Abstract
Free cooling of granular materials is analyzed on the basis of a pseudo- Liouville operator. Exchange of translational and rotational energy requires surface roughness for spherical grains, but occurs for non-spherical grains, like needles, even if they are perfectly smooth. Based on the assumption of a homogeneous cooling state, we derive an approximate analytical theory. It predicts that cooling of both rough spheres and smooth needles proceeds in two stages: An exponentially fast decay to a state with stationary ratio of translational and rotational energy and a subsequent algebraic decay of the total energy. These results are confirmed by simulations for large systems of moderate density. For higher densities, we observe deviations from the homogeneous state as well as large-scale structures in the velocity field. We study non-Gaussian distributions of the momenta perturbatively and observe a breakdown of the expansion for particular values of surface roughness and normal restitution.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
For a review on classical liquids see e.g. J.-P. Hansen and I. R. McDonald tiTheory of Simple Liquids, Academic Press (1986).
E. Thiele, J. Chem. Phys. 39, 474 (1963); M. S. Wertheim, Phys. Rev. Lett. 10, 321 (1963); J. Math. Phys. 5, 634 (1964); L. Verlet and D. Levesque, Mol. Phys. 46, 969 (1982).
J. L. Lebowitz, J. K. Percus and J. Sykes, Phys. Rev. 188, 487 (1996); H. H. U. Konijnendijk and J. M. van Leeuwen, Physica 64, 342 (1973); P. M. Furtado, G. F. Mazenko and S. Yip, Phys. Rev. A12, 1653 (1975); H. van Beijeren and M. H. Ernst, J. Stat. Phys. 21, 125 (1979).
B. J. Alder, D. M. Gass and T. E. Wainwright, J. Chem. Phys. 53, 3813 (1970).
J. J. Erpenbeck and W. W. Wood, J. Stat. Phys. 24, 455 (1981).
B. J. Alder and T. E. Wainwright, Phys. Rev. 127, 359 (1962).
J. T. Jenkins and S. B. Savage, J. Fluid Mech.130, 187 (1983); C. K. K. Lun, S. B. Savage, D. J. Jeffrey and N. Chepurniy, J. Fluid Mech.140, 223 (1984). A. Goldshtein and M. Shapiro, J. Fluid Mech.282, 75 (1995).
J. T. Jenkins and M. W. Richman, Phys. of Fluids 28, 3485 (1985).
C. K. K. Lun and S. B. Savage, J. Appl. Mech. 54, 47 (1987).
C. K. K. Lun, J. Fluid Mech. 233, 539 (1991).
A. Goldshtein and M. Shapiro, J. Fluid Mech. 282, 75 (1995).
H. Grad Principles of the kinetic theory of gases in Handbuch der Physik, ed. S. F. Fluegge, Springer (1958).
L. Waldmann Transporterscheinungen in Gasen von mittlerem Druck in Handbuch der Physik, ed. S. F. Fluegge, Springer (1958).
S. Chapman and T. G. Cowling, The Mathematical Theory of Nonuniform Gases, Cambridge University Press, London (1960).
I. Goldhirsch and G. Zanetti, Phys. Rev. Lett. 70, 1619 (1993).
S. McNamara and S. Luding, Phys. Rev. E 58, 2247 (1998).
M. Huthmann and A. Zippelius, Phys. Rev. E 56, R6275 (1997).
S. Luding, M. Huthmann, S. McNamara, A. Zippelius, Phys. Rev. E 58, 3416 (1998).
O. Walton, in Energy and Technology Review, edited by A. J. Poggio, (Lawrence Livermore National Laboratory, Livermore, CA) (1988); M. A. Hopkins and H. Shen in Micromechanics of Granular Materials, eds. M. Satake and J. T. Jenkins, Amsterdam, Elsevier, (1987).
G. W. Baxter and R. P. Behringer, Phys. Rev. A42, 1017 (1990).
B. Brogliato, Nonsmooth Impact mechanics, Springer 1996.
M. Huthmann, T. Aspelmeier and A. Zippelius, Phys. Rev. E 60, 654 (1999).
M. H. Ernst, J. R. Dorfmann, W. R. Hoegy, and J. M. J. van Leeuwen, Physica 45, 127 (1969).
P. Resibois and J. L. Lebowitz, J. Stat. Phys. 12, 483 (1975); P. Resibois, J. Stat. Phys. 13, 393 (1975); E. Leutheusser, J. Phys. C15, 2801 (1982).
D. Frenkeland J. F. Maguire, Mol. Phys. 49, 503 (1983).
P. K. Ha., Journ. of Fluid. Mech. 134, 401 (1983).
T. P. C. van Noije, M. H. Ernst, R. Brito, Physica A 251 266 (1998).
S. McNamara and W. Young, Phys. Rev. E 53, 5089 (1996).
R. Brito and M. H. Ernst, Europhysics Letters 43, 497 (1998).
W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Springer (1966).
O. Herbst, M. Huthmann, and A. Zippelius (preprint) (1999).
T. P. C. van Noije, M. H. Ernst, Granular Matter, 1, 57 (1998).
M. Huthmann, J. A. G. Orza, and R. Brito (unpublished).
J. Javier Brey, M. J. Ruiz Montero and D. Cubero, Phys. Rev. E, 54 3664 (1996).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Aspelmeier, T., Huthmann, M., Zippelius, A. (2001). Free Cooling of Particles with Rotational Degrees of Freedom. In: Pöschel, T., Luding, S. (eds) Granular Gases. Lecture Notes in Physics, vol 564. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44506-4_2
Download citation
DOI: https://doi.org/10.1007/3-540-44506-4_2
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-41458-2
Online ISBN: 978-3-540-44506-7
eBook Packages: Springer Book Archive