A fluid mechanistic approach to slow, frictional flow of powders

Chemical and mechanical engineers are quite familiar with concepts of fluid flow which include mass and momentum balances and constitutive equations which contain such material characteristics as viscosity and density. Combination of the above correlations yields the well-known Navier-Stokes equations which have to be solved to obtain details of the flow field; a large number of analytical solutions of these equations exist. Somewhat similar equations have been developed for granular materials moving in the rapid granular flow regime, i.e. in the regime in which particle-particle contacts are not very extensive and hence friction is not prevalent. These equations result from a similarity between the movement of molecules and the flow of small elastic (or elasto-plastic) particles and resemble, for certain limiting cases, the fluid flow equations. Slow, frictional flows of powders are different in that particles are in contact for extended periods of time and friction is the overwhelming interaction. Flow in this regime is governed by the same mass and momentum balances but the constitutive equations are different. Only very recently have some of these equations been combined to yield a set of differential equations which take the place of the Navier-Stokes equations; only very few solutions of these equations exist. The present paper describes common features and major dissimilarities between different equations of motion for the above systems and some of their solutions and presents examples and comparisons of powder and fluid flows in identical geometries. Furthermore, general equations of motion for compressible powders in slow flow are developed and an example of an analytical solution is given.