Advertisement

Continuum Mechanics and Thermodynamics

, Volume 6, Issue 2, pp 81–139 | Cite as

On flows of granular materials

  • K. Hutter
  • K. R. Rajagopal
Review Article

Abstract

This article reviews the behavior of materials made up of a large assemblage of solid particles under rapid and quasi static deformations. The focus is on flows at relatively high concentrations and for conditions when the interstitial fluid plays an insignificant role. The momentum and energy exchange processes are then primarily governed by interparticle collisions and Coulomb-type frictional contact. We first discuss some physical behavior —dilatancy, internal friction, fluidization and particle segregation — that are typical to the understanding of granular flows. Bagnold's seminal Couette flow experiments and his simple stress analysis are then used to motivate the first constitutive theories that use a microstructural variable — the fluctuation energy or granular temperature — governing the subscale fluctuating motion. The kinetic theories formalize the derivation of the field equations of bulk mass, momentum and energy, and permit derivation of constitutive relations for stress, flux of fluctuation energy and its dissipation rate for simple particle assemblages and when frictional rubbing contact can be ignored. These statistical considerations also show that formulation of boundary conditions needs special attention. The frictional-collisional constitutive behavior in which both Coulomb-type rubbing contact and collisional encounters are significant are discussed. There is as yet no rigorous formulation. We finally present a phenomenological approach that describes rapid flows of granular materials under simultaneous transport of heat and close with a summary of stability analyses of the basic flow down an inclined plane.

Keywords

Granular Material Couette Flow Incline Plane Frictional Contact Constitutive Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ahmadi, G., 1982: A continuum theory of smectic a liquid crystals.J. Rheology,26, 535–566Google Scholar
  2. 2.
    Ahmadi, G., 1983: A generalized continuum theory for flow of granular materials. InAdvances in the mechanics and the flow of granular materials,2. (ed. M. Shahinpoor). II Gulf Publishing Co., 497–527Google Scholar
  3. 3.
    Ahmadi, G.; Ma, D. N., 1986.Int. J. Bulk Solid Storage in Silos, 2, p. 8Google Scholar
  4. 4.
    Ahmadi, G.; Shahinpoor, M., 1983: Towards a turbulent modeling of rapid flow of granular materials.Powder Tech., 35, 241–248Google Scholar
  5. 5.
    Ahn, H., 1989: Experimental and analytical investigation of granular materials.Ph. D. Thesis. CaltechGoogle Scholar
  6. 6.
    Ahn, H.; Brennen, C. E.; Sabersky, R. H., 1992: Analysis of the fully developed chute flow of granular materials.J. App. Mech. 59, 109–119Google Scholar
  7. 7.
    Augenstein, D. A.; Hogg R., 1978: An experimental study of the flow of dry powders over inclined surfaces.Powder Techn., 19, 205–15Google Scholar
  8. 8.
    Babic, M., 1993: On the stability of rapid granular flows.J. Fluid Mech., 254, 127–150Google Scholar
  9. 9.
    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, 49–63Google Scholar
  10. 10.
    Bagnold, R. A., 1966: The shearing and dilatation of dry sand and the singing mechanism.Proc. R. Soc. London, A 295, 219–232Google Scholar
  11. 11.
    Bailard, J., 1978: An experimental study of granular-fluid flow.Ph. D. Thesis, Univ. of Calif., San Diego, 172 ppGoogle Scholar
  12. 12.
    Bashir, Y. M.; Goddard, J. D., 1990: Experiments on the conductivity of suspensions of ionically-conductive spheres.Ai. Ch. Engng. J., 36, 387–396Google Scholar
  13. 13.
    Batchelor, G. K.; O'Brien, R. W., 1977: Thermal or electrical conduction through a granular material.Proc. R. Soc., London., A 355, 313–333Google Scholar
  14. 14.
