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Timetable (GPFW07)

IMA Conference on Dense Granular Flows

Monday 5th January 2009 to Friday 9th January 2009

Monday 5th January 2009
10:20 to 11:00 Plenary: investigating the intermediate granular flow regime
Hydrodynamic modelling of gas-solid systems and granular materials traditionally focus on two flow regimes: slow dense flow, usually referred to as quasi-static flow, and fast dilute flow, usually referred to as rapid flow. In the first regime, dry particles interact via enduring contacts and the flow is often described through the Coulomb frictional law. In the second regime, dry particles interact via fast collisional contacts and the flow displays mainly viscous stresses usually described according to the kinetic theory of granular flow. Between the rapid flow and the quasi-static flow, a less understood intermediate flow might exist; in the window of intermediate flow regime, strong evidences support the argument that flow induced fluctuations and enduring contacts coexist. Researchers have attempted three different modelling approaches of the intermediate regime: (1) the kinetic-frictional approach which uses an ad hoc patching together of the stress from the two limiting regimes at a specific solid fraction (e.g., Johnson and Jackson, J. of Fluid Mech., 210, 501, 1978); (2) the switching from one regime to another using different solid stress formulations (e.g., Makkawi and Ocone, Kona, 23, 49, 2005); (3) the fluid mechanic approach allowing for smooth transition from one regime to another using a unified model (e.g., Tardos et al., Powder Tech., 131, 23, 2003). In this talk, a one-dimensional fully developed gas-solid flow model for horizontal flow is used to review the various treatments of the solid stresses; the sensitivity of the flow predictions to the frictional stress is assessed. In addition, the boundaries between rapid-intermediate-dense flow regimes are established, based on the dimensionless shear rate (ë) and a modified Reynolds number (Re).
11:45 to 12:10 The flow of fluidised particles
A gas flow through an ensemble of relatively heavy particles has a significant effect on their collective properties. When the vertical gas flow is sufficiently strong, it is possible for the weight of the particles to be supported fully and the granular material is fluidised. At this transition, there is a reduction in frictional effects generated by collisions and longer sustained contacts between the particles as the drag between the solids and the gas becomes the dominant physical process. Although there has been much recent progress in understanding aspects of dry granular flows down slopes, little attention has been paid to the interactions between the grains and the interstitial gas, but in the scenario investigated by this study, this process dominates the dynamics. There are many applications in which the interaction between the phases is central to the bulk motion, ranging from pneumatic conveying of industrial materials to the runout of volcanic particulate flows, in which the fluidising process substantially reduces the drag and increases the mobility of the material. In this work we present new experimental results for the propagation of relatively fine glass particles along a sloping, rigid, but porous surface, through which gas flows and fluidises the particles. We introduce the particles at a constant rate and measure their propagation through the apparatus. Typically they flow as a relatively thin, but highly mobile layer. For a range of angles of inclination, we measure the bulk characteristics of the motion, such as the temporal development of the depth of the layer and the flow speed, and from high-speed videography and particle tracking, we determine the velocity profiles of the grains. These feature slip at the base, a region of shear and then a plug-flow throughout the rest of the layer. We analyse these results in terms of a two-phase mathematical model. A key component of this model is the inter-phase drag that leads to the support of the excess weight of the granular layer. However the results indicate that there is also a drag that balances the gravitationally-related force driving the motion parallel to the underlying boundary. We present a mathematical model for this force, which is due to granular interactions, and demonstrate that it may account for both the fully-developed velocity profiles and the temporally evolving behaviour measured in the experiments.
12:10 to 12:35 Bed-load transport by laminar shearing flows
When beds constituted of sediment particles are submitted to shearing flows, the particles at the surface of the stream-bed are able to move as soon as hydrodynamic forces acting on them exceed a fraction of their apparent weight. Bed-load refers to the sediment in transport that is carried by intermittent contact with the stream-bed by rolling, sliding, and bouncing. This situation occurs in a wide variety of natural phenomena, such as sediment transport in rivers, and in industrial processes, such as problems due to hydrate formation in pipeline that are encountered in oil production and granular transport in food or pharmaceutical industries (Ouriemi, 2007). We propose to use a two-phase model having a Newtonian rheology for the fluid phase and Coulomb friction for the particulate phase. The model equations have been solved numerically in one-dimension and analytically in asymptotic cases. The model results have been compared with experimental data for a bed composed of spherical particles in laminar pipe flows (Ouriemi et al., 2008a). This continuum approach is able to provide an onset of motion for the particle phase and a description of the flow of the mobile granular layer. At some distance from threshold, we obtain the simpler analytical result of the particle flux varying cubically with the Shields number. This algebraic formulation seems quite satisfactory for describing experimental observations of bed-load transport in pipe flows. Based on this particle flux a simple linear stability analysis has been performed to predict the threshold for dune formation. This basic analysis accounts reasonable well for the experimental observations for the small dune formation (Ouriemi et al., 2008b). We are working on the implementation of the two-phase equations in a three dimensional numerical model to study the bed instability. This model will be used for better understanding of the coupling between the granular media and the fluid in terms of the bed instability.
12:35 to 13:00 MA Koenders & C Koenders ([Kingston])
Agitated medium dense slurries in a dead-end filtration geometry
In a dead-end filtration set-up a constant mean fluid flow is maintained. The slurry is agitated by vibrating the permeable septum at the bottom of the geometry. The agitation causes a particle pressure which expands the slurry, thus keeping the septum clear. Experimentally it is shown that a steady-state solidosity profile is created. To describe this experiment the McTigue-Jenkins granular temperature theory is used. This is the only particle-fluid model that takes explicit account of fluctuations and therefore can handle situations in which there is no mean overall motion of the slurry. The many parameters of the theory are derived from a cell model in a mean-field approximation, which is validated/justified to some extent by verification against viscosity data. The model employs the lubrication interaction for rough particles. The set-up has many practical applications, but it is also beneficial for the understanding of medium dense slurry flow (by medium dense it is meant that no enduring contacts form in the slurry, so say, solidosity less than 0.6 and greater than 0.3).
14:00 to 14:40 Plenary: Multiphase flow enigmas in inhalation delivery
The delivery of drugs to the lungs relies on multiphase flows, that are in want of being understood. Inhalation products, those intended for the treatment of Asthma for instance, are based on complex mixtures of cohesive fine particles (typically 1 to 5 ìm). The interactions between the particles, and as a consequence their flow properties, are poorly understood and have rarely been modelised. 3 examples of inhalers will be reviewed to illustrate the problems encountered: -a pMDI (pressure metered dose inhaler), a product based on a drug suspension in a pressurised non aqueous propellant, -a passive DPI (dry powder inhaler), based on a particle mixture whose aerosolisation is triggered by the patient inhalation, -an active DPI, similar to a passive DPI, although in this case aerosolisation is provided by the vibration of a piezo-electric crystal, and trough what is known as synthetic jetting. Sedimentation/creaming profiles in pMDIs give rise to a rather odd phenomenon of surface reflux. In passive DPIs the difficulty is to understand the point of fluidisation of the powder bed in relation to particulates interactions. In active DPIs the issue is to understand how the energy for dispersing the powder, in this case the energy transmitted from a piezo-electric crystal, leads to aerosolisation. The 3 issues highlighted will serve as examples to highlight the need of understanding powder flows in the pharmaceutical formulation industry. Some attempts have been made to elucidate interaction mechanisms, mostly through measurements via AFM (atomic force microscopy) and powder rheology, but as will be shown there is a need to tie up what could be seen as coincidental measurements with deterministic models.
14:40 to 15:05 Modelling oosator performance case study of solving In dense granular flow
generally similar principles) are widely used for dosing products (as capsules, tablets, so on) that are delivered to the customer in powder form. It is a big challenge for the pharmaceutical industry to deal with dose weight control (in the order of doses between 20 to 1000mg) of the products with the wide range of existing powder blends. Producing a model of this process, to allow exploration of the various factors which have been seen to have an effect, is an interesting case study in dense granular flow. Currently a work-in-progress, it is being approached using an analytically-based finite-difference technique; the constitutive inputs include a combination of various well known powder mechanics phenomenological characterisation techniques [1] [2], a proven analytical model (the “Differential Slice” method by Janssen [3]), together with the introduction of some novel phenomenological characterisation techniques regarding powder response to stress and air permeability effects. To validate and calibrate the model, the Wolfson Centre has built a “Dosator Test-Rig” which measures the forces from the powder against the dosator at filling and ejection stages of the process for a range of powders. This paper gives an example of how some problems in dense granular flow can be approached using a mix of established analytical models, augmented by some novel extensions and well-chosen novel phenomenological characterisation techniques at a multiple-particle level. Acknowledgements This research project is financially supported, technically assisted and material provided by GSK.
15:05 to 15:30 Power-law rheology and the growing correlation length at the jamming transition
Rheology of dilute granular flow, which is represented in the form of Bagnold’s scaling, is well understood in the sense that we can predict stationary flow profiles in a quantitative manner. Quite contrastingly, little is known about dense granular flow, in which the stiffness of particles comes into play. One of the main difficulties is the occurrence of shear–banding, which is generally accompanied by large temporal fluctuations in the shear stress. Our aim is to rule out such spatiotemporal heterogeneities to obtain a simple phenomenology for stationary and uniform flow and to clarify the underlying physics. Along the line of thought, we adopt one of the simplest models for granular matter: inelastic but frictionless spheres, in which we can carefully realize the spatiotemporal homogeneity. By massive numerical simulations, we obtain rheology in a wide range of shear rate and density. It is found that the system acquires the yield stress above a certain density, which is known as the random close packing (approximately 0.64 in the volume density). The acquisition of the yield stress is recently referred to as the jamming transition, which describes solidification/fluidization of granular media. This transition involves a critical point, at which the relaxation time and the correlation length diverges. We investigate these length and time scales via correlation functions and finite-size scaling to find that they diverge as the shear rate tends to zero. As a result, just like conventional critical phenomena, a scaling law which describes dense granular rheology exists.
