Index of content:
Volume 30, Issue 5, October 1986
30(1986); http://dx.doi.org/10.1122/1.549874View Description Hide Description
Mechanisms contributing to stresses in flowing granular materials are reviewed, with an indication of the present state of constitutive theories of both frictional and collisional stress contributions. The application of equations of motion based on these models to problems is then illustrated by three examples: plane shearing, hopper discharge, and standpipe flow.
30(1986); http://dx.doi.org/10.1122/1.549875View Description Hide Description
We give a physical and heuristic discussion of the kinetic model of granular fluids, wherein the grain plays the role of a molecule. A consideration of the details of grain‐grain and grain‐wall interactions leads naturally to the equations of motion and to suitable boundary conditions. Examples from a Couette flow geometry are used to support the argument that the energy and momentum equations must be treated on an equal footing. The introduction of the energy equation leads to the appearance of a new length scale Λ, the conduction length, which describes the competition between viscous heating and collisional energy absorption and whose value determines the distribution of “granular temperature” and hence the flow field in the fluid.
Viscosity, granular‐temperature, and stress calculations for shearing assemblies of inelastic, frictional disks30(1986); http://dx.doi.org/10.1122/1.549893View Description Hide Description
Employing nonequilibrium molecular‐dynamics methods the effects of two energy loss mechanisms on viscosity, stress, and granular‐temperature in assemblies of nearly rigid, inelastic frictional disks undergoing steady‐state shearing are calculated. Energy introduced into the system through forced shearing is dissipated by inelastic normal forces or through frictional sliding during collisions resulting in a natural steady‐state kinetic energy density (granular‐temperature) that depends on the density and shear rate of the assembly and on the friction and inelasticity properties of the disks. The calculations show that both the mean deviatoric particle velocity and the effective viscosity of a system of particles with fixed friction and restitution coefficients increase almost linearly with strain rate. Particles with a velocity‐dependent coefficient of restitution show a less rapid increase in both deviatoric velocity and viscosity as strain rate increases. Particles with highly dissipative interactions result in anisotropicpressure and velocity distributions in the assembly, particularly at low densities. At very high densities the pressure also becomes anisotropic due to high contact forces perpendicular to the shearing direction. The mean rotational velocity of the frictional disks is nearly equal to one‐half the shear rate. The calculated ratio of shear stress to normal stress varies significantly with density while the ratio of shear stress to total pressure shows much less variation. The inclusion of surface friction (and thus particle rotation) decreases shear stress at low density but increases shear stress under steady shearing at higher densities.
High‐Speed Motion Pictures of Nearly Steady, Uniform, Two‐Dimensional, Inertial Flows of Granular Material30(1986); http://dx.doi.org/10.1122/1.549900View Description Hide Description
Nearly steady, uniform, essentially two‐dimensional (except for grain spins) flows of 6‐mm‐diameter cellulose‐acetate spheres were generated in an inclined glass‐walled channel 3.7 m long and just 6.25 mm wide. The bed consisted of spheres like those in the flows, which were glued with random spacing to an aluminum bar. Filming was done with a 16‐mm camera at rates from 40 to 120 times normal. The flows were almost entirely inertial. They typically consist of three zones: a basal grain‐layer gliding zone, in which irregular layers of grains slide over one another in card‐deck shear fashion and free paths are short; a middle chaotic zone, in which grain motions are highly random and homogeneous, as in a dense gas, and free paths are about a grain diameter; and a superincumbent saltational zone, in which grain motions are also gas‐like but free paths are long and noticeably curved. Steepening the inclination or reducing the discharge can eliminate the grain‐layer gliding and chaotic zones entirely, whereas the converse can diminish markedly the saltational zone. Translational energy loss in impacts is approximately normally distributed about a mean of 27% and is independent of relative velocity of impact. Profiles of mean downstream velocity, granular temperature (mean‐square fluctuation velocity), and bulk density in a flow almost wholly saltational show slip at the bed of 20% of the mean flow velocity, temperature a maximum midway in the flow, and density decreasing almost linearly with distance from the bed.
