Three-dimensional shear in granular flow

a r X i v :c o n d -m a t /0507469v 3 [c o n d -m a t .s o f t ] 9 D e c 2005

Three-dimensional shear in granular ?ow

Xiang Cheng,1Jeremy B.Lechman,2Antonio F.Barbero,1,?Gary S.Grest,2Heinrich

M.Jaeger,1Greg S.Karczmar,3Matthias E.M¨o bius,1and Sidney R.Nagel 1

1

The James Franck Institute and Department of Physics,The University of Chicago,Chicago,IL 60637

2

Sandia National Laboratories,Albuquerque,NM 87185

3

Department of Radiology,The University of Chicago,Chicago,IL 60637

(Dated:February 2,2008)The evolution of granular shear ?ow is investigated as a function of height in a split-bottom Couette https://www.doczj.com/doc/eb11879249.html,ing particle tracking,magnetic-resonance imaging,and large-scale simulations we ?nd a transition in the nature of the shear as a characteristic height H ?is exceeded.Below H ?there is a central stationary core;above H ?we observe the onset of additional axial shear associated with torsional failure.Radial and axial shear pro?les are qualitatively di?erent:the radial extent is wide and increases with height while the axial width remains narrow and ?xed.

PACS numbers:45.70.Mg,45.70.-n,83.50.Ax

Shear bands in dense granular materials are localized regions of large velocity gradients;they are the antithesis of the broad uniform ?ows seen in slowly-sheared New-tonian ?uids [1,2,3,4,5,6].Until recently it was gen-erally assumed that all granular shear bands were nar-row.However,in 2003Fenistein et al.[7]discovered that in modi?ed Couette cells granular shear bands can be made arbitrarily broad.In this geometry,the bottom of a cylindrical container is split at radius r =R s and shear is produced by rotating both the outer ring and the cylindrical boundary of the container while keeping the central disk (r

FIG.1:Surface ?ow as function of ?lling height,H .(a)Angular velocity,ω0at the cell axis.(b)Center of the shear band,R c .Solid line is a gaussian ?t to guide the eye;dotted line is the ?tting function of Fenistein et al.[7];dashed line is the theoretical result [8].Vertical dash-dot line indicates H =0.6R s .Inset:Schematic of split-bottom Couette cell.

Our setup is similar to that of Fenistein et al.[7]ex-cept that we rotate the inner disk instead of the outer ring and cylinder (Fig.1b inset).In the absence of iner-tial e?ects,this makes no di?erence to the results.For surface observations with high-speed video we use a cylin-drical cell with radius R out =72.25mm whose bottom is split at R s =55.5mm .For the MRI experiments a cell with R out =43.9mm and R s =32.5mm is used.The inner disk is rotated at angular velocities,?,in the

2

shear-rate-independent regime.For the data presented here?=1.96rad/s and?=1.01rad/s for the surface and MRI experiments,respectively.We?ll the cell with granular material to a total height h=H,measured from the cell bottom.A layer of grains glued to the cell walls and bottom assures controlled friction at the boundaries. For the surface?ow measurements we used spherical mus-tard seeds(d=1.9mm)and tracked their motion with high-speed video at frame rates ranging from250s?1to 0.027s?1.To follow the motion of particles inside the pack using MRI,we use a mixture of poppy and rajagara seeds.

Rajagara seeds are more spherical than poppy seeds(Fig.2b inset),but have nearly the same average di-ameter(d=0.85mm)and the same density(ρ=1.1g/cm3). Poppy seeds contain more oil than rajagara seeds pro-viding a clear contrast in MRI signal which allows for particle tracking(Fig.2a inset).

The simulations are carried out with a discrete ele-ment method in which grains interact only upon con-tact through assumed point forces in normal and tan-gential directions and elastic tangential displacements

are truncated as necessary to satisfy the Coulomb cri-teria at the contact.Details of the speci?c implementa-tion can be found elsewhere[9].We use mono-disperse hertzian spheres with a layer of frozen particles at the bottom.The relevant parameters describing the ma-terial properties of the spheres are the normal sti?ness k n=2×105mg/d,the tangential sti?ness k t=2/7k n, the normal and the tangential viscous damping coe?-cientsγn=50

d/g?1.39rad/s.

Evidence for a change in?ow behavior with increasing ?lling height,H,can already be found by tracking par-ticle motion at the free top surface(Fig.1).The angular velocity at the center of the cell,ω0=ω(r=0,h=H) is independent of H for shallow packs but begins to de-crease beyond H??0.6R s.At a similar height Fenistein et al.found that the radial shear pro?le at the surface begins to deviate from the universal error-function shape describing the pro?le for shallow packs[7].Dramatic deviations are also seen in the evolution of the center position of the shear band,R c(H),with?lling height (Fig.1b).Our results for R c,de?ned as the radial posi-tion where the shear rate has its maximum,are consistent with previous experiments[7]as well as with models[8] in the shallow pack regime.However,whileω0decreases for H>H?,the shear zone does not disappear at the surface as predicted by the theory(dashed line).Instead, in both experiments and simulations R c asymptotically approaches a nonzero value.FIG.2:Angular velocity pro?lesω(r)at di?erent vertical positions h for(a)H=0.4R s and(b)H=0.88R s.Symbols for MRI experiments:(a)h=0.21H( ),h=0.43H(?) and h=0.65H(▽);(b)h=0.10H( ),h=0.21H(?), h=0.31H(△),h=0.42H(▽)and h=0.73H(?).Lines for simulations:(a)from right to left h=0.19H,h=0.50H and h=0.85H;(b)from top to bottom h=0.11H,h=0.23H, h=0.30H,h=0.43H and h=0.70H.(a)Inset:MRI image for one layer inside the bulk.Bright spots are poppy seeds, dark background are rajagara seeds;poppy seeds glued to the wall of the cell show up as a bright circle.(b)Inset:Optical micrographs of rajagara(top)and poppy seeds(bottom). These results imply that,beyond H?,velocity gradi-ents must also exist in the vertical,axial direction near the center of the cell.Our MRI experiments and simu-lations explore this shear?ow inside the bulk.We pre-pare our MRI samples by mixing5%(by volume)poppy seeds(MRI positive seeds)uniformly with rajagara seeds (Fig.2a inset).Images before rotation and after an in-terval of rotation are taken.By performing a cross-correlation of the two images as a function of radius,we obtain velocity pro?lesω(r)as a function of h(Fig.2). This method enables us to measure velocities with sev-eral orders of magnitude di?erence(for details,see[12]). For both experiments and simulations care was taken to assure that the systems are in the steady state by rotat-ing long enough before any measurement were performed. As an additional check we made sure that stopping and restarting the system did not change the velocity pro?le, which is consistent with the previous study[10].

For H

FIG.3:(color).Contours of constant angular velocity,ω/?,for di?erent ?lling height H .Upper panel:MRI experiments.ω/?=0.84(?),0.24( ),2.4×10?2(?),2.4×10?3( ),2.4×10?4(▽).Dashed lines indicate H and dotted lines are guides to the eye.Lower panel:simulations.Color is used to identify velocity ranges.Dark red:ω/?∈[0.84,1];orange:[0.24,0.84];yellow:[2.4×10?2,0.24];green:[2.4×10?3,2.4×10?2];blue:[2.4×10?4,2.4×10?3];white:[0,2.4×10?4].

Fig.2a.There is no slip between di?erent layers near the center of the cell,and the pro?les are ?t well by an error function.However,when we increase H above H ?,axial slip occurs in both MRI experiments and simulations:the bottom layer rotates at the same rate as the bottom disk,while the layer near the surface hardly rotates at all (Fig.2b)[11].Thus the decrease in surface ?ow velocity for H >H ?is caused by shear between horizontal layers inside the bulk.For H

?h |r =0=0.To visualize the resulting shear pro?les,we plot cross-sections of the system with contours of constant angular velocity (Fig.3).For H well above H ?,the high-velocity contours close into dome shapes (Fig.3d),which gradu-ally open as H decreases.The contours for smaller ve-locities open up earlier and eventually touch the surface (Fig.3b,c).When H

FIG.4:Scaled angular velocity ((ω/?)?a )/(1?a )for dif-ferent heights h/σ.Insets show the corresponding unscaled data from (a)MRI and (b)simulations.The main panel uses the same symbols as the insets.Solid line is a gaussian exp(?h 2/(2σ2))with σ=0.18R s .Inset (a):H =0.97R s ( ),H =0.88R s (?),H =0.78R s ( ),H =0.74R s ( )and H =0.63R s ( ).Inset (b):H =1.02R s ( ),H =0.90R s (?),H =0.78R s (△),H =0.72R s (▽)and H =0.66R s (?).Solid lines are gaussian ?ts introduced in the text.

and surface measurement with mono-disperse spherical grains are larger than those from the MRI experiments with poppy and rajagara seeds [11].Extrapolating each set of data to ω/?=1suggests that the onset of axial shear begins at H ?=0.60±0.02R s .

The data in Fig.5,together with those from Fig.1a,demonstrate that the transition in shearing behavior at H ?occurs in a continuous manner.This di?ers from the model by Unger et al.[8]which predicts a ?rst-order transition at H ?=0.7R s .While the Unger model,based on the idea of minimum dissipation of energy,includes the essential elements for the transition in the shape of the shear band,a key aspect not considered is the axial slip below H ?.The necessity for such slip emerges from

FIG.5:Width of the shear pro?les,σ,for di?erent?lling heights H.Data showσboth along the axial direction in the bulk( and from MRI experiments using di?erent methods [13],?from simulations)and along the radial direction in the surface layer(×).Dashed line isσ=0.18R s.Inset:o?set a vs.H from MRI( , )and simulations(?).Surface data from Fig.1are shown for comparison(△).Solid line indicates exponential behavior.

considering the torque balance in shallow packs.When the torque on the surface of the inner core exceeds the frictional strength at the bottom,slip will occur.The total torque on the surface of the inner core and the tor-sional strength of the contact between the bottom of the cell and the granular material above it can be calculated if the shape of the inner core h core(r)is https://www.doczj.com/doc/eb11879249.html,ing the approximate h core(r)given in[8],it can be shown that this already occurs when H>0.5R s.Thus,already within the Unger model a torsional failure mode near the bottom should preempt any?rst-order transition within the bulk.Our data demonstrate that this torsional fail-ure is associated with a well-de?ned axial shear band at r=0that exhibits a gaussian pro?le.

Gaussian shear pro?les have been observed in previous studies using traditional Couette cells[5].However,in the present geometry we?nd such a pro?le along the axial direction and an approximate error-function pro?le along the radial direction.The width of this axial shear zone appears to be independent of H,with an average value σ=0.18R s,implyingσ/d=6.9for the experiment and 5.4for the simulation(Fig.5).This is in contrast to the width of the radial shear pro?le(Fig.5),which strongly depends on H and approaches0as H→0[7].

The ubiquitous presence of shear bands is one of the crucial di?erences between granular materials and ordi-nary?uids.Understanding what gives rise to the shear pro?les is one of the outstanding puzzles in granular dy-namics.The modi?ed Couette geometry produces two distinct forms of shear:a radial component whose width grows with height as shown by Fenistein et al.[7],and an axial component with a small constant width that only appears when the?lling height exceeds a threshold.The di?erent character of the shear bands in the radial and axial directions shows that boundary conditions are es-sential for determining shear localization.Because the onset of axial shear in this geometry is continuous and controlled simply by the height of the pack,these studies have allowed for detailed observation of how shear can be initiated in the bulk.Very recent surface measurement of the central core procession[14]also corroborates our three dimensional results.

We acknowledge E.Corwin,X.Fan,D.Fenistein,M. van Hecke and J.River.Collaboration was performed under the auspices of DOE Center of Excellence for the Synthesis and Processing of Advanced Materials.Sandia is a multiprogram laboratory operated by Sandia Cor-poration,a Lockheed Martin Company,for the US DOE National Nuclear Security Administration under contract DE-AC04-94AL85000.Work at the UofC was supported by NSF MRSEC DMR-0213745,NSF CTS-0405619, DOE DE-FG02-03ER46088and MAT2003-03051-C03-01 (Spanish Government).AFB thanks the Spanish Minis-terio de Educaci′o n y Ciencia for a grant to enable an extended stay at the UofC.

?Permanent address:Complex Fluids Physics Group, Department of Applied Physics,University of Almeria, Almeria04120,Spain.

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93,088001(2004).

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Phys.Rev.Lett.92,094301(2004).

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(2004).

[11]We believe that the quantitative di?erence between MRI

experiments and simulations is due to the di?erent cell geometry and particle shape used in the two situations.

However,they all show the same trend.

[12]X.Cheng et al.,to be published.

[13]( )is measured with the method introduced in the text.

( )is obtained by placing a thin cross of poppy seeds at the center of the cell and measuring its distortion after rotation,smiliar to[7].For a detailed discussion of the experimental methods,see[12].

[14]D.Fenistein,J.W.van de Meent,and M.van Hecke,

cont-mat/0507442(2005).

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反向工作 峰值电压 URM/V 额定正向 整流电流 整流电流 IF/A 正向不重 复浪涌峰 值电流 IFSM/A 正向 压降 UF/V 反向 电流 IR/uA 工作 频率 f/KHZ 外形 封装
型 号
1N4000 1N4001 1N4002 1N4003 1N4004 1N4005 1N4006 1N4007 1N5100 1N5101 1N5102 1N5103 1N5104 1N5105 1N5106 1N5107 1N5108 1N5200 1N5201 1N5202 1N5203 1N5204 1N5205 1N5206 1N5207 1N5208 1N5400 1N5401 1N5402 1N5403 1N5404 1N5405 1N5406 1N5407 1N5408
25 50 100 200 400 600 800 1000 50 100 200 300 400 500 600 800 1000 50 100 200 300 400 500 600 800 1000 50 100 200 300 400 500 600 800 1000
1
30
≤1
＜5
3
DO-41
1．5
75
≤1
＜5
3
DO-15
2
100
≤1
＜10
3
3
150
≤0.8
＜10
3
DO-27
常用二极管参数： 05Z6.2Y 硅稳压二极管 Vz=6~6.35V,Pzm=500mW,

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Starter Unit Good to see you again知识总结 一. 短语 1. dance with me 和我一起跳舞 2. sing with me 和我一起唱歌 3. clap your hands 拍拍你的手 4. jump up high 高高跳起 5.shake your arms and your legs晃晃你的胳膊和腿 6. bend your knees 弯曲你的膝盖 7. touch your toes 触摸你的脚趾8. stand nose to nose鼻子贴鼻子站 二. 句子 1. ---Good morning. 早上好。 ---Good morning, Mr Li. 早上好，李老师。 2. ---Good afternoon. 下午好。 ---Good afternoon, Mr Brown. 下午好，布朗先生。 3. ---Good evening，Lisa. 晚上好,丽莎。 ---Good evening, Bob. 晚上好，鲍勃。 4. ---Good night. 晚安。 ----Good night. 晚安。 5. ---What’s your name? 你叫什么名字？ ---I’m Bob./ My name is Bob. 我叫鲍勃。 6. ---Open the window, please. 请打开窗户。 ---Yes ,Miss. 好的，老师。 7. ---What colour is it? 它是什么颜色？ 它是蓝红白混合的。 ---It’s blue, red and white. 皮特的桌子上是什么？ 8. ---What’s on Pit’s table? ---A schoolbag, an eraser and two books. 一个书包，一个橡皮和两本书。 9. ---What time is it? 几点钟？ 两点钟。 ---It’s two. 10.---What’s this? 这是什么？ ---My guitar. 我的吉他。

JOININ英语三年级下册知识点

JOIN IN英语三年级下册 Start unit 1 Words morning afternoon evening night 2 Sentences Good morning !早上好Good afternoon ！下午好Good evening！晚上好 Good night！晚安 3 Phrases clap your hands 拍拍手 jump up high 往高跳 shake your arms and your legs 晃动你的胳膊和腿 bend your knees 弯曲你的膝盖 touch your toes 摸摸你的脚指 stand nose to nose 鼻子对鼻子站着 Unit 1 Pets 1 Words cat猫dog狗bird 鸟mouse老鼠fish鱼rabbit 兔子frog青蛙hamster仓鼠 budgie鹦鹉tiger老虎monkey 猴子panda熊猫giraffe 长颈鹿elephant 大象bear 熊run跑sit坐fly飞swim游泳roar吼叫eat吃 2 Grammar

★名词的复数：一般在词尾直接加s,不规则变化要牢记: fish-----fish mouse------mice 3 Sentences 1.Have you got a pet ? 你有宠物吗？ Yes ,I have. 是的,我有。/No, I haven’t. 不，我没有。 2.What have you got ? 你有什么宠物吗？I’ve got a dog . / A dog. 我有一只狗。 3.What colour is the cat ? 你的猫是什么颜色的？It’s black. 它是黑色的。 What is wizard’s pet? 巫师的宠物是什么？ 4.What is it ? 它是什么? It’s a rabbit .它是只兔子。 5.How many budgies /mice are there? 这里有多少只鹦鹉/老鼠？ There are + 数字budgies/mice. 这里有------只鹦鹉/老鼠。 6.Fly like a budgie. 像鹦鹉一样飞。Run like a rabbit. 像兔子一样跑。 Swim like a fish. 像鱼一样游泳。Eat like a hamster. 像仓鼠一样吃东西。 Sit like a dog. 像狗一样坐。Roar like a tiger. 像老虎一样吼叫。 7.What are in the pictures. 图片里面是什么？Animals. 动物。 8. What animals? 什么动物？ 9.How many pandas (elephants /bears/ giraffes/ monkeys/ budgies) are there?有多少.？ How many + 可数名词的复数形式 Unit 2 The days of the week