Abstract
In this paper we show how, under certain restrictions, the hydrodynamic equations for the freely evolving granular fluid fit within the framework of the time dependent Landau–Ginzburg (LG) models for critical and unstable fluids. The granular fluid, which is usually modeled as a fluid of inelastic hard spheres (IHS), exhibits two instabilities: the spontaneous formation of vortices and of high density clusters. We suppress the clustering instability by imposing constraints on the system sizes, in order to illustrate how LG-equations can be derived for the order parameter, being the rate of deformation or shear rate tensor, which controls the formation of vortex patterns. From the shape of the energy functional we obtain the stationary patterns in the flow field. Quantitative predictions of this theory for the stationary states agree well with molecular dynamics simulations of a fluid of inelastic hard disks.
Similar content being viewed by others
REFERENCES
S. R. Nagel, Rev. Mod. Phys. 64:321 (1992).
C. S. Campbell, Annu. Rev. Fluid Mech. 22:57 (1990).
W. Losert, L. Bocquet, T. C. Lubensky, and J. P. Gollub, Phys. Rev. Lett. 85:1428 (2000). L. Bocquet, W. Losert, D. Schalk, T. C. Lubensky, and J. P. Gollub, Phys. Rev. E 65:U11307 (2001).
P. B. Umbanhowar, F. Melo, and H. L. Swinney, Nature 382:793 (1996).
G. P. Collins, Sci. Am. 284 (1):17 (2001).
E. R. Nowak, J. B. Knight, E. Ben-Naim, H. M. Jaeger, and S. R. Nagel, Phys. Rev. E 57:1971 (1998). J. S. Olafsen and J. S. Urbach, Phys. Rev. Lett. 81:4369 (1998).
D. R. Williams and F. C. MacKintosh, Phys. Rev. E 54:R9 (1996). G. Peng and T. Ohta, Phys. Rev. E 58:4737 (1998). C. Bizon, M. D. Shattuck, J. B. Swift, and H. L. Swinney, Phys. Rev. E 60:4340 (1999). A. Puglisi, V. Loreto, U. Marini Bettolo Marconi, A. Petri, and A. Vulpiani, Phys. Rev. Lett. 81:3848 (1998). A. Puglisi, V. Loreto, U. Marini Bettolo Marconi, and A. Vulpiani, Phys. Rev. E 59:5582 (1999). T. P. C. van Noije, M. H. Ernst, E. Trizac, and I. Pagonabarraga, Phys. Rev. E 59:4326 (1999).
J. T. Jenkins and M. W. Richman, Phys. Fluids 28:3485 (1985). J. T. Jenkins and S. B. Savage, J. Fluid Mech. 130:187 (1983).
I. Goldhirsch and G. Zanetti, Phys. Rev. Lett. 70:1619 (1993). I. Goldhirsch, M-L. Tan, and G. Zanetti, J. Scient. Comp. 8:1 (1993).
R. Soto, M. Mareschal, and M. Malek Mansour, Phys. Rev. E 62:3836 (2000).
P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49:435 (1977).
L. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon Press, 1959).
J. J. Brey, J. W. Dufty, and A. Santos, J. Stat. Phys. 87:1051 (1997).
J. A. G. Orza, R. Brito, T. P. C. van Noije, and M. H. Ernst, Int. J. Mod. Phys. C 8:953 (1997).
T. P. C. van Noije, M. H. Ernst, R. Brito, and J. A. G. Orza, Phys. Rev. Lett. 79:411 (1997).
T. P. C. van Noije, M. H. Ernst, and R. Brito, Phys. Rev. E 57:R4891 (1998).
P. K. Haff, J. Fluid Mech. 134:401 (1983).
S. McNamara, Phys. Fluids A 5:3056 (1993).
P. Deltour and J.-L. Barrat, J. Phys. I France 7:137 (1997).
J. J. Brey, F. Moreno, and J. W. Dufty, Phys. Rev. E 54:445 (1996)
S. E. Esipov and T. Pöschel, J. Stat. Phys. 86:1385 (1997).
T. P. C. van Noije and M. H. Ernst, Phys. Rev. E 61:1765 (2000).
R. Brito and M. H. Ernst, Europhys. Lett. 43:497 (1998). R. Brito and M. H. Ernst, Int. J. Mod. Phys. C 9:1339 (1998).
J. J. Brey, M. J. Ruiz-Montero, and D. Cubero, Phys. Rev. E 60:3150 (1999).
E. Ben-Naim, S. Y. Chen, G. D. Doolen, and S. Redner, Phys. Rev. Lett. 83:4069 (1999).
J. Wakou, to be published.
G. K. Batchelor, The Theory of Homogeneous Turbulence (Cambridge University Press, 1970).
U. Frisch, Turbulence: The legacy of A. N. Kolmogorov (Cambridge University Press, 1996).
E. Trizac and A. Barrat, Eur. Phys. J. E. 3:291 (2000).
S. Chen, Y. Deng, X. Nie, and Y. Tu, Phys. Lett. A 269:218 (2000).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Wakou, J., Brito, R. & Ernst, M.H. Towards a Landau–Ginzburg-Type Theory for Granular Fluids. Journal of Statistical Physics 107, 3–22 (2002). https://doi.org/10.1023/A:1014590000158
Issue Date:
DOI: https://doi.org/10.1023/A:1014590000158