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In 1923, the Philosophical Transactions published G. I. Taylor’s seminal paper on the stability of what we now call Taylor–Couette flow. In the century since the paper was published, Taylor’s ground-breaking linear stability analysis of fluid flow between two rotating cylinders has had an enormous impact on the field of fluid mechanics. The paper’s influence has extended to general rotating flows, geophysical flows and astrophysical flows, not to mention its significance in firmly establishing several foundational concepts in fluid mechanics that are now broadly accepted. This two-part issue includes review articles and research articles spanning a broad range of contemporary research areas, all rooted in Taylor’s landmark paper. This article is part of the theme issue ‘Taylor–Couette and related flows on the centennial of Taylor’s seminal Philosophical Transactions paper (part 1)’.

In 1923, thePhilosophical Transactionspublished G. I. Taylor’s seminal paper on the stability of what we now call Taylor–Couette flow. In the century since the paper was published, Taylor’s ground-breaking linear stability analysis of fluid flow between two rotating cylinders has had an enormous impact on the field of fluid mechanics. The paper’s influence has extended to general rotating flows, geophysical flows and astrophysical flows, not to mention its significance in firmly establishing several foundational concepts in fluid mechanics that are now broadly accepted. This two-part issue includes review articles and research articles spanning a broad range of contemporary research areas, all rooted in Taylor’s landmark paper.This article is part of the theme issue ‘Taylor–Couette and related flows on the centennial of Taylor’s seminalPhilosophical Transactionspaper (part 2)’.

The drag force on a spherical intruder in dense granular shear flows is studied using discrete element method simulations. Three regimes of the intruder dynamics are observed depending on the magnitude of the drag force (or the corresponding intruder velocity) and the flow inertial number: a fluctuation-dominated regime for small drag forces; a viscous regime for intermediate drag forces; and an inertial (cavity formation) regime for large drag forces. The transition from the viscous regime (linear force-velocity relation) to the inertial regime (quadratic force-velocity relation) depends further on the inertial number. Despite these distinct intruder dynamics, we find a quantitative similarity between the intruder drag in granular shear flows and the Stokesian drag on a sphere in a viscous fluid for intruder Reynolds numbers spanning five orders of magnitude. Beyond this first-order description, a modified Stokes drag model is developed that accounts for the secondary dependence of the drag coefficient on the inertial number and the intruder size and density ratios. When the drag model is coupled with a segregation force model for intruders in dense granular flows, it is possible to predict the velocity of gravity-driven segregation of an intruder particle in shear flow simulations.

We present a continuum approach to model segregation of size-bidisperse granular materials in unsteady bounded heap flow as a prototype for modeling segregation in other time varying flows. In experiments, a periodically modulated feed rate produces stratified segregation like that which occurs due to intermittent avalanching, except with greater layer-uniformity and higher average feed rates. Using an advection-diffusion-segregation equation and characterizing transient changes in deposition and erosion after a feed rate change, we model stratification for varying feed rates and periods. Feed rate modulation in heap flows can create well-segregated layers, which effectively mix the deposited material normal to the free surface at lengths greater than the combined layer-thickness. This mitigates the strong streamwise segregation that would otherwise occur at larger particle-size ratios and equivalent steady feed rates and can significantly reduce concentration variation during hopper discharge. Coupling segregation, deposition and erosion is challenging but has many potential applications.

Flow of size-bidisperse particle mixtures in a spherical tumbler rotating alternately about two perpendicular axes produces segregation patterns that track the location of nonmixing islands predicted by a dynamical systems approach. To better understand the paradoxical accumulation of large particles in regions defined by barriers to transport, we perform discrete element method (DEM) simulations to visualize the three-dimensional structure of the segregation patterns and track individual particles. Our DEM simulations and modeling results indicate that segregation pattern formation in the biaxial spherical tumbler is due to the interaction of size-driven radial segregation with the weak spanwise component of the advective surface flow. Specifically, we find that after large particles segregate to the surface, slow axial drift in the flowing layer, which is inherent to spherical tumblers, is sufficient to drive large particles across nominal transport barriers and into nonmixing islands predicted by an advective flow model in the absence of axial drift. Axial drift alters the periodic dynamics of nonmixing islands, turning them into "sinks" where large particles accumulate even in the presence of collisional diffusion. Overall, our results indicate that weak perturbation of chaotic flow has the potential to alter key dynamical system features (e.g., transport barriers), which ultimately can result in unexpected physical phenomena.

