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Windows-Yule CRK, Seville JPK, Ingram A, Parker DJ. Positron Emission Particle Tracking of Granular Flows. Annu Rev Chem Biomol Eng 2020; 11:367-396. [PMID: 32228041 DOI: 10.1146/annurev-chembioeng-011620-120633] [Citation(s) in RCA: 21] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Abstract
Positron emission particle tracking (PEPT) is a noninvasive technique capable of imaging the three-dimensional dynamics of a wide variety of powders, particles, grains, and/or fluids. The PEPT technique can track the motion of particles with high temporal and spatial resolution and can be used to study various phenomena in systems spanning a broad range of scales, geometries, and physical states. We provide an introduction to the PEPT technique, an overview of its fundamental principles and operation, and a brief review of its application to a diverse range of scientific and industrial systems.
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Affiliation(s)
- C R K Windows-Yule
- School of Chemical Engineering, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom;
| | - J P K Seville
- School of Chemical Engineering, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom;
| | - A Ingram
- School of Chemical Engineering, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom;
| | - D J Parker
- School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom
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2
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Experimental study and analysis of velocity correlation and intermittency of very slow and dense pebble flow in a silo bed. NUCLEAR ENGINEERING AND DESIGN 2016. [DOI: 10.1016/j.nucengdes.2016.05.036] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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3
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Asencio K, Bramer-Escamilla W, Gutiérrez G, Sánchez I. Electrical capacitance sensor array to measure density profiles of a vibrated granular bed. POWDER TECHNOL 2015. [DOI: 10.1016/j.powtec.2014.10.003] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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4
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Windows-Yule C, Parker D. Self-diffusion, local clustering and global segregation in binary granular systems: The role of system geometry. POWDER TECHNOL 2014. [DOI: 10.1016/j.powtec.2014.04.009] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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5
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Huntley JM, Tarvaz T, Mantle MD, Sederman AJ, Gladden LF, Sheikh NA, Wildman RD. Nuclear magnetic resonance measurements of velocity distributions in an ultrasonically vibrated granular bed. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2014; 372:20130185. [PMID: 24711488 PMCID: PMC3982650 DOI: 10.1098/rsta.2013.0185] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
We report the results of nuclear magnetic resonance imaging experiments on granular beds of mustard grains fluidized by vertical vibration at ultrasonic frequencies. The variation of both granular temperature and packing fraction with height was measured within the three-dimensional cell for a range of vibration frequencies, amplitudes and numbers of grains. Small increases in vibration frequency were found--contrary to the predictions of classical 'hard-sphere' expressions for the energy flux through a vibrating boundary--to result in dramatic reductions in granular temperature. Numerical simulations of the grain-wall interactions, using experimentally determined Hertzian contact stiffness coefficients, showed that energy flux drops significantly as the vibration period approaches the grain-wall contact time. The experiments thus demonstrate the need for new models for 'soft-sphere' boundary conditions at ultrasonic frequencies.
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Affiliation(s)
- J. M. Huntley
- Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Loughborough LE11 3TU, UK
| | - T. Tarvaz
- Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Loughborough LE11 3TU, UK
| | - M. D. Mantle
- Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge CB2 3RA, UK
| | - A. J. Sederman
- Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge CB2 3RA, UK
| | - L. F. Gladden
- Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge CB2 3RA, UK
| | - N. A. Sheikh
- Department of Mechanical Engineering, Muhammad Ali Jinnah University, Islamabad, Pakistan
| | - R. D. Wildman
- Department of Chemical and Environmental Engineering, University Park, Nottingham NG7 2RD, UK
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6
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Chandratilleke G, Yu A, Bridgwater J. A DEM study of the mixing of particles induced by a flat blade. Chem Eng Sci 2012. [DOI: 10.1016/j.ces.2012.05.010] [Citation(s) in RCA: 31] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/28/2022]
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7
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Comparison of flow microdynamics for a continuous granular mixer with predictions from periodic slice DEM simulations. POWDER TECHNOL 2012. [DOI: 10.1016/j.powtec.2012.01.021] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
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8
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Fiege A, Aspelmeier T, Zippelius A. Long-time tails and cage effect in driven granular fluids. PHYSICAL REVIEW LETTERS 2009; 102:098001. [PMID: 19392566 DOI: 10.1103/physrevlett.102.098001] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/25/2008] [Indexed: 05/27/2023]
Abstract
We study the velocity autocorrelation function of a driven granular fluid in the stationary state in three dimensions. As the critical volume fraction of the glass transition in the corresponding elastic system is approached, we observe pronounced cage effects in the velocity autocorrelation function as well as a strong decrease of the diffusion constant, depending on the inelasticity. At moderate densities the velocity autocorrelation function is shown to decay algebraically in time, like t(-3/2), if momentum is conserved locally, and like t(-1), if momentum is not conserved by the driving. A simple scaling argument supports the observed long-time tails.
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Affiliation(s)
- Andrea Fiege
- Georg-August-Universität Göttingen, Institut für Theoretische Physik, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
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Liu B, Goree J. Superdiffusion and non-Gaussian statistics in a driven-dissipative 2D dusty plasma. PHYSICAL REVIEW LETTERS 2008; 100:055003. [PMID: 18352381 DOI: 10.1103/physrevlett.100.055003] [Citation(s) in RCA: 33] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/01/2007] [Indexed: 05/26/2023]
Abstract
Anomalous diffusion and non-Gaussian statistics are detected experimentally in a two-dimensional driven-dissipative system. A single-layer dusty plasma suspension with a Yukawa interaction and frictional dissipation is heated with laser radiation pressure to yield a structure with liquid ordering. Analyzing the time series for mean-square displacement, superdiffusion is detected at a low but statistically significant level over a wide range of temperatures. The probability distribution function fits a Tsallis distribution, yielding q, a measure of nonextensivity for non-Gaussian statistics.
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Affiliation(s)
- Bin Liu
- Department of Physics and Astronomy, The University of Iowa, Iowa City, Iowa 52242, USA
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10
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Particle dynamics in a dense vibrated fluidized bed as revealed by diffusing wave spectroscopy. POWDER TECHNOL 2008. [DOI: 10.1016/j.powtec.2007.12.007] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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11
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12
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Hsiau S, Lu L, Tai C. Experimental investigations of granular temperature in vertical vibrated beds. POWDER TECHNOL 2008. [DOI: 10.1016/j.powtec.2007.09.015] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
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13
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Huntley J, Martin T, Mantle M, Shattuck M, Sederman A, Wildman R, Gladden L, Halliwell N. NMR measurements and hydrodynamic simulations of phase-resolved velocity distributions within a three-dimensional vibrofluidized granular bed. Proc Math Phys Eng Sci 2007. [DOI: 10.1098/rspa.2007.1885] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
We report the results of nuclear magnetic resonance imaging experiments on vertically vibrated granular beds of mustard grains. A novel spin-echo velocity profiling technique was developed that allows granular temperature, mean velocity and packing fraction distributions within the three-dimensional cell to be measured as a function of both vertical position and vibration phase. Bimodal velocity distributions were observed at certain portions of the vibration cycle, and in general the ability to acquire time-resolved data demonstrated the significant distortions to the velocity distributions and the systematic errors in calculated temperature distributions that may arise with time-averaged measurements. The experimental behaviour was compared with predictions from a time-varying one-dimensional hydrodynamic model using the experimental parameters as input to the code. In both cases, damping of longitudinal sound waves was linked to significant volume heating effects, which contrasts with the usual heat transport mechanism (i.e. diffusion from the boundaries) currently assumed in most steady-state models. This leads to a new explanation for the counterintuitive upturn in granular temperature in vibrofluidized granular beds, based on amplification and damping of sound waves in the high-altitude region.
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Affiliation(s)
- J.M Huntley
- Wolfson School of Mechanical and Manufacturing Engineering, Loughborough UniversityLoughborough LE11 3TU, UK
| | - T.W Martin
- Wolfson School of Mechanical and Manufacturing Engineering, Loughborough UniversityLoughborough LE11 3TU, UK
| | - M.D Mantle
- Department of Chemical Engineering, University of CambridgePembroke Street, Cambridge CB2 3RA, UK
| | - M.D Shattuck
- Levich Institute, City College, City University of New YorkNew York, NY 10031, USA
| | - A.J Sederman
- Department of Chemical Engineering, University of CambridgePembroke Street, Cambridge CB2 3RA, UK
| | - R.D Wildman
- Wolfson School of Mechanical and Manufacturing Engineering, Loughborough UniversityLoughborough LE11 3TU, UK
| | - L.F Gladden
- Department of Chemical Engineering, University of CambridgePembroke Street, Cambridge CB2 3RA, UK
| | - N.A Halliwell
- Wolfson School of Mechanical and Manufacturing Engineering, Loughborough UniversityLoughborough LE11 3TU, UK
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14
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Götzendorfer A, Tai CH, Kruelle CA, Rehberg I, Hsiau SS. Fluidization of a vertically vibrated two-dimensional hard sphere packing: a granular meltdown. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:011304. [PMID: 16907086 DOI: 10.1103/physreve.74.011304] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/13/2006] [Revised: 05/05/2006] [Indexed: 05/11/2023]
Abstract
We report measurements of the fluidization process in vertically vibrated two-dimensional granular packings. An initially close packed granular bed is exposed to sinusoidal container oscillations with gradually increasing amplitude. At first the particles close to the free surface become mobile. When a critical value of the forcing strength is reached the remaining crystal suddenly breaks up and the bed fluidizes completely. This transition leads to discontinuous changes in the density distribution and in the root mean square displacement of the individual particles. Likewise the vertical center of mass coordinate increases by leaps and bounds at the transition. It turns out that the maximum container velocity v0 is the crucial driving parameter determining the state of a fully fluidized system. For particles of various sizes the transition to full fluidization occurs at the same value of v 2 0/gd, where d is the particle diameter and g is the gravitational acceleration. A discontinuous fluidization transition is only observed when the particles are highly elastic.
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15
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Kiyono S, Taguchi YH. Temperature measurement under convection and segregation in a vibrated bed of powder: A numerical study. GRANULAR MATTER 2005. [DOI: 10.1007/s10035-005-0215-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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16
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Huan C, Yang X, Candela D, Mair RW, Walsworth RL. NMR experiments on a three-dimensional vibrofluidized granular medium. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:041302. [PMID: 15169012 DOI: 10.1103/physreve.69.041302] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/27/2003] [Revised: 10/16/2003] [Indexed: 05/24/2023]
Abstract
A three-dimensional granular system fluidized by vertical container vibrations was studied using pulsed field gradient NMR coupled with one-dimensional magnetic resonance imaging. The system consisted of mustard seeds vibrated vertically at 50 Hz, and the number of layers N(l)<or=4 was sufficiently low to achieve a nearly time-independent granular fluid. Using NMR, the vertical profiles of density and granular temperature were directly measured, along with the distributions of vertical and horizontal grain velocities. The velocity distributions showed modest deviations from Maxwell-Boltzmann statistics, except for the vertical velocity distribution near the sample bottom, which was highly skewed and non-Gaussian. Data taken for three values of N(l) and two dimensionless accelerations Gamma=15,18 were fitted to a hydrodynamic theory, which successfully models the density and temperature profiles away from the vibrating container bottom. A temperature inversion near the free upper surface is observed, in agreement with predictions based on the hydrodynamic parameter micro which is nonzero only in inelastic systems.
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Affiliation(s)
- Chao Huan
- Physics Department, University of Massachusetts, Amherst, Massachusetts 01003, USA
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17
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Utter B, Behringer RP. Self-diffusion in dense granular shear flows. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:031308. [PMID: 15089287 DOI: 10.1103/physreve.69.031308] [Citation(s) in RCA: 47] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/22/2003] [Indexed: 05/24/2023]
Abstract
Diffusivity is a key quantity in describing velocity fluctuations in granular materials. These fluctuations are the basis of many thermodynamic and hydrodynamic models which aim to provide a statistical description of granular systems. We present experimental results on diffusivity in dense, granular shear flows in a two-dimensional Couette geometry. We find that self-diffusivities D are proportional to the local shear rate gamma; with diffusivities along the direction of the mean flow approximately twice as large as those in the perpendicular direction. The magnitude of the diffusivity is D approximately gamma;a(2), where a is the particle radius. However, the gradient in shear rate, coupling to the mean flow, and strong drag at the moving boundary lead to particle displacements that can appear subdiffusive or superdiffusive. In particular, diffusion appears to be superdiffusive along the mean flow direction due to Taylor dispersion effects and subdiffusive along the perpendicular direction due to the gradient in shear rate. The anisotropic force network leads to an additional anisotropy in the diffusivity that is a property of dense systems and has no obvious analog in rapid flows. Specifically, the diffusivity is suppressed along the direction of the strong force network. A simple random walk simulation reproduces the key features of the data, such as the apparent superdiffusive and subdiffusive behavior arising from the mean velocity field, confirming the underlying diffusive motion. The additional anisotropy is not observed in the simulation since the strong force network is not included. Examples of correlated motion, such as transient vortices, and Lévy flights are also observed. Although correlated motion creates velocity fields which are qualitatively different from collisional Brownian motion and can introduce nondiffusive effects, on average the system appears simply diffusive.
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Affiliation(s)
- Brian Utter
- Department of Physics and Center for Nonlinear and Complex Systems, Box 90305, Duke University, Durham, North Carolina 27708, USA.
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19
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Blair DL, Kudrolli A. Collision statistics of driven granular materials. PHYSICAL REVIEW E 2003; 67:041301. [PMID: 12786356 DOI: 10.1103/physreve.67.041301] [Citation(s) in RCA: 60] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/13/2002] [Indexed: 11/07/2022]
Abstract
We present an experimental investigation of the statistical properties of spherical granular particles on an inclined plane that are excited by an oscillating side wall. The data is obtained by high-speed imaging and particle tracking techniques. We identify all particles in the system and link their positions to form trajectories over long times. Thus, we identify particle collisions to measure the effective coefficient of restitution and find a broad distribution of values for the same impact angles. We find that the energy inelasticity can take on values greater than one, which implies that the rotational degrees of freedom play an important role in energy transfer. We also measure the distance and the time between collision events in order to directly determine the distribution of path lengths and the free times. These distributions are shown to deviate from expected theoretical forms for elastic spheres, demonstrating the inherent clustering in this system. We describe the data with a two-parameter fitting function and use it to calculate the mean free path and collision time. We find that the ratio of these values is consistent with the average velocity. The velocity distributions are observed to be strongly non-Gaussian and do not demonstrate any apparent universal behavior. We report the scaling of the second moment, which corresponds to the granular temperature, and higher order moments as a function of distance from the driving wall. Additionally, we measure long-time correlation functions in both space and in the velocities to probe diffusion in a dissipative gas.
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Affiliation(s)
- Daniel L Blair
- Department of Physics, Clark University, Worcester, Massachusetts 01610, USA
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20
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Hsiau SS, Yang SC. Numerical simulation of self-diffusion and mixing in a vibrated granular bed with the cohesive effect of liquid bridges. Chem Eng Sci 2003. [DOI: 10.1016/s0009-2509(02)00519-5] [Citation(s) in RCA: 35] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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21
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Wildman R. Measurement of the first and second moments of the velocity distribution in two-dimensional vibro-fluidised granular beds. POWDER TECHNOL 2002. [DOI: 10.1016/s0032-5910(02)00146-8] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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22
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Vrhovac SB, Arsenović D, Belić A. Transport theory of granular swarms. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:051302. [PMID: 12513480 DOI: 10.1103/physreve.66.051302] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/12/2002] [Indexed: 05/24/2023]
Abstract
The transport of trace granular gas (swarm) in a carrier granular fluid is studied by means of the Boltzmann-Lorentz kinetic equation. Time-dependent perturbation theory is used to follow the evolution of the granular swarm from an arbitrary initial distribution. A nonhydrodynamic extension of the diffusion equation is derived, with transport coefficients that are time dependent and implicitly depend on the wave vector. Transport coefficients of any order are obtained as velocity moments of the solutions of the corresponding kinetic equations derived from the Boltzmann-Lorentz equation. For the special case of the initial distribution of swarm particles, transport coefficients are identified as time derivatives of the moments of the number density. Finally the granular particle transport theory is extended by the introduction of the concept of non-particle-conserving collisions.
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Affiliation(s)
- S B Vrhovac
- Institute of Physics, P.O. Box 68, 11080 Zemun, Belgrade, Yugoslavia
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23
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Wildman RD, Parker DJ. Coexistence of two granular temperatures in binary vibrofluidized beds. PHYSICAL REVIEW LETTERS 2002; 88:064301. [PMID: 11863810 DOI: 10.1103/physrevlett.88.064301] [Citation(s) in RCA: 44] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/24/2001] [Revised: 11/28/2001] [Indexed: 05/23/2023]
Abstract
An investigation into the granular temperature distributions of a binary vibrofluidized granular bed has been conducted using positron emission particle tracking. By repeating each experiment with the tracer selected in turn from the two size components, the granular temperature and packing fraction distributions for each phase were determined. It was found that, for a range of size fractions, the granular temperature of the larger particles was higher than that of the smaller diameter grains, a result which was supported by a simple theoretical analysis based on the steady state energy equation.
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Affiliation(s)
- R D Wildman
- School of Mechanical and Manufacturing Engineering, Loughborough University, Loughborough, Leicestershire, LE11 3TU, United Kingdom
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24
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Alexeev A, Goldshtein A, Shapiro M. The liquid and solid states of highly dissipative vibrated granular columns: one-dimensional computer simulations. POWDER TECHNOL 2002. [DOI: 10.1016/s0032-5910(01)00436-3] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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25
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Yang X, Huan C, Candela D, Mair RW, Walsworth RL. Measurements of grain motion in a dense, three-dimensional granular fluid. PHYSICAL REVIEW LETTERS 2002; 88:044301. [PMID: 11801123 DOI: 10.1103/physrevlett.88.044301] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/16/2001] [Indexed: 05/23/2023]
Abstract
We have used an NMR technique to measure the short-time, three-dimensional displacement of grains in a system of mustard seeds vibrated vertically at 15 g. The technique averages over a time interval in which the grains move ballistically, giving a direct measurement of the granular temperature profile. The dense, lower portion of the sample is well described by a recent hydrodynamic theory for inelastic hard spheres. Near the free upper surface the mean free path is longer than the particle diameter and the hydrodynamic description fails.
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Affiliation(s)
- Xiaoyu Yang
- Physics Department, University of Massachusetts, Amherst, Massachusetts 01003, USA
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26
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Wildman RD, Huntley JM, Hansen JP, Parker DJ. Numerical solution of the Smoluchowski equation for a vibrofluidized granular bed. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:051304. [PMID: 11735915 DOI: 10.1103/physreve.64.051304] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/02/2001] [Indexed: 05/23/2023]
Abstract
A stochastic approach, similar to that used to describe Brownian motion, was used to model the displacement probability of grains in a three-dimensional vibrofluidized granular bed. As neither an analytical description nor measurements of the diffusion coefficients were available, the governing partial differential equation, namely, the Smoluchowski equation, was solved numerically using an iterative procedure, modifying the granular temperature profile at each step. The results of this stochastic model were compared to experimental measurements of the displacement probability density made using positron emission particle tracking. The results indicate that methods based on hard elastic systems such as the Smoluchowski equation are appropriate to granular systems, particularly over timescales greater than the mean collision time.
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Affiliation(s)
- R D Wildman
- Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Loughborough, Leicstershire LE11 3TU, United Kingdom
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27
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Wildman RD, Huntley JM, Parker DJ. Granular temperature profiles in three-dimensional vibrofluidized granular beds. PHYSICAL REVIEW E 2001; 63:061311. [PMID: 11415096 DOI: 10.1103/physreve.63.061311] [Citation(s) in RCA: 97] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/26/2000] [Indexed: 11/07/2022]
Abstract
The motion of grains in a three-dimensional vibrofluidized granular bed has been measured using the technique of positron emission particle tracking, to provide three-dimensional packing fraction and granular temperature distributions. The mean square fluctuation velocity about the mean was calculated through analysis of the short time mean squared displacement behavior, allowing measurement of the granular temperature at packing fractions of up to eta approximately 0.15. The scaling relationship between the granular temperature, the number of layers of grains, and the base velocity was determined. Deviations between the observed scaling exponents and those predicted by recent theories are attributed to the influence of dissipative grain-sidewall collisions.
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Affiliation(s)
- R D Wildman
- Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom
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28
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Wildman RD, Huntley JM, Parker DJ. Convection in highly fluidized three-dimensional granular beds. PHYSICAL REVIEW LETTERS 2001; 86:3304-3307. [PMID: 11327956 DOI: 10.1103/physrevlett.86.3304] [Citation(s) in RCA: 26] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/30/2000] [Indexed: 05/23/2023]
Abstract
Free, buoyancy-driven convection has been observed experimentally in three-dimensional highly fluidized granular flows for the first time. Positron emission particle tracking was used to determine the position of a tracer grain in a vibrofluidized bed, from which packing fraction distributions as well as the velocity fields could be determined. The convection rolls, although small compared to the magnitude of velocity fluctuations (<5%), were consistently observed for a range of grain numbers and shaker amplitudes. Density variations are a signature of free convection and, with negative temperature gradients also present, were interpreted as the mechanism by which the convection rolls were initiated.
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Affiliation(s)
- R D Wildman
- Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Loughborough, Leics. LE11 3TU, United Kingdom
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29
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Valverde JM, Castellanos A, Sanchez Quintanilla MA. Self-diffusion in a gas-fluidized bed of fine powder. PHYSICAL REVIEW LETTERS 2001; 86:3020-3023. [PMID: 11290097 DOI: 10.1103/physrevlett.86.3020] [Citation(s) in RCA: 26] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/15/2000] [Indexed: 05/23/2023]
Abstract
We have investigated the self-diffusion in a stable gas-fluidized bed of fine powder. Two regimes have been observed: for gas velocities v(g) above the minimum fluidization velocity v(m) and below a critical gas velocity v(c) smaller than the minimum bubbling velocity v(b) the powder does not mix. Experimental measurements show the existence of yield stresses in this regime which are responsible for the static behavior of the bed. For v(g)>v(c) the yield stress vanishes; the bed behaves like a fluid and displays a diffusive dynamics. In this region we have found that the diffusion coefficient D increases with gas velocity until the bed expansion approaches its maximum value.
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Affiliation(s)
- J M Valverde
- Departamento de Electrónica y Electromagnetismo, Facultad de Fisica, Avenida Reina Mercedes s/n, 41012 Seville, Spain
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30
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Soto R, Mareschal M. Statistical mechanics of fluidized granular media: short-range velocity correlations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:041303. [PMID: 11308834 DOI: 10.1103/physreve.63.041303] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/05/2000] [Indexed: 05/23/2023]
Abstract
A statistical mechanical study of fluidized granular media is presented. Using a special energy injection mechanism, homogeneous fluidized stationary states are obtained. Molecular dynamics simulations and theoretical analysis of the inelastic hard-disk model show that there is a large asymmetry in the two-particle distribution function between pairs that approach and separate. Large velocity correlations appear in the postcollisional states due to the dissipative character of the collision rule. These correlations can be well-characterized by a state dependent pair correlation function at contact. It is also found that velocity correlations are present for pairs that are about to collide. Particles arrive at collisions with a higher probability that their velocities are parallel rather than antiparallel. These dynamical correlations lead to a decrease of the pressure and of the collision frequency as compared to their Enskog values. A phenomenological modified equation of state is presented.
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Affiliation(s)
- R Soto
- CECAM, ENS-Lyon, 46 Allée d'Italie, 69007 Lyon, France
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Wildman RD, Huntley JM, Hansen JP, Parker DJ, Allen DA. Single-particle motion in three-dimensional vibrofluidized granular beds. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:3826-3835. [PMID: 11088900 DOI: 10.1103/physreve.62.3826] [Citation(s) in RCA: 36] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/30/2000] [Indexed: 05/23/2023]
Abstract
A technique to probe the interior of three-dimensional dynamic granular systems is presented. Positron emission particle tracking (PEPT) allows a single tracer particle to be followed around a three dimensional vibrofluidized granular bed for periods up to six hours. At present the technique is able to resolve the position of the grains to +/-4 mm, with an average temporal resolution of about 7 ms. Packing fraction profiles are calculated by making use of the ergodicity of the system, and granular temperature profiles are obtained, in the dilute case, from the short time behavior of the mean squared displacement. At longer times, the mean squared displacement shows a range of behavior which can be explained by the presence of strong gradients in the packing fraction. Convection currents were observed, but were sufficiently small in magnitude to be ignored during the analysis of grain motion. The system was modeled using the Smoluchowski equation, which was solved numerically, and the results compared with the experimentally determined displacement probability density functions. Good agreement between experiment and numerical results was achieved using Brownian motion relationships modified to accommodate differences between granular systems and thermal systems.
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Affiliation(s)
- RD Wildman
- Department of Mechanical Engineering, Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom
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Soto R, Mareschal M. Nonlinear analysis of the shearing instability in granular gases. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:3836-3842. [PMID: 11088901 DOI: 10.1103/physreve.62.3836] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/25/2000] [Indexed: 05/23/2023]
Abstract
It is known that a finite-size homogeneous granular fluid develops a hydrodynamiclike instability when dissipation crosses a threshold value. This instability is analyzed in terms of modified hydrodynamic equations: first, a source term is added to the energy equation which accounts for the energy dissipation at collisions and the phenomenological Fourier law is generalized according to previous results. Second, a rescaled time formalism is introduced that maps the homogeneous cooling state into a nonequilibrium steady state. A nonlinear stability analysis of the resulting equations is done which predicts the appearance of flow patterns. A stable modulation of density and temperature is produced that does not lead to clustering. Also a global decrease of the temperature is obtained, giving rise to a decrease of the collision frequency and dissipation rate. Good agreement with molecular dynamics simulations of inelastic hard disks is found for low dissipation.
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Affiliation(s)
- R Soto
- CECAM, ENS-Lyon, 46 Allee d'Italie, 69007 Lyon, France
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