16:10 to 16:35 Velocity and concentration profiles measurements in concentrated particle suspensions
An optical visualisation apparatus has been designed to measure the particle-velocity and solid-concentration profiles of highly concentrated coarse-particle suspensions in a wide-gap Couette rheometer. The main objective is to investigate the frictional-viscous transition, a phenomenon that has been already be reported in recent papers [6, 3, 1, 4], but still remains partially understood. A related issue is the Couette problem, which underpine the rheometrical treatment for viscometric flows in coaxial-cylinder rheometers; we compare shear-rate computations obtained by solving the Couette problem (bulk estimate) and by differentiating the velocity profile (local measurement). Ancey [1] showed that for concentrated particle suspensions there is a transition from a frictional to a viscous behaviour that occurs at a given critical shear rate, which depends a great deal on the particle diameter. He suggested that particle lubrication is the key mechanism responsible for this transition: at sufficiently high shear rates, fluid inertia increases; part of the fluid can then break and lubricate contacts between particles, which leads to a "fluidisation of the material". Another interpretation has been suggested by geophysicists [5]: a concentrated suspension of coarse non-buoyant particles behaves like a soil and according to Coulomb theory, shear strength drops to zero when pore fluid pressure is sufficiently high to balance paraticle buoyancy forces, which results in a "liquefactions" of the material. To gain insight into this delicate problem, we are conducting experiments, where particle buoyancy can be controlled. By adjusting the fluid refraction index, we can make also our suspensions transparent and use non-invasive techniques (Fluorescent Particle Image Velocimetry) to probe both velocity and density profiles within the suspension. We will present our preliminary results obtained with a PMMA-particle suspension. Another interesting aspect of this experimental setup concerns flow curve derivation. For wide-gap viscometers and complex fluids, the flow curve must be computed by solving the Couette inverse problem [7, 2]. An alternative way of obtaining the flow curve is to measure the velocity profile across the gap, then differentiate it to derive the local shear rate. The locally derived measurements (shear rate, concentration) can finally be used as benchmark data to test the various techniques developed for solving the Couette inverse problem (e.g., Tikhonov regularisation, spline interpolation, wavelet-vaguelette decomposition). We will present the results of this benchmark.
Tuesday 6th January 2009
09:00 to 09:40 B Behringer ([Duke])
Plenary: statistics of dense granular materials
Dense granular materials present a number of interesting challenges. They are many-body systems with dissipative interactions. As suggested by Edwards et al., one would like to have a statistical approach analogous to Boltzmann statistics for energy-conserving systems. But the dissipative character of dense granular materials challenges conventional notions. New approaches require a careful examination, and suggest a number of experiments. This talk will focus on recent experiments that seek to characterize the statistics of dense granular systems and to test recent models. In particular, we have carried out experiments to determine the microscopic distributions of contact forces, the nature of jamming, the effect of stress anisotropy, the role of affine and non-affine motion in sheared granular systems, the effect of particle rotation, and the importance of particle shape. Many of these experiments use photoelastic particles. This allows the tracking of particle motion, and perhaps most importantly, the determination of the contact forces by the solution of a nonlinear inverse problem. Our results are consistent with predictions from the force ensemble approach, but much more needs to be done.
09:40 to 10:05 The modelling of dense, slow granular flow using a new plasticity model
Flows of dense granular materials exhibit material behaviour which may be considered as either solid-like or fluid-like. Such flows have proven to be remarkably resistant to successful modelling. There is an emphasis by researchers on discrete models such as DEM to provide numerical simulations. It is, perhaps, natural that such manifestly discrete systems should be modelled in a discrete manner. However, as the magnitude of the macroscopic dimensions of the system increase relative to the grain size it becomes a desirable goal to implement a continuum model. The lack of an adequate continuum model for such flows is a major reason for the lack of understanding of this regime. The two major models coming from a solid-like formulation, namely the rigid- or elasto-plastic model incorporating a non-associated flow rule and the rigid-plastic double shearing model are both ill-posed in the sense that solutions to an initial value problem do not depend continuously on the initial data. In this talk we show how to combine certain aspects of the double-shearing model with the plastic potential model in the context of Cosserat, reduced-Cosserat and classical continua in way which resolves the issue of ill-posedness. Applications of the model are considered and it is demonstrated how the model may be used to solve problem of dense granular flow.
10:05 to 10:30 Axisymmetric granular collapse: a transient 3D flow test of visco-plasticity
A viscoplastic continuum theory has recently been proposed to model dense, cohesion-less granular flows (Jop et al. Nature, 441, 727, 2006). We confront this theory for the first time with a transient, 3-dimensional flow situation - the simple collapse of a cylinder of granular matter onto a horizontal plane - by extracting stress and strain rate tensors directly from soft particle simulations. These simulations faithfully reproduce the different flow regimes and capture the observed scalling laws for the final deposit. Remarkably, the theoretical hypothesis that there is a simple stress-strain rate tensorial relationship does seem to hold over the whole flow during the evolution even close to the rough boundary provided the flow is dense enough. These encouraging results suggest viscoplastic theory is more generally applicable to transient, multi-directional, dense flow and open the way for quantitative predictions in real applications.
10:30 to 10:55 J Crassous & D Beladjine & A Valance ([Rennes])
Percussion of granular material by one grain: experiment and model
We studied experimentally and theoretically the process of erosion of a granular material submitted to the collision of one projected grain. A model experiment consisting of the projection of one bead on a half space filled of a packing of beads has been designed[1]. The effect of the impact is to produce the ejection of many different beads. The kinetics of the ejection process is measured with a rapid camera. The kinetic energy and the orientation of the ejected beads are computed and analysed as a function of the kinetic energy of the impacting bead and of the angle of incidence. We described accurately the full set of experimental observations with a simple model where collisions in the dense packing of spheres are treated as purely binaries[2,3]. In this description, every binary collision splits the momentum in two components, and re-direct their orientations. Momentum is then propagated following a chain of binary collisions. Chains ending with a bead placed on the top surface then allow the ejection of one bead. The successions of collisions and their statistical properties are calculated numerically on 3D packing of spheres. The distribution of the velocities of the ejected spheres, their locus of ejection, and their orientations are well described with this model without the any free parameter. [1] Djaoued Beladjine, Madani Ammi, Luc Oger, and Alexandre Valance Collision process between an incident bead and a three-dimensional granular packing Phys. Rev. E 75, 061305 (2007) [2] Jérôme Crassous, Djaoued Beladjine, and Alexandre Valance, Impact of a Projectile on a Granular Medium Described by a Collision Model, Phys. Rev. Lett. 99, 248001 (2007) [3] Alexandre Valance and Jérôme Crassous, in preparation (2008)
10:55 to 11:20 Washboard road
We report laboratory experiments on rippled granular surfaces formed under rolling wheels. Ripples appear above a critical speed and drift slowly in the driving direction. Ripples coarsen as they saturate and exhibit ripple creation and destruction events. All of these effects are captured qualitatively by 2D softparticle simulations in which a disk rolls over smaller disks in a periodic box. The simulations show that compaction and segregation are inessential to the ripple phenomenon. We describe a simplified scaling model which gives some insight into the mechanism of the instability.
11:45 to 12:00 An exact solution for a model of plasticity for powder materials in an un-steady plane strain process
Plastically compressible material obeying a pressure-dependent yield condition and its associated flow rule is confined between two plates which are inclined at an angle 2á and which intersect in a line. The line of intersection is a hinge and the angle á slowly decreases with increasing time. The material undergoes a plane strain deformation in planes perpendicular to the hinged line of intersection. There is no material flux at the line of intersection. Velocity and stress boundary conditions are prescribed on the plates, including the maximum friction law. The maximum friction law postulates that the friction stress is equal to the maximum possible shear stress supported by the material. The porosity is uniformly distributed at the initial instant. An exact analytic solution (the solution is reduced to subsequent calculation of several ordinary integrals) to the problem formulated is obtained by an inverse method. The qualitativebehaviour of the solution depends on the angle between the plates. In particular, a rigid zone can occur in the vicinity of the friction surface and the porosity can vanish after a certain amount of deformation. The solution is compared with the corresponding solutions based on the classical theory of rigid perfectly plastic solids and several models of incompressible pressure-dependent materials. In particular, in contrast to other solutions, the regime of sticking always occurs at the friction surface.
12:00 to 12:15 Rapid granular chute flow - multiple steady flow solutions and their linear stability
The free-surface flow of highly agitated particles on an inclined chute is analysed using a continuum model that adopts the kinetic theory of rapid granular flows. Steady, fully developed profiles of the concentration of particle, the velocity of the flow down the slope and the granular temperature, which measures the agitation of the system, may be found as solutions to the governing equations. These are calculated using a Chebyshev pseudospectral method, which exploits the asymptotic form of the solutions at large heights to obtain highly accurate approximations. The character of the steady solutions is determined by a relatively small number of dimensionless parameters, which includes the slope inclination, coefficients of restitution, and roughness of the chute surface. An asymptotic analysis appropriate to high temperature flows has been developed to obtain the boundaries in parameter space where steady solutions exist. The pseudospectral approximation lends itself to parametric continuation, which allows us to efficiently track the form of the solutions as we vary the controlling parameters. In particular, we investigate the depth of steady flows, here defined as the centre of mass, as the volume flux of material is varied and find that, in some parameter regimes, three flow depths occur for a given volume flux of material. We consider the linear stability of the multiple solutions to small perturbations in both the cross-slope and downslope directions. We find regions in which flows are unstable to small perturbations, and show there is no correlation between the region of instability and the existence of density inversions in the steady flow profile.
