Original paper Granular Matter 3, 205–214 c© Springer-Verlag 2001

A segregation mechanism in a vertically shaken bedKurt Liffman, Kanni Muniandy, Martin Rhodes, David Gutteridge, Guy Metcalfe

Abstract Metal disks of different size and density wereplaced at the bottom of a bed of monodisperse granu-lar material. The system was vibrated sinusoidally in thevertical direction. It was observed that, if the angular ac-celeration of the shaking was slightly greater than that ofgravity, the metal disks rose to the top of the bed. Thisresult has been known for over sixty years, but a basicunderstanding of the mechanism responsible for the riseof the disks is still a subject of debate. Our experimentsand theoretical model show that the ascent speed of thedisk is proportional to the square root of the disk densi-ty, approximately proportional to the disk size, and is afunction of the disk’s depth in the bed. We also investi-gated the speed of ascent of the disk as a function of theshaking frequency, fs. We found that the effective frictionor drag coefficient, �, between the disk and the granularbed, is proportional to a functional form of the frequency:� ∝ (fs − fc)−4, where fc is the critical shaking frequencyfor the disk to start moving through the bed. We discusshow such a dependency may arise.

Keywords Size segregation, density segregation,vibration

1IntroductionWhen a bed of dry granular material is subject to ver-tical vibration, the granular material tends to segregatewith larger particles at the top and smaller particles atthe bottom of the bed. This behaviour, often called the“Brazil Nut Effect” [1], has been documented since at least1939 [2] and has been the subject of quantitative labora-tory study since at least 1963 [3]. During the thirty-yearperiod between 1963 and 1993, the industrial relevanceof this problem, coupled with its intrinsic academic in-terest, inspired a number of experimental studies [3–9]and computer simulations [1, 10–14]. These latter simu-lations supported the proposal of Williams [3, 4] that the

Received: 28 August 2000

K. Liffman (&), G. MetcalfeThermal & Fluids Engineering,CSIRO/BCE, Highett, Victoria, Australia

K. Muniandy, M. Rhodes, D. GutteridgeDepartment of Chemical Engineering,Monash University, Clayton, Victoria, Australia

ascent of the larger particle was due to smaller particlesfalling into the voids produced underneath the larger par-ticle after each shaking cycle.

However, it has been known since the early nineteenthcentury, that vibration can induce convective flow in gran-ular material [14–17]. In 1993, Knight et al. [18] used thisproperty to show that bulk convective flow could trap larg-er particles at the top of the bed. These authors used aconvective flow produced by a vertically vibrated, cylindri-cal container with rough walls. In this flow, the relativelywide, upward, convective flow in the middle of the con-tainer conveyed the large and small particles to the toptogether. The larger particles remained at the top of thecontainer, because they were too large to travel along thethin downward sheets of the convection rolls. The Knightet al. paper led to further work [19–25], where authorsattempted to clarify the shaking regimes for size indepen-dent segregation (which is the signature of bulk convectiveflow) from size dependent segregation. The latter segre-gation possibly arising from percolation [26], or arching[27] phenomena related to small particles moving into thevoids generated under larger particles during each cycleevent.

In this paper, we report on our experiments and sub-sequent modelling of the size and density dependent risetime of relatively large disks moving within a sinusoidallyvibrated, quasi two-dimensional bed of granular material.The experimental method and observations are presentedin §2 and §3. In contrast to previous studies, we derivean equation of motion for the intruder disks. The mainsize-dependent segregation model is derived in §4, with amodification to the model given in §5 to account for themotion of the disks as they reach the surface of the bed.The resulting relatively good agreement between theoryand experiment is discussed in §6 with the results andpossible future development given in the conclusions (§7).

2Apparatus and experimental methodA test rig, shown in Fig. 1, was constructed to examinethe movement of a large particle within a bed of small-er particles. It consisted of a hollow transparent cell withthe internal dimensions 20×20×0.6 cm. Protruding fromthe ends of this cell were perspex blocks, each with a holedrilled at the ends. Two vertical rods extended throughthese holes from a solid base, constraining the cell to moveonly in the vertical direction. The cell was vibrated byan electric motor connected to a digital frequency meter,

206

Fig. 1. Apparatus used to study segregation of disks in avertically vibrated bed of granular material

which allowed the vibrational frequency to be altered. Thefrequency of vibration was independently monitored byelectronically observing a light emitting diode, which waseclipsed once every shaking cycle. The output shaft fromthe motor was attached to the base of the cell by meansof a scotch yoke cam, which has been tested to produce atrue sinusoidal vibration. A scotch yoke cam was used inthis rig, because a wheel-rod cam does not produce a truesinusoidal motion [28].

