Theories of immiscible and structured mixtures

Publisher Summary This chapter discusses the mechanics of rapid granular flows. A “bulk solid” or “granular fluid” may be defined as an assembly of discrete solid components dispersed in a fluid, such that the solid constituents are in contact or near contact with their neighbors. Bulk solids comprise one member of a larger class of two-phase disperse systems made up of solids and fluids; dilute suspensions form another related member, which is more familiar to fluid mechanicists. The section II of this chapter presents a review of the classical papers of Bagnold and a discussion of the various modes and regimes of granular flow. A dimensional analysis provides a physical background to the detailed review of experimental and theoretical work that follows. Section III reviews flows in vertical channels and inclined chutes; it deals primarily with experimental observations of stress, velocity, and bulk-density fields. Laboratory devices and viscometric type experiments designed to determine the stress-strain-rate behavior are discussed in Section IV. Quasi-static testers, suspension viscometers, and high-shear-rate devices are described. Section V reviews theories for high-shear-rate granular flows. It includes continuum models as well as analytical and numerical microstructural models that consider the details of collisions between particles.

The Mechanics of Rapid Granular Flows

This article reviews the behavior of materials made up of a large assemblage of solid particles under rapid and quasi static deformations. The focus is on flows at relatively high concentrations and for conditions when the interstitial fluid plays an insignificant role. The momentum and energy exchange processes are then primarily governed by interparticle collisions and Coulomb-type frictional contact. We first discuss some physical behavior —dilatancy, internal friction, fluidization and particle segregation — that are typical to the understanding of granular flows. Bagnold's seminal Couette flow experiments and his simple stress analysis are then used to motivate the first constitutive theories that use a microstructural variable — the fluctuation energy or granular temperature — governing the subscale fluctuating motion. The kinetic theories formalize the derivation of the field equations of bulk mass, momentum and energy, and permit derivation of constitutive relations for stress, flux of fluctuation energy and its dissipation rate for simple particle assemblages and when frictional rubbing contact can be ignored. These statistical considerations also show that formulation of boundary conditions needs special attention. The frictional-collisional constitutive behavior in which both Coulomb-type rubbing contact and collisional encounters are significant are discussed. There is as yet no rigorous formulation. We finally present a phenomenological approach that describes rapid flows of granular materials under simultaneous transport of heat and close with a summary of stability analyses of the basic flow down an inclined plane.

On flows of granular materials

Modern mixture theory is an outgrowth of classical thermochemistry, a discipline which developed from early attempts to describe mixtures of gases. It is based on the concept that a mixture may be represented1 by “a sequence of bodies \(\mathop B\limits_a \), all of which … occupy regions of space … simultaneously”. However, it was commonly recognized at an early stage that so strong an assumption of intermiscibility was not appropriate to all physical situations. For example, soils, porous rock, suspensions, packed powders, granular propellants, etc., consist of identifiable solid particles surrounded by one or more fluids, or an identifiable solid matrix through which one or more fluids is dispersed. Motions of the individual components are possible and, as long as there are no chemical reactions, each consistituent retains its integrity. We call such materials multiphase mixtures. They are more complicated than classical mixtures in the sense that they have geometrical structure, but less complicated in the sense that the constituents are not intimately intermixed. A theory describing them should reflect these facts.

A Theory of Multiphase Mixtures

High-speed motion pictures document a series of essentially two-dimensional, free-surface flows of 6-mm diameter plastic spheres generated in an inclined glass-walled chute 3.7 m long and 6.7 mm wide. Flows exhibit a diverse suite of structural features, depending on particle flux, chute inclination, and geometry of the fixed bed. Flows at sufficiently low inclination or high particle flux can generally be divided into factional and collisional regions. In the frictional region, intergranular momentum transfer occurs by enduring, frictional contacts; kinetic theories of granular flow do not apply. The frictional region typically consists of a quasi-static zone extending up from the fixed bed and an overlying block-gliding zone, in which coherent blocks of grains move parallel to the bed. The collisional region overlies the contact region in sufficiently deep flows; otherwise it extends to the bed. In it, momentum transfer is dominated by collisions. It typically consists of three zones: a lower grain-layer-gliding zone, in which grains appear to slide over one another; a middle chaotic zone, in which grain motions are highly random, as in a dense gas; and an overlying saltational zone, in which grains trace long, curved paths. Zones grade smoothly into one another, and some are missing in certain flows. In none of the flows does a frictional region overlie a collisional region. Zone thicknesses and block dimensions are previously unrecognized length scales in granular flows. The no-slip velocity boundary condition applies only when a frictional region adjoins a geometrically rough fixed bed, otherwise significant slip occurs (up to 50% of the mean flow velocity). Binary collisions dominate only in the collisional region and there perhaps only in the chaotic and saltational zones. Because many collision-dominated geophysical flows start from friction-dominated ones and end by reverting to the frictional regime as the energy supply diminishes, kinetic theories of granular flow, which unanimously employ the binary-collision assumption, can at best describe only a portion of the phenomena. Microstructural descriptions of the relatively slow, compact, and frictional regions evident in most geophysical flows are required to complement existing kinetic theories.

Structural features in granular flows

A theory of flowing granular materials with incompressible grains is presented which is similar to one proposed by Goodman and Cowin in 1972. Using a nonlinear version of the theory, boundary‐value problems corresponding to gravity flow in a closed vertical channel and in a channel with a free surface inclined at various angles from the vertical are considered in detail. In analyzing these problems, the dependence of the free energy and viscosity on the volume fraction are specialized to agree with those proposed as a result of recent experimental work. Solutions of these problems demonstrate many of the characteristics normally assumed in other treatments of granular materials. In particular, for vertical flow, when the volume fraction is greater than (less than) the critical volume fraction, a compressive (tensile) force must be exerted on the walls to maintain isochoric flow. Pluglike regions in which the velocity is almost constant develop, and there is a relatively thin shear region near the wall. For the flow with a free surface, as grain size or angle of inclination increases, the solutions tend to two rigid motions separated by a thin shear layer.

J. P. Thomas

Papers

Gravitational flows of granular materials with incompressible grains