12:15 to 12:30 A comparative study of models for incompressible granular materials
Models of pressure-dependent plasticity are used for describing deformation of granular materials, among other materials. In contrast to classical metal plasticity, there is no commonly accepted model of pressure-dependent plasticity, though most of such models reduce to classical plasticity at a specific set of parameters. Nevertheless, the solution behaviour can essentially depend on the pressure-dependent model chosen, independently of how close the input parameters are to these specific values. Therefore, it is of interest to carry out a systematic study for understanding the difference in solution behaviour. First, several analytic solutions based on different models of pressure-dependent plasticity (the coaxial model, the double-shearing model, and the double-slip and rotation model) are compared to each other and to the corresponding solutions based on the classical metal plasticity. It is concluded that the qualitative difference in solution behaviour can occur in the vicinity of maximum friction surfaces. One typical difference occurs in flows where the classical rigid plastic solutions are singular (some strain rate components approach infinity in the vicinity of maximum friction surfaces). The same qualitative behaviour demonstrates solutions based on the double-shearing model and the double-slip and rotation model as well, but not on the coaxial model. The other difference is related to the transition between the regimes of sliding and sticking at the friction surface. In this case, the same qualitative behaviour is revealed in solutions based on the double-slip and rotation model only. Using these mathematical arguments, it is concluded that the double-slip and rotation model leads to a more adequate description of material behaviour. It is a matter of special importance for materials with a large value of internal friction. In this case the pressure-dependency of material is very low and it is necessary that in applications both the classical rigid plastic model and the model of pressure-dependent plasticity give acceptable results for the same material. Obviously, the area of applicability of this or that model is rather conditional on the specific area of application. Therefore, it is advantageous that the solutions based on both models show the same qualitative behaviour everywhere.
12:30 to 12:45 D Takagi & JN McElwaine & HE Huppert ([Cambridge])
Granular flows on unconfined slopes
A variety of features are observed when sand is supplied steadily on a rough unconfined slope. Over a range of intermediate slope angles, a densely packed layer of finite width and critically shallow depth develops and flows downstream. Isolated avalanches form periodically for input fluxes below a sufficiently small value. The experimental results will be presented and explained by theoretical ideas.
12:45 to 13:00 Triggering of submarine granular avalanches
Under the sea, granular avalanches and landslides exist, in which the interstitial fluid plays a major role. A good description of such events requires understanding the coupling between a granular flow and a fluid. The goal of this work is to study this coupling in the case of the initiation of immersed granular avalanches down rough inclined planes. More precisely, we study how the initial volume fraction of a granular layer strongly influences how it starts to flow when suddenly inclined. The experiment consists in preparing a uniform layer of glass beads (160µm in diameter) in a long box full of liquid by creating a suspension and by letting the particles sediment. The initial volume fraction of the layer can be precisely controlled by imposing successive taps on the box. The box is then suddenly inclined from horizontal. We then measure the time evolution of the free surface velocity of the layer of the pore pressure below the layer and of the volume fraction in the central part of the box. The avalanches can be classified in tow categories. In the loose cases a rapid acceleration first takes place and a transient velocity higher than the final velocity can be observed. In the dense case, a first stage characterised by a very low velocity is observed, before the velocity increases and reaches the steady state value. The higher the volume fraction, the longer is the triggering time. The qualitative explanation of the phenomenon is given by the coupling between the granular skeleton and the pore pressure. When the avalanche starts, the granular medium starts to deform, which induces a variation of volume fraction (a dilatation if the sample is initially dense, a contraction otherwise). This change in volume fraction causes a variation of the pore pressure, which in turn changes the stresses that apply on the granular skeleton. Parallel to the experiments, a theoretical model is developed. We have used the depth averaged approach developed by Pitman and Lee for thin two phase flows. By introducing in the model a relevant granular rheology, which takes into account the variation of volume fraction observed during the initial deformation, a quantitative agreement with the experimental observations is obtained.
14:00 to 14:25 Y Bertho & A Seguin & P Gondret ([Paris-Sud 11])
Impact in a granular medium
The penetration of a projectile impacting a granular medium is studied experimentally in two configurations: a quasi two-dimensional (2D) case where a cylindrical projectile impacts a packing of spherical grains contained in a Hele-Shaw cell, and a three-dimensional (3D) case where a spherical projectile impacts the grains contained in a cylindrical vessel. In the 2D situation, the projectile trajectory can be extracted by using image analysis and compared to a simple model including a friction law between the projectile and the grains, a viscous dissipation in the bed, and a force from the collisions between the projectile and the granular medium. For the 3D case, a radial confinement is observed to influence strongly the penetration depth of the projectile impacting the grains contained in a finite vessel. The presence of close lateral walls reduces the penetration depth, and the characteristic distance for these wall effects has been found to be of the order of one projectile diameter [1]. In order to have a better understanding of the projectile dynamics during the impact, Particle Image Velocimetry (PIV) analysis is used in the 2D experiment to extract the velocity field of the grains in the bulk as the projectile penetrates the granular medium. Velocity profiles for the grains in the bulk will be discussed both for an unbounded sample and for confined granular medium. [1] A. Seguin, Y. Bertho, and P. Gondret, Phys. Rev. E 78 (2008), 010301(R)
14:25 to 14:50 Shaken, not stirred: granular equilibrium INI 1
14:50 to 15:15 Prediction of the hydrodynamics of slightly wet suspended particles
Suspension of slightly wet particles is a feature of many industrial processes, for example, oil-wet catalyst are undesirably produced in high temperature FCC units, water and steam are injected into fluidized catalyst to control exothermic reaction, which are then regenerated by inert gases in a fluidized bed stripper, solutions are sprayed into fluidized bed for particle coating, wet grains are fluidized in biomass gasification and drying. In a slightly wet and dense suspension, the liquid bridges between particles result in the reduction of interparticle friction (lubrication) by switching the frictional contacts to fluid shear resistance, while collisional contacts dissipate energy in the liquid bridges and particles. Therefore, in order to simulate the flow hydrodynamics in wet suspension, it is required to have constitutive equations capable of predicting the inter-particle interactions in such condition. In this paper we first present experimental evidence on the critical effects of liquid presence on the hydrodynamics of a dense bubbling fluidized bed. In the second part we present a new constitutive equation for the particle stress tensor that allows for the descriptions of the interaction between slightly wet particles, we then incorporated the developed equation into a two-fluid model based on the Kinetic Theory of Granular Flow (KTGF) to study the flow characteristics in a solid-gas suspension at slightly wet condition.
15:15 to 15:40 Modelling and experiments of wet granular flows in rotating drums
In addition to its wide usage in industry as mixers and kilns, rotating drums partially filled with granular materials is a prototypical systems for studying fundamentals of dynamics and mixing in dense granular flows. This study incorporates pendular liquid-bridge forces into a nonlinear soft-sphere discrete element method (DEM) to investigate the behavior of wet granular materials in rotating drums under the rolling flow regime. Simulations with two- and three-dimensional flow fields are compared and the importance of three-dimensional effects are evaluated. Simulated results, including the dynamic angle of repose and the time-averaged velocity profiles at the bed surface and along the bed depth, are validated against a series of experiments based on application of particle image velocimetery (PIV) to a quasi-two dimensional (i.d.120mm×10mm) rotating tumbler. The roles of rolling and sliding frictions are compared, and a model of resistance to rolling is developed for a better prediction of granular velocity. Particularly, the influence of dimensionless liquid bridge volume on behavior of wet granular materials in both the active layer and the plug-flow region is investigated.
16:10 to 16:35 Foam as a soft granular material
When considered as an assembly of bubbles rather than as a structure of films and vertices, a foam or an emulsion is analogous to a granular material. Hard grains experiencing solid friction are replaced by soft bubbles interacting elastically and through viscous forces [1]. Several experiments have recently been applied to two-dimensional flows of bubbles, and they have raised many questions: -what is the atress-strain-strain rate relation? -when does flow localisation occur? -how do disorder, polydispersity and liquid fraction influence rheology and localisation? -when does size segregation occur? These questions are addressed by a growing body of theory and simulations, based on several different models, with fascinating results, not yet fully reconciled. In particular, we have performed numerical simulations based on an elementary discrete element method developed by Durian [2], whose early results have been seen to be misleading. We show that this numerical approach is successful in reproducing the Herschel-Bulkley rheology observed in foams (and many other systems), therefore clarifying the problem of its explanation, and providing a link between local and continuum descriptions. Our simulations also account for the occurrence of localisation when friction along a wall is added (consistent with the predictions of continuum models, and attributable to wall drag), and predict the existence of a dynamic dilatancy effect [3]. Finally, we show how the model can be successfully applied to reproduce bubbles flows in various geometries, such as a rotating drum, a silo (constriction) or around an obstacle. References: [1] D. Weaire, V. Langlois, M. Saadatfar, S. Hutzler. Foam as granular matter, in Lecture notes in complex systems: Granular and complex materials, eds. T. Aste, T. Di Matteo, A. Tordesillas, World Scientific Publishing (2007). [2] D. Durian. Foam mechanics at the bubble scale, Phys. Rev. Lett., 75, 4780 - 4783 (1995). [3] V.J. Langlois, S. Hutzler, D. Weaire. Rheological properties of the soft-disk model for 2D foams, Phys. Rev. E, 78, 021401(2008).