    Bhatnagar, P. L.; Gross, P.; Krook, M., 1954: A model for collision processes in gases I: small amplitude processes in charged and in neutral one component systems.Physical Reviews, 94, 5511–5536Google Scholar
  15. 15.
    Blinowski, A., 1978: On the dynamic flow of granular media.Arch. of Mech., Arch. Mech. Stoso., 30, Nr. 1, pp. 27–34Google Scholar
  16. 16.
    Boyle, E. J.; Massoudi, M., 1989: Kinetic theories of granular materials with applications to fluidized beds.U. S. Department of Energy, Technical Note, DOE/METC-89-4088.Google Scholar
  17. 17.
    Boyle, E. J.; Massoudi, M., 1990: A theory for granular materials exhibiting normal stress effects based on Enskog's dense gas theory.Int. J. Engng. Sci. 28, 1261–1275Google Scholar
  18. 18.
    Bridgewater, J., 1972: Stress-velocity relationships for particulate solids.ASME paper 72-MH-21, 7 ppGoogle Scholar
  19. 19.
    Buggisch, H.; Stadtler, R., 1986: On the relation between shear rate and stresses in one-dimensional steady flow of moist bulk solids.Proc. World Congress Particle Technolgy, Part III, Mechanics of Pneumatic and Hydraulic Conveying and Mixing, Nürnberg, 16–18 April, 187–202Google Scholar
  20. 20.
    Campbell, C. S., 1986: Computer simulation of rapid granular flows.Proc. 10th National Congr. on Appl. Mech. Austin, Texas, 327–338, New York, ASMEGoogle Scholar
  21. 21.
    Campbell, C. S., 1990: Rapid granular flows.Ann. Rev. Fluid Mech. 22, 57–92Google Scholar
  22. 22.
    Carnahan, N. F.; Starling, K. E., 1969: Equations of state for non-attracting rigid spheres.J. Chem. Phys., 51, pp. 635–636Google Scholar
  23. 23.
    Criminale, W. O.; Jr., Ericksen, J. L.; Filbey, G. L., 1958: Steady shear flow of non-Newtonian Fluids.Arch. Rational Mech. Anal., 1, 410–417Google Scholar
  24. 24.
    Davies, T. R. H., 1986: Large debris flows: A macro-viscous phenomenon.Acta Mechanica, 63, 161–178Google Scholar
  25. 25.
    Davies, T. R. H., 1988: Debris flow surges-A laboratory investigation.Mitteilung No. 96 der Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie an der ETH Zürich, 122 ppGoogle Scholar
  26. 26.
    Eringen, A. C., 1969: Mechanics of micropolar continua.Contributions to Mechanics, (D. Abir, ed.) Pergamon Press, New York, 23–40Google Scholar
  27. 27.
    Faraday, M., 1831: On the peculiar class of accoustical figures assumed by groups of particles upon vibrating elastic surfaces.Phil. Trans, 299–340 (see alsoFaraday's Diary, 1 (T. Martin ed.), G. Bell and Sons Ltd., London, 324–359Google Scholar
  28. 28.
    Galdi, G.P.; Padula, M.; Rajagopal, K.R., 1991:Recent Advances in Mechanics of Structured Continua, AMD-117, ASME, New YorkGoogle Scholar
  29. 29.
    Goddard, J. D., 1986: Dissipative materials as constitutive models for granular media.Acta Mechanica, 63, 3–13Google Scholar
  30. 30.
    Goodman, M. A.; Cowin, S. C., 1971: Two problems in the gravity flow of granular materials.J. of Fluid Mechanics, 45, 321–339Google Scholar
  31. 31.
    Goodman, M. A.; Cowin, S. C., 1972: A continuum theory for granular materials,Arch. Rational Mech. Anal., 44, (4), 249–266Google Scholar
  32. 32.
    Grad, H., 1949: On the kinetic theory of rarified gases.Comm. Pure and Applied Math. 2, 331–407Google Scholar
  33. 33.
