Original papers Granular Matter 3, 151–164 c! Springer-Verlag 2001

The dynamics of granular flow in an hourglassChristian T. Veje, P. Dimon

Abstract We present experimental investigations of flowin an hourglass with a slowly narrowing elongated stem.The primary concern is the interaction between grainsand air. For large grains the flow is steady. For smallergrains we find a relaxation oscillation (ticking) due to thecounterflow of air, as previously reported by Wu et al.[Phys. Rev. Lett. 71, 1363 (1993)]. In addition, we findthat the air/grain interface in the stem is either stationaryor propagating depending on the average grain diameter.In particular, a propagating interface results in power-lawrelaxation, as opposed to exponential relaxation for a sta-tionary interface. We present a simple model to explainthis effect. We also investigate the long-time properties ofthe relaxation flow and find, contrary to expectations, thatthe relaxation time scale is remarkably constant. Finally,we subject the system to transverse vibrations of maxi-mum acceleration !. Contrary to results for non-tickingflows, the average flow rate increases with !. Also, therelaxation period becomes shorter, probably due to thelarger effective permeability induced by the vibrations.

Keywords Granular flows, hourglass experiment,jamming

1IntroductionAlthough hourglasses have been used to measure timeintervals since the middle ages [1], the physics of grainflow is still a complicated and poorly understood process

Received: 11 February 2000

C. T. Veje (&), P. DimonCenter for Chaos and Turbulence Studies,The Niels Bohr Institute, Blegdamsvej 17,DK-2100 Copenhagen Ø, Denmark

We greatly appreciate the assistance of S. Hørluck with thevideo system. We would like to thank F. B. Rasmussen for pro-viding the capacitance bridge and S. R. Nagel and H. M. Jaegerfor help on the design of the capacitive measuring device. Wehave had many helpful and stimulating discussions with R. P.Behringer, E. Clement, and K. J. Maløy. We would particularlylike to thank K. J. Maløy for bringing the work on pressure dif-fusion to our attention. Finally, the authors would like to thankStatens Naturvidenskabelige Forskningsrad (Danish ResearchCouncil) for support. Also, CV greatly appreciates the supportof The Leon Rosenfeld Scholarship Fund.

[2–5]. Traditionally, an hourglass consists of two glass am-poules, partly filled with grains, connected by a small holein the stem. Typically, ballotini (small glass beads), sand,marble powder, and even crushed egg shells have beenused as the granular material [1]. Recently, the hourglasshas become a source of interest to the physics communi-ty as a convenient simple system to investigate granularflows [1,6–10].

The flow of granular materials has been found to ex-hibit a wide range of different phenomena. In particular,density waves and shock waves are common in dry granu-lar flows. Density patterns due to rupture zones as grainsflow through hoppers have been observed using X-rayimaging [11–13]. Using similar geometries and techniques,Baxter et al. found density waves propagating both withand against the flow direction, depending on the openingangle of the hopper [14]. For two-dimensional funnel flowthrough very narrow outlets (3–10 grain diameters), shockwaves have been found due to arching at the outlet[15–17]. Moreover, due to the monodispersity of the grains,shock waves were created at particular locations where thewidth of the funnel would perfectly accommodate a trian-gular packing of balls [17]. Apparently, the density waveswere initiated by the intermittent jamming of balls sim-ilar to the arching at the outlet. Recently, Clement haspointed out that the propagation direction of the densitywaves in two-dimensional vertical pipes depends on thecoefficient of restitution of the balls [18].

For hourglasses in which one or both chambers areclosed (except for the outlet), air must flow from the low-er chamber to the upper during grain flow. This meansthat the system becomes a two-phase flow with a couplingbetween the grain and air flow. Theoretically, this problemremains a large challenge, but by regarding the air flow asa Darcy flow, simple models for the flow behavior are pos-sible [7]. For flow through pipes there is a tendency for thegrains to form plugs which propagate as a result of masstransfer balanced by the air flow [19–22]. The propaga-tion of the plugs was found to depend sensitively upon theallowed air flow [21]. In the same geometry, Nakahara andIsoda found similar results using either water or siliconeoil as the interstitial fluid [23].

In this article, the flow of grains through a slowlynarrowing elongated stem is examined using a speciallydesigned hourglass. The emphasis of the experiment is onthe air/grain interaction. Using different techniques, themass flow, air pressure, and density fluctuations are mea-sured to form an overall picture of the dynamics of theflow.

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Fig. 1. Left: Schematic drawing of an hourglass. Right: Sche-matic representation of the pressure difference "P = P0 " Pbetween the upper chamber and the laboratory pressure as afunction of time. The active and inactive times are indicatedby Ta and Ti respectively

2Flow in hourglassesA schematic drawing of an hourglass is shown in Fig. 1(left). For closed hourglasses, fluctuations in the grain den-sity in the stem have been found to be quite reminiscentof the plugs observed in pipes [9]. This results in a non-steady flow as first studied by Schick and Verveen [6] usinga light transmission technique. They claimed that the den-sity fluctuations had a 1/f noise power spectrum, but ithas been argued that this can be explained as a crossoverregion between characteristic time scales [10].

