Skip to main content

Knudsen Layer for Gas Mixtures

Abstract

The Knudsen layer in rarefied gas dynamics is essentially described by a half-space boundary-value problem of the linearized Boltzmann equation, in which the incoming data are specified on the boundary and the solution is assumed to be bounded at infinity (Milne problem). This problem is considered for a binary mixture of hard-sphere gases, and the existence and uniqueness of the solution, as well as some asymptotic properties, are proved. The proof is an extension of that of the corresponding theorem for a single-component gas given by Bardos, Caflisch, and Nicolaenko [Comm. Pure Appl. Math. 39:323 (1986)]. Some estimates on the convergence of the solution in a finite slab to the solution of the Milne problem are also obtained.

This is a preview of subscription content, access via your institution.

REFERENCES

  1. 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. 39:323-352 (1986).

    Google Scholar 

  2. C. Cercignani, Half-space problems 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.

    Google Scholar 

  3. C. Cercignani, The Boltzmann Equation and its Applications (Springer-Verlag, New-York, 1988).

    Google Scholar 

  4. F. Coron, F. Golse, and C. Sulem, A classification of well-posed kinetic layer problems, Comm. Pure Appl. Math. 41:409-435 (1988).

    Google Scholar 

  5. F. Golse and F. Poupaud, Stationary solutions of the linearized Boltzmann equation in a half-space, Math. Methods Appl. Sci. 11:483-502 (1989).

    Google Scholar 

  6. H. Grad, Singular and nonuniform limits of solutions of the Boltzmann equation, in Transport Theory, R. Bellman, G. Birkhoff, and I. Abu-Shumays, eds. (AMS, Providence, 1969), pp. 269-308.

    Google Scholar 

  7. N. B. Maslova, Kramers problem in the kinetic theory of gases, USSR Comp. Math. Phys. 22:208-219 (1982).

    Google Scholar 

  8. T. Ohwada, Y. Sone, and K. Aoki, Numerical analysis of the shear and thermal creep flows of a rarefied gas over a plane wall on the basis of the linearized Boltzmann equation for hard-sphere molecules, Phys. Fluids A 1:1588-1599 (1989).

    Google Scholar 

  9. Raymundo Peralta, Thesis Paris 7 (1995).

  10. 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, New York, 1969), pp. 243-253.

    Google Scholar 

  11. 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.

    Google Scholar 

  12. Y. Sone, Asymptotic theory of a steady flow of a rarefied gas past bodies for small Knudsen numbers, in Advances in Kinetic Theory and Continuum Mechanics, R. Gatignol and Soubbaramayer, eds. (Springer, Berlin, 1991), pp. 19-31.

    Google Scholar 

  13. Y. Sone, Kinetic Theory and Fluid Dynamics, Modeling and Simulation in Science, Engineering and Technology (Birkhäuser, Boston, 2002).

    Google Scholar 

  14. Y. Sone, T. Ohwada, and K. Aoki, Temperature jump and Knudsen layer in a rarefied gas over a plane wall: Numerical analysis of the linearized Boltzmann equation for hard-sphere molecules, Phys. Fluids A 1:363-370 (1989).

    Google Scholar 

  15. Y. Sone, T. Ohwada, and K. Aoki, Evaporation and condensation on a plane condensed phase: Numerical analysis of the linearized Boltzmann equation for hard-sphere molecules, Phys. Fluids A 1:1398-1405 (1989).

    Google Scholar 

  16. S. Takata, Diffusion slip for a binary mixture of hard-sphere molecular gases: Numerical analysis based on the linearized Boltzmann equation, in Rarefied Gas Dynamics, T. J. Bartel and M. A. Gallis, eds. (AIP, Melville, 2001), pp. 22-29.

    Google Scholar 

  17. S. Takata and K. Aoki, The ghost effect in the continuum limit for a vapor-gas mixture around condensed phases: Asymptotic analysis of the Boltzmann equation, Transport Theory Statist. Phys. 30:205-237 (2001); Aoki Erratum: Transport Theory Statist. Phys. 31:289-290 (2002).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Aoki, K., Bardos, C. & Takata, S. Knudsen Layer for Gas Mixtures. Journal of Statistical Physics 112, 629–655 (2003). https://doi.org/10.1023/A:1023876025363

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1023876025363

  • Knudsen layer
  • gas mixtures
  • Milne problem
  • Boltzmann equation
  • rarefied gas dynamics