Intermittent dry granular flow in a vertical pipe

Yann Bertho, Frederique Giorgiutti-Dauphine, Jean-Pierre Hulin

To cite this version:

Yann Bertho, Frederique Giorgiutti-Dauphine, Jean-Pierre Hulin. Intermittent dry granularflow in a vertical pipe. Physics of Fluids, American Institute of Physics (AIP), 2003, 15 (11),pp.3358. <10.1063/1.1615570>. <hal-00192104>

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7Intermittent dry granular flow in a vertical pipe

Yann Bertho, Frederique Giorgiutti-Dauphine and Jean-Pierre HulinLaboratoire FAST, UMR 7608, Bat. 502, Universite Paris XI, 91 405 Orsay Cedex (France)

The intermittent compact flow of glass beads in a vertical glass pipe of small diameter is studiedexperimentally by combining particle fraction, pressure, and air and grain flow rates measurementswith a spatio-temporal analysis of the flow. At the onset of the flow, a decompaction front is observedto propagate from the bottom to the top of the tube at a velocity much larger than that of thegrains. The blockage front also propagates upwards and at a still higher velocity. The decompactioninduces a decreasing pressure wave strongly amplified as it propagates upwards towards the top ofthe tube. Pressure variations of 3000 Pa or more are detected in this region while particle fractionvariations are of the order of 0.02. Grain velocities during the flow period also increase stronglyat the top of the tube while the corresponding fraction of total time decreases. A 1D numericalmodel based on a simple relation between the effective acceleration of the grains and the particlefraction variations reproduces the amplification effect and provides predictions for its dependenceon the permeability of the packing.

PACS numbers: 45.70.-n, 81.05.Rm

I. INTRODUCTION

Dry particle flows driven by gravity (as in emptyinghoppers or silos) or by a gas current (in pneumatic trans-port [1]) are encountered in many industrial processes.Modeling such flows represents a challenging fundamen-tal problem since they involve complex interactions ofmoving grains with the surrounding air, bounding wallsand other grains. Various flow regimes may be observedin these systems according to the spatial distribution ofthe particle concentration and velocity field. Examplesof such regimes include the free-fall of particles at highvelocities and low particle fractions, slow compact flowswith high particle fractions, and density waves in whichcompact and dilute zones alternate [2]. These particleflows may be stationary (in this case, particles – or den-sity waves – propagate at a constant mean velocity) ordisplay oscillations, stick-slip motions, intermittency orblockage effects. Such non-stationarities have been re-ported in the simultaneous flow of air and grains in ver-tical pipes [3, 4, 5, 6] and are encountered in a variety ofsettings. They represent a significant problem in manylarge-scale industrial facilities such as silos or pneumatictransport systems: intermittent flows may induce verylarge, potentially destructive, pressure variations.

Several approaches have been suggested to model theo-retically granular flows in pipes, each of them being appli-cable to only a part of the flow regimes. Kinetic theories[7, 8] are best suited when the particle fraction is lowerthan 0.5 and collisional effects play an important part.In this regime, the time between two successive collisionsis large compared to the duration of the collision. Athigher particle fractions, grains are in contact with theirneighbors a large fraction of the time and friction forcesbetween particles and walls become significant. Althoughattempts have been made to adapt the kinetic theory tothese frictional regimes [9], different approaches have of-ten been used. In their study of compact moving bedflows, Chen et al. [10] use the classical Janssen theory

[11, 12] to estimate friction forces on the walls from therelation between the vertical component of the stress-tensor in the grain packing and the horizontal normalstress component on the walls.

The present work deals with intermittent compactflows of grains in a vertical tube of small diameter. Inthis case, the particle fraction is very close to that of astatic random packing of particles, and the solid frictionbetween particles and walls is always important. Theobjective of this study is to analyze the spatio-temporalcharacteristics of the onset and blockage of such inter-mittent flows. Of special interest is the relation betweenair pressure, particle fraction and grain velocities and themanner in which variations of these quantities are ampli-fied as they propagate along the flow tube.

A similar flow regime is encountered in the “tickinghourglass” experiment [13, 14, 15, 16, 17] correspond-ing to a periodic intermittent granular flow between twoclosed glass containers connected by a short vertical con-striction. In this case, intermittency is believed to belargely due to the build-up of a pressure difference in-duced by the grain flow between the two containers (bothbecause air is dragged downwards and because the emptyvolume available for air increases with time in the uppercontainer and decreases in the lower one). Once the backpressure is sufficiently large to halt flow, it decays dueto air flow through the grain packing in the constrictionuntil it is low enough to permit flow. In this experiment,the key phenomena are localized in a small volume closeto the constriction; in particular, static arches of particlesare believed to build up right above it, where transientfractures of the packing are also observed.

In the intermittent compact flow down a long verticaltube considered in the present study, the flow intermit-tency phenomenon is distributed over most of the heightof the tube. This allows one to perform simultaneously alarge variety of measurements (local pressures and par-ticle fractions, air and grain flow rates, . . . ) at severallocations, and so characterized the variations of the flow

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Author manuscript, published in "Physics of Fluids 15, 11 (2003) 3358" DOI : 10.1063/1.1615570

2

FIG. 1: View of the flow experiment. - Insert: sketch of theconstriction at the bottom end of the pipe.

with height. High resolution spatio-temporal diagramsof particle motions along the tube are also obtained andallow for the characterization of propagation of the onsetand blockage of the grain flow along the tube length.

In section II, we describe the experimental set-up andprocedure. Then, in section III, the qualitative charac-teristics of the compact intermittent flows are described,and quantitative data and estimates of air pressure, par-ticle fraction and flow rate variations are presented. Insection IV, we discuss appropriate volume and mass con-servation equations and their bearing on the experimen-tal results. The various forces acting on the grains andthe manner in which they may induce intermittency arediscussed in section V. The equations of motion are thenwritten and solved numerically in order to model the spa-tial variations of pressure, particle fraction and grain ve-locities.

II. EXPERIMENTAL SET-UP AND

PROCEDURE

The granular flow takes place in a vertical glass tubeof radius R = (1.5± 0.025)mm and length L= 1.25m(Fig. 1). Grains flow from a spherical hopper weldedto the top of the pipe. The bottom end is fitted witha variable constriction allowing one to adjust the out-flow: this constriction is realized by a cylinder witha conical tip which can be precisely moved across thepipe (see insert of Fig. 1). The grains are sphericalglass beads of diameter 2a= (175± 25)µm and densityρ =(2.50± 0.02) 103 kgm−3. Electronic scales are placedbelow the tube and their output is transmitted to a com-puter at the rate of one sample per second. The massflow rate Fm of the grains is then determined from thetime variation of the measured mass. The grain flow rate

will be characterized by the mean superficial velocity qrepresenting the volume flow rate of grains per unit areawith:

q =Fm

ρπR2. (1)

The hopper is connected to the tube but otherwise closedexcept for an air inlet. The volume flow rate Qmeas

a of airinto the hopper is measured by an on-line sensor (theprecision is ± 1% at the largest flow rates). An addi-tional transducer measures the air pressure ph inside thehopper. Continuity requires that the inflow of air com-pensates for the volume of both air and grains leavingthrough the vertical tube. Therefore if ph remains con-stant with time, Qmeas

a is the sum of the volume flowrates of air and of the grains in the experimental tube.The superficial velocity qa of air in the tube (volume flowrate per unit area) thus satisfies:

qa =Qmeas

a

πR2− q. (2)

Four pressure sensors are placed along the tube at verticaldistances 200, 450, 700 and 950mm below the outlet ofthe hopper. They are connected to the inside of the tubethrough 0.5mm diameter holes in the tube wall. A fine-meshed grid is stretched across the holes on the externalwall to prevent grains from entering the sensors. An in-dependent experiment has been realized to estimate thepressure drop across the grid and the portholes duringtypical pressure transients encountered in the intermit-tent flows. Due to the low dead volume of the transducer(23µl), the corresponding error on the pressure measure-ment is less than 10Pa for the fastest transients: this willbe seen below to be much smaller than the pressure tobe measured.

Local mean particle fraction variations in tube sec-tions are measured by an electrical capacitance sensorusing two shielded 3mm diameter cylindrical electrodespressed against the outside tube walls and facing eachother. The sensor is connected to a GR1620 capacitancebridge (10 kHz measurement frequency) and a lock-in de-tector with a 680µs time constant. The noise level isof order 10−5 pF, which corresponds to particle fractionvariations of order 10−3.

In order to test the device, an air-grain interface wasmoved through the sensor and the output was recordedas a function of the location of the interface. This allowedone to determine that the measurement is averaged overa slice of the sample of typical thickness 2mm parallel tothe tube axis. The variation of the DC output with theparticle fraction is assumed to be linear and the probesare calibrated by comparing readings obtained with anempty tube and with the same tube filled with a staticbead packing. This assumption of a linear variation isfrequently used in the literature: in the present case, it isparticularly justified by the small particle fraction varia-tions during the experiment and by the random, approx-imately isotropic structure of the packing. The particle

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fraction cmax of the static bead packing obtained afterfilling the tube is assumed to be equal to 0.63: this valuecorresponds to a Random Close Packing of spheres andis frequently obtained experimentally in similar systems.A direct measurement of cmax has been realized but itsprecision is limited by the small amount of beads in thetube. A slightly lower value cmax =0.61± 0.02 was ob-tained (this difference may be due to the small diameterof the tube increasing the influence of wall effects reduc-ing locally the particle fraction). In view of the largeuncertainty on this measurement, the value 0.63 is re-tained in the following.

The capacitance measurements can be performed atchosen heights by moving the sensors along the tube.The instantaneous outputs of four pressure and capaci-tance sensors are digitized simultaneously at a samplingfrequency of up to 1 kHz by a Spectral Dynamics SD195signal analyzer. Time-averaged values of all pressure, ca-pacitance and flow rates sensors are also recorded but at1 s intervals.

The dynamical properties of the flow are studiedfrom spatio-temporal diagrams constructed from appro-priately assembling the output of a digital linear CCDcamera. Light intensity variations along a vertical lineprecisely aligned with the tube are recorded at samplingrates of 500 lines per second with a resolution of up to2048 pixels (the equivalent pixel size on the tube rangesfrom 10µm to 500µm depending on the magnification).One thus obtains 2D images in which the x-axis repre-sents time while the z-axis corresponds to distance alongthe tube.

III. EXPERIMENTAL RESULTS ON THE

INTERMITTENT COMPACT FLOW

The flow regimes are controlled by adjusting the con-striction at the bottom of the experimental tube. Thecompact flow regime is observed for narrow constrictions,at typical superficial velocities q between 0.01m s−1 and0.08m s−1. Narrowing further the bottom constrictionleads to a blockage of the flow while density waves ap-pear at higher flow rates. The typical aperture δ of theconstriction (see Fig. 1 insert) in the compact regimeranges from 1 to 2mm (or approximately from 6 to 12bead diameters). In all cases, the particle fraction in thecompact regime does not differ by more than 0.02 fromthe particle fraction cmax in a static packing.

Decreasing the relative humidity H of the air downto 40% or using “clean beads” with a smooth surface,changes significantly the behavior of the grain flow: inthis case, a continuous flow of grains is observed (its ve-locity is roughly constant with time and of the order ofa few centimeters per second). In return, for “rougherbeads” or relative humidity H ? 50%, the flow is inter-mittent and may remain blocked for a large fraction ofthe time, particularly near the hopper (observations us-ing a SEM microscope display particles with a size of

FIG. 2: Spatio-temporal diagram of the grain flow: (a) at thetop of the tube (100mm to 110 mm from the hopper), (b) atthe bottom of the tube (850 mm to 860 mm from the hopper).Each diagram corresponds to a time lapse of 0.6 s and to afield of view of 10 mm.

up to several micrometers at the surface of the “roughbeads” with spacings increasing with the size from a fewµm up to a few 10µm). Experiments reported here havebeen performed at a relative humidity H = (55± 5)% forwhich the grain flow is intermittent. At the outlet of theconstriction, flow is modulated but does not stop com-pletely. The modulation may be periodic, particularly atlarge flow rates. In the remainder of the paper, we shallfocus on this latter intermittent compact flow regime.

