LHE on the cover page of Physical Review Letters

Segregation is a physical process that occurs in granular mixtures made up of solid particles with different properties (e.g., their diameter). When the mixture flows, grain sorting occurs: generally the larger particles quickly migrate to the top while the finer particles go downward. Who has not seen that when opening a bag of charcoal or a package of coffee beans, the size distribution is very heterogeneous? The phenomenon also occurs in natural flows such as snow avalanches and intense sediment transport (including debris flows). This produces a fairly complex coupling between flow and particle size distribution: coarse particles migrating to the free surface quickly enrich the front, then are deposited on the side to form long “levees”, which channel the flow. These levees limit flow’s spreading capacity and so, the flow dissipates less energy and can go further. This is one reason why some natural flows have remarkable mobility.

In our laboratory, Gael Chauvin-Epely (died accidentally in January 2012) and Kasper van der Vaart have analysed this process. One of the experimental setups for studying the dynamics of segregation is the shear cell, wherein an oscillating shear motion is applied to a mixture made of two sizes of beads. The device is not new (it was introduced by Bridgwater in the 1980s), but until recently, the experiments have faced a stumbling block: grain sorting was monitored by looking at what happens at the sidewall, but in the close neighbourhood of a solid boundary, the flow is disturbed and is no longer representative of what happens away from the walls. Gaël Chauvin Epely crossed this hurdle by creating a kind of laser tomography, which allowed him to view what occurs within the bulk. The idea is simple: creating a vertical laser plane that can be shifted horizontally and make vertical cuts of the mixture. The bulk is made transparent by adding an interstitial fluid (an alcohol) whose refractive index matches that of the particles. The particles are marked with a fluorescent dye which emits light of a certain wavelength. Taking images by the side and filtering the light, we can determine the position of each particle and thus reconstruct the entire volume within the shear cell.

Kasper van der Vaart conducted several series of experiments to determine the percolation rate of the finest particles depending on their concentration. The results were compared to a theory developed by prof. Nico Gray (from the mathematics department at the University of Manchester), a relatively simple theory where segregation is seen as an advection diffusion process. A remarkable result by Kasper van der Vaart is the lack of symmetry in the behaviour of particles: the finest particles go faster if they move in a medium rich in large particles and conversely large particles move more slowly when the local small particle concentration is high. Theoretically, this provides evidence that particle flows depend asymmetrically on the concentration of small particles, with the consequence that the function flow is not quadratic and convex as originally thought, but closer to a non-convex cubic function.

This change of convexity has major consequences on the behaviour of solutions to the advection diffusion equation, which turns out to be more complex than expected. Little is known about the hyperbolic problems with non-convex flux functions and part of the research conducted by prof. Gray is just exploring these solutions. The size segregation issue also reveals interesting similarities with other problems such as traffic flows, sedimentation, or diffusion through membranes.