30(1986); http://dx.doi.org/10.1122/1.549876View Description Hide Description
The macroscopic rheological behavior of suspensions of nearly monodisperse glass fibers having a mean aspect ratio, of 24.3 and a mean length, of 267 μm, and commercial ground glass fibers, and were studied. Volume fractions of 0.02, 0.05, and 0.08 were used. For Newtonian suspending fluids, the shear viscosities and the dynamic linear viscoelastic properties of the suspension showed Newtonian behavior. In a stress growth experiment, the shear stress obtained a maximum value before reaching steady state. Upon reversal of shearing, a similar stress growth pattern was retraced. The non‐Newtonian suspending fluid, a polyisobutylene in cetane solution, was found to behave as a second‐order fluid at low shear‐rates and frequencies and shear‐thinned at higher values. Suspensions in this fluid also behaved as second‐order fluids at low shear‐rates and frequencies. The dependency of two of the second‐order fluid constants upon the volume fraction of particles was determined.
30(1986); http://dx.doi.org/10.1122/1.549892View Description Hide Description
A model magnetic suspension was formulated with acicular particles dispersed in silicone oil. Four volume fractions (2.4, 5.3, 8.7, and 12.9%) were studied in steady shearing over a wide range in rate ( to using a Rheometric System Four Rheometer. At medium rates instabilities in both cone and plate and parallel plate were observed. At low rates the stress became independent of rate indicating yield behavior. Yield stresses of 0.25, 4, 5, and 20 Pa were estimated for the four samples. On the 12.9% sample a yield stress of 20 Pa was also found using small amplitude sinusoidal oscillations and with a constant stress rheometer. Below this yield value, solid‐like elastic behavior was observed.
30(1986); http://dx.doi.org/10.1122/1.549914View Description Hide Description
Falling‐ball rheometry is used to study hydrodynamic forces in concentrated suspensions. The suspensions consist of large, uniform spheres (diameter d s =0.32 cm) neutrally buoyant in a viscous, Newtonian liquid. Falling‐ball experiments are performed by dropping steel balls of various diameters (d f ) through suspensions held in cylindrical containers of several diameters. In these optically opaque suspensions, the passage of the falling ball is observed with real‐time radiography and is recorded with digitized, high‐speed video. The average terminal velocities of the falling balls are used to calculate an apparent viscosity of each suspension (η s ). Extrapolation of the data to estimate the velocity which would occur in a cylinder of infinite diameter is used to find the apparent viscosity with zero wall effects (η s ∞). Each η s ∞ agrees with independent measurements throughout the solids concentration range of the data (0.0≤φ≤0.45). In the absence of wall effects, η s ∞ is independent of d s /d f . That is, all sizes of falling balls (0.75≤d f /d s ≤12.0) experience the same mean stress field. In addition, it is seen that in dilute and moderately concentrated suspensions (0.0<φ<0.30), wall effects are identical to those in equivalent Newtonian liquids. At higher solids loadings, we observe additional wall effects that can act over much greater distances than in equivalent Newtonian liquids. The magnitude of additional wall effects increase rapidly as φ increases above 0.30.
30(1986); http://dx.doi.org/10.1122/1.549845View Description Hide Description
A continuum mixture theory for the slow flow of a dilute suspension of solid particles in a viscous fluid is outlined. The momentum exchange between the fluid and disperse particulate phases accounts for buoyancy, drag and lift forces, and the additional viscous transport associated with the presence of the particles in the fluid. Explicit constitutive equations are posed that specialize the drag to include Stokes and Faxén forces, and the lift to accommodate the ‘‘slip–shear’’ force identified by Saffman and the ‘‘disturbance–shear’’ and ‘‘disturbance–curvature’’ forces treated by Ho and Leal. The additional viscous transport is fixed by comparison with the well‐known Einstein ‘‘effective viscosity’’ correction. Finally, the pressure difference between the phases is specified by a constitutive equation that accounts for Brownian diffusion, local inertia, and bulk viscous effects. Plane Poiseuille flow is examined to illustrate the features of the model. Both approximate analytical solutions and numerical solutions for the full, coupled, nonlinear governing equations show nonuniform distributions for the particles across the channel. This affects the apparent viscosity of the mixture according to whether the particles concentrate in regions of lesser or greater mean shear rate. Finally, the inhomogeneous particle distributions cause the mixture to appear non‐Newtonian, because the form of the distribution (and, consequently, the apparent viscosity) is rate‐dependent.
30(1986); http://dx.doi.org/10.1122/1.549846View Description Hide Description
I use a representation theorem of continuum mechanics, along with a systematic approximation, to establish an exact correspondence between the momentum interaction on the solid constituent in a multiphase flow, and the Stokes drag, the Faxén force, the Saffman lift, and the Ho and Leal lift on a particle in a viscous fluid.