The cell was vibrated sinusoidally over a range of fre-quencies from 4.0 to 5.0 Hz in the direction parallel togravity with a fixed amplitude of 15 mm. Disks of diam-eters ranging from 10 to 25 mm and effective densitiesin the range 1950 to 7460 kg/m3 were placed separatelyat the base of the granular bed and their motion stud-ied as the cell was vibrated. The bulk particles used werespherical glass beads, 1 mm in diameter. These beads werepoured into the cell to make a bed height of 10 cm througha specially designed steel hopper. The disks were slightlyless thick than the internal width of the cell (6 mm). Wefound that if the disks were too thick or too thin theywould jam between the cell walls and not move. Howev-er, by varying the thickness of the disk we were able toobtain an optimal disk thickness that did not jam or haltthe disk’s ascent.

Before the start of each experiment, the metal diskwas placed at the base of bed, and the glass beads werepoured into the bed. When the experiment was completed,the disk was extracted from the bed’s surface by meansof a magnet attached to a thin rod and the bulk par-ticles were removed by using a sealed, vacuum suctiondevice. These protocols were used to ensure that mini-mal amounts of hand oil and other contaminants came incontact with the glass beads or disk. As has been shownelsewhere [29], small quantities of wetting liquid can dra-matically change the properties of granular media. Beforethe start of each experiment, the box was tapped slightlyto settle the granular material. This was done to preventthe disk from having an initial spike of high ascent dueto an initially low, unsettled bulk density in the granularbed. It was observed, in earlier experiments, that changesin humidity could affect the ascent speed of the disk by

nearly an order of magnitude. Hence, all experiments weredone in an air-conditioned laboratory where humidity wascontinuously monitored and kept relatively constant.

We have labelled this experiment as a quasi two-dimensional system, since the disk moved in a two-dimensional framework, but the distance between thewindows was equal to six glass bead diameters. We willbriefly mention one possible consequence of this quasitwo-dimensional structure in §6.4.

3Experimental observationsWhen vibrated, the typical behaviour of the disk and thebed is shown in Fig. 2. The initial configuration is shownin Fig. 2(a). The large (20 mm diameter) disk was posi-tioned in the centre, at the base of the bed, with lines ofcoloured glass beads acting as tracer particles, so that wecould observe movements of the bulk material. The testrig was set into sinusoidal motion with a fixed amplitude,r, of 1.5 cm and a frequency, fs, of 4.383 Hz. This gavethe bed a non-dimensional acceleration � = 1.16, where� = r�2/g, with �(= 2�fs) being the angular frequency,and g the acceleration due to gravity. In Fig. 2(b) and (c),we show the bed after approximately 130 and 175 shakingcycles (30 and 40 seconds of shaking), respectively. Con-vective bulk motion of the glass beads, in the top cornersof the bed, has bent the layers into “smile” shapes. Thedisk, however, is unaffected by this bulk motion and sim-ply punches through the layers, with a trail of colouredbeads in its wake. After about 285 shaking cycles (65 s),the disk arrives at the top of the bed (Fig. 2(d)). As weshall discuss in section 4, this rise time is dependent onthe size and density of the disk.

To determine the possible influence of bulk convectivemotion on the disk motion, we vibrated the bed, withoutthe disk, for over half an hour. We found that bulk con-vective flow occurred at the top corners of the bed, butthere was no discernible convective flow in the centre ofthe bed. Given that this time scale is over an order ofmagnitude larger than the typical rise time of a disk inthe bed, we concluded that bulk convective motion hadlittle or no influence on the disk motion.

4Equation of motionTo derive the equation for the upward motion of the disk,we consider a particle that is thrown into a low-densitymedium. The distance that this particle will travel intothe medium is such that the energy in moving the dis-placed material is equal to the initial kinetic energy of theparticle. As shown in Fig. 3, we assume that the impact-ing particle will displace a “plug” of material, where theforce required to accomplish this task is �mcg, where � isa granular friction coefficient, mc is the mass of displacedplug of material and g is the acceleration due to gravity.