Size segregation of polydisperse materials during hopper discharge is problematic in various industrial situations. However, due to the complexity of hopper discharge flow kinematics and limited prediction techniques for polydisperse segregation, accurate modeling of polydisperse segregation during hopper discharge has been challenging. We extend the application of a general predictive continuum model that captures the effects of segregation, diffusion, and advection to polydisperse hopper flow. Model predictions of segregation within the hopper and the average particle diameter during discharge match experimentally validated discrete element method simulations and experiments.

To apply an existing transport model to industrially relevant granular mixtures and flows, we demonstrate an approach to experimentally measure the segregation coefficient, S, and the flowing layer thickness, δ, using a modified quasi-2D bounded heap. During heap formation, the flowing layer velocity is characterized using particle image velocimetry, which determines δ and other kinematics. After formation, the heap is sampled to determine the concentration of each species along its length. The difference between measured and model-predicted concentration profiles is minimized by varying S, which gives the experimental estimate of S for the mixture of interest. The methodology is tested for bidisperse combinations of Cellets — industrially relevant and roughly spherical microcrystalline-cellulose particles. Variation of S with particle-size-ratio matches previous DEM results for glass particles. Further validation of the approach is provided by tests with different flow rates and mixtures, including glass particles and Cellets, and a tridisperse Cellets mixture.

Using simulations and a virtual-spring-based approach, we measure the segregation force, $F_{seg},$ in size-bidisperse sphere mixtures over a range of concentrations, particle-size ratios and shear rates to develop a semiempirical model for $F_{seg}$ that extends its applicability from the well-studied non-interacting intruders regime to finite-concentration mixtures where cooperative phenomena occur. The model predicts the concentration below which the single-intruder assumption applies and provides an accurate description of the pressure partitioning between species.

International audience; Friction at the endwalls of partially-filled horizontal rotating tumblers induces curvature and axial drift of particle trajectories in the surface flowing layer. Here we describe the results of a detailed discrete element method study of the dry granular flow of monodisperse particles in threedimensional cylindrical tumblers with endwalls and cylindrical wall that can be either smooth or rough. Endwall roughness induces more curved particle trajectories, while a smooth cylindrical wall enhances drift near the endwall. This drift induces recirculation cells near the endwall. The use of mixed roughness (cylindrical wall and endwalls having different roughness) shows the influence of each wall on the drift and curvature of particle trajectories as well as the modification of the free surface topography. The effects act in opposite directions and have variable magnitude along the length of the tumbler such that their sum determines both direction of net drift and the recirculation cells. Near the endwalls, the dominant effect is always the endwall effect, and the axial drift for surface particles is toward the endwalls. For long enough tumblers, a counter-rotating cell occurs adjacent to each of the endwall cells having a surface drift toward the center because the cylindrical wall effect is dominant there. These cells are not dynamically coupled with the two endwall cells. The competition between the drifts induced by the endwalls and the cylindrical wall determines the width and drift amplitude for both types of cells.

A discrete element model based on super-ellipsoids was used to simulate cylindrical particle flow. The model can describe a cylindrical particle accurately provided the shape indices of the super-ellipsoids are set to appropriate values. To achieve more rapid calculations, we implemented an “oriented bounding box algorithm” (OBBA) for the initial contact detection of cylindrical particles. Several types of simulations were performed to validate the super-ellipsoid model and the contact-detection algorithm. First, the effect of shape index of the super-ellipsoids on model accuracy was investigated through three simulations: impact of a cylindrical particle on a flat wall, flow of cylindrical particles in a rotating tumbler, and segregation of cylindrical particles of different length flowing down a bounded heap. The simulation results show that the super-ellipsoids describe cylindrical particles accurately when the shape index that specifies the sharpness of the cylinder edges is sufficiently large. The efficiency of the OBBA is measured by simulations in which a box is filled with cylindrical particles and a tumbler that contains cylindrical particles is rotated. The simulation results show that the OBBA can accelerate the calculations significantly. The effect of particle shape (such as aspect ratio and shape index) on the calculation speed was obtained. The simulation of rod-like particles tended to take more calculation time than that of disk-like particles, and the simulation time increased with an increasing particle-shape index.

Accurate continuum models of flow and segregation of dense granular flows are now possible. This is the result of extensive comparisons, over the last several years, of computer simulations of increasing accuracy and scale, experiments, and continuum models, in a variety of flows and for a variety of mixtures. Computer simulations—discrete element methods (DEM)—yield remarkably detailed views of granular flow and segregation. Conti-nuum models, however, offer the best possibility for parametric studies of outcomes in what could be a prohibitively large space resulting from the competition between three distinct driving mechanisms: advection, diffusion, and segregation. We present a continuum transport equation–based framework, informed by phenomenological constitutive equations, that accurately predicts segregation in many settings, both industrial and natural. Three-way comparisons among experiments, DEM, and theory are offered wherever possible to validate the approach. In addition to the flows and mixtures described here, many straightforward extensions of the framework appear possible.

Modeling segregation of size disperse granular materials during hopper discharge is important as hoppers are widely used in various industries. However, due to the complexity of segregation and hopper discharge flows, accurately modeling this process has been challenging. In this study, we apply a continuum transport model to predict segregation of size bidisperse granular material during the discharge of quasi-2D hoppers. We apply Discrete Element Method (DEM) simulations to reveal that segregation occurs mainly in a surface layer where particles are transported from the sidewalls to the hopper center. Velocity profiles are also developed based on a kinematic model and DEM data. The continuum model, which captures the interplay of advection, diffusion, and segregation, is then applied to predict the particle concentration distribution in the surface layer and the bulk. The continuum model accurately predicts the segregation pattern inside the hopper during discharge and the concentration profile of the discharged materials, in agreement with DEM simulations and experiments.

Most natural and industrial flows involve mixtures of nonspherical particle species. A general approach to predicting segregation over the infinite space of all possible particle shapes and sizes seems, at first glance, to be impossible. Surprisingly, we find that for a broad range of binary particle mixtures composed primarily of pairs of disks and rods with different aspect ratios, but also cylinders combined with spheres and cubes, shape plays only a minor role. Instead, the primary driver of segregation is the volume ratio of the two species.

Flowing granular materials segregate due to differences in particle size (driven by percolation) and density (driven by buoyancy). Modelling the segregation of mixtures of large/heavy particles and small/light particles is challenging due to the opposing effects of the two segregation mechanisms. Using discrete element method (DEM) simulations of combined size and density segregation we show that the segregation velocity is well described by a model that depends linearly on the local shear rate and quadratically on the species concentration for free surface flows. Concentration profiles predicted by incorporating this segregation velocity model into a continuum advection–diffusion–segregation transport model match DEM simulation results well for a wide range of particle size and density ratios. Most surprisingly, the DEM simulations and the segregation velocity model both show that the segregation direction for a range of size and density ratios depends on the local species concentration. This leads to a methodology to determine the combination of particle size ratio, density ratio and particle concentration for which a bidisperse mixture will not segregate.

Particle segregation is common in natural and industrial processes involving flowing granular materials. Complex, and seemingly contradictory, segregation phenomena have been observed for different boundary conditions and forcing. Using discrete element method simulations, we show that segregation of a single particle intruder can be described in a unified manner across different flow configurations. A scaling relation for the net segregation force is obtained by measuring forces on an intruder particle in controlled-velocity flows where gravity and flow kinematics are varied independently. The scaling law consists of two additive terms: a buoyancy-like gravity-induced pressure gradient term and a shear rate gradient term, both of which depend on the particle size ratio. The shear rate gradient term reflects a kinematics-driven mechanism whereby larger (smaller) intruders are pushed toward higher (lower) shear rate regions. The scaling is validated, without refitting, in wall-driven flows, inclined wall-driven flows, vertical silo flows, and free-surface flows down inclines. Comparing the segregation force with the intruder weight results in predictions of the segregation direction that match experimental and computational results for various flow configurations.

In dense flowing bidisperse particle mixtures varying in size or density alone, large particles rise (driven by percolation) and heavy particles sink (driven by buoyancy). When the two particle species differ from each other in both size and density, the two segregation mechanisms either enhance (large/light and small/heavy) or oppose (large/heavy and small/light) each other. In the latter case, an equilibrium condition exists in which the two mechanisms balance and the particles no longer segregate. This leads to a methodology to design non-segregating particle mixtures by specifying particle size ratio, density ratio, and mixture concentration to achieve the equilibrium condition. Using DEM simulations of quasi-2D bounded heap flow, we show that segregation is significantly reduced for particle mixtures near the equilibrium condition. In addition, the rise-sink transition for a range of particle size and density ratios matches the predictions of the combined size and density segregation model.

Flow-driven segregation of non-spherical-particle mixtures can be predicted from the particle volume ratio for a broad range of particle shapes and sizes

The force driving intruder particles to rise or sink in dense granular beds is shown to obey a general scaling law that is reminiscent of Archimedes' buoyancy, but with additional dependence on the intruder to bed particle size ratio. Predictions match observations of rise-sink behavior over a wide range of flow conditions in various numerical and experimental flow configurations.

We present a method to estimate the segregation parameter, $S,$ a key input in a continuum transport model of particulate flows. $S$ is determined by minimizing the difference between measured and model-predicted concentration profiles. To validate the approach, we conduct discrete element method simulations of size-bidisperse mixtures in quasi-2D bounded heap flow; the resulting data show that $S$ calculated from concentration profiles is consistent with the directly measured value. The method's accuracy depends critically on the velocity profile during filling, but only weakly on the diffusion coefficient. When the velocity profile is nominally spanwise invariant, the error between estimated and measured $S$ is $10\%$. This method is intended for practical application (described in Part II), so we restrict characterization of the velocity profile to that which can be readily determined experimentally, and explore the sensitivity of concentration profiles to variation of the gap between the sidewalls of the heap., Comment: 13 pages, 11 figures. Correctly formatted version of article published in Powder Technology

While polyamide-based reverse osmosis (RO) and nanofiltration (NF) membranes are widely used for desalination and water purification, the influence of membrane porosity and charge on water transport remains to be fully understood at a molecular level. Here we use molecular dynamics (MD) to build 56 distinct piperazine-based NF membranes models, which cover a membrane density range of 0.78 g cm − 3 to 1.08 g cm − 3 . These membrane models have various charge concentrations, corresponding to a pH range of 4–11. Results indicate that membrane charge is not monotonically correlated with the membrane density or the water transport. Instead, the water transport is mostly determined by the membrane’s physical properties, specifically, the membrane density, with charged membrane end groups and counterions causing swelling of the membrane, which tends to increase flux. Additionally, the diffusion coefficient of water molecules within the membrane is strongly correlated with the membrane density. The diffusivity of water is independent of the transmembrane pressure, even under the large pressures employed in molecular simulations. Thus, the transmembrane pressure biases the direction of the random walk of water molecules through the membrane resulting in a water flux but does not alter their overall mobility within the membrane. These findings shed light on the relationship between membrane properties and water transport for charged membranes, as well as providing new insights into the structure of NF membranes at a molecular scale.

