A granular material is a collection of solid particles or grains, such that most of the particles are in contact with at least some of their neighboring particles. The terms “granular materials,” “bulk solids,” “particulate solids,” and “powders” are often used interchangeably in the literature. Common examples of granular materials are sand, gravel, food grains, seeds, sugar, coal, and cement. Figure 1.1 shows the typical size ranges for some of these materials.

Granular materials are commonly encountered in nature and in various industries. For example, with reference to the chemical industry, Ennis et al. (1994) note that about 40% of the value added is linked to particle technology. Similarly, Bates (2006) notes that more than 50% of all products sold are either granular in form or involve granular materials in their production. In spite of the importance of granular materials, their mechanics is not well understood at present. Nevertheless, some progress has been made during the past few decades. The goal of this book is to describe some of the experimental observations and models related to the mechanical behavior of flowing granular materials. As studies in this area are increasing rapidly, our account is necessarily incomplete. However, it is hoped that the book will provide a useful starting point for the beginning student or researcher.

A material is called a dry granular material if the fluid in the interstices or voids between the grains is a gas, which is usually air. On the other hand, if the voids are completely filled with a liquid such as water, the material is called a saturated granular material.

At present there are no constitutive equations that are valid over the entire range of densities and velocities encountered in the storage and handling of granular materials. Most of the available equations fall into one of two regimes: (i) slow flow and (ii) rapid flow. In the slow flow regime, the solids fraction ν is high and forces are exerted across interparticle contacts which last for a long time compared to the contact time in the rapid flow regime. The contacts occur during the sliding and rolling of particles relative to each other. In the rapid flow regime, the solids fraction is low, and momentum is transferred mainly by collisions between particles and by free flight of particles between collisions. Consider the flow of a granular material between two parallel plates. If V is the relative velocity of the plates and H is the gap between them, the stresses are found to be approximately independent of the nominal shear rate γ ≡ V/H in the slow flow regime (small γ and high ν) and to increase strongly with γ in the rapid flow regime (large γ and low ν).

In devices such as hoppers and chutes, both the regimes can occur in different spatial regions, and there can also be transition regions where the nature of the flow changes from one regime to the other. Given a device and a set of operating conditions, it is difficult to determine a priori the type of flow regime that is likely to prevail. However, the following criterion can be used as a very rough guideline (Savage and Hutter, 1989).

In this chapter, we apply the hydrodynamic equations derived in Chapter 7 to some simple flow problems, and compare the results with experimental data, where available. In each of the problems, we shall first employ the heuristic description of Haff (1983) (see §7.2), followed by the kinetic theory model that was described in Chapter 7. The simplifications made in the heuristic high-density theory allow relatively easy solution of the equations of motion, and give physical insight into the behavior. The results of the kinetic theory, when compared with those of the heuristic high-density theory, gives us an understanding of the effects of the compressibility of the granular medium. We also see how the results of the high-density theory appear as a certain limit of the kinetic theory.

All the problems that we consider, indeed all problems of practical interest, require the specification of boundary conditions at solid walls. In some cases, such as flow in inclined chutes, we also require boundary conditions at a free surface, i.e., the interface between the granular medium and the atmosphere. For the flow of fluids, the no-slip boundary condition is usually imposed at solidwalls, thereby specifying the velocity of the fluid at thewalls. The temperature of the fluid at the walls is set by specifying the temperature of the walls, or by specifying the flux of energy at the walls. In addition, the pressure at some of the boundaries must be specified. Boundary conditions for granular materials are significantly different: they usually display considerable slip at solid boundaries, and the grain temperature clearly has little to do with the thermodynamic temperature of the walls.

PRELIMINARIES AND SCALING

A characteristic feature of slow, quasi-static flow of granular materials, considered in the previous chapters, is the rate independence of the stress. As discussed earlier, in the slow flow regime, grains are in abiding contact and friction is the dominant mechanism for generating shear forces. In this chapter we consider the contrasting regime of rapid flow, in which grains are in continuous fluctuational motion, and come into contact only during very brief collisions.We shall see that the stress in this regime of flow is rate dependent; indeed, the stress varies as the square of the shear rate for shear flow with a spatially uniform shear rate.

Our physical picture of rapid flow is that of grains in a state of constant agitation, with interactions between them occurring only through instantaneous collisions, as shown in Fig. 7.1. To simplify our analysis, we shall assume uniformity in size and shape, and consider the granular material to be composed of smooth spheres. By this, we mean that there is no tangential force exerted by one sphere on the other at the point of contact. This picture is identical to that of molecules in a gas, which is why granular materials in this state are often referred to as “granular gases” in the literature. However, there is a fundamental and crucial difference between a granular material and a gas: collisions or interactions between molecules are elastic, i.e., the net energy of a colliding pair is conserved, but collisions between grains are inelastic.