Hostname: page-component-cb9f654ff-w5vf4 Total loading time: 0 Render date: 2025-09-02T08:11:27.142Z Has data issue: false hasContentIssue false
  • English
  • Français

Boundary conditions for high-shear grain flows

Published online by Cambridge University Press:  20 April 2006

K. Hui ,
P. K. Haff ,
J. E. Ungar  and
R. Jackson
K. Hui
Affiliation:
Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, California 91125
P. K. Haff
Affiliation:
Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, California 91125
J. E. Ungar
Affiliation:
Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, California 91125
R. Jackson
Affiliation:
Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125 Present address: Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544.
Get access Rights & Permissions [Opens in a new window]

Abstract

Boundary conditions are developed for rapid granular flows in which the rheology is dominated by grain–grain collisions. These conditions are $\overline{v}_0 = {\rm const}\,{\rm d}\overline{v}_0/{\rm d}y$ and u0 = const du0/dy, where $\overline{v}$ and u are the thermal (fluctuation) and flow velocities respectively, and the subscript indicates that these quantities and their derivatives are to be evaluated at the wall These boundary conditions are derived from the nature of individual grain–wall collisions, so that the proportionality constants involve the appropriate coefficient of restitution ew for the thermal velocity equation, and the fraction of diffuse (i.e. non-specular) collisions in the case of the flow-velocity equation. Direct application of these boundary conditions to the problem of Couette-flow shows that as long as the channel width h is very large compared with a grain diameter d it is permissible to set $\overline{v} = 0$ at the wall and to adopt the no-slip condition. Exceptions occur where d/h is not very small, when the wall is not rough, and when the grain–wall collisions are very elastic. Similar insight into other flows can be obtained qualitatively by a dimensional analysis treatment of the boundary conditions. Finally, the more difficult problem of self-bounding fluids is discussed qualitatively.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 145 , August 1984 , pp. 223 - 233
Copyright
© 1984 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.)

Article purchase

Temporarily unavailable

References

Ackermann, N. L. & Shew, H. 1982 Stresses in rapidly sheared fluid—solid mixtures. J. Engng Mech. Div. ASCE 108 (EM1), 95–113.Google Scholar
Campbell, C. S. & Brennen, C. E. 1982 Computer simulations of chute flows of granular materials. In Deformation and Failure in Granular Materials (ed. P. A. Vermeer & H. J. Luger). Bulkenna.
Goldsmith, W. 1960 Impact, the Theory and Physical Behaviour of Colliding Solids. Edward Arnold.
Haff, P. K. 1983 Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech. 134, 401–430.Google Scholar
Hui, K. & Haff, P. K. 1984 Rapid grain flow in a vertical channel. Submitted to J. Fluid Mech.Google Scholar
Jenkins, J. T. & Cowin, S. C. 1979 Theories for flowing granular materials. In Mechanics Applied to the Transport of Bulk Materials (ed. S. C. Cowin), pp. 79–89. ASME, AMD-31.
Jenkins, J. T. & Savage, S. B. 1981 The mean stress resulting from interparticle collisions in a rapid granular shear flow. In Continuum Models of Discrete Systems 4 (ed. O. Brulin & R. K. T. Hsieh), pp. 356–371. North-Holland.
Jenkins, J. T. & Savage, S. B. 1983 A theory for the rapid flow of identical, smooth, nearly elastic particles. J. Fluid Mech. 130, 187–202.Google Scholar
McTigue, D. F. 1978 A model for stresses in shear flow of granular material. In Proc. US—Japan Seminar on Continuum-Mechanical and Statistical Approaches in the Mechanics of Granular Materials (ed. S. C. Cowin & M. Satake), pp. 266–271. Gakujutsu-Bunken Fukyukai.
Maxwell, J. C. 1879 On the stresses in rarefied gases arising from inequalities of temperature. Phil. Trans. R. Soc. Lond. 170, 231–256.Google Scholar
Nguyen, T., Brennen, C. & Sabersky, R. 1979 Funnel flow in hoppers. In Mechanics Applied to the Transport of Bulk Materials (ed. S. C. Cowin), pp. 25–33. ASME, AMD-31.
Ogawa, S. 1978 Multitemperature theory of granular materials. In Proc. US—Japan Seminar on Continuum-Mechanical and Statistical Approaches in the Mechanics of Granular Materials (ed. S. C. Cowin & M. Satake), 189–207. Gakujutsu-Bunken Fukyukai.
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. & Jeffrey, D. J. 1981 The stress tensor in a granular flow at high shear rates. J. Fluid Mech. 110, 255–272.Google Scholar
Shen, H. & Ackermann, N. L. 1982 Constitutive relationships for fluid—solid mixtures. J. Engng Mech. Div ASCE 108 (EM5), 748–763.Google Scholar