Granular surface flow via successive destabilization: a continuum approach. (English) Zbl 1207.76137

In this paper a 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 allows to describe localized flows of various types as seen in experiments of A. Daerr and S. Douady [Nature (London) 399, 241 (1999)]. The descriptive level takes into account the free surface and the velocity of the flow solely. The presented model is limited to cases where this surface is effectively one-dimensional so that the model equations are consequently one-dimensional in space. This kind of description makes available to account for the tight coupling between local grain velocity and local height of the so-called lonely waves. The 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 the 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, global influences of the surface shape of a flowing granular layer on Bagnold friction are found. This characteristic collisional friction between flowing layers and beds is shown to cause S-shaping of the free surface for rapid flow velocities. Finally, the model also explains the power spectrum of avalanches as experimentally detected by H. Jaeger, C. Liu and S. Nagel [Phys. Rev. Lett. 62, 40 (1989)].


76T25 Granular flows
74C99 Plastic materials, materials of stress-rate and internal-variable type
37B15 Dynamical aspects of cellular automata
Full Text: DOI


[1] Nedderman, R.M., Statics and kinematics of granular materials, (1992), Cambridge University Press
[2] Bouchaud, J.-P.; Cates, M.; Ravi Prakash, J.; Edwards, S., A model for the dynamics of sandpile surfaces, J. phys. France I, 4, 1383, (1994)
[3] Douady, S.; Andreotti, B.; Daerr, A., On granular surface flow equations, Eur. phys. J. B, 11, 131, (1999)
[4] Savage, S.B., Gravity flow of cohesionless granular materials in chutes and channels, J. fluid mech., 92, 53, (1979) · Zbl 0398.76008
[5] Savage, S.B.; Hutter, K., The motion of a finite mass of granular material down a rough incline, J. fluid mech., 199, 177, (1989) · Zbl 0659.76044
[6] Khakhar, D.; Orpe, A.; Andrésen, P.; Ottino, J., Surface flow of granular materials: model and experiments in heap formation, J. fluid mech., 441, 255, (2001) · Zbl 1002.76592
[7] Aradian, A.; Raphaël, E.; de Gennes, P.-G., Surface flows of granular materials: a short introduction to some recent models, C. R. physique, 3, 187, (2002)
[8] Aranson, I.; Tsimring, L., Continuum description of avalanches in granular media, Phys. rev. E, 64, 020301, (2001)
[9] de Gennes, P.G., Granular matter: a tentative view, Rev. modern phys., 71, 374, (1999)
[10] Jaeger, H.; Nagel, S.; Behringer, R., Granular solids, liquids, and gases, Rev. modern phys., 68, 1259, (1996)
[11] Kadanoff, L., Built upon sand: theoretical ideas inspired by granular flows, Rev. modern phys., 71, 435, (1999)
[12] Nagel, S.N., Instabilities in a sandpile, Rev. modern phys., 64, 321, (1992)
[13] Campbell, C.S., Rapid granular flows, Annu. rev. fluid mech., 22, 57, (1990)
[14] MiDi, G., Granular media: some ideas from statistical physics, Eur. phys. J. E, 14, 341, (2004)
[15] Daerr, A.; Douady, S., Two types of avalanche behaviour in granular media, Nature (London), 399, 241, (1999)
[16] Daerr, A., Dynamical equilibrium of avalanches on a rough plane, Phys. fluids, 11, 2115, (2001) · Zbl 1184.76118
[17] Rajchenbach, J., Dynamics of grain avalanches, Phys. rev. lett., 88, (2002), 014301-1
[18] Jaeger, H.; Liu, C.; Nagel, S., Relaxation at the angle of repose, Phys. rev. lett., 62, 40, (1989)
[19] Forterre, Y.; Pouliquen, O., Long-surface-wave instability in dense granular flow, J. fluid mech., 486, 21, (2003) · Zbl 1156.76458
[20] Aranson, I.; Malloggi, F.; Clément, E., Transverse instability of avalanches in granular flows down incline, Phys. rev. E, 73, 050302(R), (2006)
[21] Reynolds, O., On the dilatancy of media composed of rigid particles in contact, Philos. mag., 20, 469, (1885)