In 1993 Wu et al. reported on the ticking phenomenonof hourglasses [7]. They used an hourglass with a largeopening half-angle ! = 45! which makes the stem rela-tively short. For a minimum outlet width D and graindiameter d, they found that for small grains (D/d > 12)the flow exhibited a distinct periodic ticking. They arguedthat the ticking was due to the counterflow of air createdby the outflowing grains. They identified an active phasewith grain flow and an inactive phase with no grain flow.Fig. 1 (right) shows the qualitative time evolution of thepressure difference "P = P0 ! P between the pressurein the upper chamber P and the laboratory pressure P0.In the active phase, characterized by the active time Ta,the grain flow is sufficiently fast that air cannot flow backthrough the grains quickly enough to maintain equilibri-um in the upper chamber, resulting in an under-pressure.Eventually, the pressure difference reaches a value "Pmax

large enough to stabilize an air/grain interface in the stemand the flow stops. Now the inactive phase begins and airflows back through the now stationary grain packing foran inactive time Ti. Once the pressure difference reachesa certain minimum value "Pmin, it can no longer supportthe interface and the grains start to flow again.

We now reproduce the argument of Wu et al. [7] for thebehavior of the inactive phase since we will need to modifyit later. Consider an hourglass with an upper chamber ofvolume V0, partly filled with grains (see also Fig. 1). Thechamber is closed so air can only leave or enter through theoutlet of radius R = D/2. The bottom chamber is open tothe atmosphere. Suppose a small mass of grains "M fallsfrom the upper chamber in a relatively short time. Thenthe air volume Va in the upper chamber expands leadingto a small decrease in the air density "n. It is assumed

that the air is an ideal gas so that "P = "nkBT wherekB is the Boltzmann constant and T is the temperature,and that the process is isothermal, hence, T is constant.

At the end of the active phase the flow stops whenthe pressure difference is large enough to stabilize theair/grain interface in the stem. Now the inactive phasebegins. Air leaks through the porous packing of grainsand the pressure slowly increases in the upper chamber.From the ideal gas law, we haved"P

dt= kBT

d"n

dt. (1)

Assuming that the air flow in the stem is incompressible,which is valid for the length and time scales involved here,the change in density "n is related to the air volume flowrate q byd"n

dt=

n0q

Va(2)

where n0 is the air density at laboratory pressure. Com-bining Eqs. (1) and (2), we have

d"P

dt=

P0

Vaq (3)

where P0 = n0kBT is the laboratory pressure. The air flowrate q is given by the Darcy equation

q = "R2vair = !#"R2

$L"P (4)

where vair is the average air velocity outside the grainpacking, # is the permeability, and L is a characteris-tic length over which the pressure difference "P occurs.Using Eqs. (3) and (4), we find

d"P

dt= !"P0#R2

$VaL"P . (5)

We write the solution as

"P (t) = "Pmaxe"t/! (6)

where

% =$VaL

"P0#R2 (7)

is the characteristic decay time, and "Pmax is the pressureneeded to stabilize the air/grain interface from a movingpacking. These are the results found by Wu et al. [7].

To estimate the minimum pressure difference "Pmin

needed to sustain a stable interface, we calculate the forc-es acting on a single grain sitting at the air/grain interface.First, of course, there is gravity

Fg = !mg (8)

where m is the mass of a grain, and g is the gravitationalacceleration. If we assume that the pressure gradient inthe stem is approximately linear, then the gradient acrossa single grain results in a buoyancy force

Fb ="d3

6L"P . (9)

Finally, there is the viscous drag on a grain from the airflow. We will assume that this is given by the Stokes lawFd = 3"$dvair, but since this is not a free grain, it should

153

be regarded as a rough approximation. It is, however, aninterface grain so using vair from Eq. (4) makes sense sincewe are outside the bulk grain packing (note that # is in factdefined through Eq. (4)). Thus, we write the dragforce as

Fd =3"#d

L"P . (10)

Force balance requires that Fg + Fb + Fd = 0, from whichwe find the minimum pressure difference

"Pmin =mgL

"d(3#+ d2/6). (11)

It should be noted that for the actual values used here,the relative size of the terms in the denominator is3#/(d2/6) " 10"2 so Fd is, in fact, not that importantat least during relaxation.

For similar types of hourglasses, Pennec et al. foundthat the slight curvature of the stem would actually cre-ate a small air bubble and a grain plug during the flow[9]. They also found that the plug oscillated, briefly caus-ing the pressure in the upper chamber to oscillate duringthe active phase. Hence, the notion of a stationary inter-face is a special case only found for large opening anglegeometries or small enough grain sizes. However, the effectof the geometry of the stem has largely been unexplored.Thus, a much more common scenario is the formation ofplugs and bubbles, including oscillating and propagatinginterfaces, in the stem of an hourglass. These phenome-na may be the origins of the instabilities observed in pipeflows, although in a very short-lived state. Pipe flow maytherefore be considered as an extreme case of ticking witha non-stationary air/grain interface. The region betweenpipe flow (long stem) and stationary ticking (short stem)is the concern of this paper.

