# Mechanics of Granular Flows

## Abstract

The present paper deals with the flow of bulk solids comprised of discrete particles. Attention will be focused on understanding the interactions that take place between individual particles at the microstructural level as a means to establish the constitutive relationships (expressions for stresses, energy fluxes, energy dissipation, etc.) that govern the flow of granular materials. We begin with a review of the classical papers of Bagnold and proceed to discuss the various modes and regimes of granular flow. A dimensional analysis provides a physical background for the interpretation of subsequent presentations of theoretical and experimental work. By making analogies with gases and liquids at the molecular level it is possible to develop kinetic theories of granular flows. A simple analysis of this kind is briefly outlined. Recent attempts to consider the effects of boundary conditions, to include the effects of quasi-static stresses and interstitial fluids are also discussed. Molecular dynamics type computer simulations that are related to these kinetic theories are reviewed. As an application of granular flow theories to geophysical problems we discuss some analyses of avalanches of granular materials down inclines. Continuum approaches can be used to model rockfalls and ice avalanches in a very simple, but approximate way; they appear to yield the gross features of these flows. The time dependent evolution of avalanches and other granular flows can also be modelled by means of molecular dynamics type computer simulations. Examples of two and three—dimensional computations are described. Typically the avalanches are made up materials of different sizes. The shear that takes place over the depth of material results in a grading of particles in which larger particles move to the top and the front of the moving pile. A simple explanation of this phenomenon, based upon the segregation analysis of Savage & Lun (1988), is given.

## Keywords

Debris Flow Granular Material Kinetic Theory Granular Flow Shear Cell## Preview

Unable to display preview. Download preview PDF.

## References

- Ahn, H., Brennen, C.E., Sabersky, R.H. 1988 Experiments on chute flows of granular materials.
*Micromechanics of*Granular*Materials*, (eds. M. Satake, J.T. Jenkins ), Elsevier, 339–348.Google Scholar - Araki, S. 1991 The dynamics of particle disks, Iii Dense and spinning particle disks.
*Icarus*, 90, 139–171.MathSciNetADSGoogle Scholar - Bagnold, R.A. 1954 Experiments on a gravity free dispersion of large solid spheres in a Newtonian fluid under shear.
*Proc. R. Soc. London*,*Ser. A*225, 49–63.Google Scholar - Bagnold, R.A. 1956 The flow of cohesionless grains in fluids.
*Phil. Trans. R. Soc. London*,*Ser. A*249, 235–97.MathSciNetGoogle Scholar - Bagnold, R.A. 1966 The shearing and dilation of dry sand and the `Singing’ mechanism.
*Proc. R. Soc. London*,*Ser. A*295, 219–32.Google Scholar - Bagnold, R.A. 1973 The nature of saltation and of “Bed-Load” transport in water.
*Proc. R. Soc*.,*London*,*Ser. A*332, 473–504.Google Scholar - Balmer, R.T. 1978 The operation of sand clocks and their medieval development.
*Technol*. Culture 19, 615–632.Google Scholar - Baxter, G.W., Behringer, R.P., Fagert, T., Johnson, G.A. 1990 Pattern formation and time-dependence in flowing sand.
*Two Phase Flows and Waves*(eds. D.D. Joseph, D.G. Schaeffer), Springer, New York, 1–29.Google Scholar - Bideau, D., Dodds, J.A. (Eds.) 1992
*Physics of*Granular*Media*. Nova Science Publishers, New York.Google Scholar - Bridgwater, J., Cooke, M.H., Drahun, J.A. 1985 Strain induced percolation.
*Instn. Chem. Engrs. Symposium Series No*.*69*, 171–191.Google Scholar - Bridgwater, J., Cooke, M.H., Scott, A.M. 1978 Inter-particle percolation: Equipment development and mean percolation velocities.
*Trans. Instn Chem. Engrs*. 56, 157–167.Google Scholar - Campbell, C.S. 1989 The stress tensor for simple shear flows of a granular material.
