<p>Flow in a hopper is both a fertile testing ground for understanding fundamental granular flow rheology and industrially highly relevant. Despite increasing research efforts in this area, a comprehensive physical theory is still lacking for both jamming and flow of granular materials in a hopper. In this work, I have designed a two dimensional (2D) hopper experiment using photoelastic particles ( particles' shape: disk or ellipse ), with the goal to build a bridge between macroscopic phenomenon of hopper flow and microscopic particle-scale dynamics. Through synchronized data of particle tracking and stress distributions in particles, I have shown differences between my data of the time-averaged velocity/stress profile of 2D hopper flow with previous theoretical predictions. I have also demonstrated the importance of a mechanical stable arch near the opening on controlling hopper flow rheology and suggested a heuristic phase diagram for the hopper flow/jamming transition. Another part of this thesis work is focused on studying the impact of particle shape of particles on hopper flow. By comparing particle-tracking and photoelastic data for ellipses and disks at the appropriate length scale, I have demonstrated an important role for the rotational freedom of elliptical particles in controlling flow rheology through particle tracking and stress analysis. This work has been supported by International Fine Particle Research Institute (IFPRI) .</p> / Dissertation

When a lander approaches a dusty surface, the plume from the descent engine impinges on the ground and entrains loose regolith into a high velocity spray. This problem exhibits a wide variety of complex phenomena such as highly under-expanded plume impingement, transition from continuum to free molecular flow, erosion, coupled gas-dust motions, and granular collisions for a polydisperse distribution of aerosolized particles. The focus of this work is to identify and model the important physical phenomena and to characterize the dust motion that would result during typical lunar landings. A hybrid continuum-kinetic solver is used, but most of the complex physics are simulated using the direct simulation Monte Carlo method. A descent engine of comparable size and thrust to the Lunar Module Descent Engine is simulated because it allows for direct comparison to Apollo observations. Steady axisymmetric impingement was first studied for different thrust engines and different hovering altitudes. The erosion profiles are obtained from empirically derived scaling relationships and calibrated to closely match the net erosion observed during the Apollo missions. Once entrained, the dust motion is strongly influenced by particle-particle collisions and the collision elasticity. The effects of two-way coupling between the dust and gas motions are also studied. Small particles less than 1 µm in diameter are accelerated to speeds that exceed 1000 m/s. The larger particles have more inertia and are accelerated to slower speeds, approximately 350 m/s for 11 µm grains, but all particle sizes tend obtain their maximum speed within approximately 40 m from the lander. The maximum particle speeds and erosion rates tend to increase as the lander approaches the lunar surface. The erosion rates scale linearly with engine thrust and the maximum particle speed increases for higher thrust engines. Dust particles are able to travel very far from the lander because there is no background atmosphere on the moon to inhibit their motion. The far field deposition is obtained by using a staged calculation, where the first stages are in the near field where the flow is quasi-steady and the outer stages are unsteady. A realistic landing trajectory is approximated by a set of discrete hovering altitudes which range from 20 m to 3 m. Larger particles are accelerated to slower speeds and are deposited closer to the lander than smaller particles. Many of the gas molecules exceed lunar escape speed, but some gas molecules become trapped within the dust cloud and remain on the moon. The high velocity particulate sprays can be damaging to nearby structures, such as a lunar outpost. One way of mitigating this damage is to use a berm or fence to shield nearby structures