3The experimental setupA variety of grains have been used in this work, andare listed in Table 1. The ratio D/d determines the flowbehavior discussed in section 2. We have primarily usedgrains of smooth glass beads (A7–A30). The A10 grainswill be used as the reference grain type. The glass beadsare produced in weakly polydisperse mixtures. The poly-dispersity is a skewed distribution with an average graindiameter d, with about 80% of the grains in the larger halfof the size range. The R1 grains are crushed glass but stillvery compact and with sharp edges. An often cited type of

Table 1. The grain types that were used in the experi-ments. The diameter of the narrowest point of the outlet isD = 0.3 cm. The average grain diameter d is estimated fromthe size distribution

Type Material Range (µm) d (µm) D/d

A7 Smooth glass 70–110 #100 30A10 Smooth glass 100–200 #180 18A15 Smooth glass 150–250 #230 13A30 Smooth glass 300–400 #380 8R1 Crushed glass 200–500 #400 8Boom Rough sand #200–500 #250 12

Fig. 2. Schematic of the experimental setup including the mea-suring devices described in the text

sand is the Booming Dunes sand (Boom). They are quiterough like the R1 grains but we have no nominal valuesfor the size distribution.

A schematic drawing of the hourglass experiment isshown in Fig. 2. It consists of a glass cylinder reservoir(A) of length 35 cm and inner diameter 6 cm leading intoa 15 cm long slowly narrowing stem. The smallest innerdiameter D = 0.3 cm of the stem was at the outlet. Thehourglass was supported by a frame (B) and placed on aheavy iron plate. The counterflow of air could be turned onand o! by inserting a cork in the top of the reservoir (C).

The volume of the reservoir was V0 " 1000 cm3 so withaverage mass flow rates varying from 0.1 to 3 g/s, typi-cal runs lasted 10–100 minutes. For the smallest grains,changes in the humidity were found to seriously affect theflow. Thus, the humidity was monitored with a DicksonTH550 hygrometer and recorded for future reference. Mea-surements were only made when the humidity was between30–50%.

The hourglass was mounted so that it could rotatefreely about an axis (D). A Bruel & Kjær Mini-ShakerType 4810 (E) was mounted on a wood plate attachedto the frame. A brass rod was connected from the vi-brator to the stem of the hourglass. The rod was fab-ricated in one piece so that there were as few joints aspossible. The vibrator was driven by a power amplifierdesigned to operate as low as 1 Hz. The amplifier boosteda sine source from an HP3562A Dynamical Signal Ana-lyzer (DSA). A Bruel & Kjær Cubic Delta-Tron Acceler-ometer Type 4503 was mounted on the rod, opposite thevibrator (F). The accelerometer has a calibrated outputof 8.90 mV/g (g = 9.82 m/s2) for easy measure of thedimensionless maximum acceleration ! = A&2/g, whereA is the amplitude and & is the angular frequency of thevibration. The output from the accelerometer was mea-sured by the DSA for a direct measure of the amplitudeof vibration.

Several measuring devices were used. These aredescribed below in more detail and include a scale for mea-suring the total mass flow (G), a capacitive flow detector(H), and a pressure gauge inserted in the reservoir (I). Thevideo equipment to be described later is not shown.

154

Fig. 3. Schematic top view of the capacitance measuringdevice

The total mass flow M(t) was measured with a Sarto-rius PT1500 scale (range 0–1500 g, precision 0.1 g) placedunder the outlet. The scale was connected to a PC throughan RS-232 interface (5 Hz sampling rate). The relaxationtime constant for the scale is "1 s so the real-time resolu-tion of the measurements is actually longer than the 0.2 ssampling time.

To obtain better time resolution of the flow rate fluc-tuations, a capacitive device was used in conjunction witha capacitance bridge. A schematic top view is shown inFig. 3. The capacitor was aligned just below the outlet sothe grain flow passed through its center. The black cir-cle indicates the position of the stem outlet. The plateswere separated by 1 cm and had dimensions of 1 cm by4 cm. The capacitance with only air between the platesis "0.5 pF. (The large dimensions of the capacitor plateswere needed to make room for the horizontal vibrationof the stem.) When sand passes through the capacitor,its capacitance C changes by an amount "C roughly pro-portional to the quantity of sand between the plates. Forexample, using the known polarization for a sphere in aconstant electric field, with a diameter much smaller thanthe distance between the capacitor plates, it is found that

"C

C=

!

1 !"

3K + 2

#2$

Vg

Vc(12)

where Vc and Vg are the volume of the capacitor and asingle grain, respectively, and K is the dielectric constantof the grain material. For glass, K " 5, so for a graindiameter d = 0.018 cm, Eq. (12) yields "C " 6 · 10"7 pFfor a single grain between the plates.

The capacitor was shielded by a metal cage to avoidexternal noise. It was connected to a General Radio1615-A Capacitance Bridge which can be balanced towithin "10"5 pF. With this resolution, we can detect"100 grains. The bridge was driven by a Stanford Re-search SR810 lock-in amplifier at 60 kHz with an ampli-tude of 5 V. The bridge was always nulled with no grainspresent. To ensure that the lock-in did not overload duringflow, grains were first poured through the hourglass with-out inserting the cork which gives the maximum possibleflow and hence the maximum capacitance change "C. Theoutput from the lock-in amplifier was recorded at 50 Hzby an ADC card in the PC. The time constant on thelock-in amplifier was set to 30 ms (24 dB rollo!) to serveas a Nyquist filter.

The pressure in the reservoir was measured withan Omega PX170 differential pressure transducer (range0–0.38 Bar) inserted in a hole in the cork. The nominal

response time of the transducer is "1 ms which was morethan adequate. The transducer was powered with a 9 Vbattery. Its output was amplified with a Stanford SR560low-noise preamplifier and Nyquist filtered at 30 Hz withthe amplifier’s low-pass filter. The signal-to-noise ratiowas "103. Amplifying the signal by 100–1000 made it pos-sible to easily distinguish pressure differences as small as0.1 mBar. However, the internal bridge in the transducerhas as low an accuracy as 0.5 mBar in the nulling, leadingto a systematic error in the value of the pressure differ-ence. The output from the amplifier was also read by theADC card, simultaneously with the capacitance measure-ments. This made it possible to completely synchronizethe capacitance and the reservoir pressure.

For visualization purposes, and in order to quantita-tively track the air/grain interface in the stem, a videomeasuring system developed for another experiment wasalso used. It is summarized here but discussed in detailelsewhere [17].

The camera was a Pulnix TM-6701AN, 8-bit grayscale, non-interlaced analog CCD camera. The resolutionwas 640 by 480 pixels. It was placed about 50 cm fromthe stem with a black background for contrast. A halogenlight source behind the camera gave the sand a brightappearance. Dark areas have little or no sand. Consecu-tive images were taken at 60 frames/s. The analog outputwas read by a Matrix Vision PCimage SGVS frame grab-ber card for easy storage on a PC. Fig. 4 shows a sequenceof frames during flow of A10 grains. An air/grain inter-face can be seen propagating upwards. In frame number180 the interface collapses. This sequence only lasts lit-tle more than one second, whereas the entire relaxationcycle takes about 6 s. After the collapse, an air bubble isformed with an initial length of "3 cm which then grad-ually shrinks while the air pressure equilibrates throughthe grain packing.

Using films like these, only up to 3 s of data could berecorded. To increase the measuring time, individualframes were averaged horizontally to give a one-dimen-sional sequence of the mean grain density at a given height

Fig. 4. Short film sequence showing the collapse of an interfaceand the accompanying creation of an air bubble. The grainsare type A10. (The width of the stem becomes larger than thewidth of the frames at y # 9.5 cm.)

155

Fig. 5. Example of an averaged space–time density map (inreverse video) of periodically relaxing flow using A10 grains.

Bright areas therefore correspond to low grain density and viceversa. Horizontal lines indicate motionless grains

y in the stem. Now measurements up to 7 minutes couldbe taken.

An example of such a density space–time diagram isshown in Fig. 5 in reverse video also using A10 grains.The gray scale in the figure is a relative measure of thelight intensity at a given height. The outlet of the stemis at y = 0 and the bright regions at the bottom reflectlow grain density. Higher up, bright areas only reflect thecurrent deviation from a certain mean gray-scale value.The bright and dark features for large y are not den-sity fluctuations but merely fluctuations in the reflectedintensity from the grain packing. The horizontal lines thusindicates that these grains are not moving. An electronictime stamp was used to synchronize the pressure signalwith the video sequence, and hence, with the motion ofthe air/grain interface.

Before measuring, the hourglass was cleaned and thegrains were poured through a large mesh sieve to elim-inate impurities or clumping caused by humidity. Oncea flow was initiated, after a small transient time (about3–4 relaxations), measurements could begin. Measure-ments were made in two different combinations. In thefirst, the pressure difference "P , capacitance change "C,and mass flow M(t) were measured simultaneously. Thefirst two were synchronized, while the third had a timelag of about 2–3 s since it was measured using a di!erentsoftware program. In the second, the pressure differ-ence and video measurements were taken together, andsynchronized using the time stamp.

4Relaxation oscillations and the air/grain interfaceA common feature of all the grains found in Table 1 isthat the flow is steady without ticking if the cork is notinserted. For the A30 grains this is also the case when thecork has been inserted. These grains are well out of theticking regime described by Wu et al. [7], namely, D/d " 8,and smooth enough that the coupling between the air andgrain flow is so weak that the flow finds a stable statewhere grains flow out and air flows in without causingsignificant density fluctuations.

For the rest of the grains we find a variety of tickingor relaxation phenomena as described above. In the fol-lowing some of these will be analyzed in detail and othersmerely presented to show the effects of slight changes ingrain size and shape.

4.1Propagating interfaces (A10 grains)A density map of the flow using these grains was shown inFig. 5. It can be seen that the flow has an almost periodicticking with a period of 5–6 s. Grains leave the stem at theoutlet (y = 0) cm for 1–2 s (the active phase) after whichan air/grain interface is created at about y = 2 cm dueto the counterflow of air. At this point grains above theinterface stop moving as indicated by the horizontal linesin the top part of the stem (see Fig. 5). The flow is now inthe inactive phase. However, the interface, although well-defined, is no longer stationary. Grains are continuouslyfalling out from the interface, causing it to propagate upthrough the stem. The movement of the interface and themass flow are then directly related through the geometryof the stem.

When the interface reaches a point ymax "7 cm, theair flow and buoyancy forces in the grains can no longersustain the interface. The interface collapses and the flowenters the active phase again. Note that during the col-lapse, or the active phase, a bubble is formed which prop-agates, first down and then up through the grain packing(see also Fig. 4). During upward propagation, the verticalextent of the bubble diminishes as air leaks through thepacking.

Fig. 6 shows the simultaneous measurements of pres-sure and capacitance (a), and mass flow (b). The pressureshows the expected qualitative behavior discussed in sec-tion 2. There is a fast time scale corresponding to theactive period and a longer relaxation time in the inactiveperiod. During the slow relaxation, the pressure differencedecreases, but not in the exponential manner expectedfrom the discussion in section 2. We will return to thispoint later. The capacitance shows that the flow occurscontinuously during the active period. We see that the

156

Fig. 6. a The pressure difference, capacitance change and btotal mass flow versus time for A10 grains. c The power spectra

of the pressure and capacitance signals. d A parametric repre-sentation of the pressure and capacitance signals

maximum flow rate is reached just before the maximumpressure difference.