*J. Fluid Mech*. 203, 449–473.ADSGoogle Scholar - Campbell, C.S., Brennen, C.E. 1985 Computer simulation of granular shear flows.
*J. Fluid Mech*. 151, 167–188.ADSGoogle Scholar - Campbell, C.S., Gong, A. 1986 The stress tensor in a two-dimensional granular shear flow.
*J. Fluid Mech*. 164, 107–125.MATHADSGoogle Scholar - Carnahan, N.F., Starling, K.E. 1969 Equations of state for non-attracting rigid spheres.
*J. Chem. Physics*42, 635–636.ADSGoogle Scholar - Chapman, S. 1916 On
*the kinetic theory of a gas. Phil. Trans. R. Soc. Lond*. A217, 115–197.ADSGoogle Scholar - Chapman, S., Cowling, T.G. 1970
*The Mathematical Theory of Non-Uniform Gases*. 3rd edn. Cambridge University Press.Google Scholar - Cooke, M.H., Bridgwater, J. 1979 Interparticle percolation: A statistical mechanical interpretation.
*Ind. Engng. Chem. Fundam*. 18, 25–27.Google Scholar - Craig, K., Buckholtz, R.H., Domato, G. 1986 An experimental study of the rapid flow of dry cohesionless metal powders.
*J. Appl. Mech*. 53, 935–942.ADSGoogle Scholar - Craig, K., Buckholtz, R.H., DoMato, G. 1987 Rapid shear flow of a dry magnetic powder in an externally applied magnetic field: An experimental study.
*Phys. Fluids*30, 1993–1999.ADSGoogle Scholar - Dames, T.R.H. 1982 Spreading of rock avalanche debris by mechanical fluidization.
*Rock Mech*., 15, 9–29.Google Scholar - Davies, T.R.H. 1986 Large debris flows: A macro-viscous phenomenon.
*Acta Mechanica*, 63, 161178.Google Scholar - Davies, T.R.H. 1988 Debris flow surges - A laboratory investigation. Mitteilung No. 96 der Versuchsanstalt für Wasserbau, Hydrologie and Glaziologie an der Eth, 122 pp.Google Scholar
- Davis, H.T. 1973 Kinetic theory of dense fluids and liquids revisited.
*Adv Chem. Phys*. 24, 257–343.Google Scholar - Dent, J.D. 1986 Flow properties of granular materials large overburden loads.
*Acta Mechanica*64, 111–122.Google Scholar - Ding, J., Gidaspow, D. 1990 A bubbling fluidization model using kinetic theory of granular flow.
*AichE Journ*. 36, 523–538.Google Scholar - Drescher, A., DE Josselin DE Jong, G. 1972 Photoelastic verification of a mechanical model for the flow of a granular
*material. J. Mech. Phys. Solids*20, 337–51.ADSGoogle Scholar - Erismann, T. 1986 Flowing, rolling, bouncing, sliding: Synopsis of basic mechanisms.
*Acta Me*chanica 64, 101–110.Google Scholar - Faber, T.E. 1972 An
*Introduction to the Theory of Liquid Metals*. Cambridge University Press, London and New York.Google Scholar - Farrell, M., Lux, C.K.K., Savage, S.B. 1986 A simple kinetic theory for granular flow of binary mixtures of smooth, inelastic, spherical particles.
*Acta Mech*. 63, 45–60.MATHGoogle Scholar - Ferziger, J.H., Kaper, H.G. 1972
*Mathematical Theory of Transport Processes in Gases*. Elsevier, Amsterdam.Google Scholar - Goguel, J. 1978 Scale dependent rockslide mechanisms. In
*Rockslides and Avalanches*,*Vol*.*1*(ed. B. Voight ), Elsevier, 167–180.Google Scholar - Haff, P.K. 1983 Grain flow as a fluid-mechanical phenomenon.
