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

Equations of motion correct to the first order of the gas concentration by volume are derived for a dispersion of gas bubbles in liquid through systematic averaging of the equations on the microlevel. First, by ensemble averaging, an expression for the average stress tensor is obtained, which is non-isotropic although the local stress tensors in the constituent phases are isotropic (viscosity is neglected). Next, by applying the same technique, the momentum-flux tensor of the entire mixture is obtained. An equation expressing the fact that the average force on a massless bubble is zero leads to a third relation. Complemented with mass-conservation equations for liquid and gas, these equations appear to constitute a completely hyperbolic system, unlike the systems with complex characteristics found previously. The characteristic speeds are calculated and shown to be related to the propagation speeds of acoustic waves and concentration waves.

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Two-phase flow equations for a dilute dispersion of gas bubbles in liquid

The method described in Part I [Phys. Fluids 31, XXXX (1988)] is extended to treat the problem of determining the permeability of random arrays of infinitely long cylinders. The results for the transverse and longitudinal permeabilities averaged over several configurations of random arrays of cylinders are presented as a function of the area fraction of the cylinders. A detailed comparison is made with the estimates of the permeability obtained by various approximate and asymptotic theories to determine their range of validity.

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Transport Processes in Random Arrays of Cylinders. II. Viscous Flow

This paper addresses a fundamental discrepancy between the suspension balance model and other two-phase flow formulations. The former was proposed to capture the shear-induced migration of particles in Stokesian suspensions, and hinges on the presence of a particle phase stress to drive particle migration. This stress is taken to be the “particle stress,” defined as the particle contribution to the suspension stress. On the other hand, the two-phase flow equations derived in several studies show only a force acting on the particle phase, but no stress. We show that the identification of the particle phase stress with the particle contribution to the suspension stress in the suspension balance model is incorrect, but there exists a well-defined particle phase stress. Following the rigorous method of volume averaging, we show that the force on the particle phase may be written as the sum of an interphase drag and the divergence of the particle phase stress. We derive exact micromechanical relations for thes...

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I.N. Shchelchkova

Papers

Flow of dense suspensions