- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Diffusion of impurities in a granular gas

Vicente Garzó*

José María Montanero†

  • Departamento de Física, Universidad de Extremadura, E-06071 Badajoz, Spain

  • Departamento de Electrónica e Ingeniería Electromecánica, Universidad de Extremadura, E-06071 Badajoz, Spain

  • *Email address: vicenteg@unex.es
  • †Email address: jmm@unex.es
PDF

Phys. Rev. E 69, 021301 – Published 23 February, 2004

DOI: https://doi.org/10.1103/PhysRevE.69.021301

Abstract

Diffusion of impurities in a granular gas undergoing homogeneous cooling state is studied. The results are obtained by solving the Boltzmann-Lorentz equation by means of the Chapman-Enskog method. In the first order in the density gradient of impurities, the diffusion coefficient D is determined as the solution of a linear integral equation which is approximately solved by making an expansion in Sonine polynomials. In this paper, we evaluate D up to the second order in the Sonine expansion and get explicit expressions for D in terms of the coefficients of restitution for the impurity-gas and gas-gas collisions as well as the ratios of mass and particle sizes. To check the reliability of the Sonine polynomial solution, analytical results are compared with those obtained from numerical solutions of the Boltzmann equation by means of the direct simulation Monte Carlo method. In the simulations, the diffusion coefficient is measured via the mean-square displacement of impurities. The comparison between theory and simulation shows in general an excellent agreement, except for the cases in which the gas particles are much heavier and/or much larger than impurities. In these cases, the second Sonine approximation to D improves significantly the qualitative predictions made from the first Sonine approximation. A discussion on the convergence of the Sonine polynomial expansion is also carried out.

View figure in article

References (46)

