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Clustering, coarsening and directed transport in a Granular gas: Lecture II

Tuesday 4th April 2006 - 11:30 to 12:30
INI Seminar Room 1

1. Clustering and Coarsening in a Granular Gas Granular gases are of great scientific and economic relevance. Scientific, because of their tendency to spontaneously separate into dense and dilute regions, which makes them fundamentally different from any textbook molecular gas. Economic, because no less than 5 per cent of the global energy budget is wasted due to problems with granular matter in conveyor belts, sorting machines, mixers, and other industrial machinery. Here we study - experimentally, numerically, and theoretically - the clustering of particles in a vertically vibrated array of N connected compartments. For strong shaking, the particles spread evenly over the compartments, but if the shaking strength is lowered beneath a critical level this uniform distribution gives way to a clustered state, consisting of a few well-filled compartments and a lot of diluted ones. In the course of time, this state coarsens: The smaller clusters are eaten by the larger ones, until finally only one big cluster remains. This coarsening process is exceptionally slow, with the mass of the surviving cluster growing only as the square root of log t.

2. Clustering and Directed Transport

In this second lecture we turn to the wonderful world of ratchets, which have become a hot topic in recent years. In order to extract mechanical work on a molecular scale (e.g., to make a muscle move), nature uses the concept of a Brownian ratchet: The stochastic forces from a noisy environment are converted into a directed motion. Here we create a "granular ratchet", exploiting the clustering phenomenon from the previous lecture in a slightly adapted array of N connected compartments: The stochastically colliding particles spontaneously generate a particle current perpendicular to the direction of energy input. This is the first practical realization of the theoretically predicted concept of a stochastic ratchet as a collective effect in a symmetric geometry.

A related problem of prime importance in modern society is the clustering of cars on the highway. We show how the formation of traffic jams on the Dutch highway A58 is well described - and predicted! - by a flux model similar to the one we use for the clustering of granular particles

3. Granular Impact Jets

A steel ball dropped onto loose, very fine sand ("dry quicksand") creates an upward jet exceeding the release height of the ball. There is a striking similarity with the impact of an object in a liquid: The jet is generated by the gravity-driven collapse of the void created by the ball, and the focused pressure pushes the sand straight up into the air. Using a 2-dimensional experimental setup and high-speed imaging, the collapse of the void is visualized. For high impact velocities the void collapse is seen to entrain air. The entrained air bubble slowly rises through the sand, and upon reaching the surface causes a granular eruption. The experimental observations are quantitatively explained by a Rayleigh-type model. Parallels are drawn with impacts on a planetary scale.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons