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Modelling size segregation of granular materials: the roles of segregation, advection and diffusion

  • Yi Fan (a1) (a2), Conor P. Schlick (a3), Paul B. Umbanhowar (a1), Julio M. Ottino (a1) (a4) (a5) and Richard M. Lueptow (a1) (a5)...


Predicting segregation of granular materials composed of different-sized particles is a challenging problem. In this paper, we develop and implement a theoretical model that captures the interplay between advection, segregation and diffusion in size bidisperse granular materials. The fluxes associated with these three driving factors depend on the underlying kinematics, whose characteristics play key roles in determining particle segregation configurations. Unlike previous models for segregation, our model uses parameters based on kinematic measures from discrete element method simulations instead of arbitrarily adjustable fitting parameters, and it achieves excellent quantitative agreement with both experimental and simulation results when applied to quasi-two-dimensional bounded heaps. The model yields two dimensionless control parameters, both of which are only functions of control parameters (feed rate, particle sizes, and system size) and kinematic parameters (diffusion coefficient, flowing layer depth, and percolation velocity). The Péclet number, $\mathit{Pe}$ , captures the interplay of advection and diffusion, and the second dimensionless parameter, $\Lambda $ , describes the interplay between segregation and advection. A parametric study of $\Lambda $ and $\mathit{Pe}$ demonstrates how the particle segregation configuration depends on the interplay of advection, segregation and diffusion. The model can be readily adapted to other flow geometries.


