Evolution of a sandpile in a thick-flow regime

We solve a one-dimensional sandpile problem analytically in a thick flow regime when the pile evolution may be described by a set of linear equations. We demonstrate that, if an income flow is constant, a space periodicity takes place while the sandpile evolves even for a pile of only one type of particles. Hence, grains are piling layer by layer. The thickness of the layers is proportional to the input flow of particles $r_0$ and coincides with the thickness of stratified layers in a two-component sandpile problem, which were observed recently. We find that the surface angle $\theta$ of the pile reaches its final critical value $\left({\theta }_f\right)$ only at long times after a complicated relaxation process. The deviation $\left({\theta }_f-\theta \right)$ behaves asymptotically as ${(t/r}_0{)}^{-1/2}.$ It appears that the pile evolution depends on initial conditions. We consider two cases: (i) grains are absent at the initial moment, and (ii) there is already a pile with a critical slope initially. Although at long times the behavior appears to be similar in both cases, some differences are observed for the different initial conditions are observed. We show that the periodicity disappears if the input flow increases with time.