Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T07:11:20.176Z Has data issue: false hasContentIssue false
  • English
  • Français

Stable solutions of a scalar conservation law for particle-size segregation in dense granular avalanches

Published online by Cambridge University Press:  01 February 2008

M. SHEARER ,
J. M. N. T. GRAY  and
A. R. THORNTON
M. SHEARER
Affiliation:
Department of Mathematics & Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695, USA email: [email protected]
J. M. N. T. GRAY
Affiliation:
School of Mathematics & Manchester Centre for Nonlinear Dynamics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
A. R. THORNTON
Affiliation:
School of Mathematics & Manchester Centre for Nonlinear Dynamics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

Dense, dry granular avalanches are very efficient at sorting the larger particles towards the free surface of the flow, and finer grains towards the base, through the combined processes of kinetic sieving and squeeze expulsion. This generates an inversely graded particle-size distribution, which is fundamental to a variety of pattern formation mechanisms, as well as subtle size-mobility feedback effects, leading to the formation of coarse-grained lateral levees that create channels in geophysical flows, enhancing their run-out. In this paper we investigate some of the properties of a recent model [Gray, J. M. N. T. & Thornton, A. R. (2005) A theory for particle size segregation in shallow granular free-surface flows. Proc. R. Soc. 461, 1447–1473]; [Thornton, A. R., Gray, J. M. N. T. & Hogg, A. J. (2006) A three-phase mixture theory for particle size segregation in shallow granular free-surface flows. J. Fluid. Mech. 550, 1–25] for the segregation of particles of two sizes but the same density in a shear flow typical of shallow avalanches. The model is a scalar conservation law in space and time, for the volume fraction of smaller particles, with non-constant coefficients depending on depth within the avalanche. It is proved that for steady flow from an inlet, complete segregation occurs beyond a certain finite distance down the slope, no matter what the mixture at the inlet. In time-dependent flow, dynamic shock waves can develop; they are interfaces separating different mixes of particles. Shock waves are shown to be stable if and only if there is a greater concentration of large particles above the interface than below. Constructions with shocks and rarefaction waves are demonstrated on a pair of physically relevant initial boundary value problems, in which a region of all small particles is penetrated from the inlet by either a uniform mixture of particles or by a layer of small particles over a layer of large particles. In both cases, and under a linear shear flow, solutions are constructed for all time and shown to have similar structure for all choices of parameters.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 19 , Issue 1 , February 2008 , pp. 61 - 86
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bagnold, R. A. (1954) Experiments on a gravity-free dispersion of large spheres in a Newtonian fluid under shear. Proc. R. Soc. Lond. A 255, 49–63.Google Scholar
[2]Dolgunin, V. N. & Ukolov, A. A. (1995) Segregation modelling of particle rapid gravity flow. Powder Technol. 83, 95–103.CrossRefGoogle Scholar
[3]Gray, J. M. N. T. & Chugunov, V. A. (2006) Particle-size segregation and diffusive remixing in shallow granular avalanches. J. Fluid Mech. 569, 365–398.CrossRefGoogle Scholar
[4]Gray, J. M. N. T. & Cui, X. (2007) Weak, strong and detached oblique shocks in gravity driven granular free-surface flows. J. Fluid Mech. 579, 113–136.CrossRefGoogle Scholar
[5]Gray, J. M. N. T. & Hutter, K. (1997) Pattern formation in granular avalanches. Contin. Mech. Thermodyn. 9, 341–345.CrossRefGoogle Scholar
[6]Gray, J. M. N. T., Shearer, M. & Thornton, A. R. (2006) Time-dependent solutions for particle-size segregation in shallow granular avalanches. Proc. R. Soc. A 462, 947–972.CrossRefGoogle Scholar
[7]Gray, J. M. N. T., Tai, Y.-C. & Noelle, S. (2003) Shock waves, dead-zones and particle-free regions in rapid granular free surface flows. J. Fluid. Mech. 491, 161–181.CrossRefGoogle Scholar
[8]Gray, J. M. N. T. & Thornton, A. R. (2005) A theory for particle size segregation in shallow granular free-surface flows. Proc. R. Soc. 461, 1447–1473.CrossRefGoogle Scholar