[22] Bouchaud, J.-P.; Cates, M.; Ravi Prakash, J.; Edwards, S., Hysteresis and metastability in a continuum sandpile model, Phys. rev. lett., 74, 1982, (1995)
[23] Boutreux, T.; Raphaël, E.; de Gennes, P.-G., Surface flows of granular materials: A modified picture for thick avalanches, Phys. rev. E, 58, 4692, (1998)
[24] Aranson, I.S.; Tsimring, Lev S., Patterns and collective behavior in granular media: theoretical concepts, Rev. modern phys., 78, 641, (2006)
[25] Aranson, I.; Tsimring, L., Continuum theory of partially fluidized granular flows, Phys. rev. E, 65, 061303, (2002)
[26] Linz, S.; Hänggi, P., Minmal model for avalanches in granular systems, Phys. rev. E, 40, 2538, (1995)
[27] Linz, S.; Hänggi, P., Effect of periodic shear on avalanches in granular systems, Physica D, 97, 577, (1996) · Zbl 1194.76272
[28] Hager, W.; Linz, S.; Hänggi, P., Spectral statistics of global avalanches along granular piles, Europhys. lett., 40, 393, (1997)
[29] Linz, S.; Hager, W.; Hänggi, P., Hysteretic transition between avalanches and continuous flow in rotated granular systems, Chaos, 9, 649, (1999) · Zbl 1013.74012
[30] Börzsönyi, T.; Halsey, T.C.; Ecke, R.E., Two scenarios for avalanche dynamics in inclined granular layers, Phys. rev. lett., 94, 208001, (2005)
[31] Tischler, M.; Bursik, M.; Pitman, E.B., Kinematics of sand avalanches using particle-image velocimetry, J. sed. res., 71, 355, (2001)
[32] Fischer, R.; Gondret, P.; Rabaud, P.B.M., Dynamics of dry granular avalanches, Phys. rev. E, 78, 021302, (2008)
[33] Caponeri, M.; Douady, S.; Fauve, S.; Laroche, C., Dynamics of avalanches in a rotating cylinder, (), 331
[34] Taberlet, N.; Richard, P.; Hinch, E.J., The S-shape of a granular pile in a rotated drum, Phys. rev. E, 73, 050301(R), (2006)
[35] Patton, J.S.; Brennen, C.E.; Sabersky, R.H., Shear flows of rapidly flowing granular materials, J. appl. mech., 54, 801, (1987)
[36] Anecy, C., Dry granular flows down an inclined channel: experimental investigations on the frictional-collisional regime, Phys. rev. E, 65, 011304, (2001)
[37] Metcalfe, G.; Tennakoon, S.G.K.; Kondic, L.; Schaeffer, D.G.; Behringer, R.P., Granular friction, Coulomb failure, and the fluid – solid transition for horizontally shaken granular materials, Phys. rev. E, 65, 031302, (2002)
[38] Aumaitre, S.; Puls, C.; McElwaine, J.N.; Gollub, J.P., Comparing flow thresholds and dynamics for oscillating and inclined granular layers, Phys. rev. E, 75, 061307, (2007)
[39] Bak, P.; Tang, C.; Wiesenfeld, K., Self-organized criticality: an explanation of 1/f noise, Phys. rev. lett., 59, 381, (1987)
[40] Hoffmann, A.; Linz, S.J., A domino model for granular surface flow, (), 167
[41] Pouliquen, O.; Forterre, Y., Friction law for dense granular flows: application to the motion of a mass down a rough inclined plane, J. fluid mech., 453, 133, (2002) · Zbl 0987.76522
[42] Pouliquen, O.; Renaut, N., Onset of granular flows on an inclined rough surface: dilatancy effects, J. phys. II, 6, 923, (1996)
[43] Malloggi, F.; Lanuza, J.; Andreotti, B.; Clément, E., Erosion waves: transverse instabilities and fingering, Europhys. lett., 75, 825, (2006)
[44] Rajchenbach, J., Flow in powders: from discrete avalanches to continuous regime, Phys. rev. lett., 65, 2221, (1990)
[45] Orpe, A.; Khakhar, D., Scaling relations for granular flow in quasi-two-dimensional rotating cylinders, Phys. rev. E, 64, 031302, (2001)
[46] Lajeunesse, E.; Mangeney-Castelnau, A.; Vilotte, J.P., Spreading of a granular mass on a horizontal plane, Phys. fluids, 16, 2371, (2004) · Zbl 1186.76304
[47] Siavoshi, S.; Kudrolli, A., Failure of a granular step, Phys. rev. E, 71, 051302, (2005)
[48] Daerr, A.; Douady, S., Sensitivity of granular surface flows to preparation, Europhys. lett., 47, 324, (1999)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.