To describe the subsequent movement of the particle,as it moves through the granular bed, we assume thatfrictional forces decrease its velocity. This implies:

mpd�dt

= −� mcg = −� Apx�mg , (1)

207

Fig. 2a–d. Upward motion of a 20 mm diameter steel disk in avertically vibrated bed

Fig. 3. Schematic diagram of a projectile displacing a plug ofgranular material

where � is the speed of the particle, t the time, mp themass of the particle, Ap the cross sectional area of theparticle (where the normal of the cross sectional area isparallel to the direction of motion), x is the distance theparticle moves into the bed and �m is the density of thediffuse medium. We note that, in our model, � is notthe standard solid-on-solid Coulomb friction coefficient.Rather, as we shall discuss, � relates the resistance of theentire bed to the displacement of a small part of the bed.

To determine the total distance, �l, the disk will pen-etrate into the diffuse, granular medium, Eq. (1) must beintegrated using the appropriate limits of � and x:

12

0∫�20

d(�2) = −Ap��mg

mp

�l∫0

x dx , (2)

where �0 is the initial speed of the particle as it interceptsthe medium. The solution of Eq. (2) is an equation thatgives the penetration distance as a function of the initialvelocity

�l = �0

√mp

Ap��mg. (3)

In our experiment, the impacting object is a disk, so mp =�a2

d�dL, and Ap = 2Lad, where ad is the radius of the disk,L the thickness of the bed (see Fig. 4) and �d the massdensity of the disk. So, one has mp/Ap = �ad�d/2. Notingthat �0 ≈ r2�fs, the penetration distance becomes

�l = r2�fs

√�ad�d

2��mg. (4)

To determine �l, we need to find a suitable value for �m.At first glance, one could simply take �m as the massdensity of the medium. In a granular medium, however, alarge intruder particle will not only affect the particles incontact with its surface, but also other particles through-out the bed. This is because one particle is quite oftenconnected via stress chains to many other particles in thebed [30]. Such stress chains may produce dislocations thatcan propagate through the bed from an intruder parti-cle [31]. So the large intruder particle is subject to an“effective” mass density which must take into account theextra mass from these chains of particles. In this analysiswe assumed that the disk must move a ‘wedge’ of material(see Fig. 4). This assumption is consistent with observa-tions of dislocations in a granular bed due to the presenceof a large intruder particle [31]. Thus, the effective massdensity of the medium, which we now denote by �e, is de-fined to be the mass of the wedge divided by the volume ofthe ‘cap’ of particles on the disk surface. The disk cannotmove downwards because the bottom of the bed cannot bedisplaced and so there is an infinite effective mass densityfor downward motion.

The volume of the wedge can be determined from ananalysis of its geometry asVw = y2L tan � + 2|y|adL(1 + tan �)

+a2dL

(tan � + 2 − �

2

), (5)

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Fig. 4. The mass of the wedge of material above the disk givesa large effective mass density in the cap of material directlyabove and adjacent to the disk. The value y is the distancebetween the top of the disk and the surface of the bed. Thesurface is defined to be at y = 0

where Vw is the volume of the wedge, y the distance fromthe top of the disk to the surface of the bed, and � the an-gle that the side of the wedge makes with the vertical. Bymultiplying the volume of the wedge by the bulk densityof the bed, �b, one can obtain the mass of the wedge.

Mw = Vw�b . (6)

The width of the cap at the top half of the disk is taken tobe the thickness of the layer of particles in contact withthe disk at the disk’s surface. We calculate this thicknessto be 2ap

√2/3, where ap is the average radius of the bulk

particles. This is the distance between the surface of thedisk and the second layer of particles away from the disk,where we assume that the first layer is in contact with thedisk and the second layer is in contact with the first layer.We also assume that the glass beads, to some approxima-tion, are hexagonal close packed at the surface of the disk.Thus, the volume of the cap of beads in contact with thedisk, Vs, is

Vs = 2

√23�adapL (7)

By definition, the effective mass density encountered bythe disk, �e, is given by combining Eqs. (6) and (7)

