Behringer, RP; Doorn, EV; Hartley, RR; Pak, HK, *Making a rough place "plane": Why heaping of vertically shaken sand must stop at low pressure*,
Granular Matter, vol. 4 no. 1
(2002),
pp. 9-15 [doi] .
**Abstract:**

*The heaping of a granular material subject to vertical vibration vanishes abruptly as the pressure of the surrounding gas, P, is lowered below a critial value ̃10 Torr, depending on particle diameter etc. We show that the vanishing of the heap is consistent with two different effects. One of these is the onset of a Knudsen regime where the mean free path of a gas molecule becomes comparable to or larger than the typical distance to a grain. The usual Darcian gas flow models fail in this regime, and a Knudsen replacement predicts a vanishing of gas effects as P → 0. The other is that at low enough pressures, there is not enough gas to sustain flow under the usual linearized flow scenario. A detailed description of this regime is beyond the present analysis. © Springer-Verlag 2002.*