Granular surface flow via successive destabilization: A continuum approach

Global and localized granular surface flow is analyzed in a model that takes into account the basic forces between a flowing granular layer and an underlying granular bed. Starting from a quantitatively correct description of global stick-slip avalanches on the surfaces of heaps we are able to describe localized flows of various types as seen in experiments of Daerr and Douady [A. Daerr, S. Douady, Two types of avalanche behavior in granular media, Nature (London) 399 (1999) 241]. The descriptive level takes into account the free surface and the velocity of the flow solely. We limit our model to cases where this surface is effectively one-dimensional so that our model equations are consequently one-dimensional in space. With this kind of description we are able to account for the tight coupling between local grain velocity and local height of the so-called lonely waves. Our model shows the nature of a wave of mobilization and deposition. Granular material is set into motion in a domino-like manner. In addition destabilization of material behind an avalanche is taken into account. The front velocities are found to be constant and in general head and rear front travel with the same velocity in our model. The front velocities depend on the tilt angle of the heap that those localized structures run down on. A critical angle is observed above which the rear front velocity abruptly switches sign from positive (downslope) to negative (upslope) values. Furthermore, we find global influences of the surface shape of a flowing granular layer on Bagnold friction. This characteristic collisional friction between flowing layers and beds is shown to cause S-shaping of the free surface for rapid flow velocities. Finally, our model also explains the power spectrum of avalanches as experimentally detected by Jaeger et al. [H. Jaeger, C. Liu, S. Nagel, Relaxation at the angle of repose, Phys. Rev. Lett. 62 (1989) 40].