Hostname: page-component-77c89778f8-rkxrd Total loading time: 0 Render date: 2024-07-19T21:53:17.467Z Has data issue: false hasContentIssue false
  • English
  • Français

Boundary conditions for plane flows of smooth, nearly elastic, circular disks

Published online by Cambridge University Press:  21 April 2006

J. T. Jenkins  and
M. W. Richman
J. T. Jenkins
Affiliation:
Department of Theoretical & Applied Mechanics, Cornell University, Ithaca, NY 14853, USA
M. W. Richman
Affiliation:
Department of Theoretical & Applied Mechanics, Cornell University, Ithaca, NY 14853, USA Present address: Department of Mechanical Engineering, Worcester Polytechnic Institute, Worcester, MA 01609, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider plane flows of identical, smooth, nearly elastic, circular disks interacting with a boundary formed by attaching halves of similar disks at equal intervals along a flat wall. The roughness of the boundary is given in terms of the diameters of the two types of disks and the spacing of the wall disks. We suppose that the velocity distribution of the flow disks is Maxwellian and calculate the rates at which momentum and energy are supplied to the flow disks in collisions over a unit length of the boundary. At the boundary we balance these supplies with the stress and the total flux of energy in the flow and obtain boundary conditions on the shear stress, pressure, and flux of fluctuation energy. We find that the boundary can either supply fluctuation energy to the flow or absorb it, depending on the relative magnitudes of the rate of working of the boundary tractions through the slip velocity and the rate at which energy is dissipated in collisions. As an example we solve the boundary-value problem for the steady shearing flow maintained by the relative motion of parallel plates a fixed distance apart. When the dimensions and properties of the flow disks and the boundary are given, the specification of the distance between the plates and their relative velocity determines the slip velocity, the shear stress and pressure necessary to maintain the flow, and the distributions of mean velocity, fluctuation energy, and density.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 171 , October 1986 , pp. 53 - 69
Copyright
© 1986 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

Ackermann, N. L. & Shen H. H.1983 Rapid shear flow of densely packed granular solids. In Mechanics of Granular Materials: New Models and Constitutive Relations (ed. J. T. Jenkins and M. Satake), pp. 295–304. Elsevier.
Bagnold R. A.1954 Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear. Proc. R. Soc. Lond. A 225, 49–63.Google Scholar
Campbell, C. S. & Brennen C. E.1983 Computer simulation of shear flows of granular material. In Mechanics of Granular Materials: New Models and Constitutive Relations (ed. J. T. Jenkins & M. Satake), pp. 313–326. Elsevier.
Campbell, C. S. & Brennen C. E.1985 Chute flow of granular material: some computer simulations. Trans. ASME E: J. Appl. Mech. 52, 172–178.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
Chapman, S. & Cowling T. G.1970 The Mathematical Theory of Non-Uniform gases, 3rd edn. Cambridge University Press.
Haff P. K.1983 Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech. 134, 401–430.Google Scholar
Hanes, D. M. & Inman D. L.1985 Observations of rapidly flowing granular-fluid mixtures J. Fluid Mech. 150, 357–380.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
Jenkins, J. T. & Mancini F.1986 Balance laws and constitutive relations for plane flows of a dense, binary mixture of smooth, nearly elastic, circular disks. Trans. ASMEE: J. Appl. Mech. (in press).
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.Google 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.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.1986 A simple kinetic theory for granular flow of rough, inelastic, spherical prticles. Trans. ASME E: J. Appl. Mech. (in press).Google 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 flow field. J. Fluid Mech. 140, 223–256.Google Scholar
Ogawa S., Umemura, A. & Oshima N.1980 On the equations of fully fluidized granular materials. Z. angew. Math. Phys. 31, 483–493.Google Scholar
Savage S. B.1978 Experiments on shear flows of cohesionless granular materials. In Continuum Mechanical and Statistical Approaches in the Mechanics of Granular Materials (ed. S. C. Cowin & M. Satake), pp. 241–254. Gakujutsu Bunken Fukyu-kai.
Savage, S. B. & Jeffrey D. J.1981 The stress tensor in a granular flow at high shear rates. J. Fluid Mech. 110, 255–272.Google Scholar
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, 453–472.Google Scholar
Savage, S. B. & Sayed M.1984 Stresses developed by dry cohesionless granular materials sheared in an annular shear cell. J. Fluid Mech. 42, 391–430.Google Scholar
Shen H. H.1984 Stresses in rapid deformations of spherical solids with two sizes. J. Particulate Sci. Tech. 2, 37–56.Google Scholar
Shen, H. H. & Ackerman N. L.1984 Constitutive equations for a simple shear flow of disk-shaped granular mixtures. Intl J. Engng Sci. 22, 829–843.Google Scholar
Walton O. R.1983 Particle dynamics calculations of shear flow. In Mechanics of Granular Materials: New Models and Constitutive Relations (ed. J. T. Jenkins & M. Satake), pp. 327–338. Elsevier.