    Gudhe, R., 1993: A theoretical and numerical study of the flow of granular materials down an inclined plane.Ph. D. Thesis, University of Pittsburgh, PittsburghGoogle Scholar
  34. 34.
    Gudhe, R.; Rajagopal, K.R.; Massoudi, M., 1994: Fully developed flow of granular materials down an inclined plane.Acta Mechanica, 103, 63–78Google Scholar
  35. 35.
    Gudhe, R.; Yalamanchili, R. C.; Massoudi, M., 1993: The flow of granular materials in a pipe: numerical solutions.Rec. Adv. Mech. Strctd. Continua, AMD-160/MD-41, 41–53Google Scholar
  36. 36.
    Gupta, G., 1993: Flow of granular materials and non-newtonian fluids.Ph. D. Thesis, University of Pittsburgh, PittsburghGoogle Scholar
  37. 37.
    Haff, P. K., 1983: grain flow as a fluid-mechanical phenomenon.J. Fluid Mech., 134,401–430Google Scholar
  38. 38.
    Haff, P. K., 1986: A physical picture of kinetic granular flows.J. Rheology, 30.Google Scholar
  39. 39.
    Hanes, D. M.; Inman, D. L., 1985: Observation of rapidly flowing granular-fluid mixtures.J. Fluid Mech., 150, 357–380Google Scholar
  40. 40.
    Heim, A., 1882: Der Bergsturz von Elm.Deutsch. Geol. Gesell. Zeitschrift, 34, 74–115Google Scholar
  41. 41.
    Heim, A., 1932: Bergsturz und Menschenleben, Beiblatt zur Vierteljahresschrift derNatf. Ges. Zürich, 20, 1–218Google Scholar
  42. 42.
    Heisenberg, W., 1948: Zur statistischen Theorie der Turbulenz.Z. für Physik, 124, 628–657Google Scholar
  43. 43.
    Huber, A., 1980: Schwallwellen in Seen als Folge von Felsstürzen.Mitteilung No. 47 der Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie an der ETH Zürich, 1–122Google Scholar
  44. 44.
    Hui, K.; Haff, P. K.; Ungar, J. E.; Jackson, R., 1984, Boundary conditions for high-shear grain flows.J. Fluid Mech. 145, 223–233Google Scholar
  45. 45.
    Hungr, O.; Morgenstern, N. R. 1984: High velocity ring shear tests on sand.Gotechnique, 34, 415–421Google Scholar
  46. 46.
    Hutter, K., 1989: A continuum model for finite mass avalanches having shear-flow and plug-flow regime.Internal Report, Federal Institute of Snow and Avalanche Research, Weissfluhjoch, Ch-7620 DavosGoogle Scholar
  47. 47.
    Hutter, K., 1994: Avalanche dynamics, a review. In:Hydrology of Disasters (V. P. Singh, ed.) Kluwer Academic Publishers, Amsterdam. (in press)Google Scholar
  48. 48.
    Hutter, K.; Szidarovskiy, F.; Yakowitz, S., 1986a: Plane steady shear flow of a cohesionless granular material down an inclined plane: A model for flow avalanches, Part I: Theory.Acta Mechanica, 63, 87–112Google Scholar
  49. 49.
    Hutter, K.; Szidarovskiy, F.; Yakowitz, S., 1986b: Plane steady shear flow of a cohesionless granular material down an inclined plane: A model for flow avalanches, Part II: Numerical results.Acta Mechanica, 65, 239–261Google Scholar
  50. 50.
    Ishida, M.; Shirai, T., 1979: Velocity, distributions in the flow of solid particles in an inclined open channel.J. Chem. Eng. of Japan, 12, 46–50Google Scholar
  51. 51.
    Ishida, M.; Hatano, H.; Shirai, T., 1980: The flow of solid particles in an aerated inclined channel.Powder Techn., 27, 7–12Google Scholar
  52. 52.
    Iverson, R. M.; Denlinger, R. P., 1987: The physics of debris flows-A conceptual assessment. In:Erosion and Sedimentation in the Pacific Rim (Proceedings of the Corvallis Symposium), IAHS Publ. No 165, 155–165Google Scholar
  53. 53.
    Jenkins, J. T. 1975: Static equilibrium of granular materials.J. of Appl. Mech. 42, 603–606Google Scholar
  54. 54.
    Jenkins, J. T., 1987: Balance laws and constitutive relations for rapid flows of granular materials. In:Proc. Army Research Office Workshop on Constitutive Relations (Chandra, J. and Srivastava, R., eds.), PhiladelphiaGoogle Scholar
  55. 55.
    Jenkins, J. T.; Cowin, S. C., 1979: Theories for flowing granular materials.The Joint ASME-CSME Appl. Mech. Fluid Engng. and Bioengng. Conf., AMD-Vol. 31, pp. 79–89Google Scholar
  56. 56.
    Jenkins, J.T.; Richman, M.W., 1985a: Grad's 13-moment system for a dense gas of inelastic spheres.Arch. Rat. Mech. and Anal., 87, 355–377Google Scholar
  57. 57.
    Jenkins, J.T.; Richman, M.W., 1985b: Kinetic theory for plane flows of a dense gas of identical, rough, inelastic, circular disks.Physics of Fluids, 28, 3485–3494Google Scholar
  58. 58.
    Jenkins, J.T.; Richman, M.W., 19886. Boundary conditions for plane flows of smooth, nearly elastic, circular disks.J. Fluid Mech. 171, 53–69Google Scholar
  59. 59.
    Jenkins, J. T.; Savage, S. B., 1981: The mean stress resulting from interparticle collisions in a rapid granular shear flow. In:Proceedings of the Fourth International Conference on Continuum Models of Discrete Systems (O. Bruhin and R. K. T. Hsien, eds.), North HollandGoogle Scholar
  60. 60.
    Jenkins, J. T.; Savage, S. B., 1983: A theory for the rapid flow of identical, smooth, nearly elastic spherical particles.J. Fluid Mech., 130, 187–202Google Scholar
  61. 61.
    Johnson, G.; Massoudi, M.; Rajagopal, K. R., 1991a: Flow of a fluid-solid mixture between flat plates.Chem. Engng. Sci., 46, 1713–1723Google Scholar
  62. 62.
    Johnson, G.; Massoudi, M.; Rajagopal, K. R., 1991b: Flow of a fluid infused with solid particles through a pipe.Int. J. Engng. Sci., 29, 649–661Google Scholar
  63. 63.
    Johnson, G.; Massoudi, M., Rajagopal, K. R., 1991c: Couette flow of a fluid with entrained solid particles.Rec. Adv. Mech. Strctd. Cont., AMD- 117, 97–105Google Scholar
  64. 64.
    Johnson, P. C.; Jackson, R., 1987: Frictional-collisional constitutive relations for granular materials, with application to plane shearing.J. Fluid Mech., 176, 67–93Google Scholar
  65. 65.
    Johnson, P. C.; Nott, P.; Jackson, R., 1990: Frictional-collisional equations of motion for particulate flows and their application to chutes.J. Fluid Mech., 210, 501–535Google Scholar
  66. 66.
    Kadambi, J.R.; Chen, R.C.; Bunia, S., 1989: Laser velocimeter measurements of multiphase flow of solids.Technical Report, U.S. Department of Energy, DOE/ PETC-90961-T9Google Scholar
  67. 67.
    Kanatani, K. I., 1979: A micropolar continuum theory for the flow of granular materials.Int. J. Engng. Sci., 17, 419–432Google Scholar
  68. 68.
    Kolmogoroff, A., 1941: The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers.Comptes rend. de l'Acad. de Sci. de l'USSR, 30, 301–305Google Scholar
  69. 69.
    Liu Ko-fei; Mei, C. C., 1991: Rapid flow of a Bingham-plastic fluid down a gentle slope.J. Fluid Mechanics (submitted) Google Scholar
  70. 70.
    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 flow field.J. Fluid Mech., 140, 223–256Google Scholar
  71. 71.
    Ma, D.; Ahmadi, G., 1985: A turbulence model for rapid flows of granular materials, Part II. Simple Shear Flows.Powder Tech., 44, p. 269–279Google Scholar
  72. 72.
    Ma, D.; Ahmadi, G., 1988: A kinetic model for rapid granular flows of nearly elastic particles including interstitial fluid effects.Powder Tech., 56, p. 191Google Scholar
  73. 73.
    Massoudi, M., 1986: Application of mixture theory to fluidized beds.Ph. D. Thesis, University of PittsburghGoogle Scholar
  74. 74.
    Massoudi, M., 1988: Stability analysis of fluidized beds.Int. Journal of Engineering Science, 26, 765Google Scholar
  75. 75.
    Massoudi, M.; Boyle, E. J., 1991: A review of theories for flowing granular materials with applications to fluidized beds and solids transport.U.S. Department of energy Report, DOE/PETC/TR-91/8Google Scholar
  76. 76.
    Maxwell, B.; Chartoff, R.P., 1965: Studies of a polymer melt in an othogonal rheometer.Trans. Soc. Rheology, 9, 51Google Scholar
  77. 77.
    McQuarrie, V.V., 1976:Statistical mechanics, Haper and Row, New YorkGoogle Scholar
  78. 78.
    McTigue, D. F., 1982: A nonlinear continuum theory for flowing granular materials.Ph. D. Thesis, Dept. of Geol, Stanford University Google Scholar
  79. 79.
    McTigue, D. F., 1982: A nonlinear constitutive model for granular materials: application to gravity flow.J. of Appl. Mech., 49, 291–296Google Scholar
  80. 80.
    Middleton, G. V., 1970: Experimental studies related to problem of flysch sedimentation. In:Flysch Sedimentology in North America (Lajoie, J., ed.), Business and Economics Science Ltd., Toronto, 253–272Google Scholar
  81. 81.
    Middleton, G. V.; Hampton, M. A., 1976: Subaqueous sediment transport and deposition by sediment gravity flows. In:Marine Sediment Transport and Environmental Management (Stanley, D. J. and Swift D. J. P., eds.), Wiley, New York, 197–218Google Scholar
  82. 82.
    Naylor, M. A., 1980: The origin of inverse grading in muddy debris flow deposits-A review.J. Sedimentary Petrology, 50, 1111–1116Google Scholar
  83. 83.
    Nedderman, R. M.; Tuzun, U., Savage, S. B.; Houlsby, G. T., 1982: The flow of granular materials, I. Discharge rates from hoppers.Chem. Engng. Sci., 37, 1597–1609Google Scholar
  84. 84.
    Noll, W., 1962: Motions with constant stretch history.Arch. Rational Mech. Anal., 11, 97–105Google Scholar
  85. 85.
    Norem, H.; Irgens, F.; Schieldrop, B. A., 1987: A continuum model for calculating snow avalanches. In:Avalanche Formation. Movement and Effects, (Salm, B. and Gubler, H., eds.), IAHS Publ. No. 126, 363–379Google Scholar
  86. 86.
    Novosad, J., 1964: Studies on granular materials, II. Apparatus for measuring the dynamic angle of internal and external friction of granular materials.Collection Czechoslov. Chem. Commun., 29. No. 2697Google Scholar
  87. 87.
    Nunziato, J. W.; Passman, S. L.; Thomas Jr.; J. P., 1980: Gravitational flows of granular materials with incompressible grains.J. of Rheology, 24, 395–420Google Scholar
  88. 88.
    Ogawa, S., 1978: Multitemperature theory of granular materials. In:Proc. Continuum Mechanical and Statistical Approaches in the Mechanics of Granular Materials, (S. C. Cowin and M. Satake, eds.) US-Japan Seminar, Sendai, Japan, pp. 208–217Google Scholar
  89. 89.
    Ogawa, S.; Umemura, A.; Oshima, N. 1980: On the equations of fully fluidized granular matrials.ZAMP, 31, pp. 483–493Google Scholar
  90. 90.
    Oppenheim, I., 1991: Statistical mechanical theory of inelastic granular systems.Joint NSF-DOE workshop on Flow of Particulates and Fluids, October 22–24, Worcester Polytechnic Institute, Worcester, 236–253Google Scholar
  91. 91.
    Oppenheim, I.; McBride, J.,1990: Transport equations for suspensions of inelastic particles.Physica A, 165, 279–302Google Scholar
  92. 92.
    Passman, S. L.; Nunziato, J. W.; Bailey, P. B.; Thomas Jr., J. P., 1980: Shearing flows of granular materials.J. Engg. Mech. Div., ASCE, 106, 773–783Google Scholar
  93. 93.
    Poynting, G. H., 1905: Radiation-pressure.Phil. Magazine, 9, 335–346Google Scholar
  94. 94.
    Rajagopal, K. R., 1982: On the flow of a simple fluid in an orthogonal rheometer.Arch. Rational Mech. Anal., 79, 29Google Scholar
  95. 95.
    Rajagopal, K. R.; Gudhe, R., 1992: Stability of the flow of granular materials down an inclined plane.Fourth NSF-DOE workshop on Flow of Particulates and Fluids, September 17–18, Gaithersburg, Maryland, 167–187Google Scholar
  96. 96.
    Rajagopal, K. R.; Gudhe, R., 1993: Stability analysis for the flow of granular materials down an inclined plane using kinetic model.Fifth NSF-DOE workshop on Flow of Particulates and Fluids, Cornell Google Scholar
  97. 97.
    Rajagopal, K. R.; Massoudi, M., 1990: A method for measuring material moduli of granular materials: flow in an orthogonal rheometer. Topical Report U, Department of Energy. DOE/PETC/TR-90/3Google Scholar
  98. 98.
    Rajagopal, K. R.; Massoudi, M.; Ekmann, J.M., 1990: Mathematical modeling of fluid-solid mixtures.In Recent Developments in Structural Continua II, LongmanGoogle Scholar
  99. 99.
    Rajagopal, K. R.; Troy, W. C.; Massoudi, M., 1992: Existence of solutions to the equations governing the flow of granular materials.Eur. J. Mech.,B/Fluids, 11, 265–276Google Scholar
  100. 100.
    Ratkai, G., 1976: Particle flow and mixing in vertically vibrated beds.Powder Tech., 15, 187–192Google Scholar
  101. 101.
    Reid, F., 1965: Fundamentals of statistical mechanics and thermal physics. McGraw Hill, New YorkGoogle Scholar
  102. 102.
    Reiner, M., 1945: A mathematical theory of dilatancy.Am. J. Math, 67, 350–362Google Scholar
  103. 103.
    Reynolds, O., 1885. On the dilatancy of media composed of rigid particles in contact.Phil. Mag. Ser. 5, 20, 469–481Google Scholar
  104. 104.
    Richman, M. W., 1988: Boundary conditions based upon a modified Maxwellian velocity distribution for flows of identical, smooth, nearly elastic spheres.Acta Mech. 75, 227–240.Google Scholar
  105. 105.
    Richman, M. W.; Marciniec, R. P., 1990: Gravity-driven granular flows of smooth, inelastic spheres down bumpy inclines.J. Appl. Mech. 57, 1036–1043Google Scholar
  106. 106.
    Sallenger, A. H., 1979: Inverse grading and hydraulic equivalence in grain-flow deposits.J. Sedimentary Petrology, 49, 553–562Google Scholar
  107. 107.
    Savage, S. B., 1979: Gravity flow of cohesionless granular materials in chutes and channels.J. Fluid Mech., 92, 53–96Google Scholar
  108. 108.
    Savage, S. B., 1983: Granular flows down rough inclines-Review and extension. In:Mechanics of Granular Materials: New Models and Constitutive Relations (Jenkins, J. T. and Satake, M., eds.), Elsevier, pp. 261–82Google Scholar
  109. 109.
    Savage, S. B., 1984: The mechanics of rapid granular flows. In:Advances in Applied Mechanics, 24 (Wu, T. Y. and Hutchinson, J., eds.), Academic, 289–366Google Scholar
  110. 110.
    Savage, S. B., 1987: Interparticle percolation and segregation in granular materials: A review. In:Developments in Engineering Mechanics (Selvadurai, A.P.S., ed.), Elsevier, Amsterdam, 347–363Google Scholar
  111. 111.
    Savage, S. B., 1988. Streaming motions in a bed of vibrationally fluidized dry granular material.J. Fluid Mech., 194, 457–478Google Scholar
  112. 112.
    Savage, S. B., 1989: Flow of granular materials. In:Theoretical and Applied Mechanics (P. Germain, M. Piau and D. Caillerie eds., Elsevier Science Publishers B. V., North-Holland. iutam, 241–266Google Scholar
  113. 113.
    Savage, S. B., 1993: Mechanics of granular flows. In:Continuum mechanics in environmental sciences and geophysics (K. Hutter, ed.) CISM Courses and Lectures No. 337, Springer Verlag, Vienna-New YorkGoogle Scholar
  114. 114.
    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
  115. 115.
    Savage, S. B.; Hutter, K., 1991: The dynamics of avalanches of granular materials from initiation to runout. Part I. Analysis.Acta Mech., 86, 201–223Google Scholar
  116. 116.
    Savage, S. B.; Jeffrey, D. J., 1981: The stress tensor in a granular flow at high shear rates.J. Fluid Mech., 110, 255–272Google Scholar
  117. 117.
    Savage, S. B.; Lun, C. K. K., 1988: Particle size segregation in inclined chute flow of dry cohesionless granular solids.J. Fluid Mech., 189, 311–335Google Scholar
  118. 118.
    Savage, S. B.; McKeown, S., 1983: Shear stresses developed during rapid shear of dense concentrations of large spherical particles between concentric rotating cylinders.J. Fluid Mech., 127, 453–472Google Scholar
  119. 119.
    Savage, S. B.; Nohguchi, Y., 1988: Similartiy solutions for avalanches of granular materials down curved beds.Acta Mechanica, 75, 153–174Google Scholar
  120. 120.
    Savage, S. B.; Sayed, M., 1984: Stresses developed by dry cohesionless granular materials sheared in an annular shear cell.J. Fluid Mech., 142, 391–430Google Scholar
  121. 121.
    Savage, S. B.; Nedderman, R. M.; Tuzan, U.; Houlsby, G. T., 1983: The flow of granular materials, III. Rapid shear flows.Chem. Engng. Sci., 38, 189–195Google Scholar
  122. 122.
    Scheiwiller, T.; Hutter, K., 1982: Lawinendynamik, Übersicht über Experimente und theoretische Modelle von Fließ-und Staublawinen.Mitteilung No. 5 für Wasserbau, Hydrologie und Glaziologie an der ETH Zürich, 166 ppGoogle Scholar
  123. 123.
    Schofield, A. N.; Wroth, C. P., 1968:Critical State Soil Mechanics, McGraw-Hill, New YorkGoogle Scholar
  124. 124.
    Scott, R. F., 1963:Principles of Soil Mechanics, Addison Wesley, ReadingGoogle Scholar
  125. 125.
    Shahinpoor, M.; Ahmadi, G., 1983: A kinetic theory for the rapid flow of rough inelastic spherical particles and the evolution of fluctuations. In:Advances in the mechanics and the flow of granular materials (M. Shahimpoor, ed.), Vol. II, 641–667.Gulf Publ. Comp. Houston Google Scholar
  126. 126.
    Shahinpoor, M.; Lin, S. P. 1982: Rapid Couette flow of cohesionless granular materials.Acta Mechanica. 42, 183–196Google Scholar
  127. 127.
    Spencer, A. J. M., 1964: A theory of the kinematics of ideal solids under plane strain conditions.J. Mech. Phys. Solids, 12, 337–351Google Scholar
  128. 128.
    Squire, H. B., 1933: On the stability for three-dimensional disturbances of viscous fluid flow between parallel walls.Proc. Roy. Soc., A, 142, 621–628Google Scholar
  129. 129.
    Stadler, R., 1986: Stationäres, schnelles Fließen von dicht gepackten trockenen und feuchten Schüttgütern.Dr.-Ing. Dissertation, Univ. Karlsruhe, Karlsruhe, West GermanyGoogle Scholar
  130. 130.
    Stadler, R.; Buggisch, H., 1985: Influence of the deformation rate on shear stress in bulk solids: Theoretical aspects and experimental results. In:Reliable Flow of Particulate Solids (EFCE Publication Series No. 49, Bergen, Norway), pp. 15Google Scholar
  131. 131.
    Szidarovszky, F.; Hutter, K.; Yakowitz, S., 1987: A numerical sudy of steady plane granular chute flows using the Jenkins-Savage model and its extension.Int. J. Num. Meth. Engng., 24, 1993–2015Google Scholar
  132. 132.
    Takahashi, T., 1981: Debris flow.Ann. Rev. Fluid Mech., 13, 57–77Google Scholar
  133. 133.
    Takahashi, T., 1983: Debris flow and debris flow deposition. In:Advances in the Mechanics and Flow of Granular Materials, (Shahinpoor, M., ed.), Vol. II,Gulf Publ. Company, Houston, 57–77Google Scholar
  134. 134.
    Thomson, W.; Tait, P.G., 1867:Treatise on Natural Philosophy, Part I, CambridgeGoogle Scholar
  135. 135.
    Truesdell, C.; Noll, W., 1965: The non-linear field theories of mechanics. In:Handbuch der Physik, III/3, ed. Flugge, Springer-VerlagGoogle Scholar
  136. 136.
    Tuzan, U.; Houlsby, G. T.; Nedderman, R. M.; Savage, S. B., 1982: The flow of granular materials, II. Velocity distributions in slow flows.Chem. Engng. Sci., 37, 1691–1709Google Scholar
  137. 137.
    Van Kampen, N.G.; Oppenheim, I. 1986: Brownian motion as a problem of eliminating fast variables.Physica A, 138, 231–248Google Scholar
  138. 138.
    Walker, J., 1982: The amateur scientist.Scientific American, 247, 206–216Google Scholar
  139. 139.
    Walton, O.R., 1991: Numerical simulation of inclined chute flows of nondisperse, inelastic, frictional spheres.Joint NSF-DOE workshop on Flow of Particulates and Fluids, October 1991, Worcester Polytechnic Institute, Worcester, 347–355Google Scholar
  140. 140.
    Walton, O. R.; Braun, R. L., 1986: Stress calculations for assemblies of inelastic spheres in uniform shear.Acta Mechanica, Vol. 63, p. 73Google Scholar
  141. 141.
    Yalamanchili, R.C.; Gudhe, R.; Rajagopal, K.R., Cohesionless granular materials flow in a vertical channel under the action of gravity.In Preparation Google Scholar
  142. 142.
    Yih, C.S., 1954: Stability of two-dimensional parallel flows for three-dimensional disturbances.Quaterly of Applied Math., 12Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • K. Hutter
    • 1
  • K. R. Rajagopal
    • 2
  1. 1.Institute für Mechanik (AG III)Technische Hochschule DarmstadtDarmstadtGermany
  2. 2.Departments of Mechanical Engineering and Mathematics & StatisticsUniversity of PittsburghPittsburghUSA

Personalised recommendations