Grain flow as a fluid-mechanical phenomenon

Abstract

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.