# 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 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 at 15:50