*J. Fluid Mech*. 134, 401–430.MATHADSGoogle Scholar - Hagen, G. 1852 Druck und Bewegung des Trockenen Sandes.
*Berl Monatsb. Akad. d. Wiss*., S35 - S42.Google Scholar - Hanes, D.M., Inman, D.L. 1985 Observations of rapidly flowing granular-fluid mixtures.
*J. Fluid Mech*. 150, 357–380.ADSGoogle Scholar - Hanes, D.M., Jenkins, J.T., Richman, M.W. 1988 The thickness of steady plane shear flows of circular disks driven by identical boundaries.
*J. Appl. Mech*. 110, 969–974.Google Scholar - Heim, A. 1932 Bergsturz und Menscheleben.
*Beiblatt zur Vierteljahresschrift der Natf. Ges. Zurich*20, 1–218.Google Scholar - Hirschfelder, J.O., Curtis, C.R., Bird, R.B. 1954
*The Molecular Theory of Gases and Liquids*. Wiley, New York.MATHGoogle Scholar - Hopkins, M.A., Shen, H. 1988 A Monte-Carlo simulation of a rapid simple shear flow of granular materials.
*Micromechanics of*Granular*Materials*(eds. M. Satake, J.T. Jenkins ), Elsevier, 349–358.Google Scholar - Hsi, K. 1975 On sturzstroms–catastrophic debris streams generated by rockfalls.
*Geol. Soc. Am. Bull*. 86, 129–140.ADSGoogle Scholar - Hsi, K. 1978 Albert Heim: Observations on landslides and relevance to modern interpretations. In
*Rockslides and Avalanches*,*Vol*.*1*(ed. B. Voight ), Elsevier, 69–93.Google Scholar - Hui, K., Haff, P.K., Ungar,
*J.E*., Jackson, R. 1984 Boundary conditions for high shear grain flows.*J. Fluid Mech*. 145, 223–233.Google Scholar - Hungr, O., Morgenstern, N.R. 1984a Experiments on the flow behaviour of granular materials at high velocity in an open channel flow.
*Geotechnique*34, 405–413.Google Scholar - Hungr, O., Morgenstern, N.R. 1984b High velocity ring shear tests on sand.
*Geotechnique*34, 415–421.Google Scholar - Iverson, R.M., Denlinger, R.P. 1987 The physics of debris flows - a conceptual
*assessment. Erosion and Sedimentation in the Pacific Rim (Proceedings of the Corvallis Symposium)*Iahs Publ. No. 165.Google Scholar - Jaeger, H.M., Nagel, S.R. 1992 Physics of the granular state.
*Science*255, 1523–1531. Jeans, J. 1967*An Introduction to the Kinetic Theory of Gases*. Cambridge University Press.Google Scholar - Jenkins, J.T. 1987a Balance laws and constitutive relations for rapid flows of granular
*materials. Constitutive Models of Deformations*(eds. J. Chandra, R. Srivastav), Siam, Philadelphia, 109.Google Scholar - Jenkins, J.T. 1987b Rapid flows of granular
*materials. Non-classical Continuum Mechanics: Abstract Techniques and Applications*(eds. R.J. Kops, A.A. Lacey ), Cambridge Univ. Press, 213–224.Google Scholar - Jenkins, J.T., Mancini, F. 1987 Balance laws and constitutive relations for plane flows of a dense, binary mixture of smooth, nearly elastic circular disks.
*J. Appl. Mech*. 54, 27–34.MATHADSGoogle Scholar - Jenkins, J.T., Mctigue, D.F. 1990 Transport processes in concentrated suspensions: The role of particle fluctuations.
*Two Phase Flows and Waves*(eds. D.D. Joseph, D.G. Schaeffer), Springer, New York, 70–79.Google Scholar - Jenkins, J.T., Richman, M.W. 1985a Grad’s 13-moment system for a dense gas of inelastic spheres.
