Showing papers in "Granular Matter in 2009"

Influence of relative density on granular materials behavior: DEM simulations of triaxial tests

Abstract: The rheological behavior of non-cohesive soils results from the arrangement and complex geometry of the grains. Numerical models based on discrete element modeling provides an opportunity to understand these phenomena while considering the discrete elements with a similar shape to that of the grains the soil is composed of. However, dealing with realistic shapes would lead to a prohibitive calculation cost. In a macroscopic modeling approach, simplification of the discrete elements’ shape can be done as long as the model can predict experimental results. Since the intrinsic non-convex geometry property of real grains seems to play a major role on the response of the granular medium, it is thus possible to keep this geometrical feature by using cluster of spherical discrete elements, which has the advantage to reduce dramatically the computation cost. Since the porosities found experimentally could not always be obtained with the numerical model—owing to the huge difference in shape, the notion of relative density, which requires a search for minimum and maximum porosities for the model, was chosen to compare the experimental and numerical results. Comparing the numerical simulations with the experimental triaxial tests conducted with relative densities and different confining pressures shows that the model is able to predict the experimental results.

Cyclic loading of railway ballast under triaxial conditions and in a railway test facility

Abstract: A recently developed large-scale triaxial test apparatus for railway ballast testing comprises a double-cell arrangement for measuring volume change by differential pressure. Monotonic and cyclic tests were performed on limestone ballast samples. Axial and volumetric strains and breakage were determined from both types of test. Resilient modulus and Poisson’s ratio were obtained only from the cyclic tests. The permanent axial strain and breakage results from the cyclic tests are compared with the simulated traffic loading in the railway test facility (RTF) which comprises three sleepers embedded in ballast over a subgrade. The traffic loading in the RTF was applied by hydraulic actuators with built-in displacement transducers. A column of painted ballast was placed under a rail seat of the middle sleeper to ease sample collection for sieve analysis at the end of the test. The stress condition in the RTF is predicted by a simple calculation based on findings of previous literature. It was found that the results from the cyclic triaxial test with conditions similar to the predicted conditions in the RTF were comparable to those results from the RTF tests.

A spherical discrete element model: calibration procedure and incremental response

Abstract: When using spherical elements within the Discrete Element Method, computational costs can be kept low even for large numbers of elements. However, this oversimplification of the granular geometry has drawbacks when quantitatively assessing the model even for frictional geomaterials. To overcome this limitation, the local constitutive law must at least take into account the transfer of a moment between elements. This moment, which is added to normal and shear local interaction forces, increases the number of local parameters. Moreover, when local plastic thresholds are considered, the calibration of the model becomes tricky. With such a set of local parameters, a calibration procedure is proposed, which attempts to define the respective role of each parameter in the macroscopic behavior. A series of numerical simulations of triaxial compression tests has been performed to check the capability of this model to get good quantitative results and the incremental behavior of the numerical medium is studied by performing a series of axisymmetric stress probes with varying directions. The corresponding strain responses are measured. From different initial stress states, the results indicate that the incremental response is well described by elastoplasticity with a single mechanism, and a non-associative flow rule.

Granular solid hydrodynamics

Abstract: A complete continuum mechanical theory for granular media, including explicit expressions for the energy current and the entropy production, is derived and explained. Its underlying notion is: granular media are elastic when at rest, but turn transiently elastic when the grains are agitated—such as by tapping or shearing. The theory includes the true temperature as a variable, and employs in addition a granular temperature to quantify the extent of agitation. A free energy expression is provided that contains the full jamming phase diagram, in the space spanned by pressure, shear stress, density and granular temperature. We refer to the theory as GSH, for granular solid hydrodynamics. In the static limit, it reduces to granular elasticity, shown previously to yield realistic static stress distributions. For steady-state deformations, it is equivalent to hypoplasticity, a state-of-the-art engineering model.

On contact numbers in random rod packings

Abstract: Random packings of non-spherical granular particles are simulated by combining mechanical contraction and molecular dynamics, to determine contact numbers as a function of density. Particle shapes are varied from spheres to thin rods. The observed contact numbers (and packing densities) agree well with experiments on granular packings. Contact numbers are also compared to caging numbers calculated for sphero-cylinders with arbitrary aspect-ratio. The caging number for rods arrested by uncorrelated point contacts asymptotes towards = 9 at high aspect ratio, strikingly close to the experimental contact number ≈ 9.8 for thin rods. These and other findings confirm that thin-rod packings are dominated by local arrest in the form of truly random neighbor cages. The ideal packing law derived for random rod–rod contacts, supplemented with a calculation for the average contact number, explains both absolute value and aspect-ratio dependence of the packing density of randomly oriented thin rods.

Granular packing: numerical simulation and the characterisation of the effect of particle shape

Abstract: The packing of granular particles is investigated using a combined finite-discrete element approach One of the aims of this paper is to present an application of a recently improved numerical simulation technique for deformable granular material with arbitrary shapes Our study is focused on the influence of the effect of the particle shape on (1) the emergent properties of a granular pack (packing density, coordination number, force distribution), and on (2) the spatial distribution of the stress A set of simulations that mimick the sedimentation process is carried out, with varying input parameters, such as contact friction and particle shape It is shown that the eccentricity of the particles not only significantly influences the final density of the pack but also the distribution of the stress and the contact forces The presence of surface friction increases the amount of disorder within the granular system Stress heterogeneities and force chain patterns propagate through the particles more efficiently than for the frictionless systems The results also suggest that for the monodisperse systems investigated the coordination number is one of the factors that controls the distribution of the stress within a granular medium

Flow–obstacle interaction in rapid granular avalanches: DEM simulation and comparison with experiment

Abstract: This paper investigates the interaction between rapid granular flow and an obstacle. The distinct element method (DEM) is used to simulate the flow regimes observed in laboratory experiments. The relationship between the particle properties and the overall flow behaviour is obtained by using the DEM with a simple linear contact model. The flow regime is primarily controlled by the particle friction, viscous normal damping and particle rotation rather than the contact stiffness. Rolling constriction is introduced to account for dispersive flow. The velocity depth-profiles around the obstacles are not uniform but varying over the depth. The numerical results are compared with laboratory experiments of chute flow with dry granular material. Some important model parameters are obtained, which can be used to optimize defense structures in alpine regions.

An efficient algorithm for granular dynamics simulations with complex-shaped objects

Abstract: One of the most difficult aspect of the realistic modeling of granular materials is how to capture the real shape of the particles. Here we present a method to simulate two-dimensional granular materials with complex-shaped particles. The particle shape is represented by the classical concept of a Minkowski sum, which permits the representation of complex shapes without the need to define the object as a composite of spherical or convex particles. A well defined interaction force between these bodies is derived. The algorithm for identification of neighbor particles reduces force calculations to O(N), where N is the number of particles. The method is much more efficient, accurate and easier to implement than other models. We prove that the algorithm is consistent with energy conservation, which is numerically verified using non-dissipative granular dynamics simulations. Biaxial test simulations on dissipative granular systems demonstrate the relevance of shape in the strength and stress fluctuations at the critical state.

A geometric algorithm based on tetrahedral meshes to generate a dense polydisperse sphere packing

Abstract: In the discrete element method, the packing generation of polydisperse spheres with a high packing density value is a major concern. Among the methods already developed, few algorithms can generate sphere packing with a high density value. The aim of this paper is to present a new geometric algorithm based on tetrahedral meshes to generate dense isotropic arrangements of non-overlapping spheres. The method consists of first filling in every tetrahedron with spheres in contact (i.e., hard-sphere clusters). Then, the algorithm increases the packing density value by detecting the large empty spaces and filling them with new spheres. This new geometric algorithm can also generate a complex shape structure.

Critical state behaviour of granular materials from isotropic and rebounded paths: DEM simulations

Abstract: This paper presents the results of numerical simulations using the three-dimensional discrete element method (DEM) on the critical state behaviour of isotropically compressed and rebounded assemblies of granular materials. Drained and undrained (constant volume) numerical simulations were carried out. From these numerical simulations of drained and undrained tests, it has been shown that the steady state is same as the critical state. Critical state for both isotropically compressed and rebounded assemblies form unique curved line that can be approximated by a bilinear line as proposed by Been et al. [G,otechnique 41(3): 365-381, 1991]. Further more, evolution of the internal variables such as average coordination number and induced anisotropy coefficients during shear deformation has been studied.

Ratcheting convective cells of sand grains around offshore piles under cyclic lateral loads

Abstract: Sand densification around the pile has traditionally been regarded as an explanation for the grain migration and soil subsidence that often occur around cyclic laterally loaded piles embedded in sand Supported by new empirical evidence, this paper proposes that, additionally to some soil densification around the pile, the main cause for the continuous “steady-state” grain migration is a convective cell flow of sand grains in the vicinities of the pile head Such convective flow would be caused by a ratcheting mechanism triggered by the cyclic low-frequency lateral displacements of the pile Furthermore, the experimental results suggest that the limit between the convective cell and the static soil is marked by a distinct direct shear surface This might shed some light into the complex phenomena related to the pile-soil interaction in the upper layers of the bedding, which are normally the main contributor for the lateral load-bearing capacity of piles

A finite deformation method for discrete modeling: particle rotation and parameter calibration

Abstract: We present a finite deformation method for 3-D discrete element modeling. In this method particle rotation is explicitly represented using quaternion and a complete set of interactions is permitted between two bonded particles, i.e., normal and tangent forces, rolling and torsional torques. Relative rotation between two particles is decomposed into two sequence-independent rotations, such that an overall torsional and rolling angle can be distinguished and torques caused by relative rotations are uniquely determined. Forces and torques are calculated in a finite deformation fashion, rather than incrementally. Compared with the incremental methods our algorithm is numerically more stable while it is consistent with the non-commutativity of finite rotations. We study the macroscopic elastic properties of a regularly arranged 2-D and 3-D lattice. Using a micro-to-macro approach based on the existence of a homogeneous displacement field, we study the problem of how to choose the particle-scale parameters (normal, tangent, rolling and torsional stiffness) given the macroscopic elastic parameters and geometry of lattice arrangement. The method is validated by reproducing the wing crack propagation and the fracture patterns under uniaxial compression. This study will provide a theoretical basis for the calibration of the DEM parameters required in engineering applications.

Investigation of particle damping mechanism via particle dynamics simulations

Abstract: Damping systems using granular particles as the damping medium are promising for application in extreme temperature conditions. In particle-based damping systems, mechanical energy is dissipated through inelastic collisions and friction between particles. In this work, we use particle dynamics simulations to investigate and compare the damping mechanisms of a piston-type thrust damper and a box-type oscillation damper. The mechanisms of energy transfer and energy dissipation are investigated. The roles of friction and inelastic collisions, as well as the wall effects in energy dissipation, are examined. The simulation results provide better understanding of the particle damping mechanisms, which may help in the design of next generation particle damping devices.

Cross-sectional and axial flow characteristics of dry granular material in rotating drums

Abstract: Cross-sectional and axial flow behaviors of dry granular material in rotating drums are closely related to the dynamic characteristics and velocity distributions between the surface layer and bed material. In this study, both 2D and 3D dry granular flow patterns in horizontal rotating drums are experimentally investigated with flow imaging analysis. A dimensionless flow parameter combining the effects of Froude number, relative particle size and volume filling is proposed in this study, which controls the flow characteristics in a rational drum such as dynamic angle of repose, thickness of the flowing layer, relative free surface velocity, and the shear rates in the flowing layer. The dimensionless granular temperature exhibits linear distribution in the flowing layer, being maximum at the free surface and being negligible at the interface in the rolling regime. The measured shear rate of the plug flow departs from drum angular velocity under the wall slip conditions when the drum surface is smooth. Due to the existence of axial convection and lateral surface profile, the mass flux in the flowing layer is always less than that of the plug flow in the 3D granular flows based on sidewall particle images. One the other hand, the mass flux in the flowing layer is always equal or greater than that of the plug flow in the 2D granular flows. 2D granular flows exhibit higher angles of repose and surface velocities than those of the 3D granular flows at the same volume fillings.

The dune size distribution and scaling relations of barchan dune fields

Abstract: Barchan dunes emerge as a collective phenomena involving the generation of thousands of them in so called barchan dune fields. By measuring the size and position of dunes in Moroccan barchan dune fields, we find that these dunes tend to distribute uniformly in space and follow an unique size distribution function. We introduce an analytical mean-field approach to show that this empirical size distribution emerges from the interplay of dune collisions and sand flux balance, the two simplest mechanisms for size selection. The analytical model also predicts a scaling relation between the fundamental macroscopic properties characterizing a dune field, namely the inter-dune spacing and the first and second moments of the dune size distribution.

A new method to implement comminution functions into DEM simulation of a size reduction system due to particle-wall collisions

Abstract: The ability to design a size reduction system prior to full scale experiments and to optimize existing systems has long been a goal of designers. Such a design and optimization could be achieved by correctly simulating any system under any operating condition. In this paper we present a new and innovative procedure to implement empirical comminution functions into DEM–CFD simulations. The paper is focused on the implementation procedures and not the DEM/CFD simulations, which deserve full attention. Therefore, this paper is not aimed to study any specific mill. The comminution functions include: initial strength distribution, selection function, breakage function and fatigue function. First, the traditional comminution functions (strength distribution, selection and breakage functions) and the recently investigated fatigue function are briefly described and modified. Then a procedure for implementing the functions into a DEM–CFD model or any other source to provide impact velocities and number of impacts, is described in detail. The implementation involves converting the probability comminution functions into individual particle properties by a random method and then converting the velocity dependent comminution functions into strength dependent ones. In this way, and mainly owing to the use of the fatigue function (which defines the weakening of those particles that are not breaking), a real size reduction system, in which each particle is subjected to multiple impacts at various velocities can be simulated. Three case studies for multiple impact conditions at the same average velocity (several impacts at the same velocity, various velocities at each impact and randomly selected velocities) are presented and analyzed in order to confirm qualitatively the procedure, although the comminution functions need to be further quantitatively modified. It should be emphasized that although the new procedure presents a step towards the final goal, some limitations do exist and some questions remain open.

The second and third Sonine coefficients of a freely cooling granular gas revisited

Abstract: In its simplest statistical-mechanical description, a granular fluid can be modeled as composed of smooth inelastic hard spheres (with a constant coefficient of normal restitution α) whose velocity distribution function obeys the Enskog–Boltzmann equation. The basic state of a granular fluid is the homogeneous cooling state, characterized by a homogeneous, isotropic, and stationary distribution of scaled velocities, F(c). The behavior of F(c) in the domain of thermal velocities (c ~ 1) can be characterized by the two first non-trivial coefficients (a 2 and a 3) of an expansion in Sonine polynomials. The main goals of this paper are to review some of the previous efforts made to estimate (and measure in computer simulations) the α-dependence of a 2 and a 3, to report new computer simulations results of a 2 and a 3 for two-dimensional systems, and to investigate the possibility of proposing theoretical estimates of a 2 and a 3 with an optimal compromise between simplicity and accuracy.

A packing algorithm for three-dimensional convex particles

Abstract: Simulation of granular particles is an important tool in many fields. However, simulation of particles of complex shapes remains largely out of reach even in two-dimension. One of the major hurdles is the difficulty in representing particles in an efficient, flexible, and accurate manner. By representing particles as convex polyhedrons which are themselves the intersection of a set of half spaces, we develop a method that allows one to efficiently carry out key operations, including particle–particle and particle–container wall overlapping detection, precise identification of the overlapping region, particle shifting, particle rotation, and others. The simulation of packing 1,000 particles into a container takes only a few minutes with this approach. We further demonstrate the potential of this approach with a simulation that re-generates the “Brazil nut” phenomenon by mixing and shaking particles of two different sizes.

The role of friction in mixing and segregation of granular material

Abstract: We use discrete element modelling to investigate the processes of mixing and size segregation in a polydisperse mixture of spherical particles in a three-dimensional rectangular box and analyze the influence of friction between the particles on segregation. The packed bed is stirred by a rectangular bar moving periodically in the horizontal direction. The parameters were introduced to characterise the segregation and mixing intensities, and a differential equation was proposed to describe the evolution of segregation intensity approaching exponentially a certain steady state value. It was found that the dynamic friction coefficient has a non-monotonous influence on the processes of mixing and size segregation in poly-disperse granular systems. Critical value of the dynamic friction coefficient μcrit was identified. For the values of friction μ > μcrit, behaviour of granular material can be characterised as a “laminar” flow with dominating convective motion of packed bed. For values of friction μ < μcrit, behaviour of granular mattter can be characterised as “turbulent” flow with dominating “local” mixing inside the packed bed.

