Model for pattern formation of granular matter on vibratory conveyors

Recently, Götzendorfer et al. [Götzendorfer A, Kruelle CA, Rouijaa M. Granular surface waves in a vibratory conveyor. In: Garcia-Rojo R, Herrmann HJ, McNamara S, editors. Powders and grains, Stuttgart; 2005. p. 1185] have observed subharmonic propagating surface wave patterns if granular material on a trough is subject to a combination of vertical and horizontal periodic driving. The observed structures are non-stationary in space, drift with a constant mean velocity along the trough and oscillate subharmonically with half the driving frequency. We present a phenomenological model for the surface evolution of the granular material that qualitatively reproduces and explains important aspects of the experimentally observed patterns.     

A rapid granular chute avalanche impinging on a small fixed obstacle: DEM modeling,experimental validation and exploration of granular stress

Granular flows are constrained by applied stresses. When a granular flow moves rapidly and impinges on an obstacle, the stress is significantly increased along the contact force networks. Granular stresses are still incompletely understood. The aim of this study is to investigate a rapid avalanche of spherical glass beads in an inclined chute with a small fixed semi-cylindrical obstacle by using particle image velocimetry (PIV) technique and discrete element method (DEM). The proposed DEM model produces good agreement with the corresponding avalanche experiment in terms of the velocity profiles. The validated DEM results are then used to explore the internal flow characteristics of a granular avalanche that are not directly observable in experiments, such as the solid fraction, the average coordination number and the granular stress. Rectangular measurement cells, similar to representative volume elements, are developed to determine the spatial variation in stresses for the granular avalanche. The internal flow characteristics of a rapid granular avalanche with and without obstacles are compared. For the unobstructed flow, the normal and shear stresses decrease in the downstream direction because the solid fraction and the average coordination number decrease, resulting from the gravitational acceleration. On the other hand, granular jamming forms in front of the semi-cylindrical obstacle and results in a significant increase in the normal and shear stresses. The unobstructed flow shows slightly anisotropic stress states, giving an earth pressure coefficient of approximately 1.0, whereas the disturbed flow exhibits strongly anisotropic stress states. The simulation results show that the corresponding earth pressure coefficient can be much higher than unity and increases to a maximum value of roughly 5.0. A shear band develops at a distance of roughly twice the particle diameter above the basal surface and a stronger shear band forms in the upstream vicinity of the obstacle.     

Linear and nonlinear instabilities of a granular bed: Determination of the scales of ripples and dunes in rivers

Granular media are frequently found in nature and in industry and their transport by a fluid flow is of great importance to human activities. One case of particular interest is the transport of sand in open-channel and river flows. In many instances, the shear stresses exerted by the fluid flow are bounded to certain limits and some grains are entrained as bed-load: a mobile layer which stays in contact with the fixed part of the granular bed. Under these conditions, an initially flat granular bed may be unstable, generating ripples and dunes such as those observed on the bed of rivers. In free-surface water flows, dunes are bedforms that scale with the flow depth, while ripples do not scale with it. This article presents a model for the formation of ripples and dunes based on the proposition that ripples are primary linear instabilities and that dunes are secondary instabilities formed from the competition between the coalescence of ripples and free surface effects. Although simple, the model is able to explain the growth of ripples, their saturation (not explained in previous models) and the evolution from ripples to dunes, presenting a complete picture for the formation of dunes.     

Mathematical modelling of bottom evolution by particle deposition and resuspension

A two-dimensional numerical model for the evolution of a bottom due to particle deposition and resuspension by a fluid flow is here presented. A computational fluid dynamic approach is used to calculate the flow field and a Lagrangian particle tracking technique is applied to solve the dispersed phase. The evolution of the lower boundary is simulated taking into account the mass conservation of the solid phase and the geotechnical properties of the granular material. The model is characterized by two important features. First, fluid dynamics are coupled with the bottom evolution due to particle deposition and resuspension. This permits to use the model to simulate complex flow fields as well as complex time-evolving geometries. Second, the dispersed phase is calculated by a Lagrangian approach, which retains the discrete information of the individual particles of the granular bottom which may be of interest for some industrial processes (coating) and environmental flows (sediment stratification). First consistency checks have been performed for some deposition and resuspension test cases with fluid at rest. The model has also been tested by comparison with a physical experiment of deposition inside a cavity. Finally, as an example of possible applications of industrial and environmental interest, the model has been applied to investigate particle deposition in rectangular cavities and the evolution of a sand heap by a fluid flow.     