*Arch. Rat. Mech. Anal*. 87, 355–377.MathSciNetMATHGoogle Scholar - Jenkins, J.T., Richman, M.W. 1985b Kinetic theory for plane flows of a dense gas of identical, rough, inelastic, circular disks.
*Phys. Fluids*28, 3485–3494.MATHADSGoogle Scholar - Jenkins, J.T., Richman, M.W. 1986 Boundary conditions for plane flows of smooth nearly elastic, circular disks.
*J. Fluid Mech*. 171, 53–69.MATHADSGoogle Scholar - Jenkins, J.T., Savage, S.B. 1983 A theory for the rapid flow of identical, smooth, nearly elastic particles.
*J. Fluid Mech*.*136*, 186–202.Google Scholar - Johnson, P.C., Jackson, R. 1987 Frictional-collisional constitutive relations for granular materials, with application to plane shearing.
*J. Fluid Mech*. 176, 67–93.ADSGoogle Scholar - Johnson, P.C., Nott, P., Jackson, R. 1990 Frictional-collisional equations equations of motion for particulate flows and their application to chutes.
*J. Fluid Mech*. 210, 501–535.ADSGoogle Scholar - Kadanoff, L.P., Nagel, S.R., Wu, L., Zhou, S. 1989 Scaling and universality in avalanches.
*Phys. Rev. A*39, 6524–6537.ADSGoogle Scholar - Kent, P.E. 1965 The transport mechanism in catastrophic rockfalls.
*J. Geol*. 74, 79–83.ADSGoogle Scholar - Lang, R.M. 1992 An experimental and analytical study on gravity driven free surface flows of cohesionless granular material. Ph.D. Dissertation, Technische Hochschule Darmstadt, Germany.Google Scholar
- Lees, A.W., Edwards, S.F. 1972 The computer study of transport processes under extreme conditions.
*J. Phys*. C5, 1921–1929.ADSGoogle Scholar - LI Jian, Luo Defu 1981 The formation and characteristics of mudflow and flood in the mountain area of the Dachao River and its prevention.
*Zeit. Geomorph*. 25, 470–484.Google Scholar - LI Tianchi 1983 A mathematical model for predicting the extent of a major rockfall.
*Z. Geomorph*. 27, 473–482.Google Scholar - Lun, C.K.K., Savage, S.B. 1986 The effects of an impact velocity dependent coefficient of restitution on stresses developed by sheared granular materials.
*Acta Mech*. 63, 15–44.MATHGoogle Scholar - Lucchitta, B.K. 1978 A large landslide on Mars.
*Geol. Soc. Amer. Bull*. 89, 1601–1609.ADSGoogle Scholar - Lun, C.K.K., Savage, S.B. 1987 A simple kinetic theory for granular flow of rough, inelastic, spherical particles.
*J. Appl. Mech*. 54, 47–53.MATHADSGoogle Scholar - Lun, C.K.K., Savage, S.B., Jeffrey, D.J., Chepurniy, N. 1984 Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flowfield.
*J. Fluid Mech*. 140, 223–256.MATHADSGoogle Scholar - Mandl, G., Fernandez-Luque, R. 1970 Fully developed plastic shear flow of granular materials.
*Geotech*. 20, 277–307.MathSciNetGoogle Scholar - Mcsaveney, M.J. 1978 Sherman Glacier rock avalanche, Alaska, U.S.A., In
*Rockslides and Avalanches*,*Vol*.*1*(ed. B. Voight ), Elsevier, 197–258.Google Scholar - Middleton, G.V. 1970 Experimental studies related to problems of Flysch sedimentation. In
*Flysch Sedimentology in North America*(ed. J. Lajoie), 253–72. Geol. Assoc. Can. Spec. Rep. 7.Google Scholar - Middleton, G.V., Hampton, M.A. 1976 Subaqueous sediment transport and deposition by sediment gravity flows. In
*Marine Sediment Transport and Environmental Management*, (eds. D.J. Stanley, D.J.P. Swift ), 197–218, Wiley, N.Y.Google Scholar - Naylor, M.A. 1980 The origin of inverse grading in muddy debris flow deposits–A review.
