Global Nature of Dilute-To-Dense Transition of Granular Flows in a 2D...

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The hopper, with an open angle of 60 o , is connected to atest section of width D 24 mm and length l 500 mm .At the end of the test section, there is an exit of width dwhich is controlled by micrometers to a precision of 0.01 mm. Granular ows in the test section are initiatedby allowing the steel beads in the hopper to fall bygravity. A thin plate with a number of uniformly distrib-uted holes is inserted close to the exit of the hopper tocontrol the inow rate and ensure the uniformity of particle distribution across the test section. The totalmass M of the beads falling out of the exit is measuredas a function of time t by an electronic balance withsensitivity of 0.02 g and a weighing period of 0.02 s.The ow rate Q t is obtained by the slope of the recordedM t curve:

The ow in the test section initiated from the hopper isdilute and accelerating. The typical velocity of the steelbeads close to the exit is 1 :0 0 :1 m =s . Since d is smallerthan D , two wedges (heaps) will be formed at both sidesof the exit with a base length D d =2 . If the inow rateQ0 is small or d is not too small, there will be no netaccumulation of beads in the test section other than thetwo wedges and hQ t i Q0 . This is t he regime of diluteows as shown in (a) of Fig. 1(B). However, for a givenQ0 , t here will be a critical dc below which there will benet accumulation of beads in the test section. In otherwords, if d is decreased systematically, there will be asudden drop of Q t when dc is reached. This accumula-tion of beads will proceed until the whole test section islled with beads and this is the regime of dense ows. Theprocess of this dilute-to-dense transition induced by re-

ducing d is shown in Fig. 1(B), which is recorded by avideo camera.

Figure 2 shows the d dependence of hQ t i when d isreduced or increased systematically for a given Q0 . Whend is large (point A), hQ t i Q0 , the ow is dilute. It canbe seen that hQ t i remains practically independent of dwhen d is larger than a critical size dc ( A to B in Fig. 2).When dc is reached, the dilute ow turns to dense ow

and the ow rate can be reduced instantaneously byseveral times to drop from Q0 to Qd ( B to C ). After thetransition, Qd decreases monotonically with reducing d(C to D). The ow jams when d is about the size of four-particle diameters, where permanent arching occurs tocause jamming of the ow [12] ( D to E ).When increasingd from the jammed phase, the ow starts as dense ow,and the rate hQ t i increases gradually with increasing duntil Qd reaches Q0 and turns back to dilute ow asshown by triangle points in Fig. 2 ( D through C to A).There is no sudden increase of the ow rate at the tran-sition from dense to dilute ow. We have checked that t hedense ow rate curve Qd d (the CD part of curve ACD )follows the Beverloo empirical equation d kd0 3 =2with k 4 [18,19] and is independent of Q0 .

If the experiment of Fig. 2 is repeated with differ-ent Q0 , a family of paths similar to ABCD of Fig. 2will be formed, shown as dotted lines in Fig. 2. Animportant characteristic of this family of lines is thatdc decreases with Q0 shown as broken l ine BF . That is, fora given d, there will be a Q0 [denoted as Qc d ] at which adilute-to-dense transition will occur. Figure 2 is the resultof experiments with xed D but obviously Qc d will alsodepend on D and d0 . The Qc d curves for various D andd0 have been measured and shown in Fig. 3. Four D s,D 30 , 25, 20, and 15 mm, are tested for d0 1 mm

3 6 9 12 15 18 21 24 27

0

10

20

30

40

E

Q 0

'

Q 0

d C

Q ( g / s )

d(mm)

F

D

C

B A

FIG. 2. At a given inow rate Q0 , the outow behaves differ-ently when the ow is dilute or dense. For a dilute ow, atransition from dilute to dense occurs at a critical opening sizedc , and the outow follows curve ABCDE . For a dense ow, asd increases the ow follows curve DC and is extended to Awithno abrupt change. For different Q0 , the transition occurs atdifferent d, as shown by dashed lines. The curve BF determinesthe optimal outow rate at any given Q0 . For clarity, only twodilute-ow experimental data sets are shown as solid circlesand squares.

FIG. 1. (A) Top and side views of the inclined channel.(B) Photos of dilute-to-dense ow transition in a time sequence

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particles, and two D s, D 40 and 30 mm, are tested for2 m m par ticles. In Fig. 3 the upper six curves are Qc d sand the lower six curves are the corresponding Qd dcurves for particles d0 1 and 2 mm. If the dilute-to-dense transition scales with the system size, one mightexpect that the exit width d should be scaled by channelwidth D as Qc d =D F d=D for some scaling functionF . The inset of Fig. 4 shows such a plot and obviously such

scaling does not hold. Instead, a new scaling variabledD d

dd0 is found to collapse the scaled critical ow rate

qc Qc d = D =d0 into a single scaling curve as shownin Fig. 4. The result in Fig. 4 can be expressed as qcq0 F , where q0 is some constant. It means that themaximum ow rate of a dilute ow of a system of dand D is not the same as that of a system of 2 d and 2 Dalthough their aspect ratio is the same.

A remarkable feature of the dilute-to-dense transitionis that there are strong uctuations in Q t when d is set toclose to dc . Direct observations of the motions of thebeads close to the exit reveal that there are avalanchelikeevents taking place at the two wedges on either side of theexit. After the wedges have been built up by the incomingux, ows of surface layer in the form of avalanches willoccur. The two wedges effectively act as collectors of incoming ux and direct them to the exit through thesurface ow. Therefore, there are strong density uctua-tions (in terms of a rea fraction p ) near the exit. When thedischarge from the two wedges meet at the exit, the out-ow can sometimes be blocked, leading to dense ow.This blockage is intermittent if the incoming ux is notlarge enough and therefore producing strong uctuationsin Q t . However, if the incoming ux exceeds Qc , theblockage becomes permanent and there is accumulation

of the beads in the test section until the whole section islled. Particle density close to the exit can be deter-mined from video pictures. It is found that a dilute-to-dense transition will occur if exceeds a critical value

c . In our experiments, it seems that the value of calways corresponds to that of an area fraction of 0 :650 :03 and is independent of Qc , d, or D .

Following the observations discussed above, the owacross the test section can be divided into three regions:the central part with length d and the two wedges withbase length D d2 . If v 0 and 0 are the velocity and densityof the dilute ow initiated by the hopper just before

particles reaching the wedges, the incoming ux on thewedge is v 0 0 D d2 . The outgoing ux of the wedge will becarried away by the uidized surface layer mentioned asve c , where and v e are the depth and characteristicvelocity of the surface layer, respectively. Conservation of ux gives v 0 0v e c

D d2 . In this model, will increase

with the incoming ux or 0 . When these two top layersmeet at the exit, we have d=2 , where is somegeometric factor which takes care of the angle of repose.Therefore, at the transition, we have f c v 0 0 cve c

1 dD d or

f c f

v ev

dD d , where f v c

1 . Onecan consider v as some intrinsic velocity of the problemwhich is determined by physical properties of the systemsuch as the angle of inclination or coefcient of slidingfriction, etc. Presumably v ev is a function of system pa-rameters D and d. However, for the case of D d, it isreasonable to assume that v ev depends only on

dd0 . In such a

case, one expects to see v evdd0 because this is the rst

order expansion of ve in terms of d when d 0 givesve 0 . This later form of

vev gives

f c f

dd0

dD d

which agrees with our result in Fig. 4 for small d or .Physically v e=v d=do means that the dischargevelocity of the uidized layer of the two wedges increaseswith d. Note that v e for the dense ow, v e d with

1 =2 [19].

It can be seen from Fig. 4 that qc increases withmonotonically. Obviously, for a xed D , Qc must tend to a

0 5 10 15 20 25 30 35 40

0

20

40

60

80

100

120

140

Q (

g / s )

d (mm)

d0 = 1 mmQ

cQ

dD = 15 mm D = 15 mmD = 20 mm D = 20 mmD = 30 mm D = 30 mmD = 25 mm D = 25 mm

d0 = 2 mmQ

cQ

dD = 40 mm D = 40 mmD = 30 mm D = 30 mm

FIG. 3. The Qc and Qd vs d of particle size d0 1 mm atchannel widths D 30 , 25, 20, and 15 mm, and of particle sized0 2 mm at channel widths D 40 and 30 mm.

0 50 100 150 200

0

20

40

60

80

100

120

140

160

180

200

220

240

0.2 0.3 0.4 0.50.0

0.5

1.0

Q c

/ ( D / d

0 ) ( s e c -

1 )

(d/d0)(d/(D-d))

Q C

/ D

dC /D

D = 30 mmD = 25 mmD = 20 mmD = 15 mm

FIG. 4. The rescaled qc Qc= D =d0 vs a new scalingvariable d=d0 d= D d , while Qc=D vs d=D areshown in the inset ( d0 1 mm in this case). The dotted lineis a t of qc by qm 1 e = 0 , where qm 216 and 0 45 .


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limit for large enough d=d0 in the experiments because Qis given by v 0 0 D and there must be an upper limit in 0to still have dilute ows in the test section. If qm is themaximum dilute ow ux of the channel, we have qmq0 F 1 . One can choose F 0 0 and F 1 1 to giveqc qmF . Note that there are two ways for to goto 1 . When d D , the test section is just a straight pipe(2D), qm is obviously just the maximum dilute ow

capacity of the pipe. For xed d=D , goes to 1 wheneither d goes to 1 or d0 goes to zero. In both cases, wehave a continuum limit in which the size of t he particlescan be neglected. Therefore, our model predicts that qm isindependent of d=D when d=d0 is large enough. Thefunctional form of F 1 e = 0 , which gives thecorrect form of qc for small , has been tted to the datain Fig. 4 shown as a dotted line. The scattering of themeasured values of Qc increases as increases, whichmakes it difcult to determine experimentally the limitvalue of qc . Presumably, our model discussed above isvalid only for small d.

The phenomenological model described above is basedon the observation that the transition occurs when the owdensity near the exit reaches p 0 :65 0 :03 .The abruptchange of the ow rate at the dilute-dense transition maybe understood as the inelastic collisions of the particleswith the two piles near exit. In some aspects, our system issimilar to the wedge setup of Rericha et al. [20], whereshocks identical to those in a supersonic gas are observedwhen a steady ow passes the wedge. Our system, how-ever, with the piled heaps at the two sidewalls near theexit is equivalent to a system where particles pass twoinward soft wedges, a system more commonly seen inindustrial transport of granules. The abrupt change indensity, granular temperature, and velocity before andafter the dense area near the exit may account for the owrate drops at the dilute-dense transition, as the ow rate isa function of the product of ow density and the particlevelocity. A two-dimensional molecular dynamics simu-lation using force model similar to [21] is performed toprovide the microscopic view of the transition process.Simulation results show that the dense ow and the diluteow are two different states. When the ow densityreaches p 0 :65 0 :03 , the ow becomes dense owand the multibody collisions dominate.

In conclusion, we have experimentally obtained adilute-to-dense transition curve Qc Q0 ; dc , at whichthe ow rate Q0 drops to Q

dat d

c. Instead of the intui-

tive scaling relationship d=D , experimental results showthat a global scaling variable d= D d d=d0determines if the ow is dilute or dense at a given Q0 .The scaled qc can be tted in an empirical formqm 1 e = 0 . The ratio = 0 determines the criticalux of the system. Intuitively, 0 is determined by physi-cal properties of the system, such as the elasticity of theparticles, particle size, and the inclination angle of theplate, etc. While the physics of jamming transition isdetermined by local scales close to d0 , our result suggests

that the physics of dilute-to-dense transition is simila r tohydrodynamic instability which is controlled by globalscales such as d= D d . The discovery of this globalscaling property may provide us with ideas of betterdesigns for the transport of granules in industrial pro-cessing, and better understanding of similar ow systemssuch as systems in trafc ows.

This work was supported by the National Key Program

for Basic Research and the Chinese National ScienceFoundation P roject No. A0402-10274098. One of theauthors (W. C.) would like to thank support from thespecial funds for Major State Based Research Projects.

*To whom correspondence should be addressed.Electronic address: [email protected]

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