(170e) Experimental Investigation and Kinetic Theory Based Model of an Annular Granular Shear Flow

Granular materials are employed in a variety of guises in industrial processes to aid the transport and storage of a range of products. These materials can be induced to flow through, for example, vibration (Clement and Rajchenbach 1991), shearing (Hsiau and Shieh 1999) and gas flow (Tsuji, et al. 1984) and under these conditions, their behaviour resembles that of a fluid or a gas. This qualitative similarity can be placed on a firmer mathematical footing through the use of, for example, the kinetic theory developed originally for the description of thermal fluids (Chapman and Cowling 1970). Over the last 30 years this approach has gained in popularity and evolved in sophistication as researchers have aimed for the description of granular flow through the use of a continuum or hydrodynamic equation set (Johnson, et al. 1990, Kumaran 1998, Lu, et al. 2001, Wildman, et al. 2005, Zamankhan 1995). This method has shown great promise in describing granular flows and is in a sense successful in describing flows where strictly speaking it shouldn't be.

A notable antecedent of our work is the paper by (Savage and Sayed 1984) who constructed a pressure controlled annular shear cell and found broad agreement between their results and their proposed stress constitutive law.. However, they were unable to view the internal structure of the material or the motion of the individual grains in the bulk. Consequently, they were only able to compare their results with volume-averaged quantities and were not able to test the validity of their model in detail. This paper describes the first steps towards being able to fully probe the behaviour of sheared granular flows using a three-dimensional annular shear cell. The geometry of our shear cell is as follows. Particles in an annular channel are driven by a basal boundary moving at a speed U, with a top boundary fixed with respect to the azimuthal direction, but free to move in the vertical direction to allow it to respond to the pressure of the particles impacting on it from below. The sidewalls are made from polymethylmethacrylate (PMMA) with the inner wall of the cell situated 120 mm from the axis, whilst the outer wall is at 125 mm. The driving, basal, boundary is interchangeable, and we use a sawtooth profile base, constructed from Aluminium. The depth of the teeth at the inner wall is on average, 1.42 mm whilst the depth at the outer wall is 1.92 mm, with a linear variation between, and there are approximately 22.8 teeth per radian. The top boundary is identical to the base. The pressure pushing down on the top boundary was varied using a counterweight, creating a pressure-controlled cell. The beads used were ballotini beads of average diameter, ? = 3 mm. One layer of the grains was used in the experiments, where one loose packed layer of particles consisted of, N ~ 1740 ballotini beads. The coefficients of restitution, measured using high-speed photography, were particle-particle, e = 0.91, particle-sidewall, ew = 0.80 and particle base, eb = 0.79.

A single particle in the flow was tracked using positron emission particle tracking (PEPT). PEPT works through the detection of 511 keV ?-photons that result from the annihilation event that occurs when a positron encounters an electron. The tracer particles are formed through the irradiation of a ballotini bead with 3He particles resulting in the formation of 18F, leaving the macroscopic properties of the particle unaffected and the tracer particle is then identical to the remaining beads. 18F is an unstable radioisotope of Flourine and decays through the emission of positrons. A collision between a positron and an electron occurs rapidly and results in two ?-photons moving on the same path but in opposite directions. These are detected simultaneously, without collimation, by NaI scintillator based photon detectors. The tracer particle coordinates are then determined through triangulation of successive photon detections, allowing the coordinates to be detected at a rate up to about 1kHz, at an accuracy of up to +/- 1 mm (Parker, et al. 1997, Wildman and Parker 2002). From the single particle coordinate data, the mean velocity field and the volume fraction of the bed were determined. The mean velocity field in the r-z or cross-sectional plane indicates that secondary flows are in evidence; the particles are driven round the central axis, but also flow up the outer wall and down the inner wall resulting in a helical path round the cell. The mean velocity of the grains, as a function of distance from the base, indicates that as the pressure is reduced, the slip velocity of the particles at the base reduces in conjuction with the particles accumulating towards the bottom.

Using results from the kinetic theory method, we have developed a one dimensional model to predict the distribution of particles and their mean velocity. A number of researchers have developed constitutive relations to provide closure for the conservation equations for mass, momentum and energy and we have followed the dense granular flow derivations of Jenkins for dry granular media, providing constitutive relations for dense, but collisional flows (Jenkins 1999). The continuum expressions were solved using Matlab v. 7 employing the boundary value solver bvpc4 and using the experimental parameters such as vibration amplitude and frequency as inputs.

The comparison between the predictions of the Jenkins model and the experiment show remarkably good agreement. We have ignored the centripetal accelerations in the analysis and neglected any modifications due to the radially varying packing fraction and granular temperature as a function of r that will result from the geometry of the system. It is clear from the presence of the secondary flows, that these variations have consequences that are difficult to predict using a 1D model and suggest a two-dimensional solution of the conservation equations is required to fully describe the system. Future studies will focus on improving the ability of the PEPT facility to measure the granular temperature and on developing a two-dimensional model for predicting all three components of the velocity field.

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