�e =Mw

Vs=

�b

√6

4�apad

(y2 tan � + 2|y|ad(1 + tan �)

+a2d

(tan � + 2 − �

2

)). (8)

Substituting �e of Eq. (8) in place of �m in Eq. (4) gives

�l = 4�2adrfs

×√

�dap

2√

6�g�b

[y2 tan �+2|y|ad(1+tan �)+a2

d(tan �+2− �2 )

] . (9)

The ascent speed, �, is defined as the distance that thedisk rises per vibration cycle, so � = dy/dt ≈ �lfs. If weset

a = a2d

(tan � + 2 − �

2

),

b = 2ad(1 + tan �), and c = tan � , (10)

then � takes the form

� =dy

dt= �ad

√�dapg

2√

6��b[cy2 + b|y| + a]. (11)

From Eq. (11), it is apparent that the ascent speed is pro-portional to the square root of the disk density, approxi-mately proportional to the disk size and is a function ofthe depth of the disk in the bed. As discussed in the Ap-pendix, one can integrate Eq. (11) to find the total risetime, T , of the disk:

T =F (ad, �, y0)

�ad

√2√

6��b

�dapg, (12)

where y0 is the initial distance between the top of the diskand the surface of the bed, plus

F (ad, �, y0) =(2cy0 + b)

√a + by0 + cy2

0 − b√

a

4c

+�

8c√

cln

[2√

c(a+by0+cy20)+2cy0+b

2√

ca + b

]

(13)

Eqs. (12) and (13) suggest that the total rise time of thedisk in the bed is a function of the size and density ratiosof the disk to bed material, plus the driving acceleration,bed friction and bed depth. To compute the position ofthe disk, y, as a function of time, one has to numericallysolve for y in Eq. (A3).

Eqs. (A3), (12), and (13) are the central analytic re-sults from our model for the upward ascent of a particlethrough a granular bed. There is, however, a modificationrequired to account for an effect that slows the motion ofthe disk just as it reaches the surface of the bed.

5The Surface effectOne of the implicit assumptions that we have used in de-riving Eqs. (11), (12), (13) and (A3) is that the bed sur-face is unaffected by the motion of the disk in the bed.This assumption arises when we compute the mass of thewedge of material above the disk. As is apparent fromFig. 4 and Eq. (5) the bed surface is assumed to be flatand horizontal. However, as can be seen from Fig. 5 (seealso Fig. 2), once a disk comes within approximately 2 cmof the disk surface, the surface gains a “bump”. This smallhill of material increases the mass (and therefore the ef-fective density) of the wedge of material that has to bemoved by the disk. This extra mass of bed material slowsthe upward motion of the disk.

In deriving our model, we have assumed that the wedgeof displaced material always has a flat surface. This, inturn, assumes that as the disk moves upward, any dis-placed material quickly moves so that the bed surface is

209

Fig. 5a, b. Video pictures of a 20 mm disk moving up throughthe bed where there is a the onset of the bump on the surfaceof the bed, b a moderately well developed bump as the disknears the surface

always flat. Clearly this assumption breaks down as thedisk moves close to the bed surface. One way of accountingfor this phenomenon is to assume a “drainage” time-scalewhere it takes a finite time for the material above the diskto drain away and for the surface to flatten. This modifiedmodel is outline schematically in Fig. 6.

In the modified model, as shown in Fig. 6(a), we startwith the disk moving up through the bed until we startto observe noticeable heaping at the bed surface. In ourexperiments this seems to occur when the disk is approx-imately 2 cm below the surface of the bed. This depthappears to be approximately independent of the disk size.We arbitrarily set the time for the onset of this phenom-enon to t = 0. The mass of the wedge above the disk isdenoted by M0.

After the next cycle of shaking, we have t = Ts and thedisk has moved up a distance �l1 (see Fig. 6(b)). Byassumption, a section of the wedge is now, “instantaneous-ly”, above the previous surface. This “above the surface”layer is assumed to drain away to a flat surface on ane-folding time of �d. In this figure, the mass of the wedgebelow the surface is denoted by M1, while the mass of thewedge above the bed surface is denoted by �M1. Thus, thetotal mass that is affected by the movement of the disk isM1 +�M1. Note, that we have schematically depicted the“above surface” layer as a straight section of the wedgethat is above the surface of the bed, but the actual model

Fig. 6a–c. Mass of wedge below and above the surface as thedisk moves up a at the onset of surface heaping at t = 0. b Atthe end of the first shaking cycle (t = Ts) the disk has movedup a distance, �l1, and there is a section of the wedge below

and above surface. c At the end of the second shaking cycle(t = 2Ts) the disk has moved up an additional distance, �l2,and there is a smaller section of the wedge below the surface,but a larger section above surface

is independent of the shape of the bump above the disksurface.

At the end of the next shaking cycle, t = 2Ts, andthe disk has moved up an additional distance, so thereis an additional layer of the wedge (with a mass �M2)which is now above the surface of the bed. The mass ofthe wedge below the surface is denoted by M2. In theperiod Ts < t < 2Ts, some part of the first layer has“flattened out” and the mass of the first layer above thesurface is �M1 exp(−(t−Ts)/�d). So the total mass that isaffected by the movement of the disk is now M2 + �M2 +�M1 exp(−(t − Ts)/�d).

We can continue this analysis and show for the nthtime step that the total mass of bed material, that is af-fected by the upward movement of the disk, Mw(nTs) isgiven by the equation

Mw(nTs) = MnTs +n∑

k=1

�MkTse−(t−kTs)/�d , (14)

where MnTs is the mass of the wedge below the surfaceat time t = nTs, and �MkTs

is the mass of the layer abovethe surface produced at time t = kTs (i.e., �MkTs

=M(k−1)Ts

− M(k)Ts).

Now that we have the total mass effected by the disk,we can compute the effective density of the bed in thesame manner as for Eq. (8):

�e =Mw(nTs)

Vs. (15)

This value is substituted into the �m of Eq. (4), and soprovides us with a value for �ln+1, i.e. the upward distancethat the disk moves in the n+1th time step. Unfortunate-ly, the sum in Eq. (14) seems to remove the possibilityof an analytic solution to this system. So we numericallysolve for the subsequent motion of the disk by using thedifference equation

yn = yn−1 − �ln−1 , (16)

where yn is the disk depth at t = nTs.

210

6Comparison between predictions and measurements6.1Depth of disk versus timeWe now have developed enough mathematical machineryto start comparing theory with experiment. In the first setof experiments we considered the position of the disk as afunction of time. To do this, we placed a disk at the bot-tom of the granular bed and started shaking the bed witha vertical, sinusoidal motion at a frequency of 4.45 Hzand an amplitude of 1.5 cm (� ≈ 1.2). For the first setof experiments, we considered the motion of two differentdisks, one with a diameter of 10 mm and the other with adiameter of 20 mm. Five separate trials were run for eachthe disk. The results are given in Fig. 7.

Figure 7 (a) shows the behaviour of a 10 mm diameterdisk. The top of the 10 mm diameter disk was initiallylocated at 9 cm below the surface of the bed. The dashedlines show the maximum and minimum experimentallyobserved values, while the solid line shows the result fromEq. (A3). The surface effect is incorporated into Eq. (A3)as discussed in the previous section. The surface effect isobserved/assumed to occur when the disk is 2 cm belowthe surface of the bed. In Fig. 7 (a) the surface effect caus-es the kink in the solid line at a depth of 2 cm. At lowerdepths, �, is the only free parameter and �d = 0 s.

Fig. 7a, b. Comparison between experimental and predictedresults for a 10 mm disk and b 20 mm disk. The dashed linesshow the minimum and maximum for 5 separate experimentaltrials. The solid line shows the theoretical line deduced fromequation (A3). The “kink” in the solid line shows the posi-tion where the surface effect, described in the previous section,turns on

To obtain the solid line solution for Fig. 7 (a), we set� to an angle of 35◦ and found best fit values for � and�d to be � = 130 and �d = 11.2 s (or 50 shaking cycles).The value for � was deduced by assuming that the glassballotini were in a hexagonal close-packed arrangement,i.e., the fundamental structural unit was four spheres lo-cated in a tetrahedral shape. For such a case, the angleof 35◦ is the approximate angle between the side of thetetrahedron and a perpendicular line drawn from the baseof the tetrahedron through the apex of the tetrahedron.As such, this angle is representative of the expected anglebetween the vertical and a stress line. At first sight, thevalue of � = 130 would appear to be quite high, relativeto standard friction coefficients. However, as mentioned inthe discussion surrounding Eq. (1), � is the resistance ofthe entire bed to the displacement of a small part of thebed. So the value of � has to be large to account for theinfluence of the entire bed on the movement of the disk.Finally, we note (again) that the value for �d of 11.2 s isonly valid when the top of the disk is within 2 cm of thesurface of the bed. At greater depths, �d is set to zero.

The results for the 20 mm disk are shown in Fig. 7(b). Again, the dashed lines show the experimental re-sults for five separate experimental trials, while the un-broken line shows the theoretical result from Eq. (A3).This time, however, the curve of best fit gives � = 115with � unchanged and �d = 6.7 s (or 30 shaking cycles).The variation in � and �d between the two disks can bepartially explained by noting that, from Eq. (A3), the risetime and the position of the disk as a function of timeis dependent on the square root of �, so a 6% differencein

√� readily translates into a 13% difference in �. This

behaviour is illustrated in the next set of results.

6.2Total rise time versus disk sizeTo determine the total rise time of the disks as a functionof disk size, we obtained six steel disks with diameters of10, 13, 15, 18, 20 and 25 mm and measured the total risetime for each disk. The typical experimental procedurewas to take one of the disks, place it at the bottom of thebed, and then shake the bed sinusoidally at a frequency of4.45 Hz until the disk reached the surface of the bed. Thetotal time for this to take place would be noted and theprocedure repeated. Typically, 5 separate trials were runfor each disk. The results obtained are shown in Fig. 8,where the squares show the experimental results – withassociated error bars – while the solid line shows the the-oretical line obtained from Eqs. (12) and (13) (modifiedto account for the surface effect). To obtain the solid line,we assumed � = 122.5 and �d = 8.9 s. Both values areaverages of the results obtained from the position versustime results for the 10 and 20 mm disks given in Fig. 7. Ascan be seen, from Fig. 8, there is good agreement betweentheory and experiment.

The upper dashed line in Fig. 8 shows the theoreticalresult for the � and �d values obtained from the 10 mm di-ameter disk (� = 130 and �d = 11.2 s). The lower dashedline gives the theoretical line for � = 115 and �d = 6.7 s(the values obtained from the 20 mm diameter disk). The

211

Fig. 8. Total rise time of the disk, T , in seconds as a func-tion of the normalised size of the disk, where ad and ap arethe radii of the disk and glass spheres, respectively. The ex-perimental results are denoted by the squares. The solid lineshows the theoretically expected result, while the dashed linesshow the variation in the theoretical result due to experimentaluncertainty in the values of � and �d

difference in rise time between the upper and lower lines isapproximately 8% of the solid line value, thereby reflect-ing the

√� dependence of the ascent speed of the disk

(Eq. (11)).

6.3Total rise time versus disk densityWe now consider the relationship between the total risetime and the density of an ascending particle. As far as weare aware, there have been at least four papers that haveaddressed this issue [3, 4, 6, 8]. Unfortunately, the resultsof these studies do not appear to be directly applicable toour work, because they do not discuss, in a quantitativemanner, the low-convection acceleration regime addressedin this study.

We should expect, from Eq. (12), that the total risetime of the disk is inversely proportional to the density ofthe disk. This behaviour arises, because the form of Eq. (1)is that of a simple harmonic oscillator, where – on smalltime scales – the potential energy of the bed increasesas the square of the penetration distance of the disk in-to the bed. To determine the validity of this relationship,we made six disks of the same size (20 mm diameter), butwith different densities. We were able to do this by drillingholes in five of the six disks (Fig. 9), thereby convertingthe disks to annuli. This method decreased the mass andaverage mass density of each annulus and produced diskswith densities ranging from 1.95 to 7.46 g cm−3.