*J. Sedimentary Petrology*50, 1111–1116.Google Scholar - PLÛSS, CH. 1987 Experiments on granular avalanches. Diplomarbeit, Abt X, Eidg. Techn. Hochschule, Zürich, 113 pp.Google Scholar
- Richman, M.W. 1988 Boundary conditions based upon a modified Maxwellian velocity distribution for flows of identical, smooth nearly elastic, spheres.
*Acta Mechanica*75, 227–240.Google Scholar - Richman, M.W. 1989 The source of second moment in dilute granular flows of highly inelastic spheres.
*J. Rheology*33, 1293–1306.MATHADSGoogle Scholar - Richman, M.W., Chou, C.S. 1988 Boundary effects on granular shear flows of smooth disks.
*Journ. Appl. Math. Physics (Zamp)*39, 885–901.MATHGoogle Scholar - Reynolds, O. 1885 On the dilatancy of media composed of rigid particles in contact.
*Phil. Mag. Ser*.*5*, 20, 469–481.Google Scholar - Roscoe, K.H., Schofield, A.N., Wroth, C.P. 1958 On the yielding of soils.
*Geotech*. 8, 22–53.Google Scholar - Salencon, J. 1977
*Applications of the Theory of Plasticity in Soil Mechanics. Wiley*,*N.Y*.Google Scholar - Sallenger, A.H. 1979 Inverse grading and hydraulic equivalence in grain-flow deposits.
*J. Sedimentary Petrology*49, 553–562.Google Scholar - Sanders, J.E. 1963 Concepts of fluid mechanics provided by primary sedimentary sedimentary structures.
*J. Sedim. Petrology*33, 173–9.Google Scholar - Sanders, B.E., Hopkins, M.A., Ackermann, N.A. 1988 Physical experiments and numerical simulation of two dimensional chute flow.
*Micromechanics of Granular Materials*, (eds. M. Satake, J.T. Jenkins ), Elsevier, 359–366.Google Scholar - Savage, S. B. 1979 Gravity flow of cohesionless granular materials in chutes and channels.
*J. Fluid Mech*. 92, 53–96.MATHADSGoogle Scholar - Savage, S.B. 1983 Granular flows down rough inclines–review and extension.
*Mech. of*Granular*Materials: New Models and Constitutive Relations*(eds. J.T. Jenkins, M. Satake ), Elsevier, 261–82.Google Scholar - Savage, S.B. 1984 The mechanics of rapid granular flows.
*Advances in Applied Mechanics*24 (eds. T.Y Wu, J. Hutchinson ), Academic, 289–366.Google Scholar - Savage, S.B. 1987 Interparticle percolation and segregation in granular materials: A review. In
*Developments in Engineering Mechanics*(ed. A.P.S. Selvadurai, A.P.S.),Elsevier, Amsterdam 347–363.Google Scholar - Savage, S.B. 1989 Flow of granular materials.
*Theoretical and Applied Mechanics*(eds. P. Germain, M. Piau, D. Caillerie ), Elsevier, 241–266.Google Scholar - Savage, S.B., 1990 Symbolic computation of the flow of granular avalanches.
*Journal of Symbolic Computation*9, 515–530.MATHGoogle Scholar - Savage, S.B. 1992 Numerical simulations of Couette flow of granular materials; Spatio-temporal coherence and 1/f noise.
*Physics of*Granular*Media*, (eds. D. Bideau, J.A. Dodds), Nova Science Publishers, New York, 343–362.Google Scholar - Savage, S.B., Dai, R. 1992 Some aspects of bounded and unbounded shear flows of granular materials.
*U.S.-Japan meeting on Micromechanics of*Granular*Materials*,*Clarkson University*,*August 5–9*,*1991*(eds. M. Satake and H. Shen), Elsevier, Amsterdam, (in press).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.MathSciNetMATHADSGoogle Scholar - Savage, S.B., Hutter, K. 1991 The dynamics of avalanches of granular materials from initiation to runout. Part I: Analysis.
