Slow and intermediate flow of a frictional bulk powder in the Couette geometry
Introduction
The behavior of slowly flowing powders is very different from that of fluids. This is due to the fact that flowing powders do not exhibit viscosity and the overwhelming interaction between particles is friction. Solutions for powder flows are important, however, since they yield, just as in Fluid Mechanics, details of the flow field in the powder (velocity distributions and flow rates) and forces acting on different objects in the flow and on the boundaries. Traditionally, this information was obtained by combining the stress equilibrium equations of powder statics (with added inertial terms) with a yield condition and a flow rule. Only recently were these equations combined [1] to yield a set of general flow equations for both the incompressible and the compressible [2], [18] case. The ultimate goal of the present research is to use these more general equations to predict the flow field and forces on moving objects in a bulk powder. In contrast to the increasing amount of work in the physics literature on the flow of large grains, the focus in the present work is on smaller-sized bulk powders where a continuum approach may be more advantageous.
Fig. 1 depicts a tentative, schematic representation of the different regimes of powder flow as a function of a dimensionless shear rate. This shear rate is defined as γ°*=γ°[dp/g]1/2 and contains a gravitational term g and the particle size dp. At zero shear, the powder is in a static state and the stresses in the system can be computed using the equations of static equilibrium together with a yield condition. The yield condition defines the point where the powder starts to deform. There are a large number of analytical and numerical solutions to this case and several books have been dedicated to this regime (i.e. [5], [6]). Since the powder is in equilibrium and the equations do not contain velocities, only stresses and the condition of onset of flow can be computed using this approach.
At low dimensionless shear rates of the order of 0.2 or less, the so-called “slow, frictional regime” can be identified. During this regime, frictional forces between particles are predominant. The main assumptions underlying this regime are the uninterrupted character of the flow and the existence of a continuous shear field. This flow regime is called the “quasi-static regime” in the Soil Mechanics literature. This name is somewhat inappropriate since inertial effects can, in principle, be included in the study of this regime. A Flow Rule and an Equation of Continuity are usually added to the equilibrium equations and the Yield Condition of the Static Regime. Schaeffer [1] has combined these equations to yield a set of differential equations that characterize this regime completely.
A narrow range of very low shear in which the flow exhibits a stick-slip character separates the frictional and the static regimes. This regime is characterized by large swings in stresses as the material flows and stops repeatedly. Another characteristic of this flow regime is the formation of ‘stress-chains’ in the case of larger particles. Only very recently has this regime been studied in more depth [7], [8].
At the other end of the spectrum, i.e. at very high shear rates, the so-called “rapid granular flow regime” can be identified. Here, particles are moving so fast that friction between them can be neglected and only short collisions between particles determine the character of the flow. This regime has been studied extensively and many theoretical and experimental results are available (i.e. [9], [10], [11]).
Between the “slow” (quasi-static) and the “rapid” (granular) flows is the intermediate regime where both collisional and frictional interactions between particles must be considered. This regime is characterized by stress (and strain rate) fluctuations that however decrease as the flow slows. The boundaries between the slow-and-intermediate and the intermediate-and-rapid flow regimes are not as clear cut as shown in the figure, and much more work is needed before these boundaries are better defined. Moreover, for the case of large and/or oddly shaped granules, the slow, frictional flow regime may not take place at all and fluctuations of stress and strain rate may carry over from the stick-slip regime directly to the intermediate regime.
The present paper is concerned with the study of the “intermediate” flow regime at (dimensionless) shear rates less than about 3 (but larger than 0), specifically applied to the geometry of the Couette device. The powder is sheared between two tall, rotating cylinders that form a shear gap where gravity, acting perpendicular to the shear field, cannot be neglected. The goal is to find constitutive equations that span shear rates in the intermediate regime and tend, in the limit, to the slow regime on one side and the rapid granular regime on the other. Solutions to these equations predict both the velocity profile in the bulk powder as well as the forces acting in the powder and on the boundaries. Experimental measurements are also performed to validate some theoretical findings.
Section snippets
Background—the slow-frictional regime
Throughout these considerations, the powder is assumed to be an incompressible continuum that obeys conservation of mass and momentum and also a yield condition that specifies the onset of continuous deformation. In addition, a so-called “flow rule” is required to specify the behavior of flow after the yield condition is satisfied and the powder starts to move. We adopt the convention of normal compressive stresses to be negative as shown in Fig. 2.
To calculate the flow field and the stress
The intermediate flow regime
Equations to characterize the intermediate regime were first proposed by Hibler [16] and more recently by Savage [4]. The idea behind the new approach comes from the experimental observation that stresses (and deformations) during flow of powders are not constant and uniform in time but fluctuate significantly. It is therefore appropriate to assume that the strain rate in this regime fluctuates around an average valuewhere eij′ is the fluctuation (so that 〈eij′〉=0). These authors
Dissipation function for granular (pseudo) thermal energy
Since we have introduced fluctuations and the granular temperature T, we must also write down and solve the equation of conservation of pseudo-thermal energy.
Solution for incompressible flow in the Couette geometry
The general solution for flow in the Couette device for the intermediate flow regime can now be obtained by using the constitutive Eq. (12) (or , in Table 1) in the momentum equation and the dissipation function of Eq. (19) (or , from Table 1) in the energy equation.
Numerical experiments (simulation)
We use here earlier results generated by computer simulation by Karion and Hunt [22] and published in the Journal of Powder Technology. These results are important since these kinds of correlations can be obtained from simulations but not the kind of experiments shown below. This is due to the fact that granular temperatures are difficult to measure in a dense, non-aerated system [23]. In addition, the measurement is limited to parts of the flow domain where the grains can actually be
Velocity distribution in the powder
We compare the exponential velocity decay of Eq. (29) to experiments by Hsiau and Shieh [21] in Fig. 7. These authors measured the particle average velocities in a transparent shear cell using ultra-fast imaging. The flow in this device is slightly different from the geometry in Fig. 3 in that gravity acts parallel to the direction of shear and not perpendicular as in the Couette device (see also Savage and Sayed [13]). Nevertheless, the two flows are similar especially for shallow Couette
Conclusions
We presented in this paper a new stress–strain-rate correlation that spans the gap between the rapid (grain–inertia) and the frictional (quasi-static) regimes. The new equations, while only applicable to a plane, 2D, powder layer tend in the limit of small fluctuations to the slow regime while at large fluctuations the powder exhibits a “viscous” behavior. We presented an analytical solution to the 2D flow of a dry powder in the Couette geometry but only under the assumption of large
Acknowledgements
The authors wish to thank Dr. Anna Karion for taking time and effort to redo some calculations from her simulations [22] and to extract additional data shown in Fig. 6. We also thank Dr. J. Klausner and Drs. D. Hanes and S.-S. Hsiau for providing their data in a useful form so that we could compare them to our results. Thanks are also due to Dr. H. Kalman from the Ben-Gurion University in Israel for suggesting the visualization of particle trajectories in Fig. 10.
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