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系統識別號 U0026-2707201008405400
論文名稱(中文) 切變顆粒介質模型
論文名稱(英文) Model of sheared granular material
校院名稱 成功大學
系所名稱(中) 水利及海洋工程學系碩博士班
系所名稱(英) Department of Hydraulics & Ocean Engineering
學年度 98
學期 2
出版年 99
研究生(中文) 李政賢
研究生(英文) Cheng-Hsien Lee
電子信箱 kethenlee@gmail.com
學號 n8895109
學位類別 博士
語文別 英文
論文頁數 141頁
口試委員 指導教授-黃清哲
口試委員-蕭述三
口試委員-卡艾瑋
口試委員-羅偉誠
口試委員-蕭士俊
口試委員-戴義欽
中文關鍵字 顆粒流  準靜狀態  密集狀態  慣性狀態  膨脹定律  磁滯現象 
英文關鍵字 granular flow  quasi-static regime  dense regime  inertial regime  dilatancy law  hysteretic flow threshold 
學科別分類
中文摘要 本研究試圖發展一顆粒流模型,並希望此模型可以適用於各種不同的流動狀態。為了達到此目的,我們將應力分為動態應力與靜態應力兩個部分。動態應力源自顆粒的碰撞與慣性,其與應變率相關,本研究利用顆粒動力論所推衍出的結果計算動態應力。靜態應力成分源自於顆粒間的長時間接觸,此成分與應變率無關,我們以庫倫塑性定律計算此部分應力。另外,我們設定此兩應力所做的功會將平均動能轉變為顆粒擾動動能。這使得本模型可以預測顆粒介質靜態與動態轉變間的磁滯現象,以及顆粒介質在接近靜態時會產生的局部剪切現象。為使控制方程組閉合,我們借用了膨脹定律,並設定靜態壓力最小值為零,此條件使得本模型可以描述密集狀態與慣性狀態間的變化。
研究中發現,在顆粒溫度的傳導效應很小的假設下,此模型中的應力本構關係式可簡化成Bagnold模型。另外,比較本模式預測的動態應力與分子模擬的結果後發現,靜態應力和動態應力與部分流化理論中的固相和液相有關。
本研究利用此模型模擬在重力作用下的表面驅動顆粒流。本模型所預測的流速、體積分率、顆粒溫度和應力與使用分子動力法所模擬的結果一致。這說明了本模型可應用在不同狀態的顆粒流。
此模型還被應用在研究粗糙斜坡上流動的密集顆粒流。此模型展現的流變特徵指出此流動會存在三個特徵角度,這三個特徵角度決定了流動的穩定性。另外,流變特徵還暗示體積分率在垂直底床方向呈現均勻分佈,且體積分率會隨著坡度增加而減少。根據流變特徵與應力分佈可以推測顆粒流的流速分佈滿足Bagnold剖面。上述所推測的特徵與實驗觀察一致。另外,我們利用數值方法求解本模型,數值模型所預測的速度和體積分率與前人結果一致。本研究定義了靜態凝聚長度,靜態凝聚長度可解釋成長時間接觸顆粒形成的叢集直徑,研究中發現靜態凝聚長度與顆粒流停止高度有密切關係。
英文摘要 This work developed a model of sheared granular materials. To capture the static and kinetic features of the granular flow involving different regimes, both the shear stress and pressure are superimposed by a rate-independent component (representing the static contribution) and a rate-dependent component (representing the kinetic contribution), as determined using granular kinetic theory. The dilatancy law is utilized as the equation of state, and the constraint that static pressure is nonnegative is utilized to determine the transition between the dense regime and the inertial regime. The balance equation of granular temperature incorporates the works done by both the static and kinetic components of shear stress. This enables the proposed model to predict the hysteretic flow thresholds and the shear bands.
The constitutive model of the shear stress can be reduced to the Bagnold model as conduction effect of granular temperature is small. Comparing the predicted the kinetic component of stress with other results reveals that the static and the kinetic components of shear stress are associated with the solid phase and the liquid phase in partially fluidized theory.
This model is applied to investigate a thick, surface-driven granular flow comprising two dimensional disks under gravity. The predicted velocity, volume fraction, granular temperature, and stress are consistent with results obtained elsewhere using the molecular dynamic method. This finding demonstrates the ability of the proposed model to simulate granular flow in which the quasi-static, dense, and kinetic regimes coexist simultaneously.
This investigation applies the present model to investigate dense granular flows down a bumpy inclined plane. The rheological characteristic predicted by the present model infers that dense granular flow down a bumpy incline is characterized by three special angles determining the phase diagram. Additionally, the predicted rheological characteristic and the stress distribution together indicate that the volume fraction is uniform throughout depth for the thick flow and that thick granular flow on a bumpy plane has the Bagnold velocity profile. The present model is solved numerically. The predicted velocity and volume fraction are strongly consistent with previous discretely simulated results. Furthermore, this investigation defines the static coherence length, which is interpreted as the diameter of cluster formed with enduring-contact particles, and this investigation found that the static coherence length is comparable to the stopping height.
論文目次 摘要 I
Abstract III
Acknowledgement V
Contents VIII
Figure Caption X
Table Caption XIV
Notation XV
Chapter 1 Introduction
1.1 Motivation 1
1.2 Characteristics of granular flows 5
1.3 Simulation methods of granular flows 8
1.3.1 Molecular dynamic method 8
1.3.2 Continuum model 11
1.4 Organization of this thesis 13
Chapter 2 Averaging method
2.1 Introduction 17
2.2 Weighted time-space average 18
2.3 Derivation of macroscopic governing
equations 22
2.4 Conclusions 27
Chapter 3 Governing equations and constitutive
models derived from granular kinetic
theory
3.1 Review of kinetic theory 29
3.2 Governing equations and constitutive
models 33
3.3 Discussions 41
Chapter 4 Constitutive model of stress for two
dimensional disks
4.1 Introduction 43
4.2 Constitutive model 45
4.3 Rheological characteristics 50
4.4 Static and kinetic parts of shear
stress 56
4.5 Conclusions 59
Chapter 5 Application to surface-driven shear
granular flow under gravity
5.1 Introduction 63
5.2 Governing equations 64
5.3 Numerical scheme 66
5.4 Results 68
5.5 Conclusions 72
Chapter 6 Constitutive model of stress for three
dimensional spheres
6.1 Introduction 83
6.2 Constitutive model 84
6.3 Rheological characteristics 86
6.4 Conclusions 90
Chapter 7 Application to dense granular flows on a
bumpy inclined plane
7.1 Introduction 95
7.2 Governing equations and boundary
conditions 100
7.3 Approximate flow field 103
7.4 Determination of particle properties 107
7.5 Coherence length and stopping height 109
7.6 Numerical scheme 115
7.7 Numerical results 119
7.8 Conclusions 123
Chapter 8 Conclusions 131
Reference 135
參考文獻 張金機,黃清和,1997,台灣四周海岸沖蝕防治技術,八十六年度海岸工程研討會論文集,第1-23頁。
Aguirre, M.A., Nerone, N., Calvo, A., Ippolito, I. and Bideau, D., 2000. Influence of the number of layers on the equilibrium of a granular packing. Phys. Rev. E, 62(1): 738-743.
Aharonov, E. and Sparks, D., 2002. Shear profiles and localization in simulations of granular materials. Phys. Rev. E, 65(5): 51302.
Alder, B.J., Gass, D.M. and Wainwright, T.E., 1970. Studies in molecular dynamics. VIII. The transport coefficients for a hard sphere fluid. J. Chem. Phys., 53: 3813.
Ancey, C., Coussot, P. and Evesque, P., 1999. A theoretical framework for granular suspensions in a steady simple shear flow. J. Rheol., 43: 1673.
Aranson, I. and Tsimring, L., 2006. Patterns and collective behavior in granular media: Theoretical concepts. Rev. Mod. Phys., 78(2): 641-692.
Aranson, I.S. and Tsimring, L.S., 2002. Continuum theory of partially fluidized granular flows. Phys. Rev. E, 65(6): 61303.
Aranson, I.S., Tsimring, L.S., Malloggi, F. and Clement, E., 2008. Nonlocal rheological properties of granular flows near a jamming limit. Phys. Rev. E, 78(3): 031303.
Armanini, A., Capart, H., Fraccarollo, L. and Larcher, M., 2005. Rheological stratification in experimental free-surface flows of granular–liquid mixtures. J. Fluid Mech., 532: 269-319.
Azanza, E., Chevoir, F. and Mousheront, P., 1999. Experimental study of collisional granular flows down an inclined plane. J. Fluid Mech., 400: 199-227.
Babic, M., 1997a. Average balance equations for granular materials. Int. J. Eng. Sci., 35(5): 523-548.
Babic, M., 1997b. Unsteady Couette granular flows. Phys. Fluids, 9: 2486.
Babic, M., Shen, H.H. and Shen, H.T., 1990. The stress tensor in granular shear flows of uniform, deformable disks at high solids concentrations. J. Fluid Mech., 219: 81-118.
Bagnold, R.A., 1954. Experiments on a Gravity-Free Dispersion of Large Solid Spheres in a Newtonian Fluid under Shear. Proc. R. Soc. London, Ser. A, 225(1160): 49-63.
Baran, O., Ertas, D., Halsey, T.C., Grest, G.S. and Lechman, J.B., 2006. Velocity correlations in dense gravity-driven granular chute flow. Phys. Rev. E, 74(5): 51302.
Barrat, A. and Trizac, E., 2002. Molecular dynamics simulations of vibrated granular gases. Phys. Rev. E, 66(5): 51303.
Bisi, M., Spiga, G. and Toscani, G., 2004. Grad’s equations and hydrodynamics for weakly inelastic granular flows. Phys. Fluids, 16: 4235.
Bobylev, A.V., Carrillo, J.A. and Gamba, I.M., 2000. On some properties of kinetic and hydrodynamic equations for inelastic interactions. J. Stat. Phys., 98(3): 743-773.
Bocquet, L., Errami, J. and Lubensky, T.C., 2002. Hydrodynamic Model for a Dynamical Jammed-to-Flowing Transition in Gravity Driven Granular Media. Phys. Rev. Lett., 89(18): 184301.
Bocquet, L., Losert, W., Schalk, D., Lubensky, T.C. and Gollub, J.P., 2001. Granular shear flow dynamics and forces: Experiment and continuum theory. Phys. Rev. E, 65(1): 011307.
Borzsonyi, T. and Ecke, R.E., 2007. Flow rule of dense granular flows down a rough incline. Phys. Rev. E, 76(3).
Brey, J., Dufty, J. and Santos, A., 1997. Dissipative dynamics for hard spheres. J. Stat. Phys., 87(5): 1051-1066.
Brey, J.J. and Dufty, J.W., 2005. Hydrodynamic modes for a granular gas from kinetic theory. Phys. Rev. E, 72(1): 011303.
Brey, J.J., Dufty, J.W., Kim, C.S. and Santos, A., 1998. Hydrodynamics for granular flow at low density. Phys. Rev. E, 58(4): 4638.
Campbell, C.S., 1989. The stress tensor for simple shear flows of a granular material. J. Fluid Mech., 203: 449-473.
Campbell, C.S., 1990. Rapid Granular Flows. Annu. Rev. Fluid Mech., 22(1): 57-90.
Campbell, C.S., 2002. Granular shear flows at the elastic limit. J. Fluid Mech., 465: 261-291.
Campbell, C.S., 2005. Stress-controlled elastic granular shear flows. J. Fluid Mech., 539: 273-297.
Campbell, C.S., 2006. Granular material flows–An overview. Powder Technol., 162(3): 208-229.
Carnahan N.F. and K. E. Starling, 1969. Equation of state for non-attracting spheres. J. Chem. Phys., 51: 635-636.
Chandrasekharaiah, D. and Debnath, L., 1994. Continuum mechanics. Academic press California.
Christoffersen, J., Mehrabadi, M.M. and Nematnasser, S., 1981. A micromechanical description of granular material behavior. J. Appl. Mech., 48(2): 339-344.
Cortet, P.P. et al., 2009. Relevance of visco-plastic theory in a multi-directional inhomogeneous granular flow. Europhys. Lett., 88: 14001.
Cundall, P.A., 1979. A discrete numerical model for granular assemblies. Geotechnique, 29(1): 47-65.
da Cruz, F., Chevoir, F., Bonn, D. and Coussot, P., 2002. Viscosity bifurcation in granular materials, foams, and emulsions. Phys. Rev. E, 66(5): 51305.
da Cruz, F., Emam, S., Prochnow, M., Roux, J.N. and Chevoir, F., 2005. Rheophysics of dense granular materials: Discrete simulation of plane shear flows. Phys. Rev. E, 72(2): 21309.
Daniel, R.C., Poloski, A.P. and Eduardo Saez, A., 2007. A continuum constitutive model for cohesionless granular flows. Chem. Eng. Sci., 62(5): 1343-1350.
Delannay, R., Louge, M., Richard, P., Taberlet, N. and Valance, A., 2007. Towards a theoretical picture of dense granular flows down inclines. Nature Materials, 6(2): 99-108.
Drescher, A. and de Josselin de Jong, G., 1972. Photoelastic berification of a mechanical model for flow of a granular material J. Mech. Phys. Solids, 20(5): 337-&.
Drew, D., 1983. Mathematical Modeling of Two-Phase Flow. Annu. Rev. Fluid Mech., 15(1): 261-291.
du Pont, S.C., Gondret, P., Perrin, B. and Rabaud, M., 2003. Wall effects on granular heap stability. Europhys. Lett., 61(4): 492-498.
Ertas, D. and Halsey, T.C., 2002. Granular gravitational collapse and chute flow. Europhys. Lett.
Eu, B.C. and Farhat, H., 1997. Kinetic theory of fluidized granular matter. Phys. Rev. E, 55(4): 4187-4199.
Ferziger, J.H. and Peric, M., 2002. Computational methods for fluid dynamics. Springer New York.
Forterre, Y. and Pouliquen, O., 2008. Flows of Dense Granular Media. Annu. Rev. Fluid Mech., 40: 1-23.
Garzo, V. and Dufty, J.W., 1999. Dense fluid transport for inelastic hard spheres. Phys. Rev. E, 59(5): 5895-5911.
GDR MiDi, 2004. On dense granular flows. Eur. Phys. J. E, 14(4): 341-365.
Gombosi, T., 1994. Gaskinetic theory. Cambridge University Press.
Grad, H., 1949. On the kinetic theory of rarefied gases. Comm. Pure Appl. Math., 2(4): 331–407.
Haff, P.K. and Werner, B.T., 1986. Computer simulation of the mechanical sorting of grains. Powder Technol., 48(3): 239-245.
Hatano, T., 2007. Power-law friction in closely packed granular materials Phys. Rev. E, 75(6): 60301.
Iverson, R., 1997. The physics of debris flows. Rev. Geophys., 35(3): 245-296.
Jaeger, H.M., Liu, C.H., Nagel, S.R. and Witten, T.A., 1990. Friction in granular flows. Europhys. Lett., 11(7): 619-624.
Jean, M., 1999. The non-smooth contact dynamics method. Comput. methods Appl. Eng., 177(3-4): 235-257.
Jenkins, J.T., 2006. Dense shearing flows of inelastic disks. Phys. Fluids, 18: 103307.
Jenkins, J.T., 2007. Dense inclined flows of inelastic spheres. Granular Matter, 10(1): 47-52.
Jenkins, J.T. and Richman, M.W., 1985a. Grad's 13-moment system for a dense gas of inelastic spheres. Arch. for Rat. Mech. Anal., 87(4): 355-377.
Jenkins, J.T. and Richman, M.W., 1985b. Kinetic theory for plane flows of a dense gas of identical, rough, inelastic, circular disks. Phys. Fluids, 28: 3485-3494.
Jenkins, J.T. and Richman, M.W., 1986. Boundary conditions for plane flows of smooth, nearly elastic, circular disks. J. Fluid Mech., 171: 53-69.
Jenkins, J.T. and Savage, S.B., 1983. A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles. J. Fluid Mech., 130: 187-202.
Jenkins, J.T. and Zhang, C., 2002. Kinetic theory for identical, frictional, nearly elastic spheres. Phys. Fluids, 14: 1228.
Johnson, K., 1987. Contact mechanics. Cambridge University Press.
Johnson, P.C. and Jackson, R., 1987. Frictional–collisional constitutive relations for granular materials, with application to plane shearing. J. Fluid Mech., 176: 67-93.
Johnson, P.C., Nott, P. and Jackson, R., 1990. Frictional–collisional equations of motion for participate flows and their application to chutes. J. Fluid Mech., 210: 501-535.
Jop, P., 2008. Hydrodynamic modeling of granular flows in a modified Couette cell. Phys. Rev. E, 77(3): 32301.
Jop, P., Forterre, Y. and Pouliquen, O., 2005. Crucial role of sidewalls in granular surface flows: consequences for the rheology. J. Fluid Mech., 541: 167-192.
Jop, P., Forterre, Y. and Pouliquen, O., 2006. A constitutive law for dense granular flows. Nature, 441: 727-730.
Josserand, C., Lagree, P. and Lhuillier, D., 2006. Granular pressure and the thickness of a layer jamming on a rough incline. Europhys. Lett., 73(3): 363-369.
Kanatani, K., 1981. A theory of contact force distribution in granular-materials. Powder Technol., 28(2): 167-172.
Kim, S.R. and Woodcock, L.V., 1993. Kinetic-theory of granular shear-flow-constitutive relations for the hard-disk model. J. Stat. Mech., 71(1-2): 143-162.
Kumaran, V., 2006a. The constitutive relation for the granular flow of rough particles, and its application to the flow down an inclined plane. J. Fluid Mech., 561: 1-42.
Kumaran, V., 2006b. Kinetic theory for the density plateau in the granular flow down an inclined plane. Europhys. Lett., 73(2): 232-238.
Kumaran, V., 2008. Dense granular flow down an inclined plane: from kinetic theory to granular dynamics. J. Fluid Mech., 599: 121-168.
Lacaze, L. and Kerswell, R.R., 2009. Axisymmetric granular collapse: A transient 3D flow test of viscoplasticity. Phys. Rev. Lett., 102(10): 108305.
Lois, G., Lemaitre, A. and Carlson, J.M., 2006. Emergence of multi-contact interactions in contact dynamics simulations of granular shear flows. Europhys. Lett., 76(2): 318-324.
Louge, M., 2003. Model for dense granular flows down bumpy inclines. Phys. Rev. E, 67(6): 61303.
Louge, M. and Keast, S., 2001. On dense granular flows down flat frictional inclines. Phys. Fluids, 13: 1213.
Lun, C.K.K., 1991. Kinetic-theory for granular flow of dense, slightly inelastic, slightly rough spheres. J. Fluid Mech., 233: 539-559.
Lun, C.K.K. and Savage, S.B., 1986. The effects of an impact velocity dependent coefficient of restitution on stresses developed by sheared granular materials. Acta Mech., 63(1): 15-44.
Lutsko, J.F., 2004. Rheology of dense polydisperse granular fluids under shear. Phys. Rev. E, 70(6): 61101.
Melo, F., Umbanhowar, P. and Swinney, H.L., 1994. Transition to parametric wave patterns in a vertically oscillated granular layer. Phys. Rev. Lett., 72(1): 172-175.
Melo, F., Umbanhowar, P. and Swinney, H., 1995. Hexagons, kinks, and disorder in oscillated granular layers. Phys. Rev. Lett., 75(21): 3838-3841.
Miller, S. and Luding, S., 2004. Event-driven molecular dynamics in parallel. J. Comput. Phys., 193(1): 306-316.
Mills, P., Rognon, P.G. and Chevoir, F., 2008. Rheology and structure of granular materials near the jamming transition. Europhys. Lett., 81: 64005.
Mitarai, N. and Nakanishi, H., 2005. Bagnold scaling, density plateau, and kinetic theory analysis of dense granular flow. Phys. Rev. Lett., 94(12): 128001.
Mitarai, N. and Nakanishi, H., 2007. Velocity correlations in dense granular shear flows. Phys. Rev. E, 75: 031305.
Nasuno, S., Kudrolli, A., Bak, A. and Gollub, J.P., 1998. Time-resolved studies of stick-slip friction in sheared granular layers. Phys. Rev. E, 58(2): 2161-2171.
Nedderman, R., 1992. Statics and Kinematics of Granular Materials. Cambridge University Press, Cambridge, U.K.
Ogawa, S., 1978. Multitemperature theory of granular materials. In: S.C. Cowin and M. Satake (Editors), Proc. U. S.-Japan Symp. on Continuum Mechanics and Statistical Approaches in the Mechanics of Granular Materials, Gakujutsu Bunken Fukyu-kai, Tokyo, Japan.
Ogawa, S., Umemura, A. and Oshima, N., 1980. On the equations of fully fluidized granular materials. ZAMP, 31(4): 483-493.
Papanastasiou, T.C., 1987. Flows of Materials with Yield. J. Rheol., 31: 385.
Pouliquen, O., 1999. Scaling laws in granular flows down rough inclined planes. Phys. Fluids, 11: 542.
Pouliquen, O., 2004. Velocity correlations in dense granular flows. Phys. Rev. Lett., 93(24): 248001.
Pouliquen, O., Cassar, C., Jop, P., Forterre, Y. and Nicolas, M., 2006. Flow of dense granular material: towards simple constitutive laws. J. Stat. Mech., 2006: P07020.
Pouliquen, O. and Forterre, Y., 2002. Friction law for dense granular flows: application to the motion of a mass down a rough inclined plane. J. Fluid Mech., 453: 133-151.
Reddy, K.A. and Kumaran, V., 2007. Applicability of constitutive relations from kinetic theory for dense granular flows. Phys. Rev. E, 76(6): 61305.
Reif, F., 1965. Fundamentals of statistical and thermal physics. McGraw-Hill Science Engineering.
Renouf, M., Bonamy, D., Dubois, F. and Alart, P., 2005. Numerical simulation of two-dimensional steady granular flows in rotating drum: On surface flow rheology. Physics of Fluids, 17: 103303.
Richard, P., Nicodemi, M., Delannay, R., Ribiere, P. and Bideau, D., 2005. Slow relaxation and compaction of granular systems. Nature Materials, 4(2): 121-128.
Rosato, A., Strandburg, K.J., Prinz, F. and Swendsen, R.H., 1987. Why the Brazil nuts are on top: Size segregation of particulate matter by shaking. Phys. Rev. Lett., 58(10): 1038-1040.
Rycroft, C.H., Orpe, A.V. and Kudrolli, A., 2009. Physical test of a particle simulation model in a sheared granular system. Phys. Rev. E, 80(3): 31305.
Saitoh, K. and Hayakawa, H., 2007. Rheology of a granular gas under a plane shear. Phys. Rev. E, 75(2): 21302.
Savage, S., 1998. Analyses of slow high-concentration flows of granular materials. J. Fluid Mech., 377: 1-26.
Savage, S. and Jeffrey, D., 1981. The stress tensor in a granular flow at high shear rates. J. Fluid Mech., 110: 255-272.
Savage, S.B., 1984. The mechanics of rapid granular flows. Adv. Appl. Mech., 24: 289-366.
Seville, J., Tuzun, U. and Clift, R., 1997. Processing of particulate solids. Chapman & Hall.
Silbert, L., Landry, J. and Grest, G., 2003. Granular flow down a rough inclined plane: Transition between thin and thick piles. Phys. Fluids, 15: 1.
Silbert, L.E. et al., 2001. Granular flow down an inclined plane: Bagnold scaling and rheology. Phys. Rev. E, 64(5).
Staron, L., 2008. Correlated motion in the bulk of dense granular flows. Phys. Rev. E, 77(5): 051304.
Takahashi, T., 1991. Debris Flows: IAHR. AIRH monograph, AA Balkema, Rotterdam.
Tennekes, H. and Lumley, J., 1972. A first course in turbulence. MIT press.
Thompson, P.A. and Grest, G.S., 1991. Granular flow: Friction and the dilatancy transition. Phys. Rev. Lett., 67(13): 1751-1754.
Volfson, D., Tsimring, L.S. and Aranson, I.S., 2003a. Order Parameter Description of Stationary Partially Fluidized Shear Granular Flows. Phys. Rev. Lett., 90(25): 254301.
Volfson, D., Tsimring, L.S. and Aranson, I.S., 2003b. Partially fluidized shear granular flows: Continuum theory and molecular dynamics simulations. Phys. Rev. E, 68(2): 021301.
Volfson, D., Tsimring, L.S. and Aranson, I.S., 2004. Stick-slip dynamics of a granular layer under shear. Phys. Rev. E, 69(3): 031302.
Walton, O.R. and Braun, R.L., 1986. Stress calculations for assemblies of inelastic spheres in uniform shear. Acta Mech., 63(1-4): 73-86.
Wang, Z.T., 2004. A note on the velocity of granular flow down a bumpy inclined plane. Granular Matter, 6(1): 67-69.
Yalin, M., 1977. Mechanics of sediment transport. Pergamon.
Yoon, D.K. and Jenkins, J.T., 2005. Kinetic theory for identical, frictional, nearly elastic disks. Phys. Fluids, 17: 083301.
Zhang, Y. and Campbell, C.S., 1992. The interface between fluid-like and solid-like behaviour in two-dimensional granular flows. J. Fluid Mech., 237: 541-568.
Zhu, H.P. and Yu, A.B., 2002. Averaging method of granular materials. Phys. Rev. E, 66(2): 21302.
Zhu, H.P., Yu, A.B. and Wu, Y.H., 2006. Numerical investigation of steady and unsteady state hopper flows. Powder Technol., 170(3): 125-134.
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