Hostname: page-component-77c89778f8-gvh9x Total loading time: 0 Render date: 2024-07-17T18:58:45.111Z Has data issue: false hasContentIssue false
  • English
  • Français

On the stability of rapid granular flows

Published online by Cambridge University Press:  26 April 2006

Marijan Babić
Marijan Babić
Affiliation:
Department of Civil Engineering and Geological Sciences, University of Notre Dame, Notre Dame, IN 46556, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Numerical simulations of rapid granular flows in large periodic domains have demonstrated the existence of an ‘inelastic microstructure’ characterized by agglomerations of particles into clusters. In the present work the phenomenon of particle clustering is considered to be a manifestation of hydrodynamic instability of the equations governing the dynamics of rapid granular flows. A linear stability analysis of the simple shear flow of smooth, slightly, inelastic disks and spheres is carried out utilizing the governing equations formulated by the kinetic theory. The results indicate that disturbances of large wavelengths initially grow at an exponential rate for any value of the restitution coefficient. The results also demonstrate that this phenomenon will not be exposed in numerical simulations within small periodic domains, because the range of possible disturbance wavelengths is limited by the size of the periodic domain.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 254 , September 1993 , pp. 127 - 150
Copyright
© 1993 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

Babić, M., Shen, H. H. & Shen, H. T. 1990 The stress tensor in granular shear flows of uniform, deformable disks at high solids concentrations. J. Fluid Mech. 219, 81–118.Google Scholar
Campbell, C. S. 1988 Boundary interactions for two-dimensional granular flows: asymmetric stresses and couple stresses. In Micromechanics of Granular Materials (ed. M. Satake & J. T. Jenkins). Elsevier.
Campbell, C. S. 1990 Rapid granular flows. Ann. Rev. Fluid Mech. 22, 57–92.Google Scholar
Campbell, C. S. & Gong, A. 1986 The stress tensor in a two-dimensional granular shear flow. J. Fluid Mech. 164, 107–125.Google Scholar
Carnahan, N. F. & Starling, T. G. 1969 Equation of state for non-attracting rigid spheres. J. Chem. Phys. 635–636.Google Scholar
Chapman, S. & Cowling, T. G. 1970 The Mathematical Theory of Non-Uniform Gases, 3rd Edn. Cambridge University Press.
Drazin, P. G. & Reid, W. 1981 Hydrodynamic Stability. Cambridge University Press.
Hopkins, M. A., Jenkins, J. T. & Louge, M. Y. 1991 On the structure of 3D shear flows. In Micromechanics of Granular Materials (ed. H. H. Shen & M. Satake). Elsevier.
Hopkins, M. A. & Louge, M. Y. 1991 Inelastic microstructure in rapid granular flows of smooth disks. Phys. Fluids A 3, 47–57.Google Scholar
Jenkins, J. T. & Richman, M. W. 1985a Kinetic theory for plane shear flows of a dense gas of identical, rough, inelastic, circular disks. Phys. Fluids 28, 3485–3494.Google Scholar
Jenkins, J. T. & Richman, M. W. 1985b Grad's 13-moment system for a dense gas of inelastic spheres. Arch. Rat. Mech. Anal. 87, 355–377.Google Scholar
Jenkins, J. T. & Richman, M. W. 1988 Plane simple shear flow of smooth, inelastic, circular disks: the anisotropy of the second moment in the dilute and dense limit. J. Fluid Mech. 192, 313–328.Google Scholar
Jenkins, J. T. & Savage, S. B. 1983 A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles. J. Fluid Mech. 130, 187–202.Google Scholar
Lun, C. K. K., Savage, S. B., Jeffrey, D. J. & Chepurney, N. 1984 Kinetic theories for granular flow: inelastic particles in a Couette flow and slightly inelastic particles in a general flow field. J. Fluid Mech. 140, 223–256.Google Scholar
Sanders, B. E. & Ackermann, N. L. 1991 Instability in granular chute flows. In Mechanics Computing in 1990's and Beyond (ed. H. Adeli & R. L. Sierakowski), pp. 1244–1248.
Verlet, L. & Levesque, D. 1982 Integral equations for classical fluids iii. The hard discs system. Molec. Phys. 46, 969–980.Google Scholar
Walton, O. R. & Braun, R. L. 1985 Viscosity, granular temperature and stress calculations for shearing assemblies of inelastic, frictional disks. J. Rheol. 30, 949–980.Google Scholar
Walton, O. R. & Braun, R. L. 1985 Stress calculations for assemblies of inelastic spheres in uniform shear. Acta Mech. 63, 73–86.Google Scholar
Walton, O. R., Kim, H. & Rosato, A. D. 1991 Microstructure and stress differences in shearing flows. In Mechanics Computing in 1990s and Beyond (ed. H. Adeli and R. L. Sierakowski), pp. 1249–1253.
Wolfram, S. 1988 Mathematica: A System for Doing Mathematics by Computer. Addison-Wesley.