Analysis of flow characteristics of granular material unloaded on nonlinear vibration inclined platform

In this study, the repeated discontinuous friction between granular material and contact platform and structural nonlinearity of inclined vibration platform giving rise to the vibration flow-aiding unloading is a complicated process, which has significant effects on the dynamic behaviors and flow characteristics of granular material. A simplified mathematical model of the inclined vibration platform and granular material is deduced by mechanical properties. Based on the equations of motion and a good degree of accuracy and applicability of the process with calculated data reported in the literature, the approximate analytical solution and flow properties are investigated by using the modified incremental harmonic balance method and numerical integration method. Moreover, the influences of friction coefficient, excitation amplitude, nonlinear stiffness and inclined angle on the complicated dynamic behaviors are explored and discussed. It is shown that the different motion paths of granular material on inclined vibration platform are observed depending on the different parameters. The increasing friction coefficient has complicated effects on the nonlinear dynamic behaviors of the granular material. The excitation amplitude and nonlinear stiffness can effectively control the flow characteristics of granular material at low excitation frequency but the inclined angle presents opposite property. The research may contribute to improve unloading efficiency, predict the motion state of granule and provide a theoretic foundation for further design the unloading system.     

Prediction of the bed-load transport by gas–liquid stratified flows in horizontal ducts

Solid particles can be transported as a mobile granular bed, known as bed-load, by pressure-driven flows. A common case in industry is the presence of bed-load in stratified gas–liquid flows in horizontal ducts. In this case, an initially flat granular bed may be unstable, generating ripples and dunes. This three-phase flow, although complex, can be modeled under some simplifying assumptions. This paper presents a model for the estimation of some bed-load characteristics. Based on parameters easily measurable in industry, the model can predict the local bed-load flow rates and the celerity and the wavelength of instabilities appearing on the granular bed.     

Non‐homogeneous Navier–Stokes systems with order‐parameter‐dependent stresses

We consider the Navier–Stokes system with variable density and variable viscosity coupled to a transport equation for an order‐parameter c. Moreover, an extra stress depending on c and ?c, which describes surface tension like effects, is included in the Navier–Stokes system. Such a system arises, e.g. for certain models of granular flows and as a diffuse interface model for a two‐phase flow of viscous incompressible fluids. The so‐called density‐dependent Navier–Stokes system is also a special case of our system. We prove short‐time existence of strong solution in L^q‐Sobolev spaces with q>d. We consider the case of a bounded domain and an asymptotically flat layer with a combination of a Dirichlet boundary condition and a free surface boundary condition. The result is based on a maximal regularity result for the linearized system. Copyright © 2010 John Wiley & Sons, Ltd.     

剪切平行板间密集颗粒流的接触力分布及各向异性分析

研究了剪切平行板间密集颗粒流的接触力分布规律、接触力网络的各向异性、颗粒摩擦因数对宏观流变特性及细观力链分布的影响等．为了研究以上内容,应用计算机建立了离散元数值分析模型．数值分析结果表明,颗粒之间的接触力分布按幂函数规律变化;接触角分布服从指数函数规律,平均法向接触力随平均接触角任意上下振荡变化;波动速度大小为宏观流变顺畅与否的关键性评价指标,而在细观力链方面,当剪切平行板间颗粒流变不畅时会伴随着超强力链数目显著增加．     

