Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-26T08:19:14.810Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymmetric flux models for particle-size segregation in granular avalanches

Published online by Cambridge University Press:  19 September 2014

P. Gajjar  and
J. M. N. T. Gray
P. Gajjar*
Affiliation:
School of Mathematics and Manchester Centre for Nonlinear Dynamics, University of Manchester, Manchester M13 9PL, UK
J. M. N. T. Gray
Affiliation:
School of Mathematics and Manchester Centre for Nonlinear Dynamics, University of Manchester, Manchester M13 9PL, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Particle-size segregation commonly occurs in both wet and dry granular free-surface flows through the combined processes of kinetic sieving and squeeze expulsion. As the granular material is sheared downslope, the particle matrix dilates slightly and small grains tend to percolate down through the gaps, because they are more likely than the large grains to fit into the available space. Larger particles are then levered upwards in order to maintain an almost uniform solids volume fraction through the depth. Recent experimental observations suggest that a single small particle can percolate downwards through a matrix of large particles faster than a large particle can be levered upwards through a matrix of fines. In this paper, this effect is modelled by using a flux function that is asymmetric about its maximum point, differing from the symmetric quadratic form used in recent models of particle-size segregation. For illustration, a cubic flux function is examined in this paper, which can be either a convex or a non-convex function of the small-particle concentration. The method of characteristics is used to derive exact steady-state solutions for non-diffuse segregation in two dimensions, with an inflow concentration that is (i) homogeneous and (ii) normally graded, with small particles above the large. As well as generating shocks and expansion fans, the new asymmetric flux function generates semi-shocks, which have characteristics intersecting with the shock just from one side. In the absence of diffusive remixing, these can significantly enhance the distance over which complete segregation occurs.

JFM classification

Mixing: Mixing Nonlinear Dynamical Systems: Pattern formation
Type
Papers
Information
Journal of Fluid Mechanics , Volume 757 , 25 October 2014 , pp. 297 - 329
Copyright
© 2014 Cambridge University Press 

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

Bagnold, R. A. 1954 Experiments on gravity-free dispersion of large solid spheres in a Newtonian fluid under shear. Proc. R. Soc. Lond. A 225, 49–63.Google Scholar
Bartelt, P. & McArdell, B. W. 2009 Granulometric investigations of snow avalanches. J. Glaciol. 55 (193), 829–833.CrossRefGoogle Scholar
Batchelor, G. K. 1972 Sedimentation in a dilute dispersion of spheres. J. Fluid Mech. 52, 245–268.CrossRefGoogle Scholar
Berryman, J. G. 1983 Random close packing of hard spheres and disks. Phys. Rev. A 27, 1053–1061.Google Scholar
Branney, M. J. & Kokelaar, B. P. 1992 A reappraisal of ignimbrite emplacement: progressive aggradation and changes from particulate to non-particulate flow during emplacement of high-grade ignimbrite. Bull. Volcanol. 54, 504–520.Google Scholar
Bridgwater, J. 1994 Mixing and segregation mechanisms in particle flow. In Granular Matter (ed. Mehta, A.), pp. 161–193. Springer.Google Scholar
Bridgwater, J., Foo, W. & Stephens, D. 1985 Particle mixing and segregation in failure zones – theory and experiment. Powder Technol. 41, 147–158.Google Scholar
Bridgwater, J. & Ingram, N. D. 1971 Rate of spontaneous inter-particle percolation. Trans. Inst. Chem. Engrs 49 (3), 163–169.Google Scholar
Buckley, S. E. & Leverett, M. C. 1942 Mechanism of fluid displacement in sands. Trans. AIME 146, 107–116.Google Scholar
Calder, E. S., Sparks, R. S. J. & Gardeweg, M. C. 2000 Erosion, transport and segregation of pumice and lithic clasts in pyroclastic flows inferred from ignimbrite at Lascar volcano, Chile. J. Volcanol. Geotherm. Res. 104, 201–235.CrossRefGoogle Scholar
Chadwick, P. 1976 Continuum Mechanics. Concise Theory and Problems. George Allen & Unwin.Google Scholar
Courant, R. & Hilbert, D. 1962 Methods of Mathematical Physics, vol. II. Interscience.Google Scholar
Dasgupta, P. & Manna, P. 2011 Geometrical mechanism of inverse grading in grain-flow deposits: an experimental revelation. Earth-Sci. Rev. 104 (1–3), 186–198.Google Scholar
Dingler, J. R. & Anima, R. J. 1987 Subaqueous grain flows at the head of Carmel submarine canyon, California. J. Sedim. Petrol. 59 (2), 280–286.Google Scholar
Dolgunin, V. N. & Ukolov, A. A. 1995 Segregation modelling of particle rapid gravity flow. Powder Technol. 83, 95–103.CrossRefGoogle Scholar
Dyer, F. C. 1929 The scope for reverse classification by crowded settling in ore-dressing practice. Eng. Min. J. 127 (26), 1030–1033.Google Scholar
Fan, Y., Boukerkour, Y., Blanc, T., Umbanhowar, P. B., Ottino, J. M. & Lueptow, R. M. 2012 Stratification, segregation, and mixing of granular materials in quasi-two-dimensional bounded heaps. Phys. Rev. E 86, 051305.CrossRefGoogle ScholarPubMed
Fan, Y. I. & Hill, K. M. 2011 Theory for shear-induced segregation of dense granular mixtures. New J. Phys. 13 (9), 095009.Google Scholar
Fan, Y., Schlick, C. P., Umbanhowar, P. B., Ottino, J. M. & Lueptow, R. M. 2014 Modelling size segregation of granular materials: the roles of segregation, advection and diffusion. J. Fluid Mech. 741, 252–279.CrossRefGoogle Scholar
Fisher, R. V. & Mattinson, J. M. 1968 Wheeler gorge turbidite–conglomerate series California – inverse grading. J. Sedim. Petrol. 38 (4), 1013–1023.Google Scholar
Forterre, Y. & Pouliquen, O. 2008 Flows of dense granular media. Annu. Rev. Fluid Mech. 40 (1), 1–24.Google Scholar
GDR MiDi 2004 On dense granular flows. Eur. Phys. J. E 14, 341–365.Google Scholar
Golick, L. A. & Daniels, K. E. 2009 Mixing and segregation rates in sheared granular materials. Phys. Rev. E 80 (4), 042301.CrossRefGoogle ScholarPubMed
Gray, J. M. N. T. 2001 Granular flow in partially filled slowly rotating drums. J. Fluid Mech. 441, 1–29.Google Scholar
Gray, J. M. N. T. & Ancey, C. 2009 Segregation, recirculation and deposition of coarse particles near two-dimensional avalanche fronts. J. Fluid Mech. 629, 387–423.Google Scholar
Gray, J. M. N. T. & Ancey, C. 2011 Multi-component particle-size segregation in shallow granular avalanches. J. Fluid Mech. 678, 535–588.Google Scholar
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
Gray, J. M. N. T. & Edwards, A. N. 2014 A depth-averaged $\mu (I)$ -rheology for shallow granular free-surface flows. J. Fluid Mech. 755, 503–534.Google Scholar
Gray, J. M. N. T. & Hutter, K. 1997 Pattern formation in granular avalanches. Contin. Mech. Thermodyn. 9, 341–345.Google Scholar
Gray, J. M. N. T. & Kokelaar, B. P. 2010a Large particle segregation, transport and accumulation in granular free-surface flows. J. Fluid Mech. 652, 105–137.CrossRefGoogle Scholar
Gray, J. M. N. T. & Kokelaar, B. P. 2010b Large particle segregation, transport and accumulation in granular free-surface flows – Erratum. J. Fluid Mech. 657, 539.CrossRefGoogle Scholar
Gray, J. M. N. T. & Thornton, A. R. 2005 A theory for particle size segregation in shallow granular free-surface flows. Proc. R. Soc. Lond. A 461, 1447–1473.Google Scholar
Gray, J. M. N. T., Wieland, M. & Hutter, K. 1999 Free surface flow of cohesionless granular avalanches over complex basal topography. Proc. R. Soc. Lond. A 455, 1841–1874.CrossRefGoogle Scholar
Greenberg, H. 1959 An analysis of traffic flow. Oper. Res. 7 (1), 79–85.Google Scholar
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, 11701–11706.Google Scholar
Hutter, K., Wang, Y. Q. & Pudasaini, S. P. 2005 The Savage–Hutter avalanche model: how far can it be pushed?. Phil. Trans. R. Soc. Lond. A 363 (1832), 1507–1528.Google Scholar
Iverson, R. M. 1997 The physics of debris-flows. Rev. Geophys. 35, 245–296.CrossRefGoogle Scholar
Iverson, R. M. & Vallance, J. W. 2001 New views of granular mass flows. Geology 29 (2), 115–118.Google Scholar
Jeffrey, A. 1976 Quasilinear Hyperbolic Systems and Waves. Pitman.Google Scholar
Johanson, J. R.1978 Particle segregation …and what to do about it. Chem. Engng, 8 May, pp. 183–188.Google Scholar
Johnson, C. G., Kokelaar, B. P., Iverson, R. M., Logan, M., LaHusen, R. G. & Gray, J. M. N. T. 2012 Grain-size segregation and levee formation in geophysical mass flows. J. Geophys. Res. 117, F01032.Google Scholar
Jop, P., Forterre, Y. & Pouliquen, O. 2005 Crucial role of sidewalls in granular surface flows: consequences for the rheology. J. Fluid Mech. 541, 167–192.CrossRefGoogle Scholar
Khakhar, D. V., McCarthy, J. J. & Ottino, J. M. 1999 Mixing and segregation of granular materials in chute flows. Chaos 9 (3), 594–610.Google Scholar
Kokelaar, B. P., Graham, R. L., Gray, J. M. N. T. & Vallance, J. W. 2014 Fine-grained linings of leveed channels facilitate runout of granular flows. Earth Planet. Sci. Lett. 385, 172–180.Google Scholar
Kumaran, V. 2006 The constitutive relation for the granular flow of rough particles, and its application to the flow down an inclined plane. J. Fluid Mech. 561, 1–42.Google Scholar
Kumaran, V. 2008 Dense granular flow down an inclined plane: from kinetic theory to granular dynamics. J. Fluid Mech. 599, 121–168.Google Scholar
Kynch, G. J. 1952 A theory of sedimentation. Trans. Faraday Soc. 48, 166–176.Google Scholar
Laney, C. B. 1998 Computational Gasdynamics. Cambridge University Press.Google Scholar
Lax, P. D. 1957 Hyperbolic systems of conservation laws II. Commun. Pure Appl. Maths 10 (4), 537–566.CrossRefGoogle Scholar
Lighthill, M. J. & Whitham, G. B. 1955 On kinematic waves: II. A theory of traffic flow on long crowded roads. Proc. R. Soc. Lond. A 229 (1178), 317–345.Google Scholar
Liu, T. P. 1974 The Riemann problem for general $2 \times 2$ conservation laws. Trans. Am. Math. Soc. 199, 89–112.Google Scholar
Marks, B. & Einav, I. 2011 A cellular automaton for segregation during granular avalanches. Granul. Matt. 13 (3), 211–214.Google Scholar
Marks, B., Rognon, P. & Einav, I. 2012 Grainsize dynamics of polydisperse granular segregation down inclined planes. J. Fluid Mech. 690, 499–511.Google Scholar
Marks, B., Valaulta, A., Puzrin, A. & Einav, I. 2013 Design of protection structures: the role of the grainsize distribution. In Powders and Grains 2013: Proceedings of the 7th International Conference on Micromechanics of Granular Media, Sydney, Australia, 8–12 July (ed. Yu, A., Dong, K., Yang, R. & Luding, S.), AIP Conference Proceedings, vol. 1542, pp. 658–661. American Institute of Physics.Google Scholar
May, L. B. H., Golick, L. A., Phillips, K. C., Shearer, M. & Daniels, K. E. 2010a Shear-driven size segregation of granular materials: modeling and experiment. Phys. Rev. E 81, 051301.Google Scholar
May, L. B. H., Shearer, M. & Daniels, K. E. 2010b Scalar conservation laws with non-constant coefficients with application to particle size segregation in granular flow. J. Nonlinear Sci. 20 (6), 689–707.Google Scholar
Middleton, G. V. 1970 Experimental studies related to problems of flysch sedimentation. In Flysch Sedimentology in North America (ed. Lajoie, J.), Geological Association of Canada, Special paper 7, pp. 253–272. Business and Economic Service.Google Scholar
