Following is an analysis of the small-strain nonlinear elasticity of granular media near states of zero stress, as it relates to the pressure-dependent incremental linear elasticity and wave speeds. The main object is elucidation of the p ½ dependence of incremental elastic moduli on pressure p , a dependence observed in numerous experiments but found to be at odds with the p ½ scaling predicted by various micromechanical models based on hertzian contact. After presenting a power-law continuum model for small-strain nonlinear elasticity, the present work develops micromechanical models based on two alternative mechanisms for the anomalous pressure scaling, namely: (1) departures at the single-contact level from the hertzian contact, due to point-like or conical asphericity; (2) variation in the number density of hertzian contacts, due to buckling of particle chains. Both mechanisms result in p ½ pressure scaling at low pressure and both exhibit a high-pressure transition to p ½ scaling at a characteristic transition pressure p *. For assemblages of nearly equal spheres, a non-hertzian contact model for mechanism (1) and percolation-type model for (2) yield estimates of p * of the form p * = c μ ˆ ∝ 3 . Here c is a non-dimensional coefficient depending only on granular-contact geometry, while α ≪ 1 is a small parameter representing spherical imperfections and μ ˆ is an appropriate elastic modulus of the particles. Then, with R representing particle radius and h a characteristic spherical tolerance or asperity height, it is found that α = ( h / R ) ½ for mechanism (1) as opposed to α = h / R for (2). Limited data from the classic experiments of Duffy & Mindlin on sphere assemblages tend to support mechanism (1), but more exhaustive experiments are called for. In addition to the above analysis of reversible elastic effects, a percolation model of inelastic ‘shake-down’ or consolidation is given. It serves to describe how prolonged mechanical vibration, leading to the replacement of point-like or inactive contacts by stiffer Hertz contacts may change the pressure-scaling behaviour of particulate media. The present analysis suggests that pressure-dependence of elasticity may provide a useful means of characterizing the state of consolidation and stability of dense particulate media.

An analysis is given of the collapse of a spherical cavity in a large body of an incompressible viscoelastic liquid. Proceeding from a linear rheological model for the liquid, one obtains a nonlinear integro-differential equation for the motion of the cavity. Analytical solutions are derived for certain limiting values of the parameters governing collapse, and some numerical solutions are presented for various other values. As one of the more interesting results of this work, it is found that elasticity in the liquid can significantly retard the collapse of a void and produce prolonged, oscillatory motion whenever the relaxation time of the fluid is moderately large in comparison to the Rayleigh collapse time. This is in sharp contrast to the catastrophic collapse which will aways occur for voids in purely viscous liquids. Both numerical and approximate analytical solutions are presented to demonstrate the damping effect of liquid viscosity on the cavity motion.

A method is presented in this article for deriving higher-order correction terms to the well-known asymptotic results for laminar forced-convection heat and mass transfer, and a formula is obtained for computing under fairly general conditions the first correction term to the asymptotic Nusselt number at large Péclet numbers for flows with small or moderate Reynolds numbers. This result is then applied to the problem of heat transfer from a solid, isothermal sphere in Stokes flow, to yield the asymptotic expression for the average Nusselt number, $\overline{Nu} = (Pe)^{1/3}[0\cdot 6245 + 0 \cdot 461(Pe)^{-1/3} + O(Re) + o(Pe^{-1/3})]$ for Pe→ ∞, Re→ 0, where $\overline{Nu}$ and Pe are based on the radius of the sphere.

The phrase following Eq. ( 19), together with Eq. (20) and the sentence following Eq. (20) should be corrected to read: "where the obvious relation −1 = * requires the connection between {R, * } and {L, }:

This article examines a reduced form of the 'purely dissipative' model proposed several years ago as a general continuum model for the rheology of non-colloidal particle dispersions, ranging from Stokesian suspensions to non-cohesive granular media. Essential to the model is a positive-definite viscosity tensor η, depending on the history of deformation and providing a crucial restriction on related models for anisotropic fluids and suspensions. In the present treatment, η is assumed to be as an isotropic function of a history-dependent second-rank 'texture' or 'fabric' tensor A. A formula for η(A) borrowed from the analogous theory of linear elasticity, and its subsequent expansion for weak anisotropy provides an explicit expression for the stress tensor in terms of fabric, strain-rate and eight material constants. Detailed consideration is given to the special case of Stokesian suspensions, which represent an intriguing subset of memory materials without characteristic time. For this idealized fluid one finds linear dependence of all stresses, including viscometric normal stress, on present deformation rate, with the provision for an arbitrary fabric evolution ('thixotropy') in unsteady deformations. As a concrete example, a corotational memory integral is adopted for A in terms of strain-rate history, and a memory kernel with two-mode exponential relaxation gives close agreement with the rather sparse experimental data on transient shear experiments. In the proposed model, an extremely rapid mode of relaxation is required to mimic the incomplete reversal of stress observed in experiments involving abrupt reversal of steady shearing, supporting the conclusion of others that non-hydrodynamic effects, with breaking of Stokesian symmetry, may be implicated in such experiments. Qualitative comparisons are made to a closely related model, derived from a micro-mechanical analysis of Stokesian suspensions, but also involving non-Stokesian effects. The present analysis may point the way to improved micro-mechanical analysis and to further experiments. Possible extensions of the model to the viscoplasticity of dry and liquid-saturated granular media also are discussed briefly.

▪ Abstract  A review is given of the stability of complex fluids subject to homogeneous states of shearing, a research field that is scarcely two decades old. For the benefit of fluid mechanicians, a brief, somewhat historical overview is presented of material instability in elastoplastic solids, where one finds a considerable body of experiment and a rich source of theoretical concepts including Hadamard instability, strain localization, and nonlocal constitutive models. A survey is then given of recent theoretical and experimental studies of instability with shear banding in various complex fluids, including micellar solutions, particulate suspensions, and rapidly sheared granular media. Various stability analyses are encapsulated in a mathematical dynamical-systems model for constitutive equations of the rate-type, and a general linear-stability theory is given for viscoelastic fluids in unbounded homogeneous shear flows. A general form of (Kelvin) wave-vector stretching is shown to play a key role in the growth of Fourier modes, as illustrated by recent computations for granular shear flow. The Fourier description also provides an explicit representation of higher-gradient (nonlocal) effects as higher-order powers of wave number.

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Experimental Systems and CharacteristicCarrier-mediated transport in membranes as a globally nonreactive process is distinguished from film theory with chemical reaction and other facilitated diffusion phenomena. With the concept of stoichiometric and system invariants, an approach is developed for the analysis of carriermediated transport with multiple permeants involving multiple reactions in the membrane. Approximate solutions of the requisite differential equations according to the relative importance of diffusion and reaction rates are reviewed, as well as typical experimental studies. Criteria for evaluating whether a membrane is in the diffusion or equilibrium regime are given, and, in the latter case, the effects of some system parameters are given, for example, binding constants, competitive permeants. Advances in membrane technology have made it possible in recent years to manufacture membranes in diverse forms such as sheets, tubes, and hollow fibers. In most current applications to separations processes, the membrane functions as a physical diffusion barrier or simple (micro-) sieve. However, &rough recent studies on models of biological membranes, it has become evident that artificial membranes, often in the form of liquid films, can be made functionally very specific in their properties by incorporating mobile or partially mobile compounds within the membrane structure which selectively react with a restricted class of permeants, for example, in ionspecific electrodes. These compounds serve effectively as carriers which not only can render the membrane very specific in its transport properties but also can enhance the relative rate at which the preferred permeants diffuse across the barrier.While various mechanistic models have been proposed to describe carrier-mediated transport in membranes, those models based on diffusion accompanied by chemical reaction have received the most theoretical and experimental study and are the main subject of this review. Recent studies of carriepmediated membrane transport of this type has led to a basic and more precise understanding of the effect of the major parameters involved, including the reaction kinetics, equilibrium (binding) constant, membrane diffusivities, concentration gradients, membrane thickness, and solubilities of the permeants in the membrane phase.Currently available numerical methods, together with various asymptotic and approximate analytic methods, based on the respective concepts of weak gradients or fast and slow reactions, allow one to estimate with a good deal of confidence the response of a particular carriermediated membrane to a variety of operational conditions. Apart from their potential for direct applications to processes such as drug transport into cells, the concepts and methods developed in these studies, relating to reaction boundary-layer analysis, global nonreactivity, and competitive interactions with carriers, suggest similar theoretical treatments in a diversity of related phenomena, such as facilitated heat transfer, ion-...