Simulations of collision of ice particles

The objective of this paper is to develop a realistic model for ice–structure interaction. To this end, the experiments made by Bridges et al. [Bridges FG, Hatzes A, Liu DNC. Structure, stability and evolution of Saturn’s rings. Nature 1984;309:333–5] in order to measure the coefficient of restitution for ice particles are thoroughly analyzed. One particularly troublesome aspect of the aforementioned experiments is fracture of the ice particles during a collision. In the present effort, the collisional properties of the ice particles are investigated using a Finite Element approach. It is found that a major challenge in modeling collision of the ice balls is the prediction of the onset of fracture and crack propagation in them. In simulations of a block of ice collision to a structure, it is crucial that fracture is determined correctly, as it will influence the collisional properties of the ice particles. The results of the simulation, considering fracture criterion implemented into the Finite Element Model [Zamankhan P, Bordbar M-H. Complex flow dynamics in dense granular flows. Part I: experimentation. J Appl Mech (T-ASME) 2006;73:648–57; Zamankhan P, Huang J. Complex flow dynamics in dense granular flows. Part II: simulations. J Appl Mech (T-ASME) 2007;74:691–702] together with a material model for the ice, imply that most of the kinetic energy dissipation occurs as a result of fracturing at the contact surface of the ice particles. The results obtained in the present study suggest that constitutive models such as those proposed by Brilliantov et al. [Brilliantov NV, Spahn F, Hertzsch JM, Poschel T. Model for collisions in granular gases. Phys Rev E;1996;53:5382–92] for collisions of ice particles are highly questionable.     

Energy method as solution for deformation of geosynthetic-reinforced embankment on Pasternak foundation

To more accurately analyze the settlement of geosynthetic-reinforced embankments on soft soil foundation, we simplified the ground surface structures as an Euler–Bernoulli beam, and the geosynthetic-reinforced layer as a Timoshenko beam. The granular fill and the soft soil foundation were both modelled using two-parameter Pasternak foundation models. We used energy method to establish energy balance equations for the system, and then we used the principle of resident potential energy to derive the governing differential equations of the settlement of the ground surface structures and the geosynthetic-reinforced layer. The MATLAB solver bvp4c was used to obtain numerical solutions for the settlement of the ground surface structures and the geosynthetic-reinforced layer on Pasternak foundations. By comparing to test results and existing models, the validity of the proposed solution is verified. This study analyzed the effects of parameters, such as flexural rigidity of the geosynthetic-reinforced layer, shear modulus of the granular fill, thickness of the soft soil foundation, and horizontal shear modulus of the Timoshenko beam, on the settlement of the ground surface structures and the geosynthetic-reinforced layer. The results showed that the model using a Pasternak foundation and a Timoshenko beam is more accurate at predicting settlement than that using a Winkler foundation and an Euler–Bernoulli beam.     

Discrete Lagrangian algorithm for finding geodesics on triangular meshes

The present paper introduces an approximation method for finding open geodesics on triangular surfaces. The algorithm is specifically designed to be able to solve real world problems where geodesic paths are needed. We use the model of geodesic curvature flow for open curves in the Lagrangian formulation. The model is enriched with a tangential term in order to have a control over the quality of the discretization grid during the computation. The governing equation of the flow is solved by a numerical method based on a semi-implicit time discretization and a finite difference space discretization. The paper presents the numerical scheme and various implementation details as well as numerous experiments to demonstrate the performance of the method and to provide comparison with several other well known methods. We also present a Grasshopper component for Rhinoceros for finding optimal paths on surface meshes that we developed and that includes our algorithm.     

On the equations of fully fluidized granular materials

Equations for fully fluidized granular materials are proposed and are solved in a simple case. In fully fluidized granular materials, the granular particles slip or collide with each other and energy is dissipated. In describing the energy dissipation process characteristic to granular materials, a measure of random motion of granular particles is introduced as a new internal variable. We derive the constitutive equations by using a simple kinematical model of the collision of particles. The set of equations for fully fluidized granular materials obtained has properties similar to the equations that describe turbulence. For reasonable assumptions, these equations predict the results of Bagnold, namely that the shear and normal stress depend upon the square of the velocity gradient. In case of steady one-dimensional gravity flow the calculated flow profiles resemble experimental ones.