Wednesday 7th January 2009
09:00 to 09:40 TC Halsey ([ExxonMobil Upstream Research Company])
Plenary: motion of frictional grain packings
Friction plays a key role in controlling the rheology of dense granular flows. Counting of the number of constraints vs. the number of variables in frictional grain packings indicates that critical coordination numbers zc=3 (in D=2) and zc=4 (in D=3) are special, in that “gear” states in which all contacts roll without frictional sliding are naively possible at and below these average coordination numbers. We construct an explicit example of such a state in D=2, based on a honeycomb lattice. Solving for the forces in such a state, we conclude that organized shear can exist in this state only on scales l
09:40 to 10:05 EB Pitman ([Buffalo])
Volcanic hazards and uncertainty: granular flow simulations and statistics
This paper presents a methodology for developing a volcanic hazard map. Simulations of granular mass flows using the TITAN2D computational environment, combined with field observations at Colima Volcano in Mexico and the Soufriere Hills Volcano on the island of Montserrat, provide data to be assimilated. Statistical methods are used to provide a quantitative accounting of uncertainty – uncertainty in model parameters, or initial data, or Geographic Information Systems inputs – in the output of simulation. A Bayesian study of the Soufriere Hills eruptions, together with TITAN2D simulations and an analysis of extreme events, gives a probabilistic estimate of future hazard on Montserrat.
10:05 to 10:30 I Luca & YC Tai & CY Kuo (Academia Sinica)
Modelling shallow gravity-driven solid-fluid mixtures over arbitrary topography
The description of geophysical flows over arbitrary terrain is a topic comprising many challenging problems. One of them is the incorporation in the modelling equations of the influences due to the geometry of the basal surface. This issue was revigorated by the relatively recent introduction due to Bouchut and Westdickenberg of curvilinear coordinates, able to properly account for the arbitrariness of the bottom topography and for the shallowness of the avalanche mass. These coordinates are: two surface parameters on the bed surface, and the distance between the point in question and its orthogonal projection onto the topography; the depth of the avalanche is measured along the normal direction to the bottom surface. We use these coordinates and develop depth-averaged models of gravity-driven saturated mixtures of solid grains and pore fluid on an arbitrary rigid basal surface. First, by only specifying the interaction force (as deduced by Schneider and Hutter within a thermodynamic analysis, and which is different from that used by Pitman and Le) and ordering approximations in terms of an aspect ratio between a typical length perpendicular to, and a typical length parallel to the topography, we derive governing equations for the shallow flowing mixture. In doing so, the non-uniformity through the avalanche depth of the constituent velocities and of the solid volume fraction is accounted for by coefficients of Boussinesq type. Then, the rheology of both constituents is specified. One constituent is a Newtonian/non-Newtonian fluid with a viscosity so small, that in the governing equations only the basal shear stress survives, and this is parameterized by a viscous friction law. For the bulk stresses of the solid constituent we propose three models: one reduces to the inviscid fluid, and the other two are topography adapted versions of Iverson-Denlinger and Savage-Hutter models. Common to the proposed models is the assumption that two of the principal directions of the mean stress tensor are aligned with the principal directions of the mean surface stretching. The basal shear stress is parameterized by a Coulomb friction law. With these mixture models at hand, we derived equations governing the motion of two shallow layers on arbitrary rigid basal surface: the layer near the bottom topography is essentially one of the three models of solid-fluid mixture, and the upper layer is the Newtonian/non-Newtonian fluid, present in the mixture layer. The interface between the two layers is a material surface for the solid constituent, and across it the jump conditions of mass and momentum balance equations corresponding to each constituent are envisaged. Numerical results concerning the derived model equations are planned to be obtained in future. Acknowledgements The authors are very grateful to Prof. K. Hutter for his neverending support and advice; the topography-adapted version of the Savage-Hutter model was developed following his
10:30 to 10:55 Dense granular gases: Can they jam?
Dry granular matter is described naturally by a graph. Two grains in repulsive contact in a granular fluid, or just about to collide in a gas, are represented by two vertices connected by an edge. There are no attractive, cohesive forces between grains and contacts between grains are imposed by external forces (gravity or boundaries) or by collisions. In hard dry granular materials with infinite tangential friction, the forces are scalar, geometrical contacts between grains. The elementary excitations are non-slip rotations of a grain another if it is permitted. Then, the granular material flows like a three-dimensional bearing. If not, it is jammed. In a granular gas, collisions (removal of a contact and its replacement by another) are additional elementary excitations. In the graph, the vertices are the grains, and the edges, the contacts represented by the adjacency matrix. Circuits of grains in contact can be even or odd, and the material is essentially discrete. Its physical properties are given by the eigenvalues and eigenvectors of the adjacency matrix. that is by linear algebra of the graph. Thus, whereas the statics is nonlinear, the topological dynamics, and the generic physical behavior of the granular material is a problem of linear algebra, because we have replaced material elements by geometrical objects, in a structure that is a graph. A granular material is therefore a metamaterial, with generic physical behavior given by its structure rather than by the chemistry or hardness of its constituents. In fact, it behaves like soft matter: large, extended deformations with localized critical stresses. The graph structure has the essential elements of discreteness (granularity), odd circuits (''arches'') and disorder. There can be no defect-free, continuous model of granular material, no constitutive equation. The two possible states of (disordered) granular matter, dry fluid or collision-dominated gas, and jammed, rigid but fragile solid are direct results of the topological dynamics of the graph, where the elements responsible for the jammed state are odd circuits, circuits with an odd number of grains in contact. The granular material (n grains) is jammed by the c odd circuits that frustrate the non-slip rotation. The lowest eigenvalue of the dynamical matrix is 4c/n. Odd circuits are the « arches » holding the granular packing together. The odd vorticity, core of odd circuits, forms closed (R-) loops, that are large in disordered granular materials of size L. (In ordered materials, their size is limited by the periodicity of the close-packed structure). The disordered granular material is a fragile, jammed solid stabilized by c odd circuits. The ''order'' parameter 4c/n ~ 1/L is extended over the entire material, and the unjamming transition between fragile solid and dry fluid is a second-order, scaling transition with intermittence (The large R-loops have a small line tension ~ 1/L2 and cannot be driven to shrink).
10:55 to 11:20 A multiscale simulation technique for granular flow
Due to an incomplete picture of the underlying physics, the simulation of dense granular flow remains a difficult computational challenge. Currently, modeling in practical and industrial situations would typically be carried out by using the Discrete-Element Method (DEM), individually simulating particles according to Newton’s Laws. The contact models in these simulations are stiff and require very small timesteps to integrate accurately, meaning that even relatively small problems require days or weeks to run on a parallel computer [2]. These brute-force approaches often provide scant insight into the relevant collective physics, and they are infeasible for applications in real-time process control, or in optimisation, where there is a need to run many different configurations much more rapidly. In previous work, a multiscale simulation was demonstrated that was able to correctly capture flow fields, diffusion and mixing in hopper drainage [1]. The key observation behind this simulation was that particles in dense granular flows are strongly geometrically constrained. Since they form an amorphous random packing, a single particle is confined by its neighbors, and in order to flow, it must do so cooperatively with its neighbors. Thus this simulation was based on breaking down the flow into correlated group displacements on a mesoscopic length scale. In a related study, it was shown that continuum variables, while intractable at the level of a single particle, can successfully be interpreted at the same scale [3], and this information can be used to directly test and develop a continuum theory of granular materials. Drawing on these concepts, a multiscale simulation technique will be presented, that couples a macroscopic continuum theory of granular flow to a discrete microscopic mechanism for particle motion. The technique can be applied to arbitrary slow, dense granular flows, and can reproduce similar flow fields and microscopic packing structure estimates as in DEM. Since forces and stress are coarse-grained, the simulation technique runs two to three orders of magnitude faster than conventional DEM. A particular strength is the ability to capture particle diffusion, allowing for the optimization of granular mixing, by running an ensemble of different possible configurations. References [1] Chris H. Rycroft, Martin Z. Bazant, Gary S. Grest, and James W. Landry. Dynamics of random packings in granular flow. Phys. Rev. E, 73:051306, 2006. [2] Chris H. Rycroft, Gary S. Grest, James W. Landry, and Martin Z. Bazant. Analysis of granular flow in a pebble-bed nuclear reactor. Phys. Rev. E, 74:021306, 2006. [3] Chris H. Rycroft, Ken Kamrin, and Martin Z. Bazant. Assessing continuum postulates in simulations of granular flow. submitted. Preprint available at
11:45 to 12:10 A shear rate dependent critical state theory to describe the initiation of dense granular flows
It is well know that the initiation of flow of a dry granular material strongly depends on its preparation. For example, the collapse of a column of grains initially compacted under vibrations is dramatically different from the collapse of a loose column [1]. To capture the role of the initial volume fraction in hydrodynamics model of granular flows, there is a need to take into account dilatant or contractant behaviors. Critical state theories developed in soil mechanics are simple ways to describe the initial deformation of a granular sample under quasi-static deformations and to model the coupling between stresses and volume fraction [2,3]. However, such theories are shear rate independent and are thus unable to describe the development of free surface flows like avalanches. In this work we show how a recent viscoplastic model suitable to describe the viscous behavior of granular flows in various configurations [4] can be adapted to take into account dilatancy effects. The idea consists in considering the rheology given by the visco-plastic approach as a shear rate dependent critical state and in introducing a dilatancy angle to couple volume fraction and stress tensor variations. The predictions of the model are illustrated for the problem of the initiation of flow of a granular layer on an inclined plane. Depending on the initial volume fraction, the route to reach the steady state aooears to be different. [1] A. Daerr and S. Douady Sensitivity of granular surface flows on preparation Europhys. Lett. 47 (3), pp. 324-330 (1999) [2] A. Schofield and P. Wroth Critical Soil Mechanics (McGraw-Hill, London , 1968) [3] S. Roux and F. Radjai "Statistical approach to the mechanical behavior of granular media." in Mechanics for a New Millennium, H. Aref and J. W. Philips (eds), Kluwer, Netherlands, pp. 181-196 (2001). [4] P. Jop, Y. Forterre, and O. Pouliquen, a constitutive law for dense granular flows, Nature 441, 727 (2006).
