Skip to main content

Finite-difference methods for the Boltzmann equation for binary gas mixtures

  • Chapter
Modeling and Computational Methods for Kinetic Equations

Abstract

A finite-difference method for the Boltzmann equation for a binary mixture of hard-sphere gases that has been developed in the authors’ group is explained. Then, its applications to some fundamental problems of rarefied gas dynamics are presented.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aristov, V. V: Methods of Direct Solving the Boltzmann Equation and Study of Nonequilibrium Flows. Kluwer Academic, Dordrecht (2001)

    Book  Google Scholar 

  2. Abramowitz, M., Stegun, I. A.: Handbook of Mathematical Functions. Dover, New York (1965)

    Google Scholar 

  3. Aoki, K., Takata, S., Kosuge, S.: Vapor flows caused by evaporation and condensation on two parallel plane surfaces: Effect of the presence of a noncondensable gas. Phys. Fluids, 10,1519–1533 (1998)

    Article  Google Scholar 

  4. Bird, G. A.: Molecular Gas Dynamics and the Direct Simulation of Gas Rows. Oxford Univ. Press, Oxford (1994)

    Google Scholar 

  5. Bobylev, A. V., Rjasanow, S.: Numerical solution of the Boltzmann equation using a fully conservative difference scheme based on the fast Fourier transform. Transp. Ther. Stat. Phys., 29, 289–310 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Inamuro, T., Sturtevant, B.: Numerical study of discrete-velocity gases. Phys. Fluids A, 2, 2196–2203 (1990)

    Article  MATH  Google Scholar 

  7. Kosuge, S., Aoki, K., Takata, S.: Shock-wave structure for a binary gas mixture: Finite-difference analysis of the Boltzmann equation for hard-sphere molecules. Eur. J. Mech., B/Fluids, 20, 87–126 (2001)

    Article  MATH  Google Scholar 

  8. Kosuge, S., Aoki, K., Takata, S.: Heat transfer in a gas mixture between two parallel plates: Finite-difference analysis of the Boltzmann equation. In: Bartel, T. J., Gallis, M. A. (eds) Rarefied Gas Dynamics. AIP, Melville (2001), pp. 289–296

    Google Scholar 

  9. Kowalczyk, P., Platkowski, T., Walus, W.: Focusing of a shock wave in a rarefied gas: A numerical study. Shock Waves, 10,77–93 (2000)

    Article  MATH  Google Scholar 

  10. Ohwada, T: Structure of normal shock waves: Direct numerical analysis of the Boltzmann equation for hard-sphere molecules. Phys. Fluids A, 5, 217–234 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ohwada, T.: Heat flow and temperature and density distributions in a rarefied gas between parallel plates with different temperatures. Finite-difference analysis of the nonlinear Boltzmann equation for hard-sphere molecules. Phys. Fluids, 8, 2153–2160(1996)

    Article  MATH  Google Scholar 

  12. Ohwada, T.: Investigation of heat transfer problem of a rarefied gas between parallel plates with different temperatures. In: Shen, C. (ed) Rarefied Gas Dynamics. Peking Univ. Press, Beijing (1997), pp. 327–332

    Google Scholar 

  13. Ohwada, T., Sone, Y., Aoki, K.: 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)

    Article  MATH  Google Scholar 

  14. Ohwada, T, Sone, Y, Aoki, K.: Numerical analysis of the Poiseuille and thermal transpiration flows between two parallel plates on the basis of the Boltzmann equation for hard-sphere molecules. Phys. Fluids A, 1, 2042–2049 (1989)

    Article  MATH  Google Scholar 

  15. Panferov, V. A., Heintz, A. G.: A new consistent discrete-velocity model for the Boltzmann equation. Math. Methods Appl. Sci., 25, 571–593 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Pareschi, L., Russo, G.: Numerical solution of the Boltzmann equation I: Spectrally accurate approximation of the collisional operator. SIAM J. Numer. Anal., 37, 1217–1245 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Palczewski, A., Schneider, J., Bobylev, A. V.: A consistency result for a discrete-velocity model of the Boltzmann equation. SIAM J. Numer. Anal., 34,1865–1883 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Raines, A.: Study of a shock wave structure in gas mixtures on the basis of the Boltzmann equation. Eur. J. Mech., B/Fluids, 21, 599–610 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. Raines, A.: Numerical solution of one-dimensional problems in binary gas mixture on the basis of the Boltzmann equation. In: Ketsdever, A. D., Muntz, E. P. (eds) Rarefied Gas Dynamics. AIP, Melville (2003), pp. 67–74

