Skip to main content
×
Home

Boundary conditions for high-shear grain flows

  • K. Hui (a1), P. K. Haff (a1), J. E. Ungar (a1) and R. Jackson (a2) (a3)
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.

Copyright
References
Hide All
Ackermann, N. L. & Shew, H. 1982 Stresses in rapidly sheared fluid—solid mixtures. J. Engng Mech. Div. ASCE 108 (EM1), 95113.
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, 401430.
Hui, K. & Haff, P. K. 1984 Rapid grain flow in a vertical channel. Submitted to J. Fluid Mech.
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. 7989. 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. 356371. 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, 187202.
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. 266271. Gakujutsu-Bunken Fukyukai.
Maxwell, J. C. 1879 On the stresses in rarefied gases arising from inequalities of temperature. Phil. Trans. R. Soc. Lond. 170, 231256.
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. 2533. 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), 189207. Gakujutsu-Bunken Fukyukai.
Ogawa, S., Umemura, A. & Oshima, N. 1980 On the equations of fully fluidized granular materials. Z. Angew. Math. Phys. 31, 483493.
Savage, S. B. & Jeffrey, D. J. 1981 The stress tensor in a granular flow at high shear rates. J. Fluid Mech. 110, 255272.
Shen, H. & Ackermann, N. L. 1982 Constitutive relationships for fluid—solid mixtures. J. Engng Mech. Div ASCE 108 (EM5), 748763.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 29 *
Loading metrics...

Abstract views

Total abstract views: 130 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 24th November 2017. This data will be updated every 24 hours.