# Shape Created by a Pile of Granular Objects Dropped Uniformly

I've observed many times that if you drop a lot of a 'granular' substance in one place and keep the nozzle out of which the substance flows, that the shape of the pile created very much resembles a bell curve. The situation is a bit hard to explain so for example image somebody holding a small pipe vertically above the ground and dropping a large amount of sand through that small pipe. The shape created by the sand on the ground largely resembles a bell curve wrapped around the vertical axis. I come from a math background and am really unsure of how to even start proving or disproving the proposition that the shape formed by the grains is a bell curve. Any help and input would be appreciated.

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You may be interested in the paper

• G. Aronsson, L. C. Evans, Y. Wu. Fast/slow diffusion and growing sandpiles, Journal of Differential Equations, volume 131, number 2, 1996, pages 304–335

This paper uses the $p$-laplacian $\operatorname{div} (|\nabla u|^{p-2}| \nabla u)$ to model the diffusion of sand particles. If you want to know more about this operator, the dissertation

• Lundström, Niklas LP. "$p$-harmonic functions near the boundary." (2011).

might be of use. Maybe the functions in Figure 1.1 are like what you have observed?

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Yeah-- think "landslides." Maybe the OP will settle for the expected shape prior to any chaotic event? – Carl Witthoft Apr 3 '14 at 14:26

I'd be surprised if it is a Bell curve. But sand-piles are a research area, try searching "self organized criticality".

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@Carl Witthoft Good point but hard to get something resembling the tails of the bell without some sliding. – baldrik Apr 3 '14 at 15:50

I've tried a similar experiment in the past and found that the shape is essentially a cone but with "tails" at the base. In the bulk, the conical shape can be explained by a property of the granular material known as the angle of repose. In a cohesionless material (dry, largish grains) this is equal to the arctangent of the friction coefficient between grains. The angle of repose describes the angle of a slope formed by many grains piled onto one another so a granular pile without confining walls will form a symmetric cone with slopes at this angle. At the base, however, the different frictional properties of the horizontal surface on which the pile is formed mean that the conical shape diverges at the base. A "cleaner" and better-defined edge to the cone can be formed by using a rough surface such as sand paper but particles flowing or saltating down the pile always give a slightly diffuse edge. Furthermore, if you poured your particles from any height then an indentation would be made at the vertex of the cone as they impact the pile, effectively truncating the cone. If your pile wasn't very large you may have mistaken its shape for a 2d Gaussian surface.

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