Analysis of the influence of crushing on the behavior of granular materials under shear

Abstract: The behavior of granular materials mainly depends on the mechanical and engineering properties of particles in its structural matrix. Crushing or breakage of granular materials under compression or shear occurs when the energy available is sufficient to overcome the resistance of the material. Relatively little systematic research has been conducted regarding how to evaluate or quantify particle crushing and how it effects the engineering properties of the granular materials. The aim of this study is to investigate the effect of crushing on the bulk behavior of granular materials by using manufactured granular materials (MGM) rather than using a naturally occurring cohesionless granular material. MGM allow changing only one particle parameter, namely the “crushing strength”. Four different categories of MGM (with different crushing strength) are used to study the effect on the bulk shear strength, stiffness modulus, friction and dilatancy angle “engineering properties”. A substantial influence on the stress–strain behavior and engineering properties of granular materials is observed. Higher confining stress causes some non-uniformity (strong variations/jumps) in volumetric strain and a constant volumetric strain is not always observed under large shear deformations due to crushing, i.e. there is no critical state with flow regime (with constant volumetric strain).

Using MR elastography to image the 3D force chain structure of a quasi-static granular assembly

Abstract: We have developed a magnetic resonance elastography (MRE) technique to experimentally investigate the force chain structure within a densely packed 3D granular assembly. MRE is an MRI technique whereby small periodic displacements within an elastic material are measured. We verified our MRE technique using a gel phantom and then extended the method to image the force carrying chain structure within a 3D granular assembly of particles under an initial pre-stressed condition, on top of which is superimposed a small-amplitude vibration. We find that significant coherent displacements form along force chains, where spin phase accumulates preferentially, allowing visualization. This work represents the first time that the internal force chain structure of a dry assembly of granular solids has been fully acquired in three dimensions.

Non-coaxial version of Rowe’s stress-dilatancy relation

Abstract: Non-coaxiality occurs when the directions of the principal plastic strain increments and the principal stresses deviate. Extensive experimental data have now conclusively shown that plastic flow in granular soils is non-coaxial particularly during loadings involving rotation of the principal stress directions. One way to integrate the effects of non-coaxiality is by modifying the expressions for energy dissipation and stress-dilatancy used in modeling plastic deformation of granular soils. In this regard, the paper’s main objective is to derive a non-coaxial version of Rowe’s stress-dilatancy relation, thereby making it more general and applicable to loadings involving principal stress rotation. The paper also applies Rowe’s non-coaxial stress-dilatancy equation in the determination of the effects of principal stress rotation in granular soils during simple shear loading conditions. Previous experimental data from simple shear tests on sand are used to validate the proposed non-coaxial version of Rowe’s stress-dilatancy relation.

Granular fragility under thermal cycles

Abstract: We study the effect of thermal cycling of granular materials on both the granular packing fraction and the position of an intruder object within a granular sample. We demonstrate that the packing fraction consistently increases with thermal cycling, regardless of the relative coefficients of thermal expansion for the grains and their containers. The exact magnitude of the packing fraction increase appears to be influenced by factors other than the thermal expansion coefficients, with the intergrain friction and the friction between the grains and their container likely to be important. We also examined the motion of spherical intruders with thermal cycling as a function of the size and density of the intruders, and find that such intruders sink within the grains when the pressure from the intruder exceeds a threshold. These data demonstrate the fragility of dense granular ensembles to changes in the ambient temperature.

Vibration induced flow in hoppers: continuum and DEM model approaches

Abstract: A 2D and a 3D discrete element model (DEM) simulation of cohesive spherical particles are applied to assess the benefit of point source vibration to induce flow in wedge-shaped hoppers. The model is closely compared with a continuum model based on arch stability. A significant aspect of this study is the scaling of the continuum system to a discrete system of 500 particles in 2D and 2500 particles in 3D. This illustrates how such models can complement each other. The continuum model can cope with a full-scale industrial system, but is complex with significant assumptions. The discrete approach is relatively simple at the particle level with minimal assumptions but computationally demanding. The DEM model supports the basic conclusions of the continuum model. The vibration source must be located at the appropriate height above the outlet on the hopper to optimise its flow enhancement. Too low and stable arches can form above. Too high and it might not break the stable arches in the material below. The passive/active nature of the material during vibration and flow is also illustrated. The DEM model also shows that low frequency high amplitude vibration can enable flow through small orifices.

FE-investigation of shear localization in granular bodies under high shear rate

Abstract: Shear localization in granular materials under high shear rate is analysed with the finite element method and a micro-polar hypoplastic constitutive model enhanced by viscous terms. We consider plane strain shearing of an infinitely long and narrow granular strip of initially dense sand between two very rough walls under conditions of free dilatancy. The constitutive model can reproduce the essential features of granular materials during shear localization. The calculations are performed under quasi-static and dynamic conditions with different shear rates. In dynamic regime, the viscosity terms are formulated based on a modified Newtonian fluid and according to the formula by Stadler and Buggisch (Proceedings of the conference on Reliable flow of particulate solids, EFCE Pub. Series, vol 49. Chr. Michelsen Institute, Bergen, 1985). Emphasis is given to the influence of inertial and viscous forces on the shear zone thickness and mobilized wall friction angle.

Rheology of a wet, fragmenting granular flow and the riddle of the anomalous friction of large rock avalanches

Abstract: The effective friction coefficient of rock avalanches diminishes gradually as a function of the avalanche volume. Large rock avalanches can reach run-out distances as long as ten times the fall height, despite the fact that the physics of friction would indicate a run-out only a little greater than the fall height. Numerous suggestions have been put forward to explain this remarkable departure from the predictions of both small-scale experiments and basic theory. It is shown here that accounting for rock fragmentation within the avalanche in combination with the presence of water, leads to results in line with the data.

Vibration-induced arching in a deep granular bed

Abstract: Forced vertical vibration of a granular layer can drive flow phenomena such as heaping, convection, fluidization, densification, surface waves and arching. Food, mineral processing, and pharmaceuticals industries all utilize vibratory processes for the handling and transport of granular materials. Understanding how a granular material responds when subjected to vibration is essential for equipment design. Three-dimensional discrete element simulations have been used in this study to investigate the convective motion leading to arching in a vertically vibrated, deep granular bed. The undulating granular layer contains alternating regions that first compact and then relax. The dynamics of these regions may depend on material properties such as restitution and friction coefficients; as well as particle shape. The effects of these factors on the kinematics and dynamics of the arching pattern are investigated here. The arching pattern is found to arise from synchronised momentum transfer between the rise and fall of the deforming granular layer and horizontally travelling waves. The arching pattern was found to be stable across a broad range of restitution and friction levels and particle shapes. Particles with high restitution tend to disrupt the timing between the vertical and horizontal periodic flows and affect the stability of the pattern selection. Large friction results in shear resistance, higher bed pressures, lower bulk densities, and delays in the timing of the vertical and horizontal momentum transfer. Non-sphericity leads to increased dilation of the bed, slower sideways velocities, and increased loading on the floor and dissipation rate in the bed.

Experimental studies of density segregation in the 3D rotating cylinder and the absence of banding

Abstract: We experimentally investigated 3D biparticulate systems that segregate solely due to density differences in the 3D horizontal rotating drum geometry and compare these to systems which segregate due to size differences. Radial segregation was observed in all systems studied after a few drum rotations. Size induced axial segregation (banding) was observed, as expected. However, contrary to what has sometimes been reported, we found that density differences alone did not induce axial segregation for density ratios up to 4.9.

Brazil nut effect in a rectangular plate under horizontal vibration

Abstract: An intruder to a group of identical small beads enclosed in a rectangular plate will gradually migrate to either the center or one side of the plate when the plate is subjected to a horizontal vibration. By considering probabilities for a bead to move into and off the space between the intruder and the near side of the plate, we predict that the size ratio and the mass ratio of the intruder to small bead have equal but opposite effects in determining the direction of migration. The prediction is confirmed by a molecular dynamics simulation.

Stress fields of solid flow in a model blast furnace

Abstract: The computational fluid dynamics–discrete element method approach, supported by an averaging technique, has been employed to quantitatively investigate the stress distributions of solid flow in a model blast furnace (BF). The results indicate that large normal stresses are mainly observed in the lower central part of the BF, whilst small normal stresses in the vicinity of the raceway. In the upper part, the vertical normal stress varies little horizontally in the central region but reduces a bit near the wall, whereas the horizontal normal stress has a relatively uniform distribution on the whole cross section. The shear stress has its largest magnitude in two symmetrical regions close to the stagnant zone. The couple stress can be ignored except for the regions close to the walls. The stress and couple stress are both affected by gas flow rate. In particular, increasing gas flow rate will decrease the magnitude of the stress and couple stress. The internal friction coefficient is not dependent on the inertial number for the solid flow in a BF, but it may rely on the inertial number in some specific flow regions for the cases without gas and with low gas flow rates.