Clustering in granular gas I:
Birth and sudden death of a cluster

Ko van der Weele

Department of Applied Physics, University of Twente,
P.O. Box 217, 7500 AE Enschede, The Netherlands

The key feature of a granular gas, making it fundamentally different from any ordinary molecular gas, is its tendency to form clusters (Goldhirsch and Zanetti, 1993; Kudrolli et al., 1997). This can be traced back to the inelasticity of the collisions between the particles. In applications such as conveyor belts and sorting machines, the clustering is an unwanted and very costly effect. Here we study the phenomenon in the setting of the so-called Maxwel Demon experiment (Eggers, 1999).

Granular material in $N$ connected compartments is brought into a gaseous state through vertical shaking. For sufficiently strong shaking the particles are uniformly distributed over the compartments, but if the shaking intensity is lowered this uniform distribution gives way to a clustered state. The clustering transition is experimentally shown to be of 2nd order for $N = 2$ and of 1st order for $N \geq 3$. In particular, the latter is hysteretic, involves long-lived transient states, and exhibits a striking lack of time reversibility (Van der Weele et al., 2001; Van der Meer et al., 2001).

In the strong shaking regime, a cluster breaks down very abruptly (sudden death) and in its further decay shows anomalous diffusion, with the length scale going as $t^{1/3}$ rather than the standard $t^{1/2}$ (Van der Meer et al., 2002). We focus upon the self-similar nature of this process. The observed phenomena are all accounted for within a dynamical flux model.

References
Eggers J, Sand as Maxwell's demon, Phys. Rev. Lett. 83, 5322-5325 (1999).
Goldhirsch I and Zanetti G, Clustering instability in dissipative gases, Phys. Rev. Lett. 70, 1619-1622 (1993).
Kudrolli A, Wolpert M, and Gollub JP, Cluster formation due to collisions in granular material, Phys. Rev. Lett. 78, 1383-1386 (1997).
Van der Meer, Van der Weele, Lohse D, Bifurcation diagram for compartmentalized granular gases, Phys. Rev. E 63, 061304 (2001).
Van der Meer D, Van der Weele K, and Lohse D, Sudden collapse of a granular cluster, Phys. Rev. Lett., 88, 174302 (2002).
Van der Weele K, Van der Meer D, Versluis M, and Lohse D, Hysteretic clustering in granular gas, Europhys. Lett. 53, 328-334 (2001).

Clustering in granular gas II:
David vs. Goliath and other competitive effects

Ko van der Weele

Department of Applied Physics, University of Twente,
P.O. Box 217, 7500 AE Enschede, The Netherlands

There are many extensions of the Maxwell Demon experiment. First we consider a bi-disperse mixture consisting of large and small particles (Mikkelsen et al., 2002). This is done with an eye to practical applications, where granular material is rarely mono-disperse. For moderate shaking the material clusters into the compartment which initially contained most of the large particles: Goliath wins. For very mild shaking, however, the cluster goes into the compartment originally dominated by small particles: David wins. These experimental observations are quantitatively explained within a bi-disperse version of the flux model.

Second, we study a system in which the compartments are arranged in the form of a staircase, resembling an industrial conveyor belt. The central topic here is the competition between the clustering effect and the natural tendency of the particles to stream downwards. When a cluster is formed, one can get rid of it by shaking sufficiently hard. The ensuing transition to the desired uniform flow is found to be a self-similar process involving Burgers-like shockwaves (Kloosterman et al., 2002).

Finally, we discuss two related clustering phenomena from other fields: the traffic jam problem (Helbing, 2001) and the formation of sand ripples at the beach (Andersen et al., 2001). Both turn out to be well described by flux models markedly similar to our own.

References
Andersen KH, Chabol ML, and Van Hecke M, Dynamical models for sand ripples beneath surface waves, Phys. Rev. E 63, 066308 (2001).
Helbing D, Traffic and related self-driven many-particle systems, Rev. Mod. Phys. 73, 1067-1141 (2001).
Kloosterman M, Van der Meer D, and Van der Weele K, Granular shockwaves on a staircase, preprint (2002).
Mikkelsen R, Van der Meer D, Van der Weele K, and Lohse D, Competitive clustering in bi-disperse granular gas, preprint (2002).