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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.
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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.

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

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