Mixing of fluids and mixing of solids are both relatively mature fields. In contrast, mixing in systems where flowing and non-flowing regions coexist remains largely unexplored and little understood. Here we report remarkably persistent mixing and non-mixing regions in a three-dimensional dynamical system where randomness is expected. A spherical shell half-filled with dry non-cohesive particles and periodically rotated about two horizontal axes generates complex structures that vary non-trivially with the rotation angles. They result from the interplay between fluid-like mixing by stretching-and-folding, and solids mixing by cutting-and-shuffling. In the experiments, larger non-mixing regions predicted by a cutting-and-shuffling model alone can persist for a range of protocols despite the presence of stretching-and-folding flows and particle-collision-driven diffusion. By uncovering the synergy of simultaneous fluid and solid mixing, we point the way to a more fundamental understanding of advection driven mixing in materials with coexisting flowing and non-flowing regions., Understanding mixing in yield stress materials, such as paint and sand, is complicated due to the coexistence of solid-like and fluid-like regimes. Zaman et al. examine mixing in a granular material in three dimensions and find persistent complex non-mixing structures within the chaotic flowing regime.

Mixing by cutting-and-shuffling can be understood and predicted using dynamical systems based tools and techniques. In existing studies, mixing is generated by maps that repeat the same cut-and-shu...

Extending previous linear stability analyses of the instabilities developing in permeable Taylor–Couette–Poiseuille flows where axial and radial throughflows are superimposed on the usual Taylor–Couette flow, we further examine the linear behaviour and expand the analysis to consider the weakly nonlinear behaviour of convective-type instabilities by means of the derivation of the fifth-order amplitude equation together with direct numerical simulations. Special attention is paid to the influence of the radius ratio $\unicode[STIX]{x1D702}=r_{in}/r_{out}$, and particularly to wide gaps (small $\unicode[STIX]{x1D702}$) and how they magnify the effects of the radial flow. The instabilities take the form of pairs of counter-rotating toroidal vortices superseded by helical ones as the axial flow is increased. Increasing the radial inflow draws these vortices near the inner cylinder, where they shrink relative to the annular gap, when this gap is wide. Strong axial and radial flows in a narrow annular gap lead to a very large azimuthal wavenumber with steeply sloped helical vortices. Strong radial outflow in a wide annular gap results in very large helical vortices. The analytical and numerical saturated vortices match quite well. In addition, radial inflows or outflows can turn the usually supercritical bifurcation from laminar to vortical flow into a subcritical one. The radial flow above which this change occurs decreases as the radius ratio $\unicode[STIX]{x1D702}$ decreases. A practical motivation for this weakly nonlinear analysis is found in modelling dynamic filtration devices, which rely on vortical instabilities to reduce the processes of accumulation on their membranes.

Unsteady flows of granular media are ubiquitous yet remain largely unexplored. In this research, we apply unsteady flows to strongly segregating granular materials to control the segregation pattern and enhance overall mixing. Size-bidisperse granular mixtures with large size ratios flowing onto a quasi-2D bounded heap form stratified layers of large and small particles when the flow rate is modulated. These layers exhibit better average mixing than the segregated patterns generated by steady feed rates. The mechanisms of layer formation under modulated flow differ from those for spontaneous stratification and are related to changes in the composition of the flowing layer at different stages in each feed cycle. The thickness and length of the stratified layers can be controlled by changing the feed rates and feed cycle durations, which is potentially useful for reducing segregation in industrial processes.

Heap formation occurs widely in industry when particulate materials are filled into vessels such as hoppers, silos, bulk bags and bins. If the bulk materials are composed of particles with different size, shape, material density or surface properties, they tend to segregate spatially during heap formation. In most industrial settings, heap segregation is unwanted, because the inhomogeneity of the composition is detrimental for product quality and also can potentially impact subsequential processes. In this paper, we review previous efforts on understanding physical mechanisms and developing predictive models for heap segregation in the past a few decades. We also aim to provide a general picture of what have been learned from previous experimental, computational, and theoretical work and what are still missing for understanding and predicting heap segregation.

Mixing by cutting-and-shuffling can be mathematically described by the dynamics of piecewise isometries (PWIs), higher dimensional analogs of one-dimensional interval exchange transformations. In a two-dimensional domain under a PWI, the exceptional set, $\bar{E}$, which is created by the accumulation of cutting lines (the union of all iterates of cutting lines and all points that pass arbitrarily close to a cutting line), defines where mixing is possible but not guaranteed. There is structure within $\bar{E}$ that directly influences the mixing potential of the PWI. Here we provide new computational and analytical formalisms for examining this structure by way of measuring the density and connectivity of $\varepsilon$-fattened cutting lines that form an approximation of $\bar{E}$. For the example of a PWI on a hemispherical shell studied here, this approach reveals the subtle mixing behaviors and barriers to mixing formed by invariant ergodic subsets (confined orbits) within the fractal structure of the exceptional set. Some PWIs on the shell have provably non-ergodic exceptional sets, which prevent mixing, while others have potentially ergodic exceptional sets where mixing is possible since ergodic exceptional sets have uniform cutting line density. For these latter exceptional sets, we show the connectivity of orbits in the PWI map through direct examination of orbit position and shape and through a two-dimensional return plot to explain the necessity of orbit connectivity for mixing., Comment: 22 pages, 12 figures

Individual constituent balance equations are often used to derive expressions for species-specific segregation velocities in flows of dense granular mixtures. We propose a semiempirical expression for the interspecies momentum exchange in density-bidisperse granular flows as an extension of ideas from kinetic theory and compare it to a previous viscous drag approach that is analogous to particles settling in a fluid. The proposed model expands the range of the granular kinetic theory from short-duration binary collisions to the multiple enduring contacts characteristic of dense shear flows and incorporates the effects of particle friction, concentration ratio, and local flow conditions. The segregation velocities derived from the momentum balance equation using both interspecies drag models match the downward and upward segregation velocities of heavy and light particles obtained from DEM simulations through the flowing layer depth for different density ratios and constituent concentrations in confined shear flows. Predictions of the kinetic theory inspired approach are additionally compared to results from free surface heap flow simulations, and, again, a close match is observed.

Size-driven particle segregation can occur when an object such as a blade moves through an otherwise static bed of granular material. Here we use discrete element method (DEM) simulations to study segregation resulting from a subsurface blade moving through a bed of size-bidisperse spherical particles. Segregation increases with each pass of the blade until a surface layer of mostly large particles forms above a small-particle layer adjacent to the bottom wall. The rate of segregation decreases with each pass so that the degree of segregation asymptotically approaches its maximum value, and the number of passes to reach a steady segregation state increases as the bed depth is increased or the blade height decreased. In shallow beds, the characteristic number of passes for segregation, $\ensuremath{\tau}$, scales with the inverse of the granular inertial number, $I$. In deep beds with small blade heights, the effect of the blade is more localized to its immediate vicinity, resulting in many more passes of the blade to reach a steady segregation state, and a corresponding deviation from the shallow bed scaling of $\ensuremath{\tau}$ with ${I}^{\ensuremath{-}1}$.

We examine the dynamics of cutting-and-shuffling a hemispherical shell driven by alternate rotation about two horizontal axes using the framework of piecewise isometry (PWI) theory. Previous restrictions on how the domain is cut-and-shuffled are relaxed to allow for non-orthogonal rotation axes, adding a new degree of freedom to the PWI. A new computational method for efficiently executing the cutting-and-shuffling using parallel processing allows for extensive parameter sweeps and investigations of mixing protocols that produce a low degree of mixing. Non-orthogonal rotation axes break some of the symmetries that produce poor mixing with orthogonal axes and increase the overall degree of mixing on average. Arnold tongues arising from rational ratios of rotation angles and their intersections, as in the orthogonal axes case, are responsible for many protocols with low degrees of mixing in the non-orthogonal-axes parameter space. Arnold tongue intersections along a fundamental symmetry plane of the system reveal a new and unexpected class of protocols whose dynamics are periodic, with exceptional sets forming polygonal tilings of the hemispherical shell., Comment: 22 pages, 15 figures; added additional inscribed solid that was missed

We explore the segregation of two size-polydisperse particle species with overlapping size distributions using an experimentally validated continuum segregation model and discrete element method simulations. The continuum approach is extended to successfully model segregation for two species with overlapping size distributions. Nevertheless, the impact of species size dispersity on species segregation is weak. Consequently, the local species concentration can be accurately modeled as a mixture of two size-monodisperse species, even if the distributions of the two species overlap. However, the local mean particle size can be influenced by size dispersity in some regions of the flow, particularly for broad size distributions. The segregation length scale, which characterizes the propensity of the two species to segregate, can be measured for mixtures of two polydisperse species as well, and closely follows the value associated with the mean diameters of the two species.

Particle segregation in geophysical and industrial granular flows is typically driven by gravity and shear. While gravity-induced segregation is relatively well understood, shear-induced segregation is not. In particular, what controls segregation in the absence of gravity and the interplay between shearand gravity-driven segregation remain unclear. Here, we explore the shear-induced segregation force on an intruder particle in controlled-velocity granular flows where the shear profile is systematically varied. The shear-induced segregation force is found to be proportional to the shear rate gradient, which effectively pushes the large intruder from lower to higher shear rate regions. A scaling law is developed for the segregation force that is accurate over a wide range of overburden pressures and shear rates, and hence inertial numbers.

Segregation of polydisperse granular materials occurs in many natural and industrial settings, but general theoretical modelling approaches with predictive power have been lacking. Here we describe a model capable of accurately predicting segregation for both discrete and continuous particle size distributions based on a generalized expression for the percolation velocity. The predictions of the model depend on the kinematics of the flow and other physical parameters such as the diffusion coefficient and the percolation length scale, quantities that can be determined directly from experiment, simulation or theory and that are not arbitrarily adjustable. The model is applied to heap and chute flow, and the resulting predictions are consistent with experimentally validated discrete element method (DEM) simulations. Several different continuous particle size distributions are considered to demonstrate the broad applicability of the approach.

Free surface flows of granular materials in bounded axisymmetric geometries such as a cylindrical silo are poorly understood. In particular, a detailed description of the local three-dimensional velocity field and predictive models for segregation are lacking. Here, the details of the kinematics of flow in a rising conical heap formed by a centrally fed mixture of spherical particles are investigated using Discrete Element Method (DEM) simulations in a wedge-shaped geometry with periodic azimuthal boundaries. The dependence of the streamwise and surface normal velocity components on the inlet flow rate and silo radius for size-bidisperse and monodisperse mixtures of glass spheres is characterized. Compared to a wedge with frictional sidewalls, the flowing layer is much thicker on average in the axisymmetric case. Using the scalings for kinematics obtained from the DEM simulations, a modified continuum advection-diffusion-segregation model accurately predicts steady-state segregation for axisymmetric conical heap flows of size-bidisperse mixtures.

Segregation patterns of size-bidisperse particle mixtures in a fully three-dimensional flow produced by alternately rotating a spherical tumbler about two perpendicular axes are studied over a range of particle sizes and volume ratios using both experiments and a continuum model. Pattern formation results from the interaction of size segregation with chaotic regions and nonmixing islands of the flow. Specifically, large particles in the flowing surface layer are preferentially deposited in nonmixing islands despite the effects of collisional diffusion and chaotic transport. The protocol-dependent structure of the unstable manifolds of the flow surrounding the nonmixing islands provides further insight into why certain segregation patterns are more robust than others.

The effect of confining pressure (overburden) on segregation of granular material is studied in discrete element method (DEM) simulations of horizontal planar shear flow. To mitigate changes to the shear rate due to the changing overburden, a linear with depth variation in the streamwise velocity component is imposed using a simple feedback scheme. Under these conditions, both the rate of segregation and the ultimate degree of segregation in size bidisperse and density bidisperse granular flows decrease with increasing overburden pressure and scale with the overburden pressure normalized by the lithostatic pressure of the particle bed. At overburdens greater than approximately 20 times the lithostatic pressure at the bottom of the bed, the density segregation rate is zero while the size segregation rate is small but nonzero, suggesting that different physical mechanisms drive the two types of segregation. The segregation rate scales close to linearly with the inertial number for both size bidisperse and density bidisperse mixtures under various flow conditions, leading to a proposed pressure dependence term for existing segregation velocity correlations. Surprisingly, particle stiffness has only a minor effect on segregation, although it significantly affects the packing density.

We experimentally study the transition from steady flow to unsteady flow in a quasi-2D granular heap when small amounts of water are added to monodisperse glass spheres. Particles flow uniformly down both sides of the heap for low water content, but unsteady flow occurs as the water content increases. The unsteady flow mode consists of a non-depositing downslope avalanche and an upslope propagating granular jump. The transition from steady to unsteady flow occurs when the slope exceeds a critical angle as a result of water-induced cohesion. Under unsteady flow conditions, the deposited heap consists of loosely packed and densely packed layers, the formation of which is closely related to the unsteady flow dynamics., Comment: 10 pages, 8 figures

We characterize the local concentration dependence of segregation velocity and segregation flux in both size and density bidisperse gravity-driven free-surface granular flows as a function of the particle size ratio and density ratio, respectively, using discrete element method (DEM) simulations. For a range of particle size ratios and inlet volume flow rates in size-bidisperse flows, the maximum segregation flux occurs at a small particle concentration less than 0.5, which decreases with increasing particle size ratio. The segregation flux increases up to a size ratio of 2.4 but plateaus from there to a size ratio of 3. In density bidisperse flows, the segregation flux is greatest at a heavy particle concentration less than 0.5 which decreases with increasing particle density ratio. The segregation flux increases with increasing density ratio for the extent of density ratios studied, up to 10. We further demonstrate that the simulation results for size driven segregation are in accord with the predictions of the kinetic sieving segregation model of Savage and Lun.

Discrete element method simulations of confined bidisperse granular shear flows elucidate the balance between diffusion and segregation that can lead to either mixed or segregated states, depending on confining pressure. Results indicate that the collisional diffusion is essentially independent of overburden pressure. Because the rate of segregation diminishes with overburden pressure, the tendency for particles to segregate weakens relative to the re-mixing of particles due to collisional diffusion as the overburden pressure increases. Using a continuum approach that includes a pressure dependent segregation velocity and a pressure independent diffusion coefficient, the interplay between diffusion and segregation is accurately predicted for both size and density bidisperse mixtures over a wide range of flow conditions when compared to simulation results. Additional simulations with initially segregated conditions demonstrate that applying a high enough overburden pressure can suppress segregation to the point that collisional diffusion mixes the segregated particles., Comment: 17 pages double spaced, 6 figures, 1 table

In fluids, diffusive transport and fluid parameters are determined by the thermally driven movement of molecules, whereas in granular media, thermal effects play no role in transport. Despite this fundamental difference, mixing in fluids and mixing and segregation in granular solids share many similarities. The advection–diffusion equation formalism unites the two systems and can be used in both cases to predict and understand how and when mixing occurs, through the use of strategic simplifications. We illustrate this connection with a fluid mixing example (the rotated potential mixer) and a granular segregation example (the segregation of a bidisperse mixture flowing in a chute).

Quantitatively predicting segregation of size-disperse granular materials is of potential value in many industrial applications. We consider granular segregation of size-bidisperse particles in quasi-2D bounded heaps, a canonical granular flow, using an advection-diffusion transport equation with an additional term to account for particle segregation. The equation is characterized by two dimensionless parameters that are functions of control parameters (flow rate, system size, and particle sizes) and kinematic parameters (flowing layer depth, diffusion coefficient, and percolation length scale). As the kinematic parameters are usually difficult to measure in practice, their dependence on the control parameters is determined directly from discrete element method simulations. Using these relationships, it is possible to determine which values of the control parameters result in a mixed or segregated heap. The approach used here is broadly applicable to a wide range of other flow geometries and particle systems. © 2015 American Institute of Chemical Engineers AIChE J, 61: 1524–1534, 2015