, 1964.CrossRefGoogle Scholar
  19. [19]
    WELANDER, P., “On the Temperature Jump in Rarefied Gas”, Arkiv für Fysik, 7, 507–553, 1954.MATHMathSciNetGoogle Scholar
  20. [20]
    KROOK, M. “Continuum Equations in the Dynamics of Rarefied Gases”, Jnl. Fluid Mechanics, 6, 523–541, 1959.CrossRefMATHMathSciNetGoogle Scholar
  21. [21]
    CERCIGNANI, C., “The Method of Elementary Solutions for Kinetic Models with Velocity-dependent Collision Frequency”, Annals of Physics, 40, 469–481, 1966.CrossRefMathSciNetGoogle Scholar
  22. [22]
    HOLWAY, L.H., Jr. “Approximation Procedures for Kinetic Theory”, Ph. D. Thesis, Harvard, 1963.Google Scholar
  23. [23]
    CERCIGNANI, C. and TIRONI, G., “Nonlinear Heat Transfer Between Two Parallel Plates According to a Model with Correct Prandtl Number”, Rarefied Gas Dynamics, C.L. Brundin editor, vol. 1, 441–453, Academic Press, New York, 1967.Google Scholar
  24. [24]
    GROSS, E.P. and JACKSON, E.A., “Kinetic Model and the Linearized Boltzmann Equation”, Physics of Fluids, 2, 432–441, 1959.CrossRefMATHMathSciNetGoogle Scholar
  25. [25]
    SIROVICH, L., “Kinetic Modeling of Gas Mixtures”, Physics of Fluids, 5, 908, 1962.CrossRefMathSciNetGoogle Scholar
  26. [26]
    GRAD, H., “Asymptotic Theory of the Boltzmann Equation H. Rarefied Gas Dynamics, J.A. Laurmann, editor., vol. 1, 26–59, Academic Press, New York, 1963.Google Scholar
  27. [27]
    CERCIGNANI, C., “On Boltzmann Equation with Cutoff Potentials”, Physics of Fluids, 10, 2097, 1968.CrossRefGoogle Scholar
  28. [28]
    LOYALKA, S.K. and FERZIGER, J.A., “Model Dependence of the Slip Coefficient”, Physics of Fluids, 10, 1833, 1967.CrossRefMATHGoogle Scholar
  29. [29]
    GRAD, H., “Asymptotic Theory of the Boltzmann Equation I”, Physics of Fluids, 6, 147, 1963.CrossRefMATHMathSciNetGoogle Scholar
  30. [30]
    CERCIGNANI, C., “Elementary Solutions of the Linearized Gas-dynamics Boltzmann Equation and their Application to the Slip-flow Problem”, Annals of Physics, 20, 219–233, 1962.CrossRefMATHMathSciNetGoogle Scholar
  31. [31]
    MUSKHELISHVILI, N.I., “Singular Integral Equations”, Noordhoff, Groningen, 1953.MATHGoogle Scholar
  32. [32]
    CERCIGNANI, C., “Elementary Solutions of Linearized Kinetic Models and Boundary Value Problems in the Kinetic Theory of Gases”, Brown University Report, 1965.Google Scholar
  33. [33]
    MAXWELL, J.C., “Scientific Papers”, 704–705, Dover, New York, 1965.Google Scholar
  34. [34]
    ALBERTONI, S., CERCIGNANI, C., GOTUSSO, L., “Numerical Evaluation of the Slip Coefficient”, Physics of Fluids, 6, 993, 1963.CrossRefGoogle Scholar
  35. [35]
    CERCIGNANI, C., “On the General Solution of the Steady Linearized Boltzmann Equation”, presented at the 9th Symposium on Rarefied Gas Dynamics, Göttingen, July 1974.Google Scholar
  36. [36]
    CERCIGNANI, C., “Analytic Solution of the Temperature Jump Problem by Means of the BGK Model”, Transport Theory and Stat. Physics (1977).Google Scholar
  37. [37]
    CERCIGNANI, C., FORESTI, P., SERNAGIOTTO, F., “Dependence of the slip coefficient on the Form of the Collision Frequency”, Nuovo Cimento, X, 57B, 297, 1968.CrossRefGoogle Scholar
  38. [38]
    CERCIGNANI, C. and TIRONI, G., “New Boundary Conditions in the Transition Regime”, J. Plasma Physics, 2, 293, 1968.CrossRefGoogle Scholar
  39. [39]
    LANGMUIR, I., Jnl of Chemical Society, 37, 417, 1915.CrossRefGoogle Scholar
  40. [40]
    LOYALKA, S., Z. Naturforschung, 26a, 1708 (1971).MATHMathSciNetGoogle Scholar
  41. [41]
    CERCIGNANI, C. and SERNAGIOTTO, F., “The Method of Elementary Solutions for Time-dependent Problems in Linearized Kinetic Theory”, Annals of Physics, 30, 154–167, 1964.CrossRefMathSciNetGoogle Scholar
  42. [42]
    CERCIGNANI, C., “Unsteady Solutions of Kinetic Model with Velocity Dependent Collision Frequency”, Annals of Physics, 40, 454–468, 1966.CrossRefMathSciNetGoogle Scholar
  43. [43]
    GROSS, E.P., JACKSON, E.A., ZIERING, S., “Boundary Value Problems in Kinetic Theory of Gases”, Annals of Physics, 1, 141–167, 1957.CrossRefMATHMathSciNetGoogle Scholar
  44. [44]
    LEES, L., “A Kinetic Theory Description of Rarefied Gas Flows”, Caltech, Memorandum, 1959.Google Scholar
  45. [45]