[9]Gray, J. M. N. T., Wieland, M. & Hutter, K. (1999) Free surface flow of cohesionless granular avalanches over complex basal topography. Proc. R. Soc. A 455, 1841–1874.CrossRefGoogle Scholar
[10]Hill, K. M., Kharkar, D. V., Gilchrist, J. F., McCarthy, J. J. & Ottino, J. M. (1999) Segregation driven organization in chaotic granular flows. Proc. Natl. Acad. Sci. USA 96, 11689–12210.CrossRefGoogle ScholarPubMed
[11]Hutter, K. & Koch, T. (1991) Motion of a granular avalanche in an exponentially curved chute: Experiments and theoretical predictions. Phil. Trans. R. Soc. Lond. A 334, 93–138.Google Scholar
[12]Iverson, R. M. (1997) The physics of debris flows. Rev. Geophys. 35, 245–296.CrossRefGoogle Scholar
[13]Iverson, R. M. (2003) The debris-flow rheology myth. In: Rickenmann, D. & Chen, R. C. (editors), Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Millpress, Rotterdam, pp. 303–314.Google Scholar
[14]Iverson, R. M. & Vallance, J. W. (2001) New views of granular mass flows. Geology 29, 115–118.2.0.CO;2>CrossRefGoogle Scholar
[15]Jaynes, E. T. (1963) Information theory and statistical mechanics. In: Ford, K. W. (editor), Statistical Physics III. Benjamin, N.Y., pp. 181–218.Google Scholar
[16]Khakhar, D. V., McCarthy, J. J. & Ottino, J. M. (1997) Radial segregation of granular mixtures in rotating cylinders. Phys. Fluids 9, 3600–3614.CrossRefGoogle Scholar
[17]Khakhar, D. V., McCarthy, J. J. & Ottino, J. M. (1999) Mixing and segregation of granular materials in chute flows. Chaos 9, 594–610.CrossRefGoogle ScholarPubMed
[18]Kružkov, S. N. (1970) First order quasilinear equations in several independent variables. Mat. Sb. 81, 217–243.CrossRefGoogle Scholar
[19]Lax, P. D. (1957) Hyperbolic systems of conservation laws II. Comm. Pure Appl. Math. 10, 537–566.CrossRefGoogle Scholar
[20]McIntyre, M., Rowe, E., Thornton, A. R., Gray, J. M. N. T. & Shearer, M. (2008) A lens-shaped mixing zone in granular avalanches with particle-size segregation. AMRX, Vol. 2008, Article ID abm008, 12 pages, doi:10.1093/amrx/abm008.Google Scholar
[21]Middleton, G. V. & Hampton, M. A. (1976) Subaqueous sediment transport and deposition by sediment gravity flows. In: Stanley, D. J. & Swift, D. J. P. (editors), Marine Sediment Transport and Environmental Management, Wiley, N.Y. pp. 197–218.Google Scholar
[22]Phillips, J. C., Hogg, A. J., Kerswell, R. R. & Thomas, N. H. (2006) Enhanced mobility of granular mixtures of fine and coarse particles. Earth Planet. Sci. Lett. 246, 466–480.CrossRefGoogle Scholar
[23]Pouliquen, O., Delour, J. & Savage, S. B. (1997) Fingering in granular flows. Nature, 386, 816–817.CrossRefGoogle Scholar
[24]Pouliquen, O. & Vallance, J. W. (1999) Segregation induced instabilities of granular fronts. Chaos 9, 621–630.CrossRefGoogle ScholarPubMed
[25]Savage, S. B. & Hutter, K. (1989) The motion of a finite mass of granular material down a rough incline. J. Fluid Mech. 199, 177–215.CrossRefGoogle Scholar
[26]Savage, S. B. & Lun, C. K. K. (1988) Particle size segregation in inclined chute flow of dry cohesionless granular solids. J. Fluid Mech. 189, 311–335.CrossRefGoogle Scholar
[27]Serre, D. (2000) Systems of Conservation Laws, Vol. 2. Cambridge University Press. Cambridge, U.K.Google Scholar
[28]Silbert, L. E., Ertas, D., Grest, G. S., Haley, T. C., Levine, D. & Plimpton, S. J. (2001) Granular flow down an inclined plane: Bagnold scaling and rheology. Phys. Rev. E. 64, 051302.CrossRefGoogle ScholarPubMed
[29]Thornton, A. R. & Gray, J. M. N. T. (2008) Breaking size-segregation waves and particle recirculation in granular avalanches. J. Fluid Mech. 596, 261–284.CrossRefGoogle Scholar
[30]Thornton, A. R., Gray, J. M. N. T. & Hogg, A. J. (2006) A three-phase mixture theory for particle size segregation in shallow granular free-surface flows. J. Fluid. Mech. 550, 1–25.CrossRefGoogle Scholar
[31]Vallance, J. W. (2000) Lahars. In: Sigurdsson, H. (editor-in-chief), Encyclopedia of Volcanoes. Academic Press, New York, pp. 601–616.Google Scholar
[32]Vallance, J. W. & Savage, S. B. (2000) Particle segregation in granular flows down chutes. In: Rosato, A. D. & Blackmore, D. L. (editors), IUTAM Symposium on Segregation in Granular Materials. Kluwer Academic, Dordrecht, The Netherlands. pp. 31–51.CrossRefGoogle Scholar
[33]Zuriguel, I., Gray, J. M. N. T., Peixinho, J. & Mullin, T. (2006) Pattern selection by a granular wave in a rotating drum. Phys. Rev. E 73, 061302.CrossRefGoogle Scholar