2 results
Boundary conditions for high-shear grain flows
- K. Hui, P. K. Haff, J. E. Ungar, R. Jackson
-
- Journal:
- Journal of Fluid Mechanics / Volume 145 / August 1984
- Published online by Cambridge University Press:
- 20 April 2006, pp. 223-233
-
- Article
- Export citation
-
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.
Grain flow as a fluid-mechanical phenomenon
- P. K. Haff
-
- Journal:
- Journal of Fluid Mechanics / Volume 134 / September 1983
- Published online by Cambridge University Press:
- 20 April 2006, pp. 401-430
-
- Article
- Export citation
-
The behaviour of granular material in motion is studied from a continuum point of view. Insofar as possible, individual grains are treated as the ‘molecules’ of a granular ‘fluid’. Besides the obvious contrast in shape, size and mass, a key difference between true molecules and grains is that collisions of the latter are inevitably inelastic. This, together with the fact that the fluctuation velocity may be comparable to the flow velocity, necessitates explicit incorporation of the energy equation, in addition to the continuity and momentum equations, into the theoretical description. Simple ‘microscopic’ kinetic models are invoked for deriving expressions for the ‘coefficients’ of viscosity, thermal diffusivity and energy absorption due to collisions. The ‘coefficients’ are not constants, but are functions of the local state of the medium, and therefore depend on the local ‘temperature’ and density. In general the resulting equations are nonlinear and coupled. However, in the limit s [Lt ] d, where s is the mean separation between neighbouring grain surfaces and d is a grain diameter, the above equations become linear and can be solved analytically. An important dependent variable, in this formulation, in addition to the flow velocity u, is the mean random fluctuation (‘thermal’) velocity $\overline{v}$ of an individual grain. With a sufficient flux of energy supplied to the system through the boundaries of the container, $\overline{v}$ can remain non-zero even in the absence of flow. The existence of a non-uniform $\overline{v}$ is the means by which energy can be ‘conducted’ from one part of the system to another. Because grain collisions are inelastic, there is a natural (damping) lengthscale, governed by the value of d, which strongly influences the functional dependence of $\overline{v}$ on position. Several illustrative examples of static (u = 0) systems are solved. As an example of grain flow, various Couette-type problems are solved analytically. The pressure, shear stress, and ‘thermal’ velocity function $\overline{v}$ are all determined by the relative plate velocity U (and the boundary conditions). If $\overline{v}$ is set equal to zero at both plates, the pressure and stress are both proportional to U2, i.e. the fluid is non-Newtonian. However, if sufficient energy is supplied externally through the walls ($\overline{v} \ne 0$ there), then the forces become proportional to the first power of U. Some examples of Couette flow are given which emphasize the large effect on the grain system properties of even a tiny amount of inelasticity in grain–grain collisions. From these calculations it is suggested that, for the case of Couette flow, the flow of sand is supersonic over most of the region between the confining plates.