    WILLIS, D.R., “A Study of Some Nearly Free Molecular Flow Problems”, Ph. D Thesis, Princeton, 1958.Google Scholar
  46. [46]
    LIEPMANN, H.W., NARASIMHA, R., CHAHINE, M.T., “Structure of a Plane Shock Layer”, Physics of Fluids, 5, 1313–1324, 1962.CrossRefMATHGoogle Scholar
  47. [47]
    WILLIS, D.R., “Heat Transfer in a Rarefied Gas Between Parallel Plates at Large Temperature Ratios”, Rarefied Gas Dynamics, J.A. Laurmann, ed. vol. 1, 209–225, Academic Press, New York, 1963.Google Scholar
  48. [48]
    ANDERSON, D., “On the Krook Kinetic Equation”, Part 2, Jnl of Plasma Physics, 2, 55, 1967.Google Scholar
  49. [49]
    WILLIS, D.R., “Comparison of Kinetic Theory Analyses of Linearized Couette Flow”, Physics of Fluids, 5, 127–135, 1962.CrossRefMATHGoogle Scholar
  50. [50]
    CERCIGNANI, C. and TIRONI, G., “Some Applications of a Linearized Kinetic Model with Correct Prandtl Number”, Nuovo Cimento, 43, 64–78, 1966.CrossRefGoogle Scholar
  51. [51]
    CERCIGNANI, C. and DANERI, A., “Flow of a Rarefied Gas Between Two Parallel Plates”. Jnl of Applied Physics, 34, 3509–3513, 1963.CrossRefMathSciNetGoogle Scholar
  52. [52]
    CERCIGNANI, C. and SERNAGIOTTO, F., “Cylindrical Poiseuille Flow of a Rarefied Gas”, Physics of Fluids, 9, 40–44, 1966.CrossRefGoogle Scholar
  53. [53]
    CERCIGNANI, C. and TIRONI, G., “Alcune applicazioni di un nuovo modello linearizzato dell’equazione di Boltzmann”, Atti del Congresso AIDA-AIR, AIDA-AIR, Roma, 1967.Google Scholar
  54. [54]
    CERCIGNANI, C., “Reply to the Comments by A.S. Berman”, Physics of Fluids, 10, 1859, 1967.CrossRefGoogle Scholar
  55. [55]
    BASSANINI, P., CERCIGNANI, C., SERNAGIOTTO, F., “Flow of a Rarefied Gas in a Tube of Annular Section”, Physics of Fluids, 9, 1174–1178, 1966.CrossRefGoogle Scholar
  56. [56]
    BASSANINI, P., CERCIGNANI, C., SCHWENDIMANN, P., “The Problem of a Cylinder Rotating in a Rarefied Gas”, Rarefied Gas Dynamics, C.L. Brundin, editor, vol. 1, Academic Press, New York, 1967, pp 505–516.Google Scholar
  57. [57]
    CERCIGNANI, C. and SERNAGIOTTO, F., “Cylindrical Couette Flow of a Rarefied Gas”, Physics of Fluids, 10, 1200–1204, 1967.CrossRefGoogle Scholar
  58. [58]
    BASSANINI, P., CERCIGNANI, C., PAGANI, C.D., “Comparison of Kinetic Theory Analyses Between Parallel Plates”, Int. Jnl Heat and Mass Transfer, 10, 447–460, 1967.CrossRefGoogle Scholar
  59. [59]
    CERCIGNANI, C., J. of Statistical Physics, 1, 297 (1969).CrossRefGoogle Scholar
  60. [60]
    CERCIGNANI, C. and PAGANI, C.D., “Variational Approach to Rarefied Gas Dynamics”, Physics of Fluids, 9, 1167–1173, 1966.CrossRefGoogle Scholar
  61. [61]
    CERCIGNANI, C., “Flows of Rarefied Gases Supported by Density and Temperature Gradients”, University of California Report, Berkeley, 1964.Google Scholar
  62. [62]
    CERCIGNANI, C., “Plane Poiseuille Flow According to the Method of Elementary Solutions”, Jnl Mathematical Analysis and Applications, 12, 234, 1965.MathSciNetGoogle Scholar
  63. [63]
    CERCIGNANI, C., “Plane Poiseuille Flow and Knudsen Minimum Effect”, Rarefied Gas Dynamics, J.A. Laurmann ed., vol. II, 920101, 1963.Google Scholar
  64. [64]
    KNUDSEN, M., “Die Gesetze der Molekularströmung under der inneren Reibungsströmung der Gase durch Rohren”, Ann. Physik, 8, 75–130, 1909.CrossRefGoogle Scholar
  65. [65]
    CERCIGNANI, C. and PAGANI, C.D., “Variational Approach to Rarefied Flows in Cylindrical and Spherical Geometry. Rarefied Gas Dynamics”, C.L. Brundin, ed. vol. 1, 555–573, Academic Press, New York, 1967.Google Scholar
  66. [66]
    CERCIGNANI, C. and PAGANI, C.D., “Flow of a Rarefied Gas Past an Axisymmetric Body”, I General remarks, Physics of Fluids, 11, 1395, 1968.CrossRefMATHGoogle Scholar
  67. [67]
    CERCIGNANI, C., PAGAN!, C.D., BASSANINI, P., “Flow of a Rarefied Gas Past an Axisymmetric Body”, II Case of a Sphere. Physics of Fluids, 11, 1399, 1968.CrossRefMATHGoogle Scholar
  68. [68]
    BASSANINI, P., CERCIGNANI, C. PAGANI, C.D., “Influence of the Accommodation Coefficient on the Heat Transfer in a Rarefied Gas”, Int. Jnl Heat and Mass Transfer, 11, 1359, 1968.CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 1981

Authors and Affiliations

  • Carlo Cercignani
    • 1
  1. 1.Politecnico di MilanoItaly

Personalised recommendations