    Google Scholar 

  20. Rogier, F, Schneider, J.: A direct method for solving the Boltzmann equation. Transp. Ther. Stat. Phys., 23, 313–338 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sone, Y: Continuum gas dynamics in the light of kinetic theory and new features of rarefied gas flows. In: Shen, C. (ed) Rarefied Gas Dynamics. Peking Univ. Press, Beijing, (1997), pp. 3–24

    Google Scholar 

  22. Sone, Y: Rows induced by temperature fields in a rarefied gas and their ghost effect on the behavior of a gas in the continuum limit. Annu. Rev. Fluid Mech., 32,779–811(2000)

    Article  MathSciNet  Google Scholar 

  23. Sone, Y: Kinetic Theory and Fluid Dynamics. Birkhauser, Boston (2002)

    Book  MATH  Google Scholar 

  24. Sone, Y, Aoki, K., Takata, S., Sugimoto, H., Bobylev, A. V.: Inappropriateness of the heat-conduction equation for description of a temperature field of a stationary gas in the continuum limit: Examination by asymptotic analysis and numerical computation of the Boltzmann equation. Phys. Fluids, 8, 628–638 (1996); Erratum: Phys. Fluids, 8, 841 (1996)

    Article  MathSciNet  Google Scholar 

  25. Sone, Y., Oh wada, T., Aoki, K.: 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)

    Article  MATH  Google Scholar 

  26. Sone, Y., Takata, S., Ohwada, T.: Numerical analysis of the plane Couette flow of a rarefied gas on the basis of the linearized Boltzmann equation for hard-sphere molecules. Eur. J. Mech., B/Fluids, 9, 273–288 (1990)

    MATH  Google Scholar 

  27. Tcheremissine, F. G.: Conservative discrete ordinates method for solving of Boltzmann kinetic equation. In: Shen, C. (ed) Rarefied Gas Dynamics. Peking Univ. Press, Beijing (1997), pp. 297–302

    Google Scholar 

  28. Takata, S., Aoki, K.: Two-surface problems of a multicomponent mixture of vapors and noncondensable gases in the continuum limit in the light of kinetic theory. Phys. Fluids, 11,2743–2756 (1999)

    Article  MATH  Google Scholar 

  29. Takata, S., Aoki,K.,Muraki,T.: Behavior of a vapor-gas mixture between two parallel plane condensed phases in the continuum limit. In Brun, R., Campargue, R., Gatignol, R., Lengrand J.-C. (eds) Rarefied Gas Dynamics. Cepadues-Editions, Toulouse (1999), Vol. 1, pp. 479–486

    Google Scholar 

  30. Takata, S., Sone, Y: Flow induced around a sphere with a non-uniform surface temperature in a rarefied gas, with application to the drag and thermal force problems of a spherical particle with an arbitrary thermal conductivity. Eur. J. Mech., B/Fluids, 14,487–518 (1995)

    MATH  Google Scholar 

  31. Takata, S., Sone, Y, Aoki, K.: Numerical analysis of a uniform flow of a rarefied gas past a sphere on the basis of the Boltzmann equation for hard-sphere molecules. Phys. Fluids A, 5, 716–737 (1993)

    Article  MATH  Google Scholar 

  32. Takata, S., Sone, Y, Lhuillier, D., Wakabayashi, M.: Evaporation from or condensation onto a sphere: Numerical analysis of the Boltzmann equation for hard-sphere molecules. Computers Math. Applic, 35,193–214 (1998)

    Article  MATH  Google Scholar 

  33. Takata, S., Yasuda, S., Kosuge, S., Aoki, K.: Numerical analysis of thermal-slip and diffusion-slip flows of a binary mixture of hard-sphere molecular gases. Phys. Fluids, 15, 3745–3766 (2003)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer Science+Business Media New York

About this chapter

Cite this chapter

Aoki, K., Kosuge, S. (2004). Finite-difference methods for the Boltzmann equation for binary gas mixtures. In: Degond, P., Pareschi, L., Russo, G. (eds) Modeling and Computational Methods for Kinetic Equations. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8200-2_5

Download citation

  • DOI: https://doi.org/10.1007/978-0-8176-8200-2_5

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6487-3

  • Online ISBN: 978-0-8176-8200-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics