Journal of Statistical Physics

, Volume 48, Issue 3–4, pp 901–917 | Cite as

Solution of the Fokker-Planck equation for the shock wave problem

  • R. Fernandez-Feria
  • J. Fernandez de la Mora


An eigenexpansion solution of the time-independent Brownian motion Fokker-Planck equation is given for a situation in which the external acceleration is a step function. The solution describes the heavy-species velocity distribution function in a binary mixture undergoing a shock wave, in the limit of high dilution of the heavy species and negligible width of the light-gas internal shock. The diffusion solution is part of the eigenexpansion. The coefficients of the series of eigenfunctions are obtained analytically with transcendentally small errors of order exp(−1/M), whereM ≪ 1 is the mass ratio. Comparison is made with results from a hypersonic approximation.

Key words

Fokker-Planck equation shock wave Brownian motion eigentheory 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    N. Wax, ed.,Selected Papers on Noise and Stochastic Processes (Dover, New York, 1954).Google Scholar
  2. 2.
    H. A. Kramers,Physica VII(4):284 (1940).Google Scholar
  3. 3.
    P. Résibois and M. De Leener,Classical Kinetic Theory of Fluids (Wiley, New York, 1977) Chap. II.Google Scholar
  4. 4.
    R. M. Mazo,J. Stat. Phys. 1:101 (1969).Google Scholar
  5. 5.
    W. G. N. Slinn and S. F. Shen,J. Stat. Phys. 3:291 (1971).Google Scholar
  6. 6.
    C. S. Wang Chang and G. E. Uhlenbeck, inStudies in Statistical Mechanics, Vol. 5, J. de Boer and G. E. Uhlenbeck, eds. (North-Holland, Amsterdam, 1970), Chapter V.Google Scholar
  7. 7.
    J. Fernández de la Mora and J. M. Mercer,Phys. Rev. A 26:2178 (1982).Google Scholar
  8. 8.
    J. Fernández de la Mora and R. Fernández-Feria,Phys. Fluids 30:740 (1987).Google Scholar
  9. 9.
    S. Chapman and T. G. Cowling,The Mathematical Theory of Non-Uniform Gases (Cambridge University Press, Cambridge, 1970).Google Scholar
  10. 10.
    P. RésiboisElectrolite Theory (Harper and Row, New York, 1965), Appendix A4.Google Scholar
  11. 11.
    S, Harris and J. L. Monroe,J. Stat. Phys. 17:377 (1977).Google Scholar
  12. 12.
    H. A. Bethe, M. E. Rose, and L. P. Smith,Proc. Am. Philos. Soc. 78:573 (1938).Google Scholar
  13. 13.
    D. Stein and I. B. Bernstein,Phys. Fluids 19:811 (1976).Google Scholar
  14. 14.
    N. Fish and M. D. Kruskal,J. Math. Phys. 21:740 (1980).Google Scholar
  15. 15.
    M. A. Burschka and U. M. Titulaer,J. Stat. Phys. 25:569 (1981).Google Scholar
  16. 16.
    J. Fernández de la Mora,Phys. Rev. A 25:1108 (1982).Google Scholar
  17. 17.
    R. Beals and V. Protopopescu,Transp. Theory Stat. Phys. 12:109 (1983).Google Scholar
  18. 18.
    B. D. Ganapol and E. W. Larsen,Transp. Theory Stat. Phys. 13:635 (1984).Google Scholar
  19. 19.
    S. Waidenstrom, K. J. Mork, and K. Razi Naqvi,Phys. Rev. A 28:1659 (1983).Google Scholar
  20. 20.
    D. C. Sahni,Phys. Rev. A 30:2056 (1984).Google Scholar
  21. 21.
    Vinod Kumar and S. V. G. Menon,J. Chem. Phys. 82:917 (1985).Google Scholar
  22. 22.
    R. Beals and V. Protopopescu,Transp. Theory Stat. Phys. 13:43 (1984).Google Scholar
  23. 23.
    R. Fernández-Feria and J. Fernandez de la Mora,J. Fluid Mech. 179:21 (1987).Google Scholar
  24. 24.
    P. Riesco-Chueca, R. Fernández-Feria, and J. Fernández de la Mora, inRarefied Gas Dynamics, V. Boffi and C. Cercignani, eds. (Teubner, Stuttgart, 1986), Vol. 1, p. 283.Google Scholar
  25. 25.
    R. Fernández-Feria and J. Fernández de la Mora, Hypersonic expansion of the Fokker-Planck equation, in preparation. See also R. Fernandez-Feria, Ph.D. Thesis, Yale University (1987), Chap. 4.Google Scholar
  26. 26.
    M. Abramowitz and I. Stegun,Handbook of Mathematical Functions (Dover, New York, 1965), Chapter 22.Google Scholar
  27. 27.
    A. Erdélyi,Math. Z. 40:693 (1936).Google Scholar
  28. 28.
    W. N. Bailey,J. Lond. Math. Soc. 23:291 (1948).Google Scholar
  29. 29.
    R. D. Lord,J. Lond. Math. Soc. 24:101 (1949).Google Scholar
  30. 30.
    A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi,Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 2, Chapter XVI.Google Scholar
  31. 31.
    H. Buchholz,The Confluent Hypergeometric Function (Springer, New York, 1969), Section 12.Google Scholar

Copyright information

© Plenum Publishing Corporation 1987

Authors and Affiliations

  • R. Fernandez-Feria
    • 1
  • J. Fernandez de la Mora
    • 1
  1. 1.Department of Mechanical EngineeringYale UniversityNew Haven

Personalised recommendations