Each disk was placed at the bottom of the granularbed and the system was then shaken sinusoidally at a fre-quency of 4.45 Hz and with an amplitude of 1.5 cm untilthe disk reached the surface of the bed. At least 12 tri-als were run for each density and the results obtained areshown in Fig. 10. Here we plot the rise time of the diskas a function of the disk’s density, where the disk’s den-sity is normalised relative to the bulk density, �b, of theglass ballotini (1.52 g cm−3). The experimental results are

Fig. 9. The top row shows the steel disks available for thetotal rise time versus size measurements of Fig. 8. The bottomrow shows the disk and annuli used in the total rise time versusdensity measurements of Figure 10

Fig. 10. Comparison between experimental (black squares)and theoretical results (solid line) for the total rise time, T ,of the disk vs disk density �d, where we have normalised thedisk density relative to the bulk density of the glass, �b, (1.52g cm−3)

shown as filled-in squares and the theoretical result is giv-en as an unbroken line. The error bars on the experimentalresults show the standard deviation of the mean.

To obtain the theoretical line, we used Eq. (12) andset � and �d so that the line passed through the exper-imental value for the disk with the highest density (theright most experimental value in Fig. 10). As can be seen,all the other experimental values, except for the lowestdensity disk (1.95 g cm−3), lie on the line and so appearto be consistent with theory.

6.4Total rise time versus shaking frequencyFor our final experimental result, we considered the totalrise time of the disk as a function of the shaking frequen-cy. Again, we took the 20 mm disk and measured the risetime in the manner discussed in §6.1. This time, however,we set the frequency of vibration to different frequenciesin the range from 4.07 Hz to 4.99 Hz. The lower boundof the frequency range is the critical frequency, fc, for themovement of the disk, where the value for fc, is obtainedby setting � = 1. Thus fc is given by the formula

fc =12�

√g

r, (17)

212

that has the value 4.07 Hz for r = 1.5 cm. The disk willnot move when � < 1 since the bed will not be dilatedand the particles cannot move relative to one another. Theupper bound for the frequency was chosen from observingthe influence, or otherwise, of convection on the centralregions of the granular bed.

Nine separate frequencies were chosen and at least fiveseparate experimental runs were conducted for each fre-quency. The experimental results are the black squaresshown in Fig. 11, where we have plotted the average as-cent speed, 〈�〉, against the non-dimensional acceleration�. The error bars, representing the sample standard de-viation of the results, are small relative to the absolutemagnitude of the results and cannot be seen in Fig. 11.The average ascent speed is defined by the formula

〈�〉 =y0

T, (18)

where y0 is the initial distance between the top of thedisk and the bed surface, and T is the total rise time ofthe disk. We have chosen to plot 〈�〉 instead of T , becauseit provides a better indication of the disk’s behaviour forhigher vibrational frequencies.

From Eqs. (12), (13) and (18), our model gives a moredetailed equation for 〈�〉:

〈�〉 =�ady0

F (ad, �, y0)

√�dapg

2√

6��b

. (19)

If we assume that � is independent of the shaking fre-quency, then Eq. (19) predicts that 〈�〉 should be directlyproportional to �. However, as can be seen from the ex-perimental results shown in Fig. 11, there is a non-linearrelationship between 〈�〉 and � and so Eq. (19) is not cor-rect. One way to reconcile the data with our model is tosuppose that � is a function of frequency, since all the oth-er quantities in Eq. (19) – besides � – are independent ofthe shaking frequency.

To determine the possible dependency of � on f ,we set

� = �0

(f0 − fc

fs − fc

)q

, (20)

where f0 is the “normalisation” vibrational frequency(which we set to f0 = 4.45 Hz) such that when fs = f0then � = �0. In this case, �0 is our standard value of �(�0 = 122.5), and q is a free parameter. By varying q, wewere able to obtain the “best fit” line shown in Fig. 11.Via this process, it was found that q = 4 gave the best fitto the experimental data. At this stage of our analysis, itis not clear why q should have this value. One possibilityis that � is somewhat proportional to the number of con-tacts between particles in the granular bed. Due to atomicforces, (e.g., Van der Waal forces) each contact may havea small, attractive force component making the particlesslightly sticky. For the disk to move through the granularbed, some of these force contacts would have to be brokento allow particles to move past one another. As the shak-ing frequency increases the number of contacts betweenthe particles in the granular bed would decrease, therebydecreasing �. Whether or not one could obtain q = 4 fromsuch a process remains to be seen, although, it may have

Fig. 11. The average ascent speed 〈�〉 of a 20 mm diameterdisk as a function of �. The squares represent the experimentalresults and the solid line shows the line from Eq. (19), where –as is explained in the text – it is assumed that the granularfriction coefficient, �, is dependent on the shaking frequency

something to do with the quasi two-dimensional structureof our experiment. We hope to investigate this, and otherhypotheses, in later studies. For now, however, we simplynote the result.

7ConclusionsIn this paper we document our observations of the ascentof a large intruder disk in a 2D vibrating bed of granularmaterial, where the granular bed is subject to relativelylow vibrational accelerations, typically at � = 1.2. For thisvalue of �, the bulk convection of the granular bed wouldappear to have little or no influence on the motion of thelarge intruder disks.

Our experiments showed that the ascending disk ap-pears to push through the granular bed, entraining gran-ular material in the its wake. In an attempt to understandthis behaviour, we have developed a “penetration” modelfor the disk’s motion. We suggest that the disk’s upwardmotion is driven by its momentum and that for it to movethrough the bed, the disk has to interact with a wedge ofmaterial directly above the disk. This insight allows us toconstruct an analytic model for the disk’s upward motion.

In this model the disk can move in any direction with-in the granular bed, i.e. either up or down. The granularmaterial simply acts as a source of frictional drag on themovement of the disk. For the disk to move upwards, it hasto displace particles above it. This means it has to displacethe top surface since, in a granular medium, particles areinterconnected via rigid, long-range contact chains. If thedisk moves downwards, it must displace particles belowit – but this is much more difficult, since the stress chainswill terminate at the fixed base of the granular bed. Sothe granular medium, coupled with the free top and fixedbottom surfaces, imposes a bias that favours the upwardmovement of the disk.

The disk’s upward motion is subject to drag from par-ticles in the granular bed, where the drag force is propor-tional to the weight of the material displaced by the disk.To determine the weight of displaced material, we use the

213

fact that particles within the granular bed are subject tolong range interactions with other particles via long con-tact chains. Thus, to move through the bed, the disk has tointeract with a wedge of material directly above the disk.This insight allows us to construct an analytic model forthe disk’s upward motion.

The model predicts that the total time for a disk toreach the surface of the bed should be approximately in-versely proportional to disk size, and inversely proportion-al to the square root of the disk density. Both predictionsare consistent with our experimental results. The mod-el can also provide an accurate description of the depthof the disk as a function of time, but the model requiresan extra parameter to account for a deceleration in thedisk’s ascent near the surface of the bed. Finally, our ex-periments indicate that the friction coefficient between thedisk and the granular bed is a fourth order polynomial ofthe shaking frequency. We suggest that this is due to thenumber of static, “sticky” contacts between particles inthe granular bed, which may be a function of the shak-ing frequency. We hope to study this behaviour in futurework.

8Appendix: Rise time and depth of the diskas a function of timeTo determine the depth of the disk as a function of time,we integrate Eq. (11) over a time t:

t∫0

√a + by + cy2 dy

dtdt =

t∫0

�ad

√�dapg

2√

6��b

dt . (A1)

The left hand side of Eq. (A1) is

t∫0

√a + b|y| + cy2 dy

dtdt =

−y∫−y0

√a + b|y| + cy2 dy , (A2)

where y is the depth of the particle at a time t (y is takento be a positive quantity) and y0 is the initial depth of theparticle. Eqs. (A1) and (A2) imply

(2cy0 + b)√

a + by0 + cy20

4c

+�

8c√

cln

[2√

c(a + by0 + cy20) + 2cy0 + b

2√

ca + b

]

− (2cy + b)√

a + by + cy2

4c

− �8c

√c

ln

[2√

c(a + by + cy2) + 2cy + b

2√

ca + b

]

= �adt

√�dapg

2√

6��b

, (A3)

where

� = 4ac − b2 = −4a2d

[1 +

�2

tan �]

. (A4)

The disk reaches the top of the bed at the rise time t = T ,and y = 0. Substituting these values into Eq. (A3) givesthe rise time as

T =F (ad, �, y0)

�ad

√2√

6��b

�dapg, (A5)

where

F (ad, �, y0) =(2cy + b)

√a + by0 + cy2

0 − b√

a

4c

+�

8c√

cln

[2√

c(a+by0+cy20)+2cy0+b

2√

ca + b

].

(A6)

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