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Résumé Des équations pour des matériaux granulés entièrement fluides sont proposées et résolues dans un cas simple. Par le frottement et les collisions des particules entre elles, de l'énergie est dissipée. Pour décrire l'énergie de dissipation, on introduit une mesure du mouvement aléatoire des particules comme nouvelle variable intense. Un module cinématique simple de la collision des particules permet d'établir les équations. Ces équations ont des propriétés semblables aux équations de la turbulence. Sous des hypothèses raisonnables, elles prédisent les résultats de Bagnold, à savoir que l'abrasion et la tension normale dépendent du gradient de la vitesse. Pour un flux de granité stable, les profils de flux calculés ressemblent à ceux obtenus expérimentalement.

     

Modeling the Oscillations of the CO + O2 Reaction Rate in a Granular Catalyst Layer

A new point model is proposed for the oxidation reaction of CO on the surface of a Pd cluster. The model reflects the oxidation–reduction mechanism on the palladium surface and incorporates all the stages of the kinetic scheme from the previous three-component model. A new assumption based on a series of experimental facts claims that the adsorbed oxygen atoms may diffuse into deeper-lying layers of the crystal lattice and thus influence the processes in the adsorption layer due to the micro size of the palladium clusters. The new point system has been built into the distributed general model for a granular catalyst developed previously. As a result, for parameter values corresponding to experimental conditions we have managed for the first time to obtain a wide region of chaos and complex mixed modes, close to those observed in real experiments.     

The choice of constitutive relations for an ice cover

The constitutive relations between the internal stresses and the deformation parameters of a sea ice cover, which are used in the AIDJEX elastoplastic model and Hibler's non-linearly viscous model, are investigated. It is shown that the structural instability of the ice cover with respect to plastic shear deformations is a consequence of the associated flow rule used in these models. The use of constitutive relations which violate the associated flow rule, but which are in good agreement with the physical properties of granular media, is suggested. An ice cover damage parameter and an empirical equation which describes the change in this parameter are introduced into the treatment. Energy relations are investigated.     

Vibrations of a beam with a damping tip body

We investigate a mathematical model for the dynamics of a beam with a tip body that experiences damping. The damping is due to granular material which partially fills the tip body. We establish the existence of the unique solution to the model and analyze the model. Among other things, we establish exponential energy decay when damping is present.     

Set-based granular computing: A lattice model

Set-based granular computing plays an important role in human reasoning and problem solving. Its three key issues constitute information granulation, information granularity and granular operation. To address these issues, several methods have been developed in the literature, but no unified framework has been formulated for them, which could be inefficient to some extent. To facilitate further research on the topic, through consistently representing granular structures induced by information granulation, we introduce a concept of knowledge distance to differentiate any two granular structures. Based on the knowledge distance, we propose a unified framework for set-based granular computing, which is named a lattice model. Its application leads to desired answers to two key questions: (1) what is the essence of information granularity, and (2) how to perform granular operation. Through using the knowledge distance, a new axiomatic definition to information granularity, called generalized information granularity is developed and its corresponding lattice model is established, which reveal the essence of information granularity in set-based granular computing. Moreover, four operators are defined on granular structures, under which the algebraic structure of granular structures forms a complementary lattice. These operators can effectively accomplish composition, decomposition and transformation of granular structures. These results show that the knowledge distance and the lattice model are powerful mechanisms for studying set-based granular computing.     

Streak patterns in binary granular media in a rotating drum

The Discrete Element Method (DEM) is used to understand the formation of radial streak patterns produced when binary granular material (which may differ either in size, density or shape) segregate in a slowly rotating drum. Our simulations show that initial streak formation requires temporal fluctuations in the particle bed’s strength. This, in turn, creates fluctuations in the slope and shape of the upper surface of the bed which control the particle avalanches down the free surface. These ultimately lead to streak formation. We conjecture that growth and stabilisation of a regular streak pattern requires the two sets of particles to have significantly different angles of repose.