A. Spatio-temporal characteristics of the

intermittent flow

Figure 2 displays two spatio-temporal diagrams of thegrain flow over two 10mm high segments located in thetop and bottom parts of the tube. Time corresponds tothe horizontal scale. The two diagrams were obtained atslightly different times during a same experiment: dueto the excellent periodicity of the flow, they are howevercomparable. Time intervals during which grains are atrest in the tube are marked by horizontal line segmentswith constant grey levels; those during which grains aremoving down are marked by inclined striations: the slopeof which indicates the grain velocity. Starting from rest,the grains first accelerate and reach a constant velocitybefore finally stopping abruptly. Figures 2(a) and (b)show that the fraction of the total time during whichgrains are flowing is much larger at the bottom of thetube than at the top.

A quantitative comparison is made in Fig. 3 in whichthe duration Tf of the flow phase at two different dis-tances from the hopper and the period T of the processare displayed as a function of the time average q of thesuperficial velocity (T is the sum of the durations of theflow (Tf) and static phases). The time lapse Tf is 40%lower in the upper section than in the lower one; it in-creases linearly with q in both cases, while, on the con-

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FIG. 3: Duration of flow phase Tf , as a function of the timeaverage q of the superficial velocity, at two different distancesz from the top of the tube: z = 250 mm (N) and z =600 mm(△). Global period of the intermittent flow T (�).

FIG. 4: Mean grain velocity v vs. mean superficial velocityq for z = 250 mm (H, left axis) and z = 600 mm (▽, left axis).Displacement δz of the grains during a flow period as a func-tion of q (�, right axis).

trary the period T decreases with q. Spatio-temporaldiagrams also allow one to estimate qualitatively the ve-locity of the grains in the flow phase from the slope of thestriations. Figures 2(a) and (b) show that the grain speedis much higher at the top of the tube than at the bottomas required by mass conservation. The particle fractionc varies little along the tube and the global displacementof the particles during one period of the flow must bethe same at all heights. The mean value v of this veloc-ity during the flow phase can be estimated quantitativelyfrom the relation:

v =q

c

T

Tf

, (3)

The variation of v with q in both sections of interestis displayed in Fig. 4: in both cases, v increases roughlylinearly with q. The displacement δz of the grains duringone flow period is equal to v × Tf and is also plotted

FIG. 5: Spatio-temporal diagram of the grain flow at dis-tances between 0 mm and 50 mm from the hopper at the topof the tube (q =0.07 m s−1); total time lapse corresponding todiagram: 0.7 s.

on Fig. 4. For large mean flow rates, the instantaneousgrain flow rate q in the tube exceeds the maximum flowrate from the hopper. A low particle fraction bubble thusbuilds up at the top of the tube: it appears as a lightzone in the spatio-temporal diagram of Fig. 5 (the localparticle fraction in this bubble may be lower than 0.2).The abrupt initial downwards slope of the contour of thelight zone reflects the fast downwards motion of the upperboundary of the compact packing below the bubble. Assoon as the motion of the packing stops, the bubble getsfilled up from the hopper, although at a lower velocitythat decreases with time (this may be due to the buildup of an adverse pressure gradient between the bubbleand the hopper).

B. Spatial variations of the intermittency effect

Information obtained from the spatio-temporal dia-grams are complemented by measurements of the localpressure variations. Figure 6 displays time recordingsof the pressure variations δp = p − po measured on foursensors at different heights z from the top (po is the atmo-spheric pressure). These curves are nearly periodic; thepressure drops sharply while grains are flowing (by upto 3000Pa near the top of the tube). Then the pressureincreases back above po and decays slowly (or remainsconstant) while the grains are at rest. The time averageof the pressure is close to zero (at most 25Pa) for all fourtransducers used for the measurements.

The flow has been filmed in the intermittent regime us-ing a 1000 frames per second high speed camera equippedwith analog input channels connected to the pressuretransducers. The onset of the flow at a given height isobserved to coincide exactly with the beginning of the

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FIG. 6: Time recording of pressure variations δp at four dif-ferent distances z from hopper (from top to bottom, z = 200,450, 700 and 950 mm), for q =0.02 m s−1. Dashed lines corre-spond to the onset and the blockage of the flow.

FIG. 7: Processed spatio-temporal diagram over a 500 mmlength of tube (200 mm <z < 700 mm) during one period ofthe intermittent flow. Light zones correspond to the staticphase and the darker region to the flow.

pressure drop; the transition is less sharp for the block-age since pressure diffusion is still significant even af-ter the grain flow has stopped. The region between thetwo dashed lines in Fig. 6 corresponds to the flow phase(the lines are drawn, within experimental error, betweenpoints corresponding to times at which the flow is ob-served to start or stop). Comparing the widths of thisregion in the different curves confirms that the fractionof time corresponding to the flow phase increases withz. On the contrary, the amplitude of the pressure dropdecreases with z.

FIG. 8: Variation of the absolute velocities of propagation ofthe decompaction |vdf | (H) and blockage |vbf | (�) fronts withthe mean superficial grain velocity q.

The propagation of the onset and blockage of the flowwere studied more precisely on spatio-temporal diagramsof a 500mm high region of the tube (Fig. 7). The dia-gram has been processed by subtracting each linear im-age from the next one to detect more precisely the onsetand blockage of the flow. Domains corresponding to thestatic phase have a light shade (the difference is not zerobecause of the noise of the camera and the fluctuations ofthe illumination). The domain associated with the flowphase is globally darker (the fluctuations of the grey lev-els are large because one subtracts specular reflections oflight on the beads varying very much during the motion).The velocity |vdf | for the onset of the flow corresponds tothe slope of the left boundary in Fig. 7: it was roughlyconstant in that experiment but decreases slightly withdistance in others. The velocity |vbf | for the blockage isgiven by the slope of the right boundary (averaged alongthe tube) and is lower than |vdf |. Both the onset andthe blockage propagate upwards. The variations of |vbf |and |vdf | with q are plotted in Fig. 8. The velocity |vbf |for the blockage increases linearly with q and ranges be-tween 20 and 60m s−1. This variation will be discussed insection IVA. On the contrary, the velocity |vdf | of the lo-cation of the onset of the grain flow (decompaction front)is independent of the mean superficial velocity and is ofthe order of 11m s−1. This implies that the decompactionprocess is independent of the downwards structure of theflow and of the width of the constriction.

This velocity value may be related to the velocity ofsound in two-phase systems with a large density and com-pressibility contrast between the two phases (water-airbubbles for instance). In this case, by assuming an isen-tropic expansion of the gas, one may compute the velocityof sound vsound via [18]:

v2sound =

γpo

ρc(1 − c), (4)

in which c and ρ are the volume fraction and the den-sity of the dense phase, po is the atmospheric pressure

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and γ = 1.4 for air which is considered as having negligi-ble density. This amounts to considering the air-grainsystem as a homogeneous mixture of both high com-pressibility and density resulting in a low sound veloc-ity. Taking ρ to be equal to the density of the grainsand c to the mean particle fraction c ≃ 0.615 leads tovsound ≃ 15m s−1 which is indeed of the order of thevalue |vdf | ≃ 11m s−1 deduced experimentally (0.615 rep-resents a typical value of the average particle fraction asestimated from the measurements displayed in Fig. 10assuming as above that cmax = 0.63).

Data points from the four curves in Fig. 6 together withthe boundary conditions p = po at both ends of the tubeprovide the overall shape of the pressure profile along thetube. The spatial variation of δp = p − po at several dif-ferent times during one same period of variation of theflow within the periodic sequence of Fig. 6 is displayed inFig. 9. The thick solid line in Fig. 9(a) corresponds to thestatic phase. A pressure maximum subsists in the cen-tral part of the tube from the previous flow cycle and theassociated pressure gradients drive air towards both endsof the tube through the motionless packing. When flowis initiated, a low pressure zone appears at the bottomof the tube where grains are moving, and propagates up-wards. The amplitude of the pressure minimum increasesstrongly during this propagation. This amplification ef-fect is discussed below in section V. After the pressureminimum has reached the top of the tube, pressure startsto increase again [Fig. 9(b)]. At first, the local particlefraction reaches a minimum and the expansion of the gasstops. Then, when the granular flow stops in the tube(first at the bottom), the particle fraction reverts back toits static value. Thus, air gets compressed above the at-mospheric pressure since some additional air has leakedbetween the grains during the flow period (see for in-stance the curve corresponding to t =172ms in Fig. 9(b)).Finally, the pressure profile along the static packing re-laxes back to its initial distribution at the beginning ofthe flow phase and a new flow cycle may start again.

Note that when the intermittent flow regime does notappear spontaneously, it can also be triggered externallyby periodic injections and suctions of air close to theconstriction at a frequency comparable to that of thenatural oscillations. The other characteristics of the flowsuch as the amplification of pressure and particle fractionvariations as they propagate upwards are the same as inthe spontaneous intermittent flow regime.

C. Particle fraction and flow rate variations

Spatio-temporal diagrams have allowed us to identifygranular motions in the various phases of the intermit-tency process; their relation with the spatial distributionof the pressure δp has also been analyzed. We proceed bydiscussing flow rate and particle fraction measurementswhich provide additional information on the flow struc-ture.

FIG. 9: Pressure deviation (δp = p − po) variations withdistance along the tube for several measurement times, fora mean superficial velocity q = 0.02 m s−1. (a) Solid lineobtained during the static phase at t =0 s. Dashed linesare profiles obtained at various times during the flowingphase: t= 16ms, t =31 ms, t= 62ms, t =86 ms, t= 109 msand t= 133 ms (dash length and spacing increase with time).(b) Solid line obtained near the end of the flowing phaseat t =133 ms. Dashed lines are profiles obtained at var-ious times: t= 141 ms, t =148 ms, t =156 ms, t = 172 ms,t= 211 ms, t =266 ms (dash length and spacing increase withtime).

The variations of the particle fraction c and pressureδp at a fixed height, and of the flux Qmeas

a of air into thehopper are displayed in Fig. 10. Variations of the localparticle fraction c associated with the onset and blockageof the flow are only of the order of 0.015 in the present ex-ample and were never higher than 0.02. Moreover, thesevariations are evidently synchronous with those of thelocal pressure p in the flowing phase. Note the slow pres-sure decay when the grains are at rest. The amplitudes ofthe variations of c and δp decrease, while their durationincreases with distance from the hopper. In addition todemonstrate this close correlation between the variationsof p and c, our study indicates an increase of the am-plitude of the c variations with the instantaneous grainvelocities.

The time recording of Qmeasa indicates a strong increase

of the downwards air flow rate during the flow phase.

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FIG. 10: Time recordings of the pressure deviations δp andparticle fraction c variations (at z =200 mm) and of the mea-sured air flow-rate Qmeas

a into the hopper. Qmeas

a is definedas positive for air flowing downwards. The dashed line corre-sponds to the onset of the flow.

This increase reflects the entrainment of air by the gran-ular flow and the compensation of the volume of grainsleaving the hopper by the inflow of air. The variation ofQmeas

a is slightly delayed relative to those of p and c sincea time is required for the grain flow to propagate to thetop of the tube and into the hopper. A transient upwardgas flow (Qmeas

a < 0) is observed after grains stop movingin the upper part of the tube, corresponding to air beingexpelled through the grain packing.

IV. QUANTITATIVE ANALYSIS OF THE

INTERMITTENT FLOW COMPONENTS

A. Conservation equations for air and grains

In the following, z =0 corresponds to the top of thetube and the coordinate z increases downwards. Down-flow is defined as being positive. v(z, t) and u(z, t) are,respectively, the grain and air velocities (averaged overthe flow section) in the laboratory frame at a time t anda distance z from the hopper. They are related to thesuperficial velocities q(z, t) and qa(z, t) and to the localparticle fraction c(z, t) (also averaged over a flow section)by the relations:

q = c v, (5)

qa = (1 − c)u. (6)

The superficial velocity q of the grains is related to thevariation of c with time by the volume conservation equa-tion:

∂c

∂t= −∂q

∂z= −c

∂v

∂z− v

∂c

∂z. (7)

A similar relation can be written for the mass flux of airρaqa (ρa being its density):

∂

∂t[ρa(1 − c)] = − ∂

∂z(ρaqa). (8)

Assume that the variations of the gas pressure are isen-tropic so that:

p

ρaγ

=po

ρao

γ= constant, (9)

(po is the atmospheric pressure, ρaothe air density at po

and γ=1.4). Then Eq. (8) becomes:

∂c

∂t=

(1 − c)

γp

∂p

∂t+

∂qa

∂z+

qa

γp

∂p

∂z. (10)

A first prediction provided by mass conservation condi-tions is the velocity vbf at which the blockage front movesup the tube. We define cmax to be the particle fractionin the static column and c+ and v+ the particle fractionand grain velocity right above the front. vbf is obtainedby expressing the conservation of mass of the grains ina slice of thickness δz while the front moves from thebottom to the top of the slice:

|vbf | =v+ c+

cmax − c+. (11)

Since v+ increases linearly with the superficial veloc-ity q, equation (11) implies therefore that |vbf | shouldalso increase linearly with q. This agrees with the trenddisplayed in Fig. 8. Using the experimental values ofv+, c+ and cmax provides an order of magnitude of vbf

(|vbf | ≃ 20m s−1) which is in a good agreement with thatobtained in Fig. 8.

B. Relative magnitude of air flow components

Another issue regarding the flow of air through thesystem is the relative importance of the passive drag ofair by the moving grains and of the permeation of airthrough the grain packing. The first flow component isequal to q(1−c)/c and the second is assumed to be relatedto the pressure gradient by Darcy’s equation:

qDarcya = −K

ηa

∂p

∂z, (12)

where ηa is the viscosity of air and K the permeabilityof the grain packing. The superficial velocity qa of air isthen given by:

qa = −K

ηa

∂p

∂z+

q

c(1 − c). (13)

Equations (7), (10) and (13) can be combined to yield:

∂p

∂t= −v

∂p

∂z− γp

(1 − c)

∂v

∂z+ D

∂2p

∂z2, (14)

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FIG. 11: Average inflow of air as a function of q/c.

in which the coefficient D satisfies:

D =γpK

(1 − c)ηa

, (15)

(the second order term [D/(γp)](∂p/∂z)2 has been ne-glected in Eq. (14)). Considering locally the system as apacking of spherical beads of diameter d, the permeabil-ity K of the grains can be estimated from the Carman–Kozeny relation:

K =(1 − c)3 d2

180 c2. (16)

In the following, we assume that to leading order, p = po

and c = c, so that D and K are constants.The average of qDarcy

a over a flow period is determinedby plotting the time averaged value of Qmeas

a as a functionof the ratio q/c in Fig. 11 (this amounts to assumingthat the mean grain velocity is equal to q/c which isapproximately verified due to the small relative variationsof c in these experiments). Volume conservation impliesthat Qmeas

a /πR2 and the sum q + qa of the volume flowrates of air and grains are equal [see Eq. (2)]. UsingEq. (13), this leads to:

Qmeasa

πR2= q + qa = qDarcy

a +q

c. (17)

Experimentally, the variation of Qmeasa with q/c is lin-

ear with a slope equal to 1± 0.02 in Fig. 11. These re-sults indicate that the time average of qDarcy

a is negligi-ble within experimental error although its instantaneousvalue is non zero. The relative motion of air and of thegrains is therefore significant only during transient flow.

C. Pressure variation mechanisms

The dynamical properties of the system are largely de-termined by spatial and temporal variations of the air

FIG. 12: Pressure vs. particle fraction variations during 10cycles of the intermittent flow. Arrows indicate the directionof the variation on each part of the curve: the vertical phasecorresponds to the slow pressure relaxation through the mo-tionless grain packing, the lower section corresponds to theonset of the flow and the upper one to the blockage of theflow. The dashed line has the theoretical slope γpo/(1 − c).

pressure. We first assume that variations of density ρa

are only due to those of the local particle fraction c. Thisamounts to neglecting the permeation of air relative tothe grains. Then, ∂qa/∂z and [qa/(γp)] ∂p/∂z are equalto zero in which case Eq. (10) becomes:

1

(1 − c)

∂c

∂t=

1

γpo

∂p

∂t. (18)

(p and c are replaced by po and c within a first order ap-proximation). This relation is tested in Fig. 12 by plot-ting pressure variations as a function of particle fractionvariations: for variations small compared to the meanvalue, data points should be located on the dashed lineof slope γpo/(1 − c) shown on Fig. 12. The two param-eters are clearly strongly correlated and the slope of thecurves has the right order of magnitude in the initial andfinal phases of the variation. However, pressure is higherin the blockage phase than during the development ofthe flow, indicating that ∂qa/∂z cannot be neglected inEq. (10) (the last term [qa/(γp)] ∂p/∂z is of lower orderand should remain negligible). Some air gets sucked intothe lower pressure regions when grains flow; this leads toan overpressure when flow stops and the particle fractionincreases.

This overpressure decays slowly during the remainderof the static phase (short vertical part of the curve at topright) due to the finite permeability of the packing. Inthis static phase of the flow sequence, v = 0. The twofirst terms on the right hand side of Eq. (14) are equalto zero leading to the diffusion equation:

∂p

∂t= D

∂2p

∂z2. (19)

Experimentally, the particle fraction varies from 0.605to 0.625; the permeability K computed from Eq. (16)

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ranges therefore from 23 10−12 to 29 10−12 m−2. Tak-ing ηa = 1.85 10−5 m2 s−1, po =105 Pa and a mean valueof K (K =26 10−12 m−2), one obtains D≃ 0.5m2 s−1.The characteristic diffusion distance during the periodT ≃ 0.25 s of the intermittent flow is

√DT ≃ 0.35m

which is comparable to the length over which pressurefluctuations diffuse. During the onset of the flow whichtakes place over a shorter time (a few hundredths of asecond) the diffusion of the pressure variations is suffi-ciently small for Eq. (18) to be approximately valid. Onethus concludes that the permeability of the grain packingis high enough to allow for relative motions between airand the grains during the flow phase; but too low to al-low for a global mean flow of air through the packing. Inthe previous computation, the initial particle fraction ofthe static packing is taken equal to cmax =0.63. Assumethat wall effects reduce this value to 0.61 as discussed insection II so that the particle fraction varies from 0.585to 0.605 during a flow cycle. The mean correspondingvalue of K is 32 10−12 m−2 so that D would increase to≃ 0.6m2 s−1 and remain of the same order of magnitudeas for cmax = 0.63.

Our experiments suggest that the intermittent flowsresult from the amplification of pressure and velocityfluctuations appearing spontaneously or induced in theconstriction at the bottom of the tube. One expects theperiod of the intermittency to be determined by the char-acteristics of the flow over the full length of the tube andnot just at the constriction. The period of the flow is ofthe order of the characteristic diffusion time of air overa distance of half of the tube length. This diffusion timerepresents the order of magnitude of the time necessaryfor pressure perturbations induced during the flow periodto relax before flow is reinitiated. It may thus be closelyrelated to the duration of the static phase of the periodicflow.

V. MODELING OF THE INTERMITTENT

FLOW

In this section, the amplification of the decompactionwave as it propagates towards the top of the flow tube ismodeled numerically. This approach yields insight intothe dependence of the amplification on the permeabilityof the grain packing.

A. Equations of motion and force distributions

The equations of motion and force distribution in thegranular packing will first be written. They will then besolved numerically together with the conservation equa-tions for air and grains. For simplicity, one models thecase of a lower particle fraction zone propagating upwardsin a vertical channel.

The equation of motion reflecting momentum conser-vation for both air and grains contained in a vertical slice

with tube radius R can be written as:

ρcdv

dt= ρcg − ∂p

∂z− ∂σzz

∂z− 2

Rσzr, (20)

(the equation has been divided by the section S = πR2).Acceleration terms for air have been neglected due to thelarge density difference between the air and grains. Thefriction forces on the air applied at the walls are similarlynegligible due to their low area compared to that of thegrains.

Note that when Eq. (20) (excluding the gravity, σzz ,and σzr terms) is combined with Eq. (14) (neglecting thesecond order v ∂p/∂z and diffusion terms), one obtains asound wave propagation equation with a velocity givenby Eq. (4).

The left side of Eq. (20) corresponds to the accelera-tion term and the first term on the right to the weightof the grains. σzr and σzz represent, respectively, fric-tion forces on the grains at the lateral walls and verti-cal stress forces between grains in the tube section; theyare assumed to be constant around the perimeter of thetube and in each section. σzr is then estimated by ap-plying Janssen’s model [11, 12, 19] assuming first thatσzr = µσrr following Coulomb’s relation. σrr is takento be KJσzz , where the coefficient KJ depends on thestructure of the packing and of the surface properties ofthe grains. Equation (20) then becomes:

ρc∂v

∂t+ ρcv

∂v

∂z= ρcg − ∂p

∂z− ∂σzz

∂z− σzz

λ, (21)

in which the characteristic length λ is given by:

λ =R

2µKJ

. (22)

In a static packing, all terms involving the grain veloc-ity vanish: the weight of the grains is balanced by thepressure gradient and the stress forces at the walls andin the tube section. When flow starts, the local parti-cle fraction decreases [Fig. 10(b)] which, in turn, reducesfriction forces at the walls, resulting in an acceleration ofthe flow. No exact relation is available on the variationsof the stress forces. It will be assumed in the followingthat the resultant of the weight of the grains, the pressuregradient and the friction forces increases linearly with thedeviation of the particle fraction c from its static valuecmax. Equation (21) then becomes:

∂v

∂t= −v

∂v

∂z+ J(cmax − c)

(

g − 1

ρc

∂p

∂z

)

, (23)

where J is the parameter characterizing the magnitudeof the variation. This assumption represents a first orderapproximation: it will be valid in the initial phase of thedevelopment of the perturbation as long as the derivativeof the resultant force with respect to the particle fractionis non zero for c = cmax. Other types of functions may beenvisioned to replace J(cmax − c): many will also repro-duce an amplification phenomenon provided they predict

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a resultant force in the right direction with a monotonousdependence on (cmax − c) and if (as will be seen below)the prefactor equivalent to J is large enough.

B. Numerical procedure

Numerical simulations are realized on a simplified 1Dmodel in which a region of reduced particle fraction(called bubble) moves upwards inside a vertical tube filledwith an otherwise grain packing. The bubble length ismuch smaller than that of the tube and it is assumedto be far enough from either of the ends that boundaryconditions do not influence the motion. As suggestedby the experimental observations, the particle fractionand the velocity are discontinuous at the bottom end ofthe bubble (the blockage front), while they vary continu-ously at the top (the decompaction region). Air pressureis continuous in both cases but the pressure gradient isdiscontinuous at the lower front.

In order to solve the problem numerically, one uses theconservation equations for grains (7) and air (10) estab-lished in section IV and the equation of motion (23) ofthe grains. For the sake of practicality, Eq. (14), in whichDarcy’s law is used to determine qa, replaces Eq. (10). Inthe static regions of the packing, all terms in Eqs. (7) and(23) are equal to zero. Equation (14) reduces then to thediffusion equation (19) for air with a coefficient D givenby Eq. (15).

At large distances below and above the bubble, theair pressure p is equal to the atmospheric pressure po;c and v are respectively equal to cmax and zero bothfar above the bubble and everywhere below the blockagefront. Additional boundary conditions corresponding tothe mass conservation for air and for the grains mustalso be verified at the lower front (in addition to pressurecontinuity). In the reference frame moving with the frontat vbf , grain mass conservation is expressed by Eq. (11)which can be rewritten as:

(v+ − vbf )c+ = −vbfcmax. (24)

In the same way, the mass conservation of air implies:

(v+−vbf )(1−c+)−K

ηa

∂p

∂z

∣

∣

∣

∣

+

= −vbf (1−cmax)−K

ηa

∂p

∂z

∣

∣

∣

∣

−

.

(25)The indexes + and − correspond to values close to theblockage front respectively inside the bubble and in thestatic zone. Combining Eqs. (24) and (25) leads to:

v+ − K

ηa

∂p

∂z

∣

∣

∣

∣

+

= −K

ηa

∂p

∂z

∣

∣

∣

∣

−

. (26)

The initial particle fraction, velocity and pressure pro-files are assumed to follow a 1/ cosh(z) variation with acutoff at the lower front (z = zf ) for c and v. Pressureis continuous at z = zf with a faster decrease below.

The corresponding profiles are displayed in Figs. 13(a–c) (rightmost solid curves). The initial particle fractionand pressure variations are related by Eq. (9) above thefront (this amounts to assuming that the bubble hasbeen created through a local decompaction of the grainsinducing an isentropic adiabatic expansion of the air).The amplitudes of these initial variations are respectivelyδc =0.001, δp= 600Pa and δv = 310−3 ms−1.

Pressure, particle fraction and grain velocity profiles atsubsequent time steps are then computed by solving nu-merically Eqs. (7), (14) and (23) in the reference frame ofthe front using a finite difference Euler explicit scheme.For the resolution of Eq. (23), an upwind scheme wasrequired to avoid numerical instabilities due to the ad-vection term v ∂v/∂z. The global length of the simula-tion domain is 1.4m and the mesh size 2 mm. The originof distances is taken at the initial location zf(0) of thefront (the vertical dashed line on Figs. 13(a–c)). Simu-lations correspond to a time lapse of 27ms during whichthe bubble propagates over a distance of 0.4m. At eachstep of the simulation, the pressure, the particle fractionand the velocity are determined at all distances from thefront and the front velocity is also obtained. This al-lows one in particular to determine the new location ofthe front after each time step and to translate distancesfrom the front into distances in fixed coordinates whichwill be used in subsequent plots.

C. Numerical results

1. Qualitative observations

Particle fraction, velocity and pressure profiles at dif-ferent times obtained for a typical set of values of J , Kand cmax are displayed in Fig. 13. The other parametersof the simulation are given by the experimental valueslisted above or computed from J and K. A clear ampli-fication effect is observed for all three parameters. Thevelocity v reaches its maximum value faster than boththe particle fraction c and the pressure δp (typically af-ter 0.3m versus 0.5m). The ratio of the maximum andinitial amplitudes is also higher (20 for v versus 2 and1.6 for δp and c). A small localized overpressure appearsin the static region close to the initial location of thebubble: it is due to air advected by the grains into thisregion at short times [from Eq. (26)]. At later times, thepressure gradient close to the front increases so that thediffusion of air back into the bubble balances the advec-tion term. The increase of the amplitude of the variationsof δp and c is consistent with the decrease of the widthof these curves: the total area under these curves mustremain constant as required by mass conservation of airand grains. The velocity vbf of the lower front is alsoroughly constant with a value |vbf | ≃ 12.5m s−1 of thesame order of magnitude as that observed in our experi-ments.

While the computed amplitude of the pressure vari-

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FIG. 13: Numerical pressure (a), particle fraction (b) and ve-locity (c) profiles, obtained for J =450, K =20 10−12 m2 andcmax = 0.63. From right to left: solid lines: initial profile att =0 s - dashed lines: t =9ms, t =15 ms, t= 21ms, t =27 ms(dash length and spacing increases with time). The verticaldashed lines represents the position of the front zf at t =0 s.

ations is of comparable magnitude to the experimentalones, particle fraction and velocity variations are smaller.This is likely a result of the difference between the modelcase of a localized bubble and the actual intermittentflow which sometimes extends over the entire length ofthe tube.

FIG. 14: Variation of the amplification parameter ∆Ap/∆z(◦) and of the absolute front velocity |vbf | (�) with the forceparameter J , for K = 20 10−12 m2.

2. Quantitative dependence on the modelization parameters

The key adjustable parameter in the model is the co-efficient J in Eq. (23) (that characterizes the effectiveacceleration of the grains when the particle fraction cdecreases below cmax). Figure 14 displays the depen-dance on J of the front velocity |vbf | and of the param-eter ∆Ap/∆z characterizing the amplification. ∆Ap/∆zis determined by plotting the minimum value Ap of thepressure variation curves [as shown in Fig. 13(a)] as afunction of the coordinate zf of the front for each curve.After a short transient, Ap increases quickly with zf andthe variation levels off thereafter. The amplification pa-rameter ∆Ap/∆z is taken equal to the maximum ab-solute value of the slope |dAp/dzf | of this curve. Thepermeability K of the grain packing has been takenequal to the value K =20 10−12 m2). Using the valuesof (cmax − c) deduced from the simulations, the maxi-mum value J =500 in Fig. 14 corresponds in Eq. (23) toan effective acceleration of the grains of the order of g/3or lower.

For J = 0, ∆Ap/∆z is negative and initial variationsof δp and c damp out as the bubbles propagate upwards.∆Ap/∆z becomes equal to 0 for J = 40 and increasessteadily at higher values. The front velocity |vbf | alsoincreases with J in the amplification domain but neverbecomes zero. Its value is of the same order of magnitudeas the experimental front velocities, particularly when Jis large. We note that in the case of a bubble of slowlyvarying length, vdf and vbf are nearly equal while theydiffer in the experiments so that more detailed quanti-tative comparisons would be irrelevant. These resultsindicate that the simple model discussed above allowsus to reproduce the amplification effect observed exper-imentally. In the following, it is applied to study theinfluence of the permeability K of the grain packing onthe amplification process.

The permeability influences the dynamics of the sys-tem through the diffusion coefficient D which is pro-

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FIG. 15: Variation of the amplification parameter ∆Ap/∆z(◦) and of the absolute front velocity |vbf | (�) with the per-meability K for J =450.

portional to K and appears in Eq. (14). The depen-dance of |vbf | and the amplification parameter ∆Ap/∆zon K are displayed in Fig. 15. The range of perme-abilities investigated is between 0.5 and 4 times thevalue K = 2010−12 m2 assumed in Fig. 14. The frontvelocity vbf is first almost independent of K while∆Ap/∆z sharply decreases with K and vanishes forK ≃ 80 10−12 m2. Note that reducing cmax from 0.63 to0.61 assuming the influence of wall effects increases thepermeability K from 22 to 27 10−12 m2 and only reducesthe amplification parameter by roughly 15%.

The above results demonstrate that the diffusion termproportional to D (and therefore to K) in Eq. (14) ispurely dissipative and acts only to enhance the attenu-ation of the decompaction wave. This results from thefact that, in high permeability packing, pressure varia-tions induced by the decompaction quickly diffuse awayfrom the front. On the contrary, in low permeability sys-tems, these pressure variations remain localized: theirgradients accelerate nearby grains and so sustain and am-plify the motion. Intermittency effects can therefore beexpected to be stronger for flowing grains of smaller di-ameter.

VI. CONCLUSION

An experimental study has allowed us to analyze thedynamical properties of periodic intermittent verticalcompact flows in a tube. Our experiments involved si-multaneous measurements of local pressure and particlefraction fields, combined with spatio-temporal diagrams.

A key observation is the amplification of the parti-cle fraction and pressure variations as the decompactionfront marking the onset of the flow propagates upwards.This propagation takes place at a much higher veloc-ity (|vdf | > 10m s−1) than that of the grains (a few10−1 ms−1). The velocity |vbf | of the blockage front is

still faster than |vdf |: it increases with the flow rate andranges between 20 and 50m s−1.

The characteristic grain velocity during the flow phaseincreases with height above the outlet: this variation isassociated with a decrease with height of the duration Tf

of the flow phase as required by the mass conservationof the grains. Close to the top of the flow tube, Tf rep-resents only 10% of the total flow period T . The grainvelocities and amplitude of the pressure fluctuations aresufficiently high in this zone that flow from the feedinghopper gets blocked and a transient low particle fractionregion appears briefly at the top of the tube.

A characteristic feature of these flows is the small am-plitude of the particle fraction variations (0.02–0.03 atmost); this is much smaller than variations observed inthe decompaction of a granular column with an openbottom end or a flow in the density wave regime. It isnoteworthy that these small particle fraction variationsnevertheless induce large pressure drops (up to 3000Paat the top of the tube). The decompaction process isindeed fast enough for air to undergo an isentropic ex-pansion: pressure variations closely mirror the variationsof the local particle fraction. Diffusion of air through thegrain packing is too slow to influence strongly that phase;however, it plays an important role in the longer staticphase by allowing the decay of the pressure towards equi-librium before flow is reinitiated. The characteristic timefor air diffusion through the packing is of the order of thetotal period of the intermittent flow and may be a keyfactor determining its value.

The amplification effect can be reproduced by a 1Dmodel assuming an effective driving force on the grainsthat increases with the deviation of the particle fractionfrom the static value. This numerical model also indi-cates that the permeability K of the packing is a veryimportant factor. The amplification effect (and perhapsalso the intermittent flows) should be observable only ifK is sufficiently low that pressure perturbations do notdiffuse away too quickly.

Another important parameter is the relative humid-ity of the surrounding air which influence strongly theinteraction forces both between the grains and betweenthe grains and the tube walls. Preliminary results in-dicate that the intermittent flow regime is rarely ob-served at relative humidities of the order of 40% or be-low, at least with the beads used in the present study.Systematic studies in a broad range of relative humidi-ties should greatly help understand better the interactionforces which determine the characteristics of these inter-mittent flows.

ACKNOWLEDGEMENTS

We wish to thank B. Perrin and E. J. Hinch for helpfulsuggestions and discussions during this work. We thankD. Gobin and Ch. Ruyer-Quil for their help for the nu-merical simulations and G. Chauvin, Ch. Saurine and

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R. Pidoux for the realization of the experimental setup.We are grateful to J. W. M. Bush and Ph. Gondret fora thoughtful reading of the manuscript. We finally wish

to thank T. Raafat and V. Terminassian for their partic-ipation to early experiments.

[1] S. Laouar and Y. Molodtsof, Experimental character-ization of the pressure drop in dense phase pneumatictransport at very low velocity, Powder Technol. 95, 165(1998).

[2] T. Raafat, J. -P. Hulin and H. J. Herrmann, Densitywaves in dry granular media falling through a verticalpipe, Phys. Rev. E 53, 4345 (1996).

[3] L. S. Leung and P. J. Jones, Flow of gas-solid mixtures instandpipes. A review, Powder Technol. 20, 145 (1978).

[4] T. M. Knowlton, T. J. Mountziaris and R. Jackson, Theeffect of pipe length on the gravity flow of granular ma-terials in vertical standpipes, Powder Technol. 47, 115(1986).

[5] J. -L. Aider, N. Sommier, T. Raafat and J. -P. Hulin,Experimental study of a granular flow in vertical pipe: aspatio-temporal analysis, Phys. Rev. E 59, 778 (1999).

[6] Y. Bertho, F. Giorgiutti-Dauphine, T. Raafat, E. J.Hinch, H. J. Herrmann, and J. -P. Hulin, Powder flowdown a vertical pipe: the effect of air flow, J. Fluid Mech.459, 317 (2002).

[7] S. B. Savage, Gravity flow of cohesionless granular ma-terials in chutes and channels, J. Fluid Mech. 92, 53(1979).

[8] J. T. Jenkins and S. B. Savage, A theory for the rapidflow of identical, smooth, nearly elastic, spherical parti-cles, J. Fluid Mech. 187, 53 (1983).

[9] E. Azanza, F. Chevoir and P. Moucheront, Experimentalstudy of collisional granular flows down an inclined plane,J. Fluid Mech. 400, 199 (1999).

[10] Y. M. Chen, S. Rangachari and R. Jackson, Theoretical

and experimental investigation of fluid and particle flowin a vertical standpipe, Ind. Eng. Chem. Fundam. 23,354 (1984).

[11] H. A. Janssen, Versuche uber getreidedruck in silozellen,Z. Ver. Dtsch. Ing. 39, 1045 (1895).

[12] J. Duran, Sands, Powders and Grains. An introductionto the Physics of Granular Materials (Springer-Verlag,Berlin, 2000).

[13] X. -l. Wu, K. J. Maløy, A. Hansen, M. Ammi and D.Bideau, Why hour glasses tick, Phys. Rev. Lett. 71,1363 (1993).

[14] K. J. Maløy, M. Ammi, D. Bideau, A. Hansen and X.-l. Wu, Quelques experiences sur le sablier intermittent,C.R. Acad. Sci. Paris 319, 1463 (1994).

[15] T. Le Pennec, M. Ammi, J. C. Messager, B. Truffin, D.Bideau and J. Garnier, Effect of gravity on mass flowrate in an hour glass, Powder Technol. 85, 279 (1995).

[16] T. Le Pennec, K. J. Maløy, A. Hansen, M. Ammi, D.Bideau and X. -l. Wu, Ticking hour glasses: experimen-tal analysis of intermittent flow, Phys. Rev. E 53, 2257(1996).

[17] C. T. Veje and P. Dimon, The dynamics of granularflow in an hourglass, Granular matter 3 (Springer-Verlag,Berlin, 2001), pp.151-164.

[18] G. B. Wallis, One-dimensional two-phase flow (McGraw-Hill, New-York, 1969).

[19] P. G. de Gennes, Granular matter: a tentative view,Rev. Mod. Phys. 71, S374 (1999).

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