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Ames, W. F. 1977 Numerical Methods for Partial Differential Equations. 2nd edn. Academic.
Aranson, I. S. & Tsimring, L. S. 2006 Patterns and collective behaviour in granular media: theoretical concepts. Rev. Mod. Phys. 78, 641692.
Arnarson, B. Ö. & Willits, J. T 1998 Thermal diffusion in binary mixtures of smooth, nearly elastic spheres with and without gravity. Phys. Fluids 10, 13241328.
Besseling, R., Weeks, E. R., Schofield, A. B. & Poon, W. C. K. 2007 Three-dimensional imaging of colloidal glasses under steady shear. Phys. Rev. Lett. 99 (2), 28301.
Boutreux, T. 1998 Surface flows of granular mixtures: II. Segregation with grains of different size. Eur. Phys. J. B 6 (3), 419424.
Boutreux, T. & de Gennes, P.-G. 1996 Surface flows of granular mixtures: I. General principles and minimal model. J. Phys. I 6, 12951304.
Bridgwater, J. 1980 Self-diffusion coefficients in deforming powders. Powder Technol. 25 (1), 129131.
Bridgwater, J. 2012 Mixing of powders and granular materials by mechanical means – a perspective. Particuology 10 (4), 397427.
Bridgwater, J., Cooke, M. H. & Scott, A. M. 1978 Interparticle percolation: equipment development and mean percolation velocities. Trans. Inst. Chem. Engrs 56, 157167.
Campbell, C. S. 1997 Self-diffusion in granular shear flows. J. Fluid Mech. 348 (1), 85101.
Chen, P., Ottino, J. M. & Lueptow, R. M. 2008 Subsurface granular flow in rotating tumblers: a detailed computational study. Phys. Rev. E 78, 021303.
Chen, P., Ottino, J. M. & Lueptow, R. M. 2011 Granular axial band formation in rotating tumblers: a discrete element method study. New J. Phys. 13, 055021.
Christov, I. C., Ottino, J. M. & Lueptow, R. M. 2011 From streamline jumping to strange eigenmodes: bridging the Lagrangian and Eulerian pictures of the kinematics of mixing in granular flows. Phys. Fluids 23 (10), 103302.
Cundall, P. A. & Strack, O. D. L. 1979 A discrete numerical model for granular assemblies. Geotechnique 29, 4765.
Danckwerts, P. V. 1952 The definition and measurement of some characteristics of mixtures. Appl. Sci. Res., Sec. A 3 (4), 279296.
Dolgunin, V. N., Kudy, A. N. & Ukolov, A. A. 1998 Development of the model of segregation of particles undergoing granular flow down an inclined chute. Powder Technol. 96 (3), 211218.
Drahun, J. A. & Bridgwater, J. 1983 The mechanisms of free surface segregation. Power Technol. 36, 3953.
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.
Fan, Y. & Hill, K. M. 2010 Shear-driven segregation of dense granular mixtures in a split-bottom cell. Phys. Rev. E 81 (4), 041303.
Fan, Y. & Hill, K. M. 2011a Phase transitions in shear-induced segregation of granular materials. Phys. Rev. Lett. 106, 218301.
Fan, Y. & Hill, K. M. 2011b Theory for shear-induced segregation of dense granular mixtures. New J. Phys. 13, 095009.
Fan, Y., Umbanhowar, P. B., Ottino, J. M. & Lueptow, R. M. 2013 Kinematics of monodisperse and bidisperse granular flows in quasi-two-dimensional bounded heaps. Proc. R. Soc. A 469, 20130235.
Galvin, J. E., Dahl, S. R. & Hrenya, C. M. 2005 On the role of non-equipartition in the dynamics of rapidly flowing granular mixtures. J. Fluid Mech. 528, 207232.
GDR MiDi, 2004 On dense granular flows. Eur. Phys. J. E 14, 341365.
Golick, L. A. & Daniels, K. E. 2009 Mixing and segregation rates in sheared granular materials. Phys. Rev. E 80, 042301.
Goyal, R. K. & Tomassone, M. S. 2006 Power-law and exponential segregation in two-dimensional silos of granular mixtures. Phys. Rev. E 74, 051301.
Gray, J. M. N. T. & Ancey, C. 2009 Segregation, recirculation and deposition of coarse particles near two-dimensional avalanche fronts. J. Fluid Mech. 629, 387423.
Gray, J. M. N. T. & Chugunov, V. A. 2006 Particle-size segregation and diffusive remixing in shallow granular avalanches. J. Fluid Mech. 569, 365398.
Gray, J. M. N. T. & Thornton, A. R. 2005 A theory for particle size segregation in shallow granular free-surface flows. Proc. R. Soc. A 461, 14471473.
Hajra, S. K., Shi, D. & McCarthy, J. J. 2012 Granular mixing and segregation in zigzag chute flow. Phys. Rev. E 86, 061318.
Hill, K. M. & Fan, Y. 2008 Isolating segregation mechanisms in a split-bottom cell. Phys. Rev. Lett. 101, 088001.
Hill, K. M., Khakhar, D. V., Gilchrist, J. F., McCarthy, J. J & Ottino, J. M. 1999 Segregation-driven organization in chaotic granular flows. Proc. Natl Acad. Sci. 96, 1170111706.
Hsiau, S. S. & Hunt, M. L. 1996 Granular thermal diffusion in flows of binary-sized mixtures. Acta Mechanica 114 (1), 121137.
Iverson, R. M. 1997 The physics of debris flows. Rev. Geophys. 35 (3), 245296.
Jain, Nitin, Ottino, J. M. & Lueptow, R. M. 2002 An experimental study of the flowing granular layer in a rotating tumbler. Phys. Fluids 14 (2), 572582.
Jenkins, J. T. & Mancini, F. 1989 Kinetic theory for binary mixtures of smooth, nearly elastic spheres. Phys. Fluids A 1, 20502057.
Jones, S. W. 1994 Interaction of chaotic advection and diffusion. Chaos, Solitons Fractals 4, 929940.
Katsuragi, H., Abate, A. R. & Durian, D. J. 2010 Jamming and growth of dynamical heterogeneities versus depth for granular heap flow. Soft Matt. 6, 30233029.
Khakhar, D. V., McCarthy, J. J. & Ottino, J. M. 1999 Mixing and segregation of granular materials in chute flows. Chaos 9 (3), 594610.