The power spectra S"P (f) and S"C(f) of the pres-sure and capacitance signals, respectively, are shown inFig. 6(c). The peaks represent the period of the ticking.They also have a significant width, corresponding to acoherence time of about 20 s. Both the second and thirdharmonics can also be seen in the spectra, with ampli-tudes decaying as f"2 arising from the discontinuities inthe signal. At about 1 Hz there is a small shoulder corre-sponding to the fast rise time of "1 s in the active periodwhich can be seen in the time signals Fig. 6(a).

A parametric plot of the capacitance vs. pressure isshown in Fig. 6(d). The long time scale during the inac-tive phase is at the bottom of the plot. The pressure slowlydecreases with an almost constant flow rate. When theinterface collapses, the flow rate increases dramaticallywhile the pressure increases. When the pressure reaches"Pmax, the interface forms, the flow rate drops dramat-ically, and the cycle repeats. An interesting question isif there is a correlation between values of the pressureextrema. We have checked for this by using both aLomb periodogram and directly computed powerspectra and found that there were no significant cor-relations.

4.2Stationary interfaces (A7 grains)We now turn to the smallest grains (type A7), which iswell in the ticking regime (D/d " 30). Fig. 7 shows the

density map for this flow. We see that there is a sta-ble stationary interface close to the outlet. The interfacebreaks up in more or less regular intervals of "5 s, let-ting out lumps of grains, after which it stabilizes again ata new position. Note that the ejection of these lumps ofgrains does not contribute to movement of the rest of thegrains in the stem. At certain points the interface collaps-es and the entire column of grains flows downwards afterwhich the interface stabilizes at a new position close to theoutlet.

Fig. 8(a) shows the simultaneous measurements of thepressure and capacitance. We see a dramatic increase in"P as the interface collapses, accompanied by a burst ofoutflowing grains visible in the capacitance signal. Aftersuch a collapse, the pressure relaxes through the porouspacking in the stem. The intermittent ejection of smalllumps of grains does not contribute to movement of therest of the grains in the stem so the pressure continues torelax. A small kink in the pressure signal can be observedand is due to the decrease in the length L over which thepressure decays. When a lump of grains falls, L becomesslightly smaller.

This means that the balance pressure also becomessmaller and the pressure can continue relaxing until itreaches the new value of "Pmin. Once the pressure reachesthe minimum pressure the whole column of grains collaps-es and the stem refills. Thus there is an extra time scaleinvolved concerning the intermittent ejection of smalllumps of grains. The power spectra of the pressure signalin Fig. 8 shows a peak at about 0.075 Hz correspondingto a period of about 13 s.

157

Fig. 7. Space–time density plot for A7 grains

Fig. 8. a Signals of the pressure and capacitance versus timefor grain type A7. b The power spectra of the pressure andcapacitance signals

4.3Edge detection and pressure decayWe now return to the A10 grains and the non-exponen-tial pressure decay. Using an edge detection technique onthe density map (Fig. 5), we can track the air/grain inter-face as it propagates up through the stem. The techniquesimply detects the point with the largest gradient in thedensity and uses that as the location of the interface. Since

Fig. 9. The position yi of the interface (points) and the simul-taneous pressure difference (lines) versus time for A10 grains.The gaps in yi occur during the active phase when the air/graininterface is not well-defined

the interface is largely undefined during the active peri-ods, the tracking technique does not return any data forthis period.

The position of the edge yi and the simultaneous pres-sure difference is shown in Fig. 9. Clearly the decay of thepressure is not exponential as argued for stationary tickingin section 2 and as discussed in section 4.1. The departurefrom pure exponential relaxation may be primarily attrib-uted to the change in the length L over which the pressuredifference decays. Less importantly, the change in the stemdiameter may also cause deviations from exponential.

These effects can be incorporated into the theory insection 2, but let us first consider the effect of the grainsin the reservoir on the pressure drop. We will use a simpli-fied model of an hourglass regarding it as two pipes con-nected in series as shown schematically in Fig. 10 (left).The measured pressure difference is the sum of the pres-sure drops in each pipe, i.e., "P = "P1 + "P2. Since weassume that the flow is incompressible, the flow rate q isconserved, hence, using Eq. (4), we have

R21#1

L1"P1 =

R22#2

L2"P2 . (13)

158

Fig. 10. Left: Two-pipe model of the hourglass (see text).Right: Different coordinates used for the relaxation model (seetext)

If the pipes are not filled with grains, then for circularpipes #1,2 = R2

1,2/8. In this case the pressure drop in thethe reservoir is given by

"P1

"P2=

L1

L2

"

R2

R1

#4

. (14)

Thus, if L1 " L2 and R1 " 10R2, then "P1 # "P2. How-ever, if the pipes are filled with grains, then the situationchanges. In this case, the permeability only depends on thegrain size which is the same in both pipes. Thus, #1 = #2and Eq. (14) becomes

"P1

"P2=

L1

L2

"

R2

R1

#2

. (15)

This ratio is not as small as it was in Eq. (14), but even inthis case, "P1/"P2 " 1% so for the rest of the discussionwe take "P = "P2.

