Boundary conditions for high-shear grain flows

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.