Session EU: Granular III: Segregation and Mixing

Density Segregation of Granular Materials

We have studied the segregation of spherical particles of same size with different densities flowing on a rough bumpy inclined surface. An `effective temperature' ($T_{E}$) relating diffusivity and mobility of the particles, in analogy with Stokes-Einstein relation obtained from the fluctuation-dissipation theorem for jammed systems, has been defined for granular systems. Using the Stokes law for a sphere in a viscous medium, we obtain a simple expression for $T_{E}$ as a function of diffusivity $D$ and viscosity $\eta$. A balance of segregation and diffusion fluxes leads to prediction of $f$, the fraction of heavy particles in the flowing layer that depends on effective temperature apart from the density difference and inclination of the plane. The proposed theory has no fitting parameter and is able to predict the extent of segregation very well through out the layer for different mixtures, even for systems that are far away from jamming.   

The Role of Friction in the Segregation of Particles in a Chute Flow

When a granular media, compound of particles with different properties, flows de-mixing of the constituent components may occur. This phenomenon, known as segregation, has been observed in particles with different sizes (the Brazil nut effect). In spite of being a well studied process, there is not a general understanding of the mechanisms that dominate this process. It is our interest to study the conditions that determine particle segregation in avalanches. Using a two-dimensional, discrete-element simulation, we study a bidisperse flow of particles over an inclined plane with periodic boundaries. We vary the roughness of the wall by attaching fixed particles on it. A wide range of parameters were varied (slope, number of particles, size ratio, density ratio). Both ordinary and reverse segregation patterns were observed. In this talk some preliminary results and their interpretation will be presented.   

Numerical analysis of the equilibrium behavior of two-phase granular mixtures

In this talk we present a numerical analysis of the the equilibrium behavior of two-phase granular mixtures, as predicted by the model in \textit{M.V.~Papalexandris, J.~Fl.~Mech.~(2004), 517, 103-112}. The equilibrium equations consist of an overdetermined system of quasi-linear partial differential equations with respect to the pressure and the volume fraction of the granular phase. Based on the Helmholtz decomposition and Ladyzhenskaya's decomposition theorem we develop a projection-type numerical method that overcomes the overdeterminacy of the system. The proposed method is proven to be both stable and consistent, hence, convergent. Further, it is general enough and can be applied to a variety of continuum models of complex, non-Newtonian mixtures. The talk concludes with the presentation of representative numerical results.   

Bulk horizontal size segregation in circular and parallel split-bottom cell

We perform Discrete Element Method (DEM) simulations of mixtures of different sized granular materials sheared in circular and parallel split- bottom cells. Horizontal segregation patterns in bulk are observed in both systems, but it appears that the underlying dominant driving mechanisms are different. In the curved cell, a global vertical convection roll is observed whose center is located at the plane of highest shear rate. The boundary between regions dominated by larger particles and those dominated by smaller particles are separated by the centers of the convection roll and the shear zone. In the parallel system, there is no such global convection. Nevertheless, a horizontal segregation pattern emerges, in which larger particles accumulate to the middle of shear zone and smaller particles migrate to the edges of shear zone. In this case the horizontal segregation may be caused by horizontal shear gradients and associated kinematic features.   

Dynamics of Large Intruder Particles in a Split Bottom Cell

We have performed experimental studies on the behavior of a single intruder particle in an otherwise relatively uniform matrix of granular materials sheared in a split bottom cell. We study the effect of the size and density of the intruder particle relative to the particles in the matrix on the behavior of the intruder particle. When an intruder particle is sufficiently large relative to the size of the particles in the matrix, it will rise to the surface only when the ratio between the density of the intruder particle and that of the matrix particle density $d_r$ is somewhat less than a critical value $d_{cr1} < 1$. Intruder particles of a higher density move to an equilibrium distance $h$ from the bottom of the cell that varies with $d_r$. When $d_r$ is greater than a second critical value $d_{cr2}$ -- where $d_{cr1} < d_{cr2} < 1$ -- $h = 0$. We model the behavior of these large intruder particles considering effective buoyancy and volume fraction variations as well as drag forces in the shear flow.   