Khosropour, R., Zirinsky, Jessie, Pak, H. K. & Behringer, R. P. 1997 Convection and size segregation in a Couette flow of granular material. Phys. Rev. E 56, 44674473.
Knight, J. B., Jaeger, H. M. & Nagel, S. R. 1993 Vibration-induced size separation in granular media: the convection connection. Phys. Rev. Lett. 70, 37283731.
Komatsu, T. S., Inagaki, S., Nakagawa, N. & Nasuno, S. 2001 Creep motion in a granular pile exhibiting steady surface flow. Phys. Rev. Lett. 86, 17571760.
Kowalski, J. & McElwaine, J. N. 2013 Shallow two-component gravity-driven flows with vertical variation. J. Fluid Mech. 714, 434462.
Kudrolli, A. 2004 Size separation in vibrated granular matter. Rep. Prog. Phys. 67 (3), 209247.
Larcher, M. & Jenkins, J. T. 2013 Segregation and mixture profiles in dense, inclined flows of two types of spheres. Phys. Fluids 25 (11), 113301.
Makse, H. A., Havlin, S., King, P. R. & Stanley, H. E. 1997 Spontaneous stratification in granular mixtures. Nature 386, 379382.
Marks, B., Rognon, P. & Einav, I. 2011 Grainsize dynamics of polydisperse granular segregation down inclined planes. J. Fluid Mech. 690, 499511.
May, L. B. H., Golick, L. A., Phillips, K. C., Shearer, M. & Daniels, K. E. 2010 Shear-driven size segregation of granular materials: modelling and experiment. Phys. Rev. E 81, 051301.
Meier, S. W., Lueptow, R. M. & Ottino, J. M. 2007 A dynamical systems approach to mixing and segregation of granular materials in tumblers. Adv. Phys. 56 (5), 757827.
Ott, E., Du, Y., Sreenivasan, K. R., Juneja, A. & Suri, A. K. 1992 Sign-singular measures: fast magnetic dynamos, and high-Reynolds-number fluid turbulence. Phys. Rev. Lett. 69 (18), 26542657.
Ottino, J. M. & Khakhar, D. V. 2000 Mixing and segregation of granular materials. Annu. Rev. Fluid Mech. 32, 5591.
Pouliquen, O., Delour, J. & Savage, S. B 1997 Fingering in granular flows. Nature 386, 816817.
Rahman, M., Shinohara, K., Zhu, H. P., Yu, A. B. & Zulli, P. 2011 Size segregation mechanism of binary particle mixture in forming a conical pile. Chem. Engng Sci. 66, 60896098.
Rapaport, D. C. 2002 Simulational studies of axial granular segregation in a rotating cylinder. Phys. Rev. E 65, 061306.
Ristow, G. H. 2000 Pattern Formation in Granular Materials. Springer.
Rosato, A., Strandburg, K. J., Prinz, F. & Swendsen, R. H. 1987 Why the Brazil nuts are on top: size segregation of particulate matter by shaking. Phys. Rev. Lett. 58, 10381040.
Savage, S. B. & Dai, R. 1993 Studies of granular shear flows wall slip velocities, layering and self-diffusion. Mech. Mater. 16 (1), 225238.
Savage, S. B. & Lun, C. K. K. 1988 Particle size segregation in inclined chute flow of dry cohesionless granular solids. J. Fluid Mech. 189, 311335.
Schafer, J., Dippel, S. & Wolf, D. E. 1996 Force schemes in simulations of granular materials. J. Phys. I France 6, 520.
Schlick, C. P., Christov, I. C., Umbanhowar, P. B., Ottino, J. M. & Lueptow, R. M. 2013 A mapping method for distributive mixing with diffusion: Interplay between chaos and diffusion in time-periodic sine flow. Phys. Fluids 25 (5), 052102.
Shinohara, K., Shoji, K. & Tanaka, T. 1972 Mechanism of size segregation of particles in filling a hopper. Ind. Eng. Chem. Process Des. Dev. 11, 369376.
Silbert, L. E., Grest, G. S., Brewster, R. & Levine, A. J. 2007 Rheology and contact lifetimes in dense granular flows. Phys. Rev. Lett. 99, 068002.
Singh, M. K., Galaktionov, O. S., Meijer, H. E. H. & Anderson, P. D. 2009a A simplified approach to compute distribution matrices for the mapping method. Comput. Chem. Engng 33 (8), 13541362.
Singh, M. K., Speetjens, M. F. M. & Anderson, P. D. 2009b Eigenmode analysis of scalar transport in distributive mixing. Phys. Fluids 21 (9), 093601.
Socie, B. A., Umbanhowar, P., Lueptow, R. M., Jain, N. & Ottino, J. M. 2005 Creeping motion in granular flow. Phys. Rev. E 71, 031304.
Thornton, A., Weinhart, T., Luding, S. & Bokhove, O. 2012 Modelling of particle size segregation: calibration using the discrete particle method. Intl J. Mod. Phys. C 23 (08), 1240014.
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, 126.
Tripathi, A. & Khakhar, DV 2013 Density difference-driven segregation in a dense granular flow. J. Fluid Mech. 717, 643669.
Utter, B. & Behringer, R. P. 2004 Self-diffusion in dense granular shear flows. Phys. Rev. E 69, 031308.
Wandersman, E., Dijksman, J. A. & van Hecke, M. 2012 Particle diffusion in slow granular bulk flows. Europhys. Lett. 100 (3), 38006.
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.
Williams, J. C. 1963 The segregation of powders and granular materials. Univ. Sheffield Fuel Soc. J. 14, 2934.
Williams, J. C 1968 The mixing of dry powders. Powder Technol. 2, 1320.
Yoon, D. K. & Jenkins, J. T. 2006 The influence of different species granular temperatures on segregation in a binary mixture of dissipative grains. Phys. Fluids 18 (7), 073303.
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Modelling size segregation of granular materials: the roles of segregation, advection and diffusion

  • Yi Fan (a1) (a2), Conor P. Schlick (a3), Paul B. Umbanhowar (a1), Julio M. Ottino (a1) (a4) (a5) and Richard M. Lueptow (a1) (a5)...


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