Although R is not constant, its dependence on y is soweak that we will assume R = R0. Thus, the pressure stillobeys Eq. (5) but in the more general form

d"P

dt= !"P0#R2

0$VaL(t)

"P (16)

where L(t) now depends on time. The coordinates usedin the model are indicated in Fig. 10 (right). We assume(from Fig. 9) that the propagation of the interface is alinear function of time

yi(t) = ymin + v0t (17)

where the interface velocity v0 and starting point ymin aredetermined from the interface tracking data.

Hence, L(t) = L0 !yi(t) where L0 is the characteristiclength over which the pressure decays when the stem isfull (see Fig. 10). Then, using Eq. (17), Eq. (16) becomes

d"P

dt= ! "P0#R2

0$Va(L0 ! ymin ! v0t)

"P . (18)

We now define the relaxation time as (compare withEq. (7))

% =$Va(L0 ! ymin)

"P0#R20

(19)

and write Eq. (18) as

d"P

dt= !1

%

"P

(1 ! t/%0)(20)

where %0 = (L0 ! ymin)/v0. Integration now yields

"P (t) = "Pmax(1 ! t/%0)!0/! (21)

where "Pmax is the initial value of the pressure at thebeginning of the inactive phase. Thus, as a consequence ofthe propagation of the interface, the pressure now decaysalgebraically. As v0 $ 0 then %0 $ % and we correctlyrecover Eq. (6) for a stationary interface.

As stated earlier, the interface velocity v0 is deter-mined from the interface tracking data. Using the formEq. (17), we extract a mean value &v0' = 1.00±0.06 cm/sfrom a total of 48 relaxation cycles. The average valuefor ymin was &ymin' = 2.2 ± 0.3 cm which is consistentwith Figs. 5 and 9. Using these values the validity of themodel can be tested by fitting the pressure data. Fig. 11(top) shows a fit to the pressure data for a single relax-ation. The data is fitted to Eq. (21) with "Pmax, %0 and% as free parameters. Multiple fits were done to see if thefitted parameters were robust. Only decays which lastedlonger than 4 s were included to eliminate fluctuations,such as small decays (like the one at about 38 s in Fig. 9).Fig. 11 (bottom) shows the values of the fitted parame-ters for 48 relaxations as a function of the starting time.The parameters are quite constant. The average valuesare: &"Pmax' = 8.1 ± 0.2 mBar, &%0' = 7.9 ± 0.7 s and&%' = 24 ± 2 s.

Using the result for &v0' and &%0' we estimate &L0' =&%0'&v0' + &ymin' = 10.0 ± 1.7 cm. This is the point wherethe stem has become wide enough that the permeability ofthe stem and grain packing no longer plays a role, i.e., weare now in pipe 1 in Fig. 10 (left). This value is reasonablesince the interface never reaches such a large y value.

A determination of % from the constants which enterin Eq. (19) is not possible since the value of # is not knownprecisely. Using the known constants, $ = 1.7 · 10"4 g/cm s,P0 = 106 dyn/cm2 and Va =500 cm3, and the fit-ted values for &%' and &L0', we obtain a value of # =(3.9 ± 0.5) · 10"7 cm2, which is in reasonable agreementwith the expected value for grains of this size [7].

As already shown, the pressure dynamics for the A7grains are quite different from the A10 grains. As shown inFig. 8 and Fig. 11 (top) the pressure decay has a curvatureopposite to that of the A10 grains. Since in this case theinterface is stationary, we use Eq. (6). In Fig. 11 (top) thepressure decay during a single relaxation is shown. Fittingthe data to Eq. (6) with "Pmax and % as free parameters,

159

Fig. 11. Top: The measured pressure decay in a single relax-ation cycle for grain types A7 and A10. The solid lines arefits using Eqs. (6) and (21) for A7 and A10 respectively. Bot-tom: By repeating the procedure in the top figure, the fittedparameters for grain type A10 are obtained for a sequence ofrelaxation cycles (see text). Points are connected for clarity

we obtain % = 66.6 ± 0.2 s. This may again be related tothe permeability # through Eq. (7) resulting in a value# = (8 ± 2) · 10"8 cm2 which is again within the expectedrange.

4.4The oscillation mechanismWe now turn to the mechanism of the oscillation which inparticular concerns "Pmin and "Pmax. Recall the argu-ments leading to Eq. (11) which expresses the minimumpressure difference needed to sustain a stable interface.Using the numbers found above, Eq. (11) yields theresult "Pmin " 5 mBar which is very close to thevalue of "6 mBar (seen from Fig. 6(a)) when the interfacecollapses. Recall that Eq. (11) was derived using only themass of a single grain.

How can it be that grains can fall o! the interface,causing it to propagate, when the grain packing is fullystabilized? Consider a grain placed in the packing dur-ing relaxation of the pressure. The force balance is givenby Eq. (11). For grains outside the packing, however, the

pressure gradient is essentially zero since the pressurerelaxes very fast when there is no grain packing giving asufficient permeability. We will now consider again whathappens to a grain sitting at the interface. The pressuregradient will relax over the grain diameter in a character-istic time %r. This will cause the buoyancy term Eq. (9) inthe force balance to vanish. The balance equation for theinterface grain then becomes

"Pint =mgL(t)3"d#

. (22)

If "P < "Pint the grain will fall. On the other hand if"P < "Pmin the grain packing will collapse. Thus wehave a range for "P

"Pmin < "P < "Pint (23)

in which the packing is stable but the interface will prop-agate. If "P > "Pint the interface is stationary. Of courseall these values depend on time and as "P decays, thecollapse criterion "P < "Pmin is eventually reached.