  1. J.J. Brey, J.W. Dufty, and A. Santos, J. Stat. Phys. 87, 1051 (1997).
  2. S. Chapman and T. G. Cowling, The Mathematical Theory of Nonuniform Gases (Cambridge University Press, Cambridge, 1970).
  3. J.J. Brey, J.W. Dufty, C.S. Kim, and A. Santos, Phys. Rev. E 58, 4638 (1998).
  4. J. J. Brey and D. Cubero, in Granular Gases, Lecture Notes in Physics, edited by T. Pöschel and S. Luding (Springer Verlag, Berlin, 2001), p. 59.
  5. V. Garzó and J.M. Montanero, Physica A 313, 336 (2002).
  6. J.T. Jenkins and F. Mancini, Phys. Fluids A 1, 2050 (1989).
  7. P. Zamankhan, Phys. Rev. E 52, 4877 (1995).
  8. B. Arnarson and J.T. Willits, Phys. Fluids 10, 1324 (1998).
  9. J.T. Willits and B. Arnarson, Phys. Fluids 11, 3116 (1999).
  10. M. Alam, J.T. Willits, B. Arnarson, and S. Luding, Phys. Fluids 14, 4085 (2002).
  11. V. Garzó and J.W. Dufty, Phys. Rev. E 60, 5706 (1999).
  12. P.A. Martin and J. Piasecki, Europhys. Lett. 46, 613 (1999).
  13. A. Barrat and E. Trizac, Granular Matter 4, 57 (2002).
  14. U.M.B. Marconi and A. Puglisi, Phys. Rev. E 65, 051305 (2002); ibid.66, 011301 (2002).
  15. E. Ben-Naim and P.L. Krapivsky, Eur. Phys. J. E 8, 507 (2002).
  16. J.M. Montanero and V. Garzó, Granular Matter 4, 17 (2002).
  17. R. Clelland and C.M. Hrenya, Phys. Rev. E 65, 031301 (2002).
  18. S.R. Dahl, C.M. Hrenya, V. Garzó, and J.W. Dufty, Phys. Rev. E 66, 041301 (2002).
  19. R. Pagnani, U.M.B. Marconi, and A. Puglisi, Phys. Rev. E 66, 051304 (2002).
  20. D. Paolotti, C. Cattuto, U.M.B. Marconi, and A. Puglisi, Granular Matter 5, 75 (2003).
  21. P. Krouskop and J. Talbot, Phys. Rev. E 68, 021304 (2003).
  22. H. Wang, G. Jin, and Y. Ma, Phys. Rev. E 68, 031301 (2003).
  23. R.D. Wildman and D.J. Parker, Phys. Rev. Lett. 88, 064301 (2002).
  24. K. Feitosa and N. Menon, Phys. Rev. Lett. 88, 198301 (2002).
  25. V. Garzó and J.W. Dufty, Phys. Fluids 14, 1476 (2002).
  26. J.M. Montanero and V. Garzó, Phys. Rev. E 67, 021308 (2003).
  27. G. A. Bird, Molecular Gas Dynamics and the Direct Simulation Monte Carlo of Gas Flows (Clarendon, Oxford, 1994).
  28. N. Brilliantov and T. Pöschel, Phys. Rev. E 61, 1716 (2000).
  29. J.J. Brey, M.J. Ruiz-Montero, D. Cubero, and R. García-Rojo, Phys. Fluids 12, 876 (2000).
  30. J.J. Brey, J.W. Dufty, and A. Santos, J. Stat. Phys. 97, 2811 (1999).
  31. J.J. Brey, M.J. Ruiz-Montero, R. García-Rojo, and J.W. Dufty, Phys. Rev. E 60, 7174 (1999).
  32. J.W. Dufty, J.J. Brey, and J. Lutsko, Phys. Rev. E 65, 051303 (2002).
  33. J.W. Dufty and V. Garzó, J. Stat. Phys. 105, 723 (2001).
  34. J. Lutsko, J.J. Brey, and J.W. Dufty, Phys. Rev. E 65, 051304 (2002).
  35. S.B. Savage and R. Dai, Mech. Mater. 16, 225 (1993); C.S. Campbell, J. Fluid Mech. 348, 85 (1997); P. Zamankhan and P. Sarkomaa, Physica A 275, 347 (2000).
  36. V. Garzó, Phys. Rev. E 66, 021308 (2002).
  37. O. Zik and J. Stavans, Europhys. Lett. 16, 255 (1991); V.V.R. Natajaran, M.L. Hunt, and E.D. Taylor, J. Fluid Mech. 304, 1 (1995); N. Menon and D.J. Durian, Science 275, 1920 (1997); S.-S. Hsiau and W.-L. Yang, Phys. Fluids 14, 612 (2002); B. Utter and R.P. Behringer, e-print cond-mat/0309040.
  38. T.P.C. van Noije and M.H. Ernst, Granular Matter 1, 57 (1998).
  39. J.J. Brey, M. Ruiz-Montero, and D. Cubero, Phys. Rev. E 54, 3664 (1996).
  40. J.M. Montanero and A. Santos, Granular Matter 2, 53 (2000).
  41. A. Santos and J.W. Dufty, Phys. Rev. Lett. 86, 4823 (2001); Phys. Rev. E 64, 051305 (2001).
  42. J. A. McLennan, Introduction to Nonequilibrium Statistical Mechanics (Prentice-Hall, New Jersey, 1989).
  43. E.A. Mason, J. Chem. Phys. 22, 169 (1954).
  44. M. López de Haro and E.G.D. Cohen, J. Chem. Phys. 80, 408 (1984).
  45. I. Goldhirsch and G. Zanetti, Phys. Rev. Lett. 70, 1619 (1993).
  46. T. Boublik, J. Chem. Phys. 53, 471 (1970); E.W. Grundke and D. Henderson, Mol. Phys. 24, 269 (1972); ibid.L.L. Lee and D. Levesque, 26, 1351 (1973).

Outline

Information

Phys. Rev. E 69, 021301– Published 23 February, 2004
  • Received 30 September 2003

© 2004 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation