Summary
A rigid-plastic Cosserat model has been used to study dense, fully developed flow of granular materials through a vertical channel. Frictional models based on the classical continuum do not predict the occurrence of shear layers, in contrast to experimental observations. This feature has been attributed to the absence of a material length scale in their constitutive equations. The present model incorporates such a material length scale by treating the granular material as a Cosserat continuum. Thus, localized couple stresses exist, and the stress tensor is asymmetric. The velocity profiles predicted by the model are in close agreement with available experimental data. The predicted dependence of the shear layer thickness on the width of the channel is in reasonable agreement with data. In the limit of small ε (ratio of the particle diameter to the half-width of the channel), the model predicts that the shear layer thickness scaled by the particle diameter grows as ɛ-1/3.
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References
Jackson, R.: Some mathematical and physical aspects of continuum models for the motion of the granular materials. In: Theory of dispersed multiphase flow (Meyer, R. E., ed.), pp. 291–337. New York: Academic Press 1983.
Roscoe, K. H.: The influence of strains in soil mechanics. 10th Rankine Lecture. Géotechnique20, 129–170 (1970).
Nedderman, R. M., Laohakul, C.: The thickness of the shear zone of flowing granular materials. Powder Technol.25, 91–100 (1980).
Gudehus, G., Tejchman, J.: Some mechanisms of a granular mass in a silo-model test and a numerical Cosserat approach. In: Advances in continuum mechanics (Brüller, O., Mannel, V., Najar, J., eds.), pp. 178–194. Berlin Heidelberg New York: Springer 1991.
Tejchman, J., Gudehus, G.: Silo-music and silo-quake experiments and a numerical Cosserat approach. Powder Technol.76, 201–212 (1993).
Tejchman, J., Wu, W.: Numerical study of patterning of shear bands in a Cosserat continuum. Acta Mech.99, 61–74 (1993).
Mohan, L. S., Nott, P. R., Rao, K. K.: Fully developed flow of coarse granular materials through a vertical channel. Chem. Engng. Sci.52, 913–933 (1997).
Mühlhaus, H. B.: Shear band analysis in granular materials by Cosserat theory. Ing. Arch.56, 389–399 (1986).
Mühlhaus, H. B., Vardoulakis, I.: The thickness of shear bands in granular materials. Géotechnique37, 271–283 (1987).
Lun, C. K. K., Savage, S. B., Jeffrey, D. J., Chepurniy, N.: Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flow field. J. Fluid Mech.140, 223–256 (1984).
Mühlhaus, H. B.: Application of Cosserat theory in numerical solution of limit load problems. Ing. Arch.59, 124–137 (1989).
Tejchman, J., Wu, W.: Numerical study on sand and steel interfaces. Mech. Res. Comm.21, 109–119 (1994).
Jaunzemis, W.: Continuum mechanics. New York: Macmillan 1967.
Cowin, S. C.: The theory of polar fluids. Adv. Appl. Mech.14, 279–347 (1974).
Dahler, J. S.: Transport phenomena in a fluid composed of diatomic molecules. J. Chem. Phys.30, 1447–1475 (1959).
Campbell, C. S.: Boundary interaction for two-dimensional granular flows. Part 1. Flat boundaries, asymmetric stresses and couple stresses. J. Fluid Mech.247, 111–136 (1993).
Jenkins, J., T., Cundall, P. A., Ishibashi, I.: Micromechanical modeling of granular materials with the assistance of experiments and numerical simulations. In: Powders and grains (Biarez, J., Gourvés, R., eds.), pp. 257–264. Rotterdam: Balkema 1989.
Besdo, D.: Ein Beitrag zur nichtlinearen Theorie des Cosserat-Kontinuums. Acta Mech.20, 105–131 (1974).
Lippmann, H.: Cosserat plasticity and plastic spin. Appl. Mech. Rev.48, 753–762 (1995).
de Borst, R.: A generalisation of theJ 2-flow theory for polar continua. Comp. Meth. Appl. Mech. Eng.103, 347–362 (1993).
Schofield, A. N., Wroth, C. P.: Critical state soil mechanics. London: McGraw-Hill 1968.
Prakash, J. R., Rao, K. K.: Steady compressible flow of granular materials through a wedge-shaped hopper: the smooth wall radial gravity problem. Chem. Eng. Sci.43, 479–494 (1988).
Johnson, P. C., Jackson, R.: Frictional-collisional constitutive relations for granular materials, with application to plane shearing. J. Fluid Mech.176, 67–93 (1987).
Brennen, C., Pearce, J. C.: Granular material flow in two dimensional hoppers. ASME J. Appl. Mech.45, 43–50 (1978).
Nedderman, R. M.: Statics and kinematics of granular materials. Cambridge: Cambridge University Press 1992.
Petzold, L.: Automatic selection of methods for solving stiff and nonstiff systems of ordinary differential equations. SIAM J. Sci. Stat. Comput.4, 136–148 (1983).
Natarajan, V. V. R., Hunt, M. L., Taylor, E. D.: Local measurements of velocity fluctuations and diffusion coefficients for a granular material flow. J. Fluid Mech.304, 1–25 (1995).
Tüzün, U., Nedderman, R., M.: Gravity flow of granular materials round obstacles—II. Chem. Eng. Sci.40, 337–351 (1985).
Jyotsna, R., Rao, K. K.: Flow of coarse granular materials through a wedge-shaped hopper. J. Fluid Mech.346, 239–270 (1997).
Fickie, K. E., Mehrabi, R., Jackson, R.: Density variations in a granular material flowing from a wedge-shaped hopper. AIChE. J.35, 853–855 (1989).
Press, W. H., Teukolsky, S. A., Vetterling, W. T., Flannery, B. P.: Numerical recipes in Fortran. Cambridge: Cambridge University Press 1992.
Kaza, K. R.: The mechanics of flowing granular materials. Ph.D. thesis, University of Houston 1982.
Dienes, J. K.: On the analysis of rotation and stress rate in deforming bodies. Acta Mech.32, 217–232 (1979).
Van Dyke, M.: Perturbation methods in fluid mechanics. New York: Academic Press 1964.
Bender, C. M., Orszag, S. A.: Advanced mathematical methods for scientists and engineers. Singapore: McGraw-Hill 1984.
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Mohan, L.S., Nott, P.R. & Rao, K.K. A frictional Cosserat model for the flow of granular materials through a vertical channel. Acta Mechanica 138, 75–96 (1999). https://doi.org/10.1007/BF01179543
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DOI: https://doi.org/10.1007/BF01179543