12:10 to 12:35 Experimental observations of fluid-inertial behaviour of dam-break, dense granular flows and their relevance for the propagation of pyroclastic flows
Pyroclastic flows are commonly generated during volcanic eruptions by the gravitational collapse of lava domes or explosive columns, and consist of dense, hot mixtures of gas and particles. In order to investigate the physics of these flows, we carried out laboratory dam-break experiments using both granular material and water, and the flow kinematics were studied quantitatively. Unsteady granular flows of glass beads of 60-90 ƒÊm in diameter generated in a horizontal channel from the release of initially fluidized, slightly expanded (2.5-4.5%) columns behave as their inertial water counterparts for about 65% and 80% of their flow duration and run-out, respectively. For a range of initial column height to length ratios of 0.5-3, both types of flows propagate in three stages, controlled by the timescale of column free fall ~(h0/g)1/2, where h0 is the column height and g the gravitational acceleration. Flows first accelerate as the column collapses. Transition to a second, constant velocity phase occurs at a normalised time t/(h0/g)1/2~1.5, when the flow height in the channel has a maximum value of about 0.2h0. The flow velocity is then U~ã2(gh0)1/2, larger than that for dry (non-initially fluidized) granular flows. The behaviour of the initially fluidized granular flows departs from that of the water flows when they enter a last, third phase at t/(h0/g)1/2~4, as the height of the collapsing column has dropped to about that of the resulting flow. The granular flows then steadily decelerate and their front position varies as t1/3, as in dry flows, and their motion ceases at t/(h0/g)1/2~6.5 and normalized run-out x/h0~5.5-6. The equivalent behaviour of water and highly concentrated granular flows up to the end of the second (constant velocity) phase indicates a similar overall bulk resistance, although mechanisms of energy dissipation in both cases would not be of the same nature. These results show that the interstitial air can damp particle-particle interactions. Further analysis suggests that air-particle viscous interactions can be dominant and can generate pore-fluid pressure sufficient to confer a fluid-inertial behaviour to granular flows propagating at almost maximum concentration, before they enter a granular-frictional regime at late stages. This work suggests that analyses combining inertial-turbulent and granular-frictional form of resistance are the most appropriate to model the propagation of pyroclastic flows.
12:35 to 13:00 Transition from discontinuous avalanches to continuous flow
We investigate experimentally in a half-filled rotating drum set-up the transition from the regime of discontinuous avalanches at low rotation velocities to the regime of continuous flow at higher rotation velocities. For different glass bead diameters, we always observe a smooth transition between the two regimes: At a first critical rotation velocity, brief bursts of continuous flow can be observed in the predominant avalanche flow, while at a higher second critical rotation speed, the last rare events of discontinuous avalanches disappear of the predominant continuous flow. Such a smooth transition by intermittency is very different from the abrupt hysteretic transition described by Rajchenbach [1]. With a model equation describing well the dynamics of unsteady avalanches [2] and extended to non negligible rotation rate of the drum, we can reproduce the two types of transition depending on the noise level added in the deterministic model equation. The predictions of the model can then be tested: The two critical rotation velocities are measured for the different grain diameters and are shown to be related to the measured noise fluctuations for the first one and to the distribution of maximum angle of stability relative to the neutral angle for the second one. The distribution of life time of each regime have also been measured and shown to be Poissonian, with a mean life time that varies from zero to infinity across the transition zone. [1] J. Rajchenbach, “Flow in Powders: From Discrete Avalanches to Continuous Regime,” Phys. Rev. Lett. 65, 2221 (1990). [2] R. Fischer, P. Gondret, B. Perrin, and M. Rabaud, “Dynamics of Dry Granular Avalanches,” Phys. Rev. E 78, 021302 (2008)
Thursday 8th January 2009
09:00 to 09:40 Plenary: dense granular flows down inclines
We'll show how kinetic theory, slightly extended to include an additional length scale in the rate of collisional dissipation that is associated with clusters and/or correlated collisions, can predict the observed features of inclined flows of inelastic, frictional spheres over a rigid bumpy base in the absence of side walls and over the surface of a heap when side walls are present.
09:40 to 10:05 V Kumaran (Indian Institute of Science)
Rapid granular flows: from kinetic theory to granular dynamics
Rapid granular flows are defined as flows in which the time scales for the particle interactions are small compared to the inverse of the strain rate, so that the particle interactions can be treated as instantaenous collisions. We first show, using Discrete Element simulations, that even very dense flows of sand or glass beads with volume fraction between 0.5 and 0.6 are rapid granular flows. Since collisions are instantaneous, a kinetic theory approach for the constitutive relations is most appropriate, and we present kinetic theory results for different microscopic models for particle interaction. The significant difference between granular flows and normal fluids is that energy is not conserved in a granular flow. The differences in the hydrodynamic modes caused by the non-conserved nature of energy are discussed. Going beyond the Boltzmann equation, the effect of correlations is studied using the ring kinetic approximation, and it is show that the divergences in the viscometric coeffecients, which are present for elastic fluids, are not present for granular flows because energy is not conserved. The hydrodynamic model is applied to the flow down an inclined plane. Since energy is not a conserved variable, the hydrodynamic fields in the bulk of a granular flow are obtained from the mass and momentum conservation equations alone. Energy becomes a relevant variable only in thin boundary layers at the boundaries of the flow where there is a balance between the rates of conduction and dissipation. We show that such a hydrodynamic model can predict the salient features of a chute flow, including the flow initiation when the angle of inclination is increased above the friction angle, the striking lack of observable variation of the volume fraction with height, the observation of a steady flow only for certain restitution coeffecients, and the density variations in the boundary layers.
10:05 to 10:30 A Thornton & JM Gray & P Kokelaar ([Manchester])
Modelling of particle size segregation and its applications to geophysical problems
It is important to be able to predict the distance to which a hazardous natural flow (eg snow slab avalanches, debris-flows and pyroclastic flows) might travel, as this information is vital for accurate assesment of the risks posed by such events. In the high solids fraction regions of these flows the large particles commonly segregate to the surface, where they are transported to the margins to form bouldery flow fronts. In many natural flows these bouldery margins experience a much greater frictional force, leading to frontal instabilities. These instabilities create levees that channelise the flow, vastly increasing the run-out distance. A similar effect can be observed in dry granular experiments, which use a combination of small round and large rough particles. When this mixture is poured down an inclined plane, particle size segregation causes the large particles to accumulate near the margins. Being rougher, the large particles experience a greater friction force and this configuration (rougher material in front of smoother) can be unstable. This instability causes the uniform flow front to break up into a series of fingers. A recent model for particle size-segregation has been coupled, though a particle concentration dependent friction law, with existing avalance models. In this talk, numerical solutions of this coupled system are presented and compared to both large scale experiments carried out at the USGS flume and more controlled small scale laboratory experiments. These simulations and experiments show good agreement and the numerical model allows the whole of parameter space to be sampled in a systematic way. Additionally, the numerical model can be used to analyse the effect of individual assumptions, leading to a better understanding of the key mechanisms which drive the observed phenomena.
10:30 to 10:55 CS Campbell ([Southern California])
Elastic effects in granular flows
Previously, granular flows had been divided into (1) the slow, quasistatic regime using models born of metal plasticity theory and (2) the fast, rapid-flow regime using models born of the kinetic theory of gases. However it was clear that even collectively, the models were incomplete. In addition, there was evidence that many flows operate in an intermediate regime where the rheological behaviour was not understood. Furthermore, the parameter space in which either model was valid were entirely unknown. Recently. it was discovered that adding the interparticle stiffness as an additional rheological parameter allowed the entire flowmap of granular flow to be drawn. This allows one to put practical limits on the quasistatic and rapid regimes and to understand the physics of the intermediate regime. The flows could be divided into two broad regimes, Elastic and Inertial. The Elastic regime are dominated by force chains with particles in intimate contact with their neighbours. The internal forces are generated by the compression of the interparticle contacts and thus scale with the interparticle stiffness. This is divided into two subregimes, the Elastic-Quasistatic, the old quasistatic regime, and the Elastic-Inertial regime, the previously understood intermediate regime. In the Elastic-Quasistatic regime, the stresses are generated by the process of the formation, compression, rotation, and destruction of force chains. The process is shear rate independent because the formation is proportional to the shear rate, while the destruction is inversely proportional to the shear rate and the compression is largely geometrically controlled and shear rate. However, some degree of the chain's compression reflects the particle inertia and at large enough shear rates this becomes noticeable and the material enters the Elastic-Inertial regime, where, reflecting the increased inertia, the stresses increase linearly with the shear rate. This is the new "flow" regime referred to above. Inertial flows are free of force chains and demonstrate the famous Bagnold scaling where the stresses vary as the square of the shear rate. However, it too can be divided into two subregimes, the Inertial-non-Collisional regime, in which the particles interact in clusters that do not quite become force chains, and the Inertial-Collisional regime (old Rapid Flow regime) for which kinetic theory models are appropriate. Through a coordination of wave Elastic effects have also been shown to control the convection in deep vertically vibrated boxes, and are the key to understanding the convection processes. Furthermore the Elastic-Inertial regime may help explain a controversy in the fluidised bed community in which there is experimental evidence of an unexplained viscous-like behaviour that may reflect the linear behaviour in the Elastic-Inertial regime. This talk will refect the current state of the elastic flow theory and include recent work on non-round particles.