Shear-segregation and mixing of sheared bidisperse granular materials

We perform experiments on granular size-segregation in an annular Couette apparatus in which a layer of small particles mixes with, and then resegregates from, a layer of large particles beneath it. We model this process using a modification of the Gray-Thornton model in which we impose a nonlinear shear profile typical of boundary-driven, confined flows. The experimentally-measured exponential velocity profile provides an input to this one-dimensional nonlinear PDE and the resulting solution of the initial value problem is non-standard, involving curved characteristics. We further interpret these solutions by numerically connecting the segregation process to changes in packing fraction, and find qualitative agreement with experimental results. As in the experiment, mixing times are observed to be faster than segregation times. Interestingly, while the size-segregation of granular materials has generally been thought to proceed faster the greater the size difference of the particles we observe that the segregation rate is quite sensitive to both the particle-size ratio and the confining pressure on the system. As a result, we observe that particles of both dissimilar and similar sizes segregate more slowly than intermediate particle size ratios and interpret this anomalous behavior in terms of a species-dependent distribution of forces within the system.   

Particle-size segregation of granular materials under shear

Particle size segregation in avalanches occurs through shearing within the granular flow. In such a flow, large particles migrate upwards, their vacated spaces being filled by smaller particles. The Gray-Thornton continuum model is a scalar conservation law in two space variables and time, but with variable coefficients corresponding to the spatially dependent velocity in shear flow. Sharp interfaces separating different mixtures are shock wave solutions that typically form in finite time from smooth initial conditions. Shocks with more large particles below small are physically unstable, leading to time-dependent multidimensional patterns. An experiment in a Couette shear cell exhibits mixing and segregation predicted by theoretical solutions.   

Granular mixing in quasi-two-dimensional tumblers with a vanishing flowing layer

We study, numerically and analytically, a singular limit of granular tumbled flows in quasi-two-dimensional rotating drums. Focusing on two versions of the kinematic continuum model of such flows, we examine the transition to the limiting dynamics as the shear layer vanishes. The limiting behavior is shown to be the same for both versions of the continuum model. Moreover, we demonstrate that, just as in a three-dimensional spherical tumbler, the limiting no-shear-layer dynamical system belongs to a class of discrete discontinuous mappings called piecewise isometries. In doing so, we identify a new mechanism of mixing, in the absence of the usual streamline crossing mediated by the flowing layer, termed streamline jumping. This leads to complex, if not technically chaotic, dynamics as long as the tumbler and fill fraction are such that the free surface of the flow moves vertically and horizontally in time.   

Cutting and shuffling of a granular mixture in a spherical tumbler

Good mixing in a fluid system is usually associated with chaotic advection. For a granular system, good mixing can be achieved through an entirely different mechanism that is well-known to mathematicians and card-players, `cutting and shuffling,' which has theoretical foundations in a relatively new area of mathematics known as ``piecewise isometries'', PWIs. Cutting and shuffling experiments are conducted in a spherical tumbler of diameter D=14cm that is half-filled with two colors of d=1mm glass beads and can be rotated by arbitrary angles periodically about each of two horizontal, orthogonal axes. In order to connect experimental results, which have a finite thickness flowing layer, with theoretical PWI mappings, which have a zero-thickness flowing layer, a continuum model with a variable flowing layer depth is utilized. The PWI theory accurately predicts the experimental mixing in this granular system demonstrating that PWI theory captures the essential kinematic features responsible for the mixing of granular materials in a three-dimensional tumbler. Furthermore, PWI results in mixing without stretching, a characteristic of chaotic mixing.   

Coarsening in axial segregation, an entropic approach

Binary mixtures in rotating drums can segregate into a stripe pattern where the individual stripes merge on long timescales [1]. Here we present an X-ray tomography study which indicates that the driving mechanism of this coarsening process might be the increase of configurational entropy [2]. \newline [1] Finger {\it et al.} PRE {\bf 74} 031312 (2006) \newline [2] Edwards \& Oakeshott, Physica A {\bf 157} 1080 (1989)