The characteristic time %r essentially determines thevelocity of the interface through v0 = d/%r. The veloci-ty v0 is known from the edge analysis. Since %r " d/vair

we have that v0 = vair which we may estimate during apressure relaxation. Using Eq. (4) we obtain vair " 2 cm/swhich is indeed comparable to the velocity of the interface.This gives a relaxation time %r " 9 ms.

As the interface collapses, the packing dilates resultingin a larger permeability and thus a larger "Pmin. Now thegrains start flowing from the reservoir and "P increas-es until it reaches the new "Pmin. Then an interface isformed close to the outlet and grains from above settle onthe newly formed packing. This is where the air bubbleshown in Fig. 4 becomes trapped. While the air bubbledecays, the pressure difference continues to increase, onlynow at a different rate. This can be seen by close inspec-tion of the pressure signal in Fig. 9. During the activeperiod "P increases, quickly at first until the bubble isformed, and then slightly slower while the bubble decays.

4.5Results for other grain sizes and shapesOne of the most important parameters determining theflow behavior is the properties of the grains. The size andshape are especially important. In Fig. 12a–c we show thedensity maps for grain types A15 (a), Booming sand (b)and rough grains (c). Comparing with Figs. 5 and 7, wesee that the different grains show quite a diversity of flowpatterns.

For the larger grains, A15 and Booming sand, the sce-nario with the propagating interface is still pronounced.The relaxation time is faster due to the larger permeabilityof the packing. For R1 grains the active periods complete-ly dominate the flow. The flow is interrupted at regularintervals by inactive bursts as an interface is briefly creat-ed. This causes a short decrease in the pressure and flowrate, but since the grains are rough and the permeabilityhigh, the interface breaks down very fast.

Thus it may be argued that when the permeability ofthe grain packing increases, that is, when the grain sizeincreases, the relaxation time becomes shorter. At some

160

Fig. 12a–c. Space–time density plots for different grain types:a A15, b Boom and c Rough grains (R1)

point the permeability becomes large enough that the tick-ing completely vanishes. This happens for the type A30grains where the flow is continuous with no relaxationeffects. This result agrees well with the discussion in sec-tion 2 since D/d " 8 and is now out of the ticking range.For even larger grains (d " 700 µm) mechanically stablearches form in the stem and the flow stops completely.

Considering the similarity between the two grain typesR1 and Booming sand, it is quite interesting how they dif-fer in flow behavior. Figs. 12a–c (b) and (c) show there arefundamental differences in the relaxation process. This isstriking since not only the size range but also the shapeof the grains are very similar. An explanation is probablyto be found in differences in density and of the friction

properties between real sand and glass. In fact, there aremore similarities between the flow of the Booming sandand the A15 grains (Fig. 12a–c (a)) although the differ-ence in average grain size is significant.

5Non-stationary effectsIn the above analysis it was assumed that the underlyingprocess is stationary. However the process of emptying anhourglass is, in fact, inherently non-stationary. The char-acteristic relaxation time % Eq. (7) includes the volumeof air in the upper chamber Va. Obviously Va will slowlyincrease as grains flow out.

161

Fig. 13. Top: Pressure signal using A10 grains for the totaltime span from a full reservoir to an almost empty reservoir.Bottom: Power spectra of the pressure signal of #80 s segmentsof the signal for different starting times t0 (see text)

To investigate this non-stationary effect, the methodis straightforward: fill the reservoir with grains and mea-sure until it is empty. Since the average flow rate appearsconstant (see e.g. Fig. 6) one would expect the relaxationtime % , and therefore the ticking period to increase linearlywith time following Eq. (7). However, this turned out notto be the case. In Fig. 13 (top) the pressure signal is shownfor A10 grains. The time span is 1200 s which is nearly thetime it takes to empty a full reservoir for this grain type.Clearly there are non-stationary signatures in the signal,in particular, the values of "Pmax decrease with time.

As a first test of the non-stationarity, the power spec-tra of different segments of the pressure signal are consid-ered in Fig. 13 (bottom). Taking the power spectra of thefirst and last 4096 points in the pressure signal, we obtainspectra for both a nearly full and nearly empty reservoir.We focus on the peaks of the spectra. If the relaxationperiod had significantly changed, the peak positions wouldhave moved. Contrary to expectations, the peaks remainat the same frequency (within the resolution of themeasurements).

Since V0 = Va(t)+Vg(t) where Vg(t) is the total volumeof grains in the reservoir at time t, then assuming that theaverage flow rate of the grains Q is constant, we get

dVa

dt=

1'm

dM

dt=

Q

'm. (24)

Fig. 14. Top: Mass on the scale as function of time. Thedashed line is a linear fit. Bottom: The relaxation time ! asfunction of time. The data is obtained by fitting the pres-sure signal as described in section 4.3. The dashed line is theexpected !(t) from Eq. (25)

so that

%(t) =$(L0 ! ymin)'mP0"#&R'2 Qt + %(0) . (25)

The average flow rate Q is easily found by fitting the massflow signal from the scale as shown in Fig. 14 (top). A val-ue of Q = 0.65 g/s is obtained. In Fig. 14 (bottom), therelaxation time % is shown for successive relaxations. Thevalues have been fitted from the data in Fig. 13 (top)using the same method as in section 4.3. First we note, aswe already knew, that % has significant fluctuations. Sec-ond, we see that there is no linear increase as predicted.However, there is a weak tendency for the fluctuations toincrease in time. Using the numbers from section 4.3 andthe density 'm = 2.5 g/cm3, the prediction from Eq. (25)is also shown. It is remarkable how insensitive % is to thechange in Va.

It has been suggested that an extra time scale con-cerning pressure diffusion may account for this unexpect-ed result [24]. As argued by Pennec et al. [25], there is apressure diffusion constant DP in a porous medium whichgives rise to a characteristic time %P to form a pressuregradient. For a porous medium, this characteristic timeis significantly longer than the time associated with the

162

speed of sound over the same distance. Following [25] weuse DP " P0#/$ and so

%P " L20

DP" $L2

0P0#

. (26)

If %P were the slower time scale, i.e., if %P > % , then itwould dominate the relaxation process. Since it is inde-pendent of Va (or any other time dependent parameter)this would explain the constancy of the relaxation time.However, the ratio

%P%

="R2

0L0

Va(27)

is the ratio of the volume of the stem to the reservoirwhich is always small. Even for a full reservoir Va " V0/2so %P /% " 10"3. Thus, this time scale cannot explain ourresults which still remain a mystery.

6External excitation of the grain flowSome experiments have been done on horizontally vi-brated flows in hoppers [26,27]. Among other things,these experiments were concerned with the dependenceof the average flow rate on the maximum acceleration. InFig. 15 (top) we show results for the A7 type grains (seeTable 1). The flow is steady since the cork is out. The

Fig. 15. Top: Flow rate versus ! for a steady flow (no cork).The grain type is A7 and the vibration frequency is 30 Hz.Bottom: Flow rate versus ! for a relaxation flow. The resultsare for A10 grains

average flow rate Q decreases weakly as ! increases. Thisresult is attributed to the dilation effect due to the vibra-tion. As the acceleration increases, the packing dilates andthe mass flow thereby decreases.

Turning to the relaxation flow, the effects of vibra-tions on two different grain types are considered. The firstis type A10 which had a propagating interface. The sec-ond is type A7 which had a stationary interface. As inchapter 4, the mass flow, pressure and capacitance weremeasured.

Contrary to the results for a steady flow, vibrationson a relaxing flow tend to increase the flow rate. Thiscan be seen in Fig. 15 where the flow rate dependenceon ! for different frequencies is shown. Clearly there is adependence on the driving frequency. Generally we mayargue that the flow rate must increase since vibrations in-crease the permeability of the grain packing due to dilatione!ects.

The pressure and capacitance signal of the A10 grainsfor different values of ! are shown in Fig. 16. The vibra-tion frequency is f = 30 Hz. The relaxation flow becomesincreasingly irregular as ! is increased. For the largest val-ues of !, periodic relaxation has almost disappeared. Onecan imagine that the propagating interface stabilizes thegrain packing above. Once vibrations perturb the system,the interface becomes disturbed and breaks down beforethe pressure reaches the non-perturbed value "Pmin. This

Fig. 16. Pressure (top) and capacitance (bottom) signals ofA10 grains. The data are all for f = 30 Hz and have beenshifted for clarity. From bottom to top: ! = 0, 1.5, 2.3, and 3.1

163

premature collapse of the interface causes the flow to be-come more unstable, increasing the average flow rate. Thisincrease leads to a higher average pressure difference andthereby an increasing air flow. Thus the fixed frequen-cy ticking becomes disturbed and the peak disappears.This can also be seen in Fig. 17 where the power spectraof the signals in Fig. 16 are shown. As ! increases, thepeak in the non-vibrated spectrum spreads out and flat-tens the spectrum. A characteristic time scale is still leftas a cuto! in the spectrum. A general feature is that thischaracteristic time scale moves to higher frequencies as !is increased. This is reasonable since the relaxation time% (see e.g. Eq. (7)) is inversely proportional to the per-meability #. As ! is increased # also increases leading toa shorter relaxation time with a correspondingly shorterticking period.

7ConclusionsIn conclusion, we have found that the relaxation oscilla-tion flow in a gradually narrowing stem of an hourglasswill manifest itself with propagating interfaces for suffi-ciently large grain sizes. For small grains the interfaceswere found to be stationary and the pressure relaxationwas exponential. We found that the propagation of theinterfaces will result in a modification of the air pressure

Fig. 17. Power spectra of the pressure and capacitance signalsshown in Fig. 16. The data have been shifted for clarity. Frombottom to top: ! = 0, 1.5, 2.3, and 3.1

decay. The particular form of the decay is highly depen-dent on the specific geometry of the outlet. If the geometryis assumed to be a pipe of constant diameter the propa-gation results in a power-law decay. We have also estab-lished pressure conditions for stationary, propagating andcollapsing interfaces.

The long time behavior of the oscillation flow showssigns of a non-stationary process. The non-stationarityarises from the gradually emptying reservoir. However,the expected long term behavior for the relaxation timeconstant was not found to agree with measurements.

External vibrations applied to the system were foundto increase the characteristic time constants. This wasattributed to a larger permeability of the grain packinginduced by the vibrations. Likewise, we found that inthe case of relaxation flow the average mass flow rateincreased, unlike the steady flow case where the averagemass flow rate decreased.

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