from the dust spray. This work attempts to predict the effectiveness of such a fence. The effects of fence height, placement, and angle as well as the model sensitivity to the fence restitution coefficient are discussed. The expected forces exerted on fences placed at various locations are computed. The pressure forces were found to be relatively small at fences placed at practical distances from the landing site. The trajectories of particles that narrowly avoid the fence were not significantly altered by the fence, suggesting that the dust motion is weakly coupled to the gas in the near vicinity of the fence. Future landers may use multi-engine configurations that can form 3-dimensional gas and dust flows. There are multiple plume-plume and plume-surface interactions that affect the erosion rates and directionality of the dust sprays. A 4-engine configuration is simulated in this work for different hovering altitudes. The focusing of dust along certain trajectories depends on the lander hovering altitude, where at lower altitudes the dust particles focus along symmetry planes while at higher altitudes the sprays are more uniform. The surface erosion and trenching behavior for a 4-engine lander are also discussed. / text

Μελετάμε την ροή κοκκώδους σε κεκλιμένη πειραματική διάταξη δυο διαστάσεων, αποτελούμενη από Κ το πλήθος γραμμές και Μ το πλήθος στήλες δοχείων, τα οποία αναταράσσονται κάθετα. Η ροή του υλικού από δοχείο σε δοχείο περιγράφεται από ένα μοντέλο ροής [Eggers, 1999; Van der Weele, 2008]. Υποκινούμενοι από δυσλειτουργίες που παρουσιάζονται στις σύγχρονες βιομηχανικές μονάδες μεταφοράς (όπως είναι ο σχηματισμός συσσωματωμάτων), εισάγουμε σταθερή ποσότητα υλικού στην πρώτη γραμμή των δοχείων, και καθορίζουμε τις συνθήκες κάτω από τις οποίες η ροή παραμένει ομαλή και συνεχής μέχρι την τελευταία γραμμή. Ενώ στην περίπτωση μιας και μόνο σειράς δοχείων (Μ=1) η εκροή μηδενίζεται με την εμφάνιση ενός και μόνο συσσωματώματος [Κανελλόπουλος, 2008], για Μ>1 απαιτούνται περισσότερα συσσωματώματα για τον μηδενισμό της. Μελετάμε τον τρόπο με τον οποίο αυτά τα συσσωματώματα διατάσσονται στα δοχεία, ο οποίος πολλές φορές όπως βλέπουμε μπορεί να είναι ιδιαίτερα πολύπλοκος, αποτελώντας έτσι ένα εξαίρετο παράδειγμα μορφογένεσης σε δυναμικά συστήματα [Cross and Hohenberg, 1993]. Εντοπίζουμε τα βασικά χαρακτηριστικά αυτής της μορφογένεσης και εξηγούμε πως αυτά σχετίζονται με το μοντέλο ροής. Για την περαιτέρω μαθηματική και φυσική τους ερμηνεία προτείνουμε το συνεχές όριο του μοντέλου ροής, το οποίο θα αποτελέσει την απαρχή για μελλοντικές έρευνες [Van der Weele et al, 2008]. Αναφορές: •J. Eggers, Sand as Maxwell’s demon, Phys. Rev. Lett. 83, 5322 (1999). •K. van der Weele, Granular gas dynamics: How Maxwell’s demon rules in a nonequilibrium system, Contemporary Phys. 49, 157-175 (2008). •Γ. Κανελλόπουλος, Οριακή ροή κοκκώδους υλικού σε διάδρομο μεταφοράς, Διπλωματική Εργασία, Τμήμα Μαθηματικών, Πανεπιστήμιο Πατρών (2008). •M.C. Cross and P.C. Hohenberg, Pattern formation outside of equilibrium, Rev.Mod.Phys. 65, 851 (1993). •K. van der Weele, G. Kanellopoulos, C. Tsiavos, and D. van der Meer, Transient granular shock waves and upstream motion on a staircase (submitted, 2009). / We study the dynamics of granular material in a 2-dimensional tilted setup, consisting of K rows and M columns of equal sized compartments, which is brought into motion by vertical shaking. Particles are inserted into the system along the upper row of compartments, then flow downwards, and eventually exit the system from the bottom row. Similar setups are encountered frequently in industrial transport machinery and are notorious for the tendency of the particles to form dense clusters, which obstruct the flow. We model the particle flow from box to box by a flux function and determine the maximal value of the inflow rate for which the flow remains steady. If the inflow rate exceeds this value, clusters are formed. The way in which these clusters are distributed over the KxM compartments is a fine example of spontaneous pattern formation. We analyze the main characteristics of this cluster formation, its dependence on the various parameters of the system, and its sensitivity to experimental fluctuations.

<p>Granular impact, where a foreign object strikes a granular material like sand, is common in nature and industry. Due to experimental difficulties in obtaining sufficiently fast data at the scale of a single grain, a description of this process which connects to physics at the grain-scale is lacking. In this thesis, I will present data from a series of two-dimensional granular impact experiments. By cutting each grain out of a photoelastic material and using a very fast camera, we obtain data on the intruder trajectory, as well as the particle flow and force response of the granular material. Past experiments have shown that the decelerating force on an intruder moving through a granular medium is often well captured by a force law which is dominated by a velocity-squared drag force. Using the intruder trajectories, as well as the flow and force response of the granular material, I will demonstrate that, while these force laws describe the intruder trajectories on slow time scales, the instantaneous force on the intruder is highly fluctuating in space and time. I will particularly focus on the velocity-squared drag force, showing that it arises from random, locally normal collisions with chain-like clusters of particles which send energy and momentum away into the granular material. In this regime, the particles and intruder reach a kind of adiabatic steady state, where the particle motion scales linearly with the intruder speed. However, for impact velocities which are fast compared to the rate of momentum transfer within the granular material, the system response qualitatively changes, behaving like an elastic solid with a shock-like response at impact.</p> / Dissertation

Granular materials are composed of solid, discrete particles and exhibit mechanical properties that range from fluid to solid behavior. Some of the complexity exhibited by granular systems arises due to the long-range order that develops due to particle-particle contact. Inter-particle forces in granular materials often form a distributive network of filamentary force-accommodating chains (i.e. force chains), such that a fraction of the total number of particles accommodates the majority of the forces in the system. The force chain network inherent to a system composed of granular materials controls the macroscopic behavior of the granular material. Force transmission by these filamentary chains is focused (or localized) to the grain scale at boundaries such as the granular flow substrate. Recent laboratory experiments have shown that force chains transmit extreme localized forces to the substrates of free surface granular flows. In this work we combine analog and numeric experimental approaches to investigate the forces at the bed of a simplified granular flow. A photoelastic experimental approach is used to resolve discrete forces in the granular flows. We also conduct discrete element method (DEM) simulations, using input parameters derived from measureable physical material properties of experimental and natural materials, which successfully reproduce the analog experimental results. This work suggests that force chain activity may play an unexpected and important role in the bed physics of dense granular flows through substrate modification by erosion and entrainment, and that DEM numerical methods effectively treat force chain processes in simulated granular flows.

Dry Powder Inhalers (DPIs) have great potential in pulmonary drug delivery; the granular powder, used as active ingredient in DPIs, is ozone friendly and the operation of DPIs ensures coordination between dose release and patient inhalation. However, the powder fluidisation mechanisms are poorly understood which leads to low efficiency of DPIs with 10-35 % of the dose reaching the site of action. The main aim of this thesis is to study the hydrodynamics of powder fluidisation in DPIs, using experimental and computational approaches. An experimental test rig was developed to replicate the process of transient powder fluidisation in an impinging air jet configuration. The powder fluidisation chamber was scaled up resulting in a two dimensional particle flow prototype, which encloses 3.85 mm glass beads. Using optical image processing techniques, individual particles were detected and tracked throughout the experimental time and domain. By varying the air flow rate to the test section, two particle fluidisation regimes were studied. In the first fluidisation regime, the particle bed was fully fluidised in less than 0.25 s due to the strong air jet. Particle velocity vectors showed strong convective flow with no evidence of diffusive motion triggered by inter-particle collisions. In the second fluidisation regime, the particle flow experienced two stages. The first stage showed strong convective flow similar to the first fluidisation regime, while the second stage showed more complex particle flow with collisional and convective flow taking place on the same time and length scales. The continuum Two Fluid Model (TFM) was used to solve the governing equations of the coupled granular and gas phases for the same experimental conditions. Sub-models for particle-gas and particle-particle interactions were used to complete the model description. Inter-particle interactions were resolved using models based on the kinetic theory of granular flow for the rapid flow regime and models based on soil mechanics for the frictional regime. Numerical predictions of the first fluidisation regime showed that the model should incorporate particle-wall friction and minimise diffusion, simultaneously. Ignoring friction resulted in fluidisation timing mismatch, while increasing the diffusion resulted in homogenous particle fluidisation in contrast to the aggregative convective fluidisation noticed in the experiments. Numerical predictions of the second fluidisation regime agreed well with the experiments for the convection dominated first stage of flow up to 0.3 s. However, later stages of complex particle flow showed qualitative discrepancies between the experimental and the computational approaches suggesting that current continuum granular models need further development. The findings of the present thesis have contributed towards better understanding of the mechanics of particle fluidisation and dense multiphase flow in DPI in particular, and particle bed fluidisation using impinging air jet in general. The use of TFM for predicting high speed convective granular flows, such as those in DPIs, is promising. Further studies are needed to investigate the form of particle-particle interactions within continuum granular flow models.

This thesis presents an investigation on multiphase reactor hydrodynamics using magnetic resonance imaging (MRI). The study demonstrates experimental techniques by which computational and quasi-analytical fluid models may be validated. Three types of industrially-important multiphase reaction vessels are considered: a co-current upflow gas-liquid-solid bed, a co-current downward trickle bed (gas, liquid, solid), and a gas-solid fluidised bed. These reactors were selected as they commonly demonstrate local hydrodynamic anisotropy which affects the global performance of industrial units. MRI was used to obtain 2D velocity images of the gas and liquid phases in the packed beds, and of the gas and the solid phases in the fluidised bed. This study reports the first spatially resolved velocity measurements of both the gas and liquid phases in a co-current upflow bed, and the gas and solid phases of an isolated bubble in a fluidised bed. The experimental vessels were: 52 mm in diameter using 5 mm glass spheres in the upflow bed at 8 bara, 27 mm with 5 mm glass spheres in the trickle bed at 6.75 bara, and 52 mm using 1.2 mm poppy seeds as the fluidised particles at 8.5 bara. The experiments were conducted at a laboratory temperature of 25.0 ± 3.0 °C. In the upflow bed, time-averaged velocity images were acquired over a 2.5 h experimental time. This was done to capture the steady state behaviour of the vessel operating in the pulsing flow regime. The temporally-stable trickle flow state in the trickle bed was imaged over 15-100 minutes. In both packed beds, severe spatial anisotropy in the distribution of flow between pores was revealed. Furthermore, the data were used to determine classical design features such as catalyst wetting and liquid holdup which compared well with literature models. The trickle bed data were further analysed using a morphological algorithm which unambiguously identified the gas-liquid and liquid-solid interfaces. The interfacial flow fields were found to be similar to the bulk flow, with most voxels exhibiting static behaviour. The amount of interaction between the phases was found to be minimal, which is typical of the low interaction regime. A single bubble injection system was employed in the fluidised bed which allowed the injection of isolated bubbles into the incipiently fluidised bed. It also enabled the triggered acquisition of NMR data at precise time intervals. The bubble was found to be an indented ellipsoidal shape, which rose with atypical behaviour which caused it to collapse. Rise velocity was found to be consistent with theory, and the injected bubbles were sufficiently spatially reproducible to acquire 2D velocity images using single-point imaging. These velocity images showed flow behaviour characteristic of a 'fast' rising bubble, with a gas recirculation cloud 37 mm in diameter. The particle field was shown to have very high flow in the bubble wake, revealing the mechanism of bubble collapse. The flow data were compared to classical two-phase fluidisation theory, which revealed noteworthy differences in the division of flow between the particulate and bubbling regions.

In this dissertation, an experimental and numerical study of dense coarse solids-liquid flows has been performed. The experimental work mainly involved pressure drop measurements in a vertical flow loop. A limited number of measurements of solids velocity profiles were also obtained in the upward flow section of the flow loop. The numerical work involved simulations of coarse particles-in-water flows in vertical and horizontal pipes. The vertical flow simulations were performed using the commercial CFD software, ANSYS CFX-4.4, while ANSYS CFX-10 was used to simulate the flows in the horizontal pipes. The simulations were performed to investigate the applicability of current physically-based models to very dense coarse-particle flows.<p> In the experimental study, measurements of pressure drop and local solids velocity profiles were obtained. The experiments were conducted in a 53 mm diameter vertical flow loop using glass beads of 0.5 mm and 2.0 mm diameter solids for concentration up to 45%. The liquid phase was water. The measured pressure drop exhibited the expected dependence on bulk velocity and solids mean concentration. The wall shear stress was determined by subtracting the gravitational contribution from the measured pressure drop. For flow with the 0.5 mm particles at high bulk velocities, the values of the wall shear stress were essentially similar for each concentration in the upward flow sections but more variation, indicating the effect of concentration, was noted in the downward flow section. At lower bulk velocities, the wall shear stresses with the 0.5 mm glass beads-water flow showed a dependence on concentration in both test sections. This was attributed to an increase in the slip velocity. For the large particle (2.0 mm glass beads), similar observations were made but the effect of concentration was much less in the upward test section. In the downward test section, the wall shear stress for the flow of the 2.0 mm glass beads increased by almost a constant value for the bulk velocities investigated. The solids velocity profiles showed that the solids velocity gradient is large close to the wall. In addition, the solids velocity profiles indicated that the slip velocity increased at lower velocities due to increase in the bulk concentration in the upward flow section.<p> For the vertical flow simulations, different physical models based on the kinetic theory of granular flows were programmed and implemented in ANSYS CFX-4.4. These models, referred to as the kf-ef-ks-es, kf-ef-ks-es-Ts and kf-ef-ks-kfs models, were investigated by focusing on the closure laws for the solids-phase stress. The treatment of the granular temperature Ts depends on whether small- or large-scale fluctuating motion of the particles is considered. The models were implemented via user-Fortran routines. The predicted results were compared with available experimental results. The predicted solids-phase velocity profiles matched the measured data quite well close to the pipe wall but over-predicted it in the core region. The solids concentration, on the other hand, was significantly under-predicted for concentrations higher than 10%. Variations in the predictions of the phasic turbulent kinetic energy and the eddy viscosity were noted; the effect of solids concentration on them was mixed. A general conclusion drawn from the work is that a more accurate model is required for accurate and consistent prediction of coarse particle flows at high concentrations (less than 10%). In a related study, attention was given to wall boundary conditions again focusing on the effect of the solids-phase models at the wall. Comparison between numerical predictions, using some of the existing wall boundary condition models for the solids phase in particulate flows, with experimental results indicated that the physical understanding of the influence of the fluid and solids-phase on each other and their effect on frictional head loss is far from complete. The models investigated failed to reproduce the experimental results. At high solids concentration, it was apparent from the present study that the no-slip and free-slip wall boundary conditions are not appropriate for liquid-solid flows.<p> For the horizontal flow case, three-dimensional simulations were performed with a focus on the velocity and concentration distributions. Medium and coarse sand-in-water flows in three pipe diameters were considered to investigate the default solids stress models in ANSYS CFX-10. Simulations were performed for three cases by considering: 1) no additional solids-phase stress, i.e. no model for Ts; 2) a zero equation, and 3) an algebraic equilibrium model for the granular temperature. The model predictions were compared to experimental results. The effect of particle size, solids-phase concentration, and pipe diameter was explored using the algebraic equilibrium model. All the cases for the models considered exhibited the characteristic features of horizontal coarse particle slurry flows. The zero equation and the algebraic equilibrium model for the granular temperature produced similar results that were not significantly different from the prediction obtained when no solids-phase stress was considered. The comparison with experimental results was mixed. Locally, the measured solids-phase velocity distributions were over-predicted, whereas the solids concentration was reasonably reproduced in the core of all the pipes. The concentration at the bottom and top walls were over-, and under-predicted, respectively. This was attributed to the inappropriate phasic wall boundary condition models available.

In this dissertation, an experimental and numerical study of dense coarse solids-liquid flows has been performed. The experimental work mainly involved pressure drop measurements in a vertical flow loop. A limited number of measurements of solids velocity profiles were also obtained in the upward flow section of the flow loop. The numerical work involved simulations of coarse particles-in-water flows in vertical and horizontal pipes. The vertical flow simulations were performed using the commercial CFD software, ANSYS CFX-4.4, while ANSYS CFX-10 was used to simulate the flows in the horizontal pipes. The simulations were performed to investigate the applicability of current physically-based models to very dense coarse-particle flows.<p> In the experimental study, measurements of pressure drop and local solids velocity profiles were obtained. The experiments were conducted in a 53 mm diameter vertical flow loop using glass beads of 0.5 mm and 2.0 mm diameter solids for concentration up to 45%. The liquid phase was water. The measured pressure drop exhibited the expected dependence on bulk velocity and solids mean concentration. The wall shear stress was determined by subtracting the gravitational contribution from the measured pressure drop. For flow with the 0.5 mm particles at high bulk velocities, the values of the wall shear stress were essentially similar for each concentration in the upward flow sections but more variation, indicating the effect of concentration, was noted in the downward flow section. At lower bulk velocities, the wall shear stresses with the 0.5 mm glass beads-water flow showed a dependence on concentration in both test sections. This was attributed to an increase in the slip velocity. For the large particle (2.0 mm glass beads), similar observations were made but the effect of concentration was much less in the upward test section. In the downward test section, the wall shear stress for the flow of the 2.0 mm glass beads increased by almost a constant value for the bulk velocities investigated. The solids velocity profiles showed that the solids velocity gradient is large close to the wall. In addition, the solids velocity profiles indicated that the slip velocity increased at lower velocities due to increase in the bulk concentration in the upward flow section.