11:30 to 11:45 Pre-avalanche structural rearrangements in granular packing
It has long been known that, under increasing inclination, a granular medium displays a transition from a solid like (static regime) to a fluid like (flow regime) phase. A complete understanding of this transition seems yet unavailable. However, predicting this transition is important for geophysics, numerous industrial applications, and for understanding the dynamics of granular materials and the unjamming transition.An important question is whether reliable precursors can be found when approaching the critical angle. The main finding reported in previous studies is that the density of surface rearrangements gradually increases when increasing the inclination angle. For small angles (~10), events are rare and occur randomly on the surface while, close to the critical angle (~25), quasi-periodic rearrangements involving larger and larger numbers of grains were observed. Our experimental work consists in studying the dynamics of a 3D granular packing constituted by glass beads contained in a rectangular box which is inclined up to the threshold of instability. With an image processing technique and tracking methods, we obtain the number of surface rearrangements and we follow the trajectories of grains as a function of the angle inclination. When the angle increases, four types of events are observed. Weak rearrangements take place at random on the surface for small angles (
11:45 to 12:00 CJ Cawthorn & EJ Hinch & JN McElwaine ([Cambridge])
Consequences of the ?(I) constitutive law
Following the proposal of a three-dimensional constitutive law for dense granular flows by Jop, Forterre & Pouliquen, a wide range of researchers have begun to apply this so-called p(I) law to a wide range of granular flows. In this presentation, I shall highlight some of the many applications of the p(I) law, describe its strengths and weaknesses (most of which are mentioned in a recent review paper) and suggest some possible modifications that may extend its range of validity. The p(I) law was developed with free surface flows in mind. Indeed it was constructed from a friction law for granular flow down a rough inclined plane. Since its proposal, the mu(I) law has been used to successfully model a range of free surface flows, and has produced good quantitive results. As well as describing some past successes, we will discuss some new stability calculations predicting the formation of longitudinal vortices in flow down an inclined plane, lending more support to the use of the p(I) law. For confined flows, such as Coutte or Taylor-Coutte shear, it has been n0ted that the p(I) law makes predictions that are quantatively poor. We shall discuss some possible reasons for this, and propose a simple modification that should extend the contitutive law's predictive powers to encompass such confined geometrics. Finally we shall highlight the need for a proper understanding of how flowing regions can interact with static of plug regions, either by erosion or accretion. We will discuss some progress in this direction, and consider its application to experimental geometrics, such as split-walled shear cells, rotating drums, and the collapse of a granular column.
12:00 to 12:15 The flow generated by oblique impingement of granular material on an inclined plane
We consider a steady stream of granular material impinging on, and subsequently flowing down, an inclined plane. The flow exhibits a wide range of behaviours, depending on the slope angle and a non-dimensionalised fall velocity, ranging from kinetic gas-like flow to intermittant avalanche formation. As well as being of theoretical interest, the scenario is representative of many industrial granular flows. We study in detail a previously unreported steady state flow regime, which displays a teardrop-shaped hydrodynamic shock surrounding the area of impingement. We construct a depth-integrated hyperbolic flow model which predicts with good agreement the experimental location and shape of the shock. A number of the experimentally observed features cannot be predicted by current shallow-layer models of granular materials, and we discuss these with reference to the constitutive rheology of thin-layer granular flows.
12:15 to 12:30 T Börzsönyi & RE Ecke & JN McElwaine ([Research Ins for Solid State / Cambridge])
Lateral instability of dense granular flows on a rough incline
Dense granular flows are often observed to become unstable and form inhomogeneous structures in nature or industry. The nature of such instabilities is often not understood in details, and in certain cases even the existence of the instability and the structure of the developing pattern is not yet explored. We present experimental and numerical (MD) results about a longitudinal stripe state that arises from instabilities of the uniform flowing state of granular media on a rough inclined plane. It is know that longitudinal vortices can be formed in dilute flows where the mean density is less than 50% of the random close packed density. We show that a robust longitudinal stripe state is formed already in dense flows staarting at mean densitites of about 0.95. We measure the mean density of the flow, the lateral profiles of the velocity, the flow height, the transmitted light intensity and the surface fluidisation. The form of the stripes depends critically on the mean density of the flow. At high density faster flowing regions correspond to height maxima of the modulated height profile. At higher inclination where the average density becomes significantly lower we find the previously reported state, where faster flow occurs in the lower regions of the flow. Molecular dynamics simulations reveal details of the internal flow structure and confirm a transition between the dense and dilute regimes.
12:30 to 12:45 Structures formation by migration of particles in suspension flows
Flowing dense particle mixtures give rise to fascinating phenomena that are ubiquitous in nature and in engineering applications: migration, segregation, anisotropic viscosity and dilatancy. Relevant examples are quick sands, lubricants, colloids, erosion flows, magma, debris flow, biological fluids and detergents. One of the most intriguing features of dense suspension flows is particle migration. Experimental and simulation studies to date with smooth and neutrally buoyant particles which are sufficiently coarse (mm range) so that Brownian motion can be neglected show that in slow shearing channel flows, the particles migrate to the centre of the channel where the shear rates are low; see for example Morris et al (1999), Miller & Morris (2006). The migration of much smaller colloidal (micron to nano size range) are also affected by the bulk of fluid shear strain gradients but to a lesser extent due to the Brownian motion created by the thermal fluctuations in the carrier fluid; see Frank et al (2003). The ration of these two effects is given by the Peclet number. At low Peclet numbers, the Brownian motion dominates and affects the structure's formation process in such a way that destroys and diffuses away the structures whilst at much higher Peclet numbers, the migration is believed to become more pronounced. For coarse particles in suspension, Jenkins and Koenders (2005) have proposed a lubrication interaction with additional collisional mechanics with particles separated by gap sizes of the order of the roughness dimension. The lubrication interactions between micro and nano-size particles are much less understood due to the anisotropes introduced by the non-uniformity of the Brownian motion on the fluctuating temperatures and the stress and strain rates thus resulting in heterogeneity of particle structures. A hybrid numerical simulation strategy has been developed to probe the particle-particle and particle-fluid interactions in dense suspensions of fine particles with a view to aid the development of appropriate lubrication interaction models for such systems. The simulations start off with simple MD calculations using Lennard Jones potentials to simulate the flows of fluid particles incorporating thermal diffusivity effects to be followed by frictional DEM simulations of collisional dynamics flows of solid particles. The hybrid scenario is aimed to simulate the two-phase flow by the interaction of the lubrication forces between the particles and the solvent molecules. The project is funded by the Leverhulme Trust and the progress to date with numerical simulations will be presented in the early stages of the current investigations. Challenges associated with simulation strategies and the relaxing of simplifying assumptions will be discussed.
12:45 to 13:00 Simultaneous measurement of viscosity and kinetic temperature in a driven dense granular suspension
We look at thermodynamic and rheological properties of a slowly-driven dense suspension of glass spheres ($D = 250\mu m$) in water. We do so using a torsional pendulum as a rheometer and we measure simultaneously the kinetic granular temperature using speckle visibility spectroscopy. We can control the density of the suspension very precisely in our region of interest (between $\phi = 0.5$ abd $\phi = 0.56$) by changing the flow rate and viscosity of the liquid used to fluidise the grains. A spherical torsional pendulum covered in glued grains is immersed in the suspension. It is driven at small angles to measure viscosity. Simultaneously, we use a laser speckle pattern created in transmission to measure the velocities of grains as a function of the degree of fluidisation, and thence obtain a measure of the kinetic temperature.
14:00 to 14:25 JM Huntley & T Tarvaz & NA Sheikh ([Loughborough])
Nuclear Magnetic Resonance studies of an ultrasonically vibrated granular bed
A granular bed on a vibrating base (with vibration amplitude A, and angular frequency w) is normally fluidised once the peak base acceleration A0w2 exceeds that due to gravity. Once fluidised, the rate of energy input to the bed is normally controlled by the peak base velocity A0w. In the current paper we test both experimentally and numerically the validity of these assertions at vibration frequencies some 2-3 orders of magnitude higher than normally used. The experiments were performed using a Nuclear Magnetic Resonance spectrometer that provided time-averaged 1-D profiles of packing fraction and granular temperature within a bed of mustard seeds. A high power ultrasonic transducer and waveguide produced peak velocities of order 1ms -1 and peak accelerations of order 105 ms-2. A laser vibrometer was used to maintain a constant peak base velocity at a set of discrete resonant frequencies in the range 10-20 kHz. Despite the constant base velocity, dramatic reductions in granular temperature and mean height of the bed were observed as the frequency increased. The interaction between a grain and the vibrating base was modelled using a Hertzian contact law, with the stiffness coefficient measured experimentally through quasi-static compression tests of the grains. The equation of motion was integrated by a Runge-Kutta scheme, taking proper account of multiple contacts. By averaging over all vibration phases, an efficient coefficient of restitution, e, was obtained as a function of approach velocity and vibration frequency over the range 10-20 kHz, in agreement with the experimentally-observed reduction in granular temperature and mean height of the bed over the same range.The Hertzian mean impact duration of approximately 60 us lies within the 50-100 us range of the vibration period. The transition between "classical" vibrofluidised bed behaviour and the "high-frequency" behaviour studied here can therefore be interpreted as a consequence of a breakdown of the usual assumption of instantaneous collisions. Presentation of the results in non-dimensional form provides a general expression for the energy flux boundary condition, which may prove useful for the development of future applications of ultrasonic excitation.