*Acta Mechanica*86, 201–223.MathSciNetMATHGoogle Scholar - Savage, S.B., Jeffrey, D.J. 1981 The stress tensor in a granular flow at high shear rates.
*J. Fluid Mech*. 110, 255–272.MATHADSGoogle Scholar - Savage, S.B., Nohguchi, Y. 1988 Similarity solutions for avalanches of granular materials down curved beds.
*Acta Mechanica*75, 153–174.MATHGoogle Scholar - Savage, S. B., Saved, M. 1984 Stresses developed by dry cohesionless granular materials sheared in an annular shear cell.
*J. Fluid Mech*. 142, 391–430.ADSGoogle Scholar - Scheidegger, A.E. 1975
*Physical Aspects of*Natural*Catastrophies*. Elsevier. Shreve, R.L. 1966 Sherman landslide, Alaska.*Science*154, 1639–1643.Google Scholar - Shreve, R.L. 1968b Leakage and fluidization in air-lubricated avalanches.
*Geol. Soc. Am. Bull*. 79, 653–658.ADSGoogle Scholar - Sinclair, J.L., Jackson, R. 1989 Gas-particle flow in a vertical pipe with particle-particle interactions.
*AichE Journ*. 35, 1473–1486.Google Scholar - Schofield, A.N., Wroth, C.P. 1968 Critical
*State Soil Mechanics*. McGraw-Hill, New York.Google Scholar - Sinclair, J.L., Jackson, R. 1989 Gas-particle flow in a vertical pipe with particle-particle interactions.
*AichE Journ*. 35, 1473–1486.Google Scholar - Spencer, A.J.M. 1981 Deformation of an ideal granular material. In
*Mechanics of Solids*,*Rodney Hill 60th*Anniv.*Volume*. (eds. H.G. Hopkins, J.J. Sewell ). Pergamon, Oxford.Google Scholar - Stadler, R. 1986 Stationäres, schnelles Fliessen von dicht gepackten trockenen and feuchten Schüttgütern. Dr: Ing. Dissertation, Univ. Karlsruhe, Karlsruhe, West Germany.Google Scholar
- Stadler, R., Buggisch, H. 1985 Influence of the deformation rate on shear stress in bulk solids, theoretical aspects and experimental results.
*Reliable Flow of Particulate Solids*, (Efce Publication Series No. 49, Bergen, Norway ), pp. 15.Google Scholar - Szidarovszky, F., Nutter, K., Yakowitz, S. 1987 A Numerical Study of Steady Plane Granular Chute Flows Using the Jenkins-Savage Model and Its Extension.
*Int. J. Num. Meth. Engrg*. 24, 1993–2015.MATHGoogle Scholar - Takahashi, T. 1980 Debris flow on prismatic channel.
*bourn*. Hydr.*Div*.,*Asce*106, 381–395.Google Scholar - Temperley, H.N.V., Rowlinson, J.S., Rushbrooke, G.S. 1968
*Physics of Simple Liquids*. North Holland.Google Scholar - Walton, O.R., Braun, R.L. 1986 Viscosity and temperature calculations for assemblies of inelastic frictional disks.
*J. Rheology*30, 949–980.ADSGoogle Scholar - Walton, O.R., Braun, R.L., Mallon, R.G., Cervelli, D.M. 1987 Particle-dynamics calculations of gravity flow of inelastic, frictional spheres.
*Micromechanics of Granular Materials*(eds. M. Satake, J.T. Jenkins ), Elsevier, 153–162.Google Scholar - Walton, O.R., Kim, H., Rosato, A. 1991 Micro-structure and stress difference in shearing flows of granular materials. Proc. Asce Eng. Mech. Div. Conf., Columbus, Ohio, May 19–21, 1991.Google Scholar