<p> For the vertical flow simulations, different physical models based on the kinetic theory of granular flows were programmed and implemented in ANSYS CFX-4.4. These models, referred to as the kf-ef-ks-es, kf-ef-ks-es-Ts and kf-ef-ks-kfs models, were investigated by focusing on the closure laws for the solids-phase stress. The treatment of the granular temperature Ts depends on whether small- or large-scale fluctuating motion of the particles is considered. The models were implemented via user-Fortran routines. The predicted results were compared with available experimental results. The predicted solids-phase velocity profiles matched the measured data quite well close to the pipe wall but over-predicted it in the core region. The solids concentration, on the other hand, was significantly under-predicted for concentrations higher than 10%. Variations in the predictions of the phasic turbulent kinetic energy and the eddy viscosity were noted; the effect of solids concentration on them was mixed. A general conclusion drawn from the work is that a more accurate model is required for accurate and consistent prediction of coarse particle flows at high concentrations (less than 10%). In a related study, attention was given to wall boundary conditions again focusing on the effect of the solids-phase models at the wall. Comparison between numerical predictions, using some of the existing wall boundary condition models for the solids phase in particulate flows, with experimental results indicated that the physical understanding of the influence of the fluid and solids-phase on each other and their effect on frictional head loss is far from complete. The models investigated failed to reproduce the experimental results. At high solids concentration, it was apparent from the present study that the no-slip and free-slip wall boundary conditions are not appropriate for liquid-solid flows.<p> For the horizontal flow case, three-dimensional simulations were performed with a focus on the velocity and concentration distributions. Medium and coarse sand-in-water flows in three pipe diameters were considered to investigate the default solids stress models in ANSYS CFX-10. Simulations were performed for three cases by considering: 1) no additional solids-phase stress, i.e. no model for Ts; 2) a zero equation, and 3) an algebraic equilibrium model for the granular temperature. The model predictions were compared to experimental results. The effect of particle size, solids-phase concentration, and pipe diameter was explored using the algebraic equilibrium model. All the cases for the models considered exhibited the characteristic features of horizontal coarse particle slurry flows. The zero equation and the algebraic equilibrium model for the granular temperature produced similar results that were not significantly different from the prediction obtained when no solids-phase stress was considered. The comparison with experimental results was mixed. Locally, the measured solids-phase velocity distributions were over-predicted, whereas the solids concentration was reasonably reproduced in the core of all the pipes. The concentration at the bottom and top walls were over-, and under-predicted, respectively. This was attributed to the inappropriate phasic wall boundary condition models available.

[pt] O propósito deste trabalho é propor um modelo teórico axiomático, que descreva o comportamento termomecânico de materiais granulares, fundamentado em uma descrição física simples, do que entende-se por material granular. No capítulo I, faz-se uma revisão bibliográfica, objetivando uma visão geral do estado da arte e delinear a importância do estudo de materiais granulares. No capítulo II, utiliza-se um método axiomático estabelecendo-se as equações de balanço que descrevem o comportamento termodinâmico de materiais granulares. No capítulo III, utilizando-se um outro método axiomático, analisa-se algumas condições de contorno para materiais granulares. No capítulo IV, estuda-se restrições internas em materiais granulares, exemplificando-se algumas. Finalmente no capítulo V, aplica-se o embasamento teórico dos capítulos anteriores, na solução do problema de escoamento de um material granular entre placas paralelas. / [en] The aim of this work is to establish an axiomatic theoretical model, for description of the thermomechanical behavior of granular materials, based on a simple physical description of what mean granular materials. On the first chapter, it s make an overview of the acctual state of the arte, giving the real importance of the study of granular materials. On the second chapter, by using an axiomatic method, the balances equations for thermomechanical behavior of granular materials, was established. On the the III, by using another axiomatic method, some boundary condictions for granular materials, was analysed. On the chapter IV, by using a coceptual model for internal constraints, some internal constraints for granular materials, was exemplified. At least, on chapter V, the theoretical backgoung established in the previous chapter, is applied to solve the problem of granular flow between two parallel plates.