Morland, L. W. 1992 Flow of viscous fluids through a porous deformable matrix. Surv. Geophys. 13 (3), 209–268.Google Scholar
Nityanand, N., Manley, B. & Henein, H. 1986 An analysis of radial segregation for different sized spherical solids in rotary cylinders. Metall. Trans. B 17 (2), 247–257.Google Scholar
Oleinik, O. A. 1959 Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation. Usp. Mat. Nauk (NS) 14, 165–170.Google Scholar
Ottino, J. M. & Khakhar, D. V. 2000 Mixing and segregation of granular materials. Annu. Rev. Fluid Mech. 32 (1), 55–91.Google Scholar
Pitman, E. B., Nichita, C. C., Patra, A., Bauer, A., Sheridan, M. & Bursik, M. 2003 Computing granular avalanches and landslides. Phys. Fluids 15 (12), 3638–3646.Google Scholar
Pouliquen, O. 1999 Scaling laws in granular flows down rough inclined planes. Phys. Fluids 11 (3), 542–548.Google Scholar
Pudasaini, S. P. & Hutter, K. 2007 Avalanche Dynamics: Dynamics of Rapid Flows of Dense Granular Avalanches. Springer.Google Scholar
Rhee, H. K., Aris, R. & Amundson, N. R. 1986 First-order Partial Differential Equations, Theory and Applications of Single Equations, vol. 1. Prentice-Hall.Google Scholar
Rognon, P. G., Roux, J. N., Naaim, M. & Chevoir, F. 2007 Dense flows of bidisperse assemblies of disks down an inclined plane. Phys. Fluids 19 (5), 058101.Google Scholar
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.Google Scholar
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.Google Scholar
Schminck, H. U. 1967 Graded lahars in type sections of Ellensburg Formation, south-central Washington. J. Sedim. Petrol. 37 (2), 438–448.Google Scholar
Shannon, P. T., Stroupe, E. & Tory, E. M. 1963 Batch and continuous thickening – basic theory – solids flux for rigid spheres. Ind. Engng Chem. Fundam. 2 (3), 203–211.Google Scholar
Shinbrot, T. & Muzzio, F. J. 2000 Non-equilibrium patterns in granular mixing and segregation. Phys. Today 53 (3), 25–30.CrossRefGoogle Scholar
Silbert, L. E., Ertas, D., Grest, G. S., Halsey, T. C., Levine, D. & Plimpton, S. J. 2001 Granular flow down an inclined plane: Bagnold scaling and rheology. Phys. Rev. E 64 (5), 051302.Google Scholar
Sohn, Y. K. & Chough, S. K. 1993 The Udo tuff cone, Cheju Island, South Korea: transformation of pyroclastic fall into debris fall and grain flow on a steep volcanic cone slope. Sedimentology 40 (4), 769–786.CrossRefGoogle Scholar
Stock, J. D. & Dietrich, W. E. 2006 Erosion of steepland valleys by debris flows. Geol. Soc. Am. Bull. 118, 1125–1148.CrossRefGoogle Scholar
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
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.Google Scholar
Thornton, A. R., Weinhart, T., Luding, S. & Bokhove, O. 2012 Modeling of particle size segregation: calibration using the discrete particle method. Intl J. Mod. Phys. C 23 (8), 1240014.Google Scholar
Tripathi, A. & Khakhar, D. V. 2011 Rheology of binary granular mixtures in the dense flow regime. Phys. Fluids 23, 113302.Google Scholar
Tucker, M. E. 2003 Sedimentary Rocks in the Field. Wiley.Google Scholar
Vallance, J. W. & Savage, S. B. 2000 Particle segregation in granular flows down chutes. In IUTAM Symposium on Segregation in Granular Materials (ed. Rosato, A. D. & Blackmore, D. L.), Kluwer.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.Google Scholar
Wiederseiner, S., Andreini, N., Epely-Chauvin, G., Moser, G., Monnereau, M., Gray, J. M. N. T. & Ancey, C. 2011 Experimental investigation into segregating granular flows down chutes. Phys. Fluids 23, 013301.CrossRefGoogle Scholar
Williams, J. C. 1976 The segregation of particulate materials – review. Powder Technol. 15 (2), 245–251.Google Scholar
Woodhouse, M. J., Thornton, A. R., Johnson, C. G., Kokelaar, B. P. & Gray, J. M. N. T. 2012 Segregation-induced fingering instabilities in granular free-surface flows. J. Fluid Mech. 709, 543–580.Google Scholar