14:25 to 14:50 J Marshall ([Vermont])
Particle segregation in oscillating straining flows
Numerical computations performed using a discrete-element method for both adhesive and non-adhesive particles indicate that particles placed in an oscillating straining flow will drift towards the nodal points of straining - a phenomenon that we refer to as oscillaing clustering. A theoretical prediction of this phenomenon is developed which yields the particle drift rate toward the straining nodal point, which is found to increase with increase in particle Strokes number up to a stability limit. The theory also predicts that in the presence of gravitational force, particles will exhibit a limit cycle behaviour in which they oscillate under the opposing downward gravitational drift and upward drift due to oscillating clustering. Computations are performed to demonstrate the effects of this phenomenon for particles in an oscillating box, for peristaltic pumping of a particulate suspension, and for a suspension flowing through a corrugated tube.
14:50 to 15:15 MA Wheel & B Edmans ([Strathclyde])
Can cellular automata based models accurately simulate granular material flow?
Cellular automata (CA) based methods have been suggested as an alternative to continuum and discrete element models for predicting the flow of granular materials. These methods divide the flow field into cells, each of which can display a finite number of states. The current state of each cell is a function of its state and those of its neighbours at the previous timestep. This function is governed by update rules that definte the interactions that can occur between particles moving around a lattice connecting adjacent cell centres. The rules themselves can incorporate mass and momentum conservation albeit in a simple, discrete manner. CA methods offer significant potential in simulating the behaviour of systems with very large numbers of degrees of freedom: they are well suited to parallel processing, rule sets are easy to implement in a software environment and numerical stability is ensured by the local conservation attribute. While a number of potentially suitable CA methods for simulating granular flows have been described in the literature, insufficient quantitative data have, as yet, been reported to validate their predictive capabilities across a range of flow conditions. To remedy this three previously reported CA methods were implemented within a MATLAB based environment. These methods were the viscous model of Peng and Ohta, which can be regarded as a variant of the lattice gas model previously used in modelling hydrodynamic phenomena, the lattice grain model of Gutt and Haff, which is similar to the first model but incorporates continuous rather than discrete particle velocities, and the hopper flow model of Kozicki and Tejchman, which incorporates void filling update rules that are mass but not momentum conserving and is therefore non synchronus. Each of the models was used to simulate the emptying of a two dimensional hopper across a range of flow conditions. Experiments were also performed to observe the flow of glass beads in a similar hopper when emptying. Flow conditions were varied by altering the wall angle of the funnel section constituting the lower part of the hopper. The experiments revealed that as the wall angle increased (with respect to the vertical axis of the hopper) the particle flow rate at the exit reduced. However, the lattice grain model predicted that the flow rate was insensitive to the changing wall angle while results provided by the hopper flow model indicate that the flow rate will increase with wall angle. Nevertheless, the overall forms of flow fields predicted by the latter two models show sufficient similarity with the experiments to encourage more detailed assessment of the suitability of CA methods in simulating granular flows.
15:15 to 15:40 Dense granular flow: a review of the types of problems, approaches to solutions and outstanding opportunities
Dense granular flow is everywhere; in the natural world, for example in the formation of sand dunes, pyroclastic flows from volcanoes, and landslips; and in the man-made world, from the building of embankments and stockpiles, through discharge of hoppers and silos, to separation of museli in bag packing and flow of sand in an egg-timer. The common and familiar nature of the phenomena, and their superficial simplicity, belie the extreme complexity of the physics within the systems and the consequent severe difficulty which is faced when trying to model them. In spite of well over a hundred years of scientific and engineering effort, our ability to predict such systems is still in its infancy. The authors have a long experience of developing and applying both new and established techniques for predicting the behaviour of dense granular flows, mainly for the purposes of designing and troubleshooting industrial particle processing systems across a huge range of industries. This paper sets out to build on this experience and the observed experience of many others, firstly by reviewing a wide range of practical activities involving dense granular flow which need to be modelled for the purposes of prediction and design. It explores the full range of different techniques which are available for modelling them, and evaluates the success of various different approaches which have been tried in many different classes of problems. It makes an attempt classify dense granular flow problems, and propose a philosphy for choosing techniques which are likely to be successful, based on these classifications. Finally, it identifies a number of key areas and types of problem which present major opportunities for furthering the understanding of the area.
16:10 to 16:35 K Hill & Y Fan ([Minnesota])
Sorting out segregation mechanisms at the interface between densely creeping and energetic granular flows
Densely flowing granular mixtures segregate die to difference in the size, material density, and other particle properties. In some situations the segregation is simple: smaller particles tend to sink compared to their larger, equal-density counterparts; denser particles tend to sink compared with their equal-sized lighter counterparts. These are often attributed to kinetic sieving and buoyancy, respectively. However, in some situations the segregation is more complex: particles poured into a pile may segregate into stratified layers, and in drums some mixtures will segregate into radial stripes and axial bands. It is difficult to determine the dominant segregation mechanism(s) in each case. In most experimental segregation studies of dense flowing granular mixtures, velocity gradients, volume fraction gradients and gravity simultaneously act on the granular mixture.To isolate the segregation mechanisms, we study segregation in densely sheared granular mixtures experimentally and computationally in two systems: a rotating drum and in a split-bottom cell. In a free surface flow that occurs in the rotating drum, we find the relative importance of each mechanism depends on the position in the flow, the importance likely governed by the local volume fraction. On the other hand, when shear is perpendicular to gravity as in a split-bottom cell, we find gravity-driven segregation by size is significantly reduced; partially where the volume fraction gradient is eliminated. In this regime, segregation mechanisms associated with a variable shear rate dominate. We discuss these results and their relative importamce for segregation in different regimes of relatively dense granular mixtures.
16:35 to 17:00 P Viot & A Burdeau & K Combs ([UPMC])
Quasi-gaussian velocity distribution of a vibrated granular bilayer system
We present a Molecular Simulation study of a bilayer of vibrated granular bidisperse spheres which exhibits the striking feature that the horizontal velocity distribution of the top layer particles has a quasi-Gaussian shape, whereas that of the bottom layer is far from Gaussian. We investigate the relevance of all physical parameters (acceleration of the bottom plate, mass ratio, layer coverage) and compare our results to the experiments of Baxter and Olafsen. In addition, a microscopic analysis of the trajectories of the particles of the top layer as well as of the collision statistics provides information on the mechanism of randomisation at the origin of this effect. In collaboration with K Combs and J Olafsen, we have also found that the velocity statistics for a top layer consisting of a single particle are deeply modified in the presence of mobile defects in the first layer. We have developed a model which is able to capture the role of the defects.
Friday 9th January 2009
09:40 to 10:05 On the role of numerical diffusion on QMOM and DQMOM simulation of fluidized beds
When the behaviour of fluidized polydisperse powders is simulated by using an Eulerian-Eulerian framework, the population balance must be solved along with other governing equations. Different formulations for the population balance model exist, and in this work only particle size dependencies will be considered. Under this hypothesis, particles with the same size are assumed to move with the same velocity, calculated with specific momentum balance equations, leaving to the population balance model the tast of tracking changes in particle size only. The population balance equation can be conveniently solved with computational fluid dynamics (CDF) codes by using quadrature methods, such as the quadrature method of moments (QMOM) and the direct quadrature method of moments (DQMOM). QMOM has been mainly used in the past assuming that all the particles move with the same velocity, resulting in strong limitations, for example, when it comes to the description of particle segregation by size. DQMOM has been presented as an evolution of QMOM, being able to track the differences in particle size as well as those in particle velocity (and therefore able to descibe segregation). In this work, QMOM and DQMOM are formulated in a new form in terms of a volume density function (VDF), rather than the original number desnity function (NDF). Both methods use a quadrature approximation of order N, coupled with a multifluid model including N dispered phases plus a continuous phase. For the first time in this work QMOM has been implemented in such a form in a commercial CDF code (ie Fluent), by means of user-defined functions and scalars for the solution of the 2N transport equation for the moments, which must be convected with the proper velocity. Also DQMOM has been implemented in the same commerical CFD code, resulting in the same formulation already reported by other authors. Although the two methods are theoretically equivalent (as it is very easy to show by simple manipulation of the governing equations) they lead to completely different results because of numerical diffusion. This is demonstrated on a very simple test case, a two-dimensional fluidised bed containing two polydispersed powders, characterised by a very wide particle size distribution ranging from 70 to 400 micrometers. The system is fluidised under very different operating conditions, resulting in segregated and completely mixed configurations and the performances of QMOM and DQMOM are compared. Eventually, simulations are also compared with experimental results. Results clearly show that, in this particular case where no specific particulate processes occur (no particle aggregation nor breakage), because of numerical diffusion DQMOM leads to the wrong solution, and that for these cases it could be more convenient to solve the problem with QMOM, directly tracking the moments.
10:05 to 10:30 Orientational ordering in sheared inelastic dumbbells
Using event driven simulations, we show that homogeneously sheared inelastic dumbbells in 2D are randomly orientated at low packing fraction, but show an increasingly preferred alignment with the shear direction as the packing fraction increases. The orientational order parameter exhibits a continuous increase with packing fraction and does not appear to exhibit a universal scaling with elongation. Except at the highest packing fractions, the orientational distribution function can be reconstructed with a few coefficients of the Fourier expansion. We also present results for the translational and rotational granular temperatures and the stress tensor.
10:30 to 10:55 Stick slip dynamics in a sheared granular system
We have investigated the response of a granular medium to shear both experimentally and theoretically. The experiments consisted of a granular medium confined to a circular channel and subjected to the shear exerted by an overhead horizontal plate. On the theoretical side we devised a 1-D Langevin equation capable of describing the dynamics. We explored the parameter space and characterised the statistics of the system response by measuring several quantities, in particular: the reaction torque exerted by the medium against the shear (reaction), the plate velocity, the duration and extension of the slip events in the stick slip regime. Here we summarise the main results: (1) Fluctuations of response stress (the torque exerted by the system against the motion of the plate) are Gaussian in the steady sliding regime, but not in the stick slip regime; (2) The dynamics of the plate is quantitatively well described by a Langevin equation in which the reaction force performs Brownian motion; (3) For thin granular beds, the system behaviour strongly depends on the number of grain layers. An asymptotic state is approached at about five layers, where transition between quasi-solid and quasi-liquid behaviour becomes clearly indetifiable.
11:45 to 12:10 Important thermodynamic aspects in the formulation of solid-fluid debris flow models (dense and particle laden)
The present literature on continuous modeling of static and dynamic solid-fluid interactions is fraught with different forms of balance laws that are based on mixture and multi-phase theories (balance laws with a priori estimates for constituent pressure) respectively. They are characterised by different postulates of the partial stresses and on this basis, it is claimed eg that the multi-phase concept is superior to the mixture theory concept. It is my conjecture that such claims are premature without a proper thermodynamic analysis and a comparison of the final field equations. We perform a thermodynamic analysis for a mixture of n solid-fluid constituents with thermo-visco-elastic properties and account for rubbing frictional effects by an internal symmetric internal tensor variable, which models hypo-plastic behaviour. Volume fraction dependence is accounted for by a scalar internal variable, mixture saturation is taken into account by a constraint condition and incompressibility is interpreted as preserving the true of the respective constituents. The thermodynamic analysis is conducted with Muller's form of the entropy principle. The analysis also requires a number of ad hoc assumptions to be able to deduce definite results. These results deliver an explicit expression of the Gibbs relation and the entropy flux and yield, via the integrability conditions, explicit forms for the contituent equilibrium stresses, constituent interaction forces, entropy and equilibrium heat flux. These expressions are determined by the prescription of the Helmholtz free energy, Gibbs free energy, the saturation constraint variable and the extra entropy flux. The structure of the formulae shows how these equilibrium variables depend via partial derivatives of the free energy upon the independent constitutive variables. Three different pressure terms arise: thermodynamic pressures are the responses to constituent density variations, configuration pressures those to volume fraction dependences and saturation pressure that to the saturation constraint. It follows from these results that the hypothesis of pressure equilibriium, according to which the mixture pressure is distributed among the constituents with the weights of the volume fractions, is almost never correct except when the hypo-plastic friction and constituent volume fraction dependences are ignored as independent constitutive variables. Non-equilibrium stresses and interaction forces are dominantly governed by viscous postulates via stress parameterisations known in rheology of polymers and foods, and Darcy type relations for the interaction forces. The work suggests that debris flow modeling for solid-fluid mixtures along arbitrary terrain should be anticipated by a thermodynamic analysis to guarantee correctness of the equation.
12:10 to 12:35 Rheology of confined granular flows: scale invariance, glass transition and friction weakening
We study fully developed, steady granular flows confined between parallel flat frictional sidewalls using numerical simulations and experiments. Above a critical rate, sidewall friction on the flow stabilises the underlying heap at an inclination larger than the angle of repose. The shear rate is constant and independent of inclination over much of the flowing layer. In the direction normal to the free surface, the solid volume fraction increases on a characteristic scale equal to half the flowing layer depth. Beneath a crictical depth at which internal friction is invariant, grains exhibit creeping and intermittent cage motion similar to that in glasses, causing gradual weakening of friction at the walls.
12:35 to 13:00 A new convection scenario in granulates under geometrical restriction
An experiment is presented that extends the diversity of pattern forming phenomena found in granular media. A flat container (Hele-Shaw cell) is filled with a granular mixture and slowly rotated about its horizontal long axis. The filling fraction is crucial for the observed effects. At partial filling of the container, the material can be fluidised during rotation and patterns of axially segregated stripes appear which undergo slow coarsening. On longer timescales the onset of periodically travelling stripes are observed. This effect resembles stripe patterns commonly found in rotating drums. A novel interesting phenomenon emerges under geometrical restrictions when the container is nearly filled. Although the particles are on the brink of jamming, and their mobility is almost inhibited, we observe regular convection rolls that are accompanied by, and decorated by a conspicuous serpentine segregation pattern. In contrast to the loosely moving beads at partial filling, the particles move in collective clusters. Furthermore the number of convection rolls is long-term stable and only related to the container geometry. Even though there are some superficial similarities to well known convection rolls in vibrated granular systems, there are striking differences concerning driving forces, segregation patterns, and number of rolls. Our system complements convection phemonena found in agitated granulates and brings up new questions that are discussed in the study.
14:00 to 14:25 Size segregation in granular fluid flows
Particle segregation has important implications both for natural phenomena and industrial activities, inducing important effects on the rheology of the mixture and on transport dynamics. Despite the importance of particle segregation, the theoretical framework developed for the description of this phenomenon is still incomplete. For these reasons, the present contribution is aimed at the presentation of the development of a flow model, based on the mass balance of the particles coupled with the momentum balance of fluid and sediments, in order to describe in a realistic way segregation phenomena that are important in environmental problems, such as debris flow or wind blown sand. We focus on steady fully developed flows of a mixture of water and spherical particles of two different sizes made of the same material. The hyrodynamic model is an evolution of that employed by Jenkins and Hanes (J. Fluid Mech. 370, 29-53, 1998) and the segregation model is that introduced by Arnarson and Jenkids (Phys. Fluids 16, 4543-4550, 2004). We consider flows down inclines that are driven both by gravity and the shear stress of a clear turbulent fluid above the particles. We take the fluid to be either air or water. Profiles of mixture volume fraction, particle velocity, fluid velocity, and species volume fraction are predicted and, where possible, compared to experimental measurements.
14:25 to 14:50 Free shear zones in granular bulk flow
We study shear localisation of granular media in slow three-dimensional shear flows. We focus on free shear zones which arise in the bulk of the material far from the confining walls of the shear cell. Free shear zones have many remarkable properties. Eg, their shapes exhibit nontrivial open or closed forms found in modified Couette experiments, their widths show sublinear scaling with the system size, shear zones are refracted at material interfaces similarly to light refraction. The shear zones are investigated in quasi-static and stationary flows where the behaviour is very robust: there is no dependence on the driving rate or on the preparation history. Our goal is to learn more about the puzzling mechanism that sets the rheology in quasi-static shear flows. We analyse the inner structure, the deformation and stress field of shear zones based on computer simulations using a discrete element method. The behaviour is also discussed within the framework of a recent mesoscopic model of slow shear flows, where the smooth deformation profile is provided by a series of thin shear hands.
14:50 to 15:15 Shock waves and snow avalanches
Many geophysical granular flows, such as snow slab avalanches, occur as dense free-surface flows that are driven downslope under the action of gravity. In mountainous regions, avalanche defences are often built to deflect the avalanche away from people and infrastructure, or, to stop it before it reaches them. As the avalanche is deflected there are rapid changes in the avalanche height and velocity. We apply classical oblique shock theory and use small scale experiments to investigate how weak, strong and detached shock waves are generated by a wedge and compare shock capturing numerical simulations on realistic topography to field observations from a deflecting dam in Flateyri, Iceland. These show that there is no one single set of upstream conditions that parameterises the flow behaviour, but the solution evolves as the avalanche propagates along the dam in response to the deceleration imposed by the slope. Nevertheless the classical theory still yields important order of magnitude estimates for the flow velocity and thickness immediately upstream of the shock. The numerical method is extended to handle the flow around arrays of rectangular obstacles, which are often placed in the run-out zone to slow the avalanche down. The results are also compared with small scale experiments.
15:15 to 15:40 PJ Thomas & E Guyez ([Warwick])
The spatiotemporal dynamics of segregation-band drift in particle- laden rimming flow
We described a new pattern formation process associated with flow inside a horizontal rotating cylinder that is partially filled with a particle-laden fluid. The pattern is observed to form as the particles segregate and agglomerate to form circumferential bands with high particle concentrations. Any two neighbouring granule bands are separated from one another by regions of liquid with low particle concentrations or, in some cases, regions entirely free of particles. Out most recent experiments have revealed that the system can display an extremely rich, often highly symmetric, spatiotemporal behaviour that emerges over days or weeks as the patterns drift very slowly along the axis of rotation. The plot was assembled by extracating pixel lines from successive photos and composing those into a single figure. The presence of a band is indetified in black whereas regions of low granule concentration are represented in white. It can be seen that the bands drift from location L>0 towards the cylinder end walls at L=0 cm and L= 27 cm. For other experimental conditions bands can drift from the end walls towards the cylinder centre, they may display entirely irregular drift characteristics, or they may not drift at all. We discuss results from long-term observations of the system for different values of the involved relevant physical variables. The key non-dimensional parameters governing the system are identified.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons