Kurt Liffman, Kanni Muniandy, Martin Rhodes, David Gutteridge, Guy Metcalfe. Abstract Metal disks of different size and density were placed at the bottom of a ...
Original paper
c Springer-Verlag 2001 Granular Matter 3, 205–214
A segregation mechanism in a vertically shaken bed Kurt Liffman, Kanni Muniandy, Martin Rhodes, David Gutteridge, Guy Metcalfe
Abstract Metal disks of different size and density were placed at the bottom of a bed of monodisperse granular material. The system was vibrated sinusoidally in the vertical direction. It was observed that, if the angular acceleration of the shaking was slightly greater than that of gravity, the metal disks rose to the top of the bed. This result has been known for over sixty years, but a basic understanding of the mechanism responsible for the rise of the disks is still a subject of debate. Our experiments and theoretical model show that the ascent speed of the disk is proportional to the square root of the disk density, approximately proportional to the disk size, and is a function of the disk’s depth in the bed. We also investigated the speed of ascent of the disk as a function of the shaking frequency, fs . We found that the effective friction or drag coefficient, , between the disk and the granular bed, is proportional to a functional form of the frequency: ∝ (fs − fc )−4 , where fc is the critical shaking frequency for the disk to start moving through the bed. We discuss how such a dependency may arise.
Keywords Size vibration
segregation,
density
segregation,
1 Introduction When a bed of dry granular material is subject to vertical vibration, the granular material tends to segregate with larger particles at the top and smaller particles at the bottom of the bed. This behaviour, often called the “Brazil Nut Effect” [1], has been documented since at least 1939 [2] and has been the subject of quantitative laboratory study since at least 1963 [3]. During the thirty-year period between 1963 and 1993, the industrial relevance of this problem, coupled with its intrinsic academic interest, inspired a number of experimental studies [3–9] and computer simulations [1, 10–14]. These latter simulations supported the proposal of Williams [3, 4] that the Received: 28 August 2000 K. Liffman (&), G. Metcalfe Thermal & Fluids Engineering, CSIRO/BCE, Highett, Victoria, Australia K. Muniandy, M. Rhodes, D. Gutteridge Department of Chemical Engineering, Monash University, Clayton, Victoria, Australia
ascent of the larger particle was due to smaller particles falling into the voids produced underneath the larger particle after each shaking cycle. However, it has been known since the early nineteenth century, that vibration can induce convective flow in granular material [14–17]. In 1993, Knight et al. [18] used this property to show that bulk convective flow could trap larger particles at the top of the bed. These authors used a convective flow produced by a vertically vibrated, cylindrical container with rough walls. In this flow, the relatively wide, upward, convective flow in the middle of the container conveyed the large and small particles to the top together. The larger particles remained at the top of the container, because they were too large to travel along the thin downward sheets of the convection rolls. The Knight et al. paper led to further work [19–25], where authors attempted to clarify the shaking regimes for size independent segregation (which is the signature of bulk convective flow) from size dependent segregation. The latter segregation possibly arising from percolation [26], or arching [27] phenomena related to small particles moving into the voids generated under larger particles during each cycle event. In this paper, we report on our experiments and subsequent modelling of the size and density dependent rise time of relatively large disks moving within a sinusoidally vibrated, quasi two-dimensional bed of granular material. The experimental method and observations are presented in §2 and §3. In contrast to previous studies, we derive an equation of motion for the intruder disks. The main size-dependent segregation model is derived in §4, with a modification to the model given in §5 to account for the motion of the disks as they reach the surface of the bed. The resulting relatively good agreement between theory and experiment is discussed in §6 with the results and possible future development given in the conclusions (§7). 2 Apparatus and experimental method A test rig, shown in Fig. 1, was constructed to examine the movement of a large particle within a bed of smaller particles. It consisted of a hollow transparent cell with the internal dimensions 20 × 20 × 0.6 cm. Protruding from the ends of this cell were perspex blocks, each with a hole drilled at the ends. Two vertical rods extended through these holes from a solid base, constraining the cell to move only in the vertical direction. The cell was vibrated by an electric motor connected to a digital frequency meter,
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nearly an order of magnitude. Hence, all experiments were done in an air-conditioned laboratory where humidity was continuously monitored and kept relatively constant. We have labelled this experiment as a quasi twodimensional system, since the disk moved in a twodimensional framework, but the distance between the windows was equal to six glass bead diameters. We will briefly mention one possible consequence of this quasi two-dimensional structure in §6.4.
Fig. 1. Apparatus used to study segregation of disks in a vertically vibrated bed of granular material
which allowed the vibrational frequency to be altered. The frequency of vibration was independently monitored by electronically observing a light emitting diode, which was eclipsed once every shaking cycle. The output shaft from the motor was attached to the base of the cell by means of a scotch yoke cam, which has been tested to produce a true sinusoidal vibration. A scotch yoke cam was used in this rig, because a wheel-rod cam does not produce a true sinusoidal motion [28]. The cell was vibrated sinusoidally over a range of frequencies from 4.0 to 5.0 Hz in the direction parallel to gravity with a fixed amplitude of 15 mm. Disks of diameters ranging from 10 to 25 mm and effective densities in the range 1950 to 7460 kg/m3 were placed separately at the base of the granular bed and their motion studied as the cell was vibrated. The bulk particles used were spherical glass beads, 1 mm in diameter. These beads were poured into the cell to make a bed height of 10 cm through a specially designed steel hopper. The disks were slightly less thick than the internal width of the cell (6 mm). We found that if the disks were too thick or too thin they would jam between the cell walls and not move. However, by varying the thickness of the disk we were able to obtain an optimal disk thickness that did not jam or halt the disk’s ascent. Before the start of each experiment, the metal disk was placed at the base of bed, and the glass beads were poured into the bed. When the experiment was completed, the disk was extracted from the bed’s surface by means of a magnet attached to a thin rod and the bulk particles were removed by using a sealed, vacuum suction device. These protocols were used to ensure that minimal amounts of hand oil and other contaminants came in contact with the glass beads or disk. As has been shown elsewhere [29], small quantities of wetting liquid can dramatically change the properties of granular media. Before the start of each experiment, the box was tapped slightly to settle the granular material. This was done to prevent the disk from having an initial spike of high ascent due to an initially low, unsettled bulk density in the granular bed. It was observed, in earlier experiments, that changes in humidity could affect the ascent speed of the disk by
3 Experimental observations When vibrated, the typical behaviour of the disk and the bed is shown in Fig. 2. The initial configuration is shown in Fig. 2(a). The large (20 mm diameter) disk was positioned in the centre, at the base of the bed, with lines of coloured glass beads acting as tracer particles, so that we could observe movements of the bulk material. The test rig was set into sinusoidal motion with a fixed amplitude, r, of 1.5 cm and a frequency, fs , of 4.383 Hz. This gave the bed a non-dimensional acceleration = 1.16, where = r2 /g, with (= 2fs ) 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 shaking cycles (30 and 40 seconds of shaking), respectively. Convective bulk motion of the glass beads, in the top corners of the bed, has bent the layers into “smile” shapes. The disk, however, is unaffected by this bulk motion and simply punches through the layers, with a trail of coloured beads in its wake. After about 285 shaking cycles (65 s), the disk arrives at the top of the bed (Fig. 2(d)). As we shall discuss in section 4, this rise time is dependent on the size and density of the disk. To determine the possible influence of bulk convective motion on the disk motion, we vibrated the bed, without the disk, for over half an hour. We found that bulk convective flow occurred at the top corners of the bed, but there was no discernible convective flow in the centre of the bed. Given that this time scale is over an order of magnitude larger than the typical rise time of a disk in the bed, we concluded that bulk convective motion had little or no influence on the disk motion. 4 Equation of motion To derive the equation for the upward motion of the disk, we consider a particle that is thrown into a low-density medium. The distance that this particle will travel into the medium is such that the energy in moving the displaced material is equal to the initial kinetic energy of the particle. As shown in Fig. 3, we assume that the impacting particle will displace a “plug” of material, where the force required to accomplish this task is mc g, where is a granular friction coefficient, mc is the mass of displaced plug 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 that frictional forces decrease its velocity. This implies: mp
d = − mc g = − Ap xm g , dt
(1)
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Fig. 2a–d. Upward motion of a 20 mm diameter steel disk in a vertically vibrated bed
In our experiment, the impacting object is a disk, so mp = a2d d L, and Ap = 2Lad , where ad is the radius of the disk, L the thickness of the bed (see Fig. 4) and d the mass density of the disk. So, one has mp /Ap = ad d /2. Noting that 0 ≈ r2fs , the penetration distance becomes ad d . (4) l = r2fs 2m g Fig. 3. Schematic diagram of a projectile displacing a plug of granular material
where is the speed of the particle, t the time, mp the mass of the particle, Ap the cross sectional area of the particle (where the normal of the cross sectional area is parallel to the direction of motion), x is the distance the particle moves into the bed and m is the density of the diffuse medium. We note that, in our model, is not the standard solid-on-solid Coulomb friction coefficient. Rather, as we shall discuss, relates the resistance of the entire bed to the displacement of a small part of the bed. To determine the total distance, l, the disk will penetrate into the diffuse, granular medium, Eq. (1) must be integrated using the appropriate limits of and x: 1 2
0 20
d(2 ) = −
Ap m g mp
l x dx ,
(2)
0
where 0 is the initial speed of the particle as it intercepts the medium. The solution of Eq. (2) is an equation that gives the penetration distance as a function of the initial velocity mp l = 0 . (3) Ap m g
To determine l, we need to find a suitable value for m . At first glance, one could simply take m as the mass density of the medium. In a granular medium, however, a large intruder particle will not only affect the particles in contact with its surface, but also other particles throughout the bed. This is because one particle is quite often connected via stress chains to many other particles in the bed [30]. Such stress chains may produce dislocations that can propagate through the bed from an intruder particle [31]. So the large intruder particle is subject to an “effective” mass density which must take into account the extra mass from these chains of particles. In this analysis we assumed that the disk must move a ‘wedge’ of material (see Fig. 4). This assumption is consistent with observations of dislocations in a granular bed due to the presence of a large intruder particle [31]. Thus, the effective mass density of the medium, which we now denote by e , is defined to be the mass of the wedge divided by the volume of the ‘cap’ of particles on the disk surface. The disk cannot move downwards because the bottom of the bed cannot be displaced and so there is an infinite effective mass density for downward motion. The volume of the wedge can be determined from an analysis of its geometry as Vw = y 2 L tan + 2|y|ad L(1 + tan ) , (5) +a2d L tan + 2 − 2
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The ascent speed, , is defined as the distance that the disk rises per vibration cycle, so = dy/dt ≈ lfs . If we set a = a2d tan + 2 − , 2 b = 2ad (1 + tan ), and c = tan , (10) then takes the form d ap g dy √ = ad . = dt 2 6b [cy 2 + b|y| + a]
Fig. 4. The mass of the wedge of material above the disk gives a large effective mass density in the cap of material directly above and adjacent to the disk. The value y is the distance between the top of the disk and the surface of the bed. The surface is defined to be at y = 0
where Vw is the volume of the wedge, y the distance from the top of the disk to the surface of the bed, and the angle that the side of the wedge makes with the vertical. By multiplying the volume of the wedge by the bulk density of the bed, b , one can obtain the mass of the wedge. M w = V w b .
(6)
The width of the cap at the top half of the disk is taken to be the thickness of the layer of particles in contact with the disk at the disk’s surface. We calculate this thickness to be 2ap 2/3, where ap is the average radius of the bulk particles. This is the distance between the surface of the disk and the second layer of particles away from the disk, where we assume that the first layer is in contact with the disk and the second layer is in contact with the first layer. We also assume that the glass beads, to some approximation, are hexagonal close packed at the surface of the disk. Thus, the volume of the cap of beads in contact with the disk, Vs , is 2 (7) Vs = 2 ad ap L 3 By definition, the effective mass density encountered by the disk, e , is given by combining Eqs. (6) and (7) √ b 6 2 Mw e = y tan + 2|y|ad (1 + tan ) = Vs 4ap ad . +a2d tan + 2 − 2
(8)
Substituting e of Eq. (8) in place of m in Eq. (4) gives l = 42 ad rfs × √
d ap
.
2 6gb y 2 tan +2|y|ad (1+tan )+a2d (tan +2− 2 )
(9)
(11)
From Eq. (11), it is apparent that the ascent speed is proportional to the square root of the disk density, approximately proportional to the disk size and is a function of the depth of the disk in the bed. As discussed in the Appendix, one can integrate Eq. (11) to find the total rise time, T , of the disk: √ F (ad , , y0 ) 2 6b , (12) T = ad d ap g where y0 is the initial distance between the top of the disk and the surface of the bed, plus √ (2cy0 + b) a + by0 + cy02 − b a F (ad , , y0 ) = 4c
2 c(a+by0 +cy02 )+2cy0 +b √ + √ ln 8c c 2 ca + b (13) Eqs. (12) and (13) suggest that the total rise time of the disk in the bed is a function of the size and density ratios of the disk to bed material, plus the driving acceleration, bed friction and bed depth. To compute the position of the disk, y, as a function of time, one has to numerically solve for y in Eq. (A3). Eqs. (A3), (12), and (13) are the central analytic results from our model for the upward ascent of a particle through a granular bed. There is, however, a modification required to account for an effect that slows the motion of the disk just as it reaches the surface of the bed. 5 The Surface effect One of the implicit assumptions that we have used in deriving Eqs. (11), (12), (13) and (A3) is that the bed surface is unaffected by the motion of the disk in the bed. This assumption arises when we compute the mass of the wedge of material above the disk. As is apparent from Fig. 4 and Eq. (5) the bed surface is assumed to be flat and horizontal. However, as can be seen from Fig. 5 (see also Fig. 2), once a disk comes within approximately 2 cm of the disk surface, the surface gains a “bump”. This small hill of material increases the mass (and therefore the effective density) of the wedge of material that has to be moved by the disk. This extra mass of bed material slows the upward motion of the disk. In deriving our model, we have assumed that the wedge of displaced material always has a flat surface. This, in turn, assumes that as the disk moves upward, any displaced material quickly moves so that the bed surface is
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Fig. 5a, b. Video pictures of a 20 mm disk moving up through the bed where there is a the onset of the bump on the surface of the bed, b a moderately well developed bump as the disk nears the surface
always flat. Clearly this assumption breaks down as the disk moves close to the bed surface. One way of accounting for this phenomenon is to assume a “drainage” time-scale where it takes a finite time for the material above the disk to drain away and for the surface to flatten. This modified model is outline schematically in Fig. 6. In the modified model, as shown in Fig. 6(a), we start with the disk moving up through the bed until we start to observe noticeable heaping at the bed surface. In our experiments this seems to occur when the disk is approximately 2 cm below the surface of the bed. This depth appears to be approximately independent of the disk size. We arbitrarily set the time for the onset of this phenomenon to t = 0. The mass of the wedge above the disk is denoted by M0 . After the next cycle of shaking, we have t = Ts and the disk has moved up a distance l1 (see Fig. 6(b)). By assumption, a section of the wedge is now, “instantaneously”, above the previous surface. This “above the surface” layer is assumed to drain away to a flat surface on an e-folding time of d . In this figure, the mass of the wedge below the surface is denoted by M1 , while the mass of the wedge above the bed surface is denoted by M1 . Thus, the total mass that is affected by the movement of the disk is M1 + M1 . Note, that we have schematically depicted the “above surface” layer as a straight section of the wedge that is above the surface of the bed, but the actual model
Fig. 6a–c. Mass of wedge below and above the surface as the disk moves up a at the onset of surface heaping at t = 0. b At the end of the first shaking cycle (t = Ts ) the disk has moved up a distance, l1 , and there is a section of the wedge below
is independent of the shape of the bump above the disk surface. At the end of the next shaking cycle, t = 2Ts , and the disk has moved up an additional distance, so there is an additional layer of the wedge (with a mass M2 ) which is now above the surface of the bed. The mass of the wedge below the surface is denoted by M2 . In the period Ts < t < 2Ts , some part of the first layer has “flattened out” and the mass of the first layer above the surface is M1 exp(−(t−Ts )/d ). So the total mass that is affected by the movement of the disk is now M2 + M2 + M1 exp(−(t − Ts )/d ). We can continue this analysis and show for the nth time step that the total mass of bed material, that is affected by the upward movement of the disk, Mw (nTs ) is given by the equation Mw (nTs ) = MnTs +
n
MkTs e−(t−kTs )/d ,
(14)
k=1
where MnTs is the mass of the wedge below the surface at time t = nTs , and MkTs is the mass of the layer above the 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 the same manner as for Eq. (8): e =
Mw (nTs ) . Vs
(15)
This value is substituted into the m of Eq. (4), and so provides us with a value for ln+1 , i.e. the upward distance that the disk moves in the n+1th time step. Unfortunately, the sum in Eq. (14) seems to remove the possibility of an analytic solution to this system. So we numerically solve for the subsequent motion of the disk by using the difference equation yn = yn−1 − ln−1 ,
(16)
where yn is the disk depth at t = nTs .
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
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6 Comparison between predictions and measurements 6.1 Depth of disk versus time We now have developed enough mathematical machinery to start comparing theory with experiment. In the first set of experiments we considered the position of the disk as a function of time. To do this, we placed a disk at the bottom of the granular bed and started shaking the bed with a vertical, sinusoidal motion at a frequency of 4.45 Hz and an amplitude of 1.5 cm ( ≈ 1.2). For the first set of experiments, we considered the motion of two different disks, one with a diameter of 10 mm and the other with a diameter of 20 mm. Five separate trials were run for each the disk. The results are given in Fig. 7. Figure 7 (a) shows the behaviour of a 10 mm diameter disk. The top of the 10 mm diameter disk was initially located at 9 cm below the surface of the bed. The dashed lines show the maximum and minimum experimentally observed values, while the solid line shows the result from Eq. (A3). The surface effect is incorporated into Eq. (A3) as discussed in the previous section. The surface effect is observed/assumed to occur when the disk is 2 cm below the surface of the bed. In Fig. 7 (a) the surface effect causes the kink in the solid line at a depth of 2 cm. At lower depths, , is the only free parameter and d = 0 s.
Fig. 7a, b. Comparison between experimental and predicted results for a 10 mm disk and b 20 mm disk. The dashed lines show the minimum and maximum for 5 separate experimental trials. The solid line shows the theoretical line deduced from equation (A3). The “kink” in the solid line shows the position 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 glass ballotini were in a hexagonal close-packed arrangement, i.e., the fundamental structural unit was four spheres located in a tetrahedral shape. For such a case, the angle of 35◦ is the approximate angle between the side of the tetrahedron and a perpendicular line drawn from the base of the tetrahedron through the apex of the tetrahedron. As such, this angle is representative of the expected angle between the vertical and a stress line. At first sight, the value of = 130 would appear to be quite high, relative to standard friction coefficients. However, as mentioned in the discussion surrounding Eq. (1), is the resistance of the entire bed to the displacement of a small part of the bed. So the value of has to be large to account for the influence of the entire bed on the movement of the disk. Finally, we note (again) that the value for d of 11.2 s is only valid when the top of the disk is within 2 cm of the surface 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 results for five separate experimental trials, while the unbroken line shows the theoretical result from Eq. (A3). This time, however, the curve of best fit gives = 115 with unchanged and d = 6.7 s (or 30 shaking cycles). The variation in and d between the two disks can be partially explained by noting that, from Eq. (A3), the rise time and the position of the disk as a function of time is dependent on the square root of , so a 6% difference in readily translates into a 13% difference in . This behaviour is illustrated in the next set of results.
6.2 Total rise time versus disk size To determine the total rise time of the disks as a function of disk size, we obtained six steel disks with diameters of 10, 13, 15, 18, 20 and 25 mm and measured the total rise time for each disk. The typical experimental procedure was to take one of the disks, place it at the bottom of the bed, and then shake the bed sinusoidally at a frequency of 4.45 Hz until the disk reached the surface of the bed. The total time for this to take place would be noted and the procedure repeated. Typically, 5 separate trials were run for each disk. The results obtained are shown in Fig. 8, where the squares show the experimental results – with associated error bars – while the solid line shows the theoretical line obtained from Eqs. (12) and (13) (modified to account for the surface effect). To obtain the solid line, we assumed = 122.5 and d = 8.9 s. Both values are averages of the results obtained from the position versus time results for the 10 and 20 mm disks given in Fig. 7. As can be seen, from Fig. 8, there is good agreement between theory and experiment. The upper dashed line in Fig. 8 shows the theoretical result for the and d values obtained from the 10 mm diameter disk ( = 130 and d = 11.2 s). The lower dashed line gives the theoretical line for = 115 and d = 6.7 s (the values obtained from the 20 mm diameter disk). The
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Fig. 9. The top row shows the steel disks available for the total rise time versus size measurements of Fig. 8. The bottom row shows the disk and annuli used in the total rise time versus density measurements of Figure 10
Fig. 8. Total rise time of the disk, T , in seconds as a function of the normalised size of the disk, where ad and ap are the radii of the disk and glass spheres, respectively. The experimental results are denoted by the squares. The solid line shows the theoretically expected result, while the dashed lines show the variation in the theoretical result due to experimental uncertainty in the values of and d
difference in rise time between the upper and lower lines is approximately 8% of the solid line value, thereby reflect ing the dependence of the ascent speed of the disk (Eq. (11)). 6.3 Total rise time versus disk density We now consider the relationship between the total rise time and the density of an ascending particle. As far as we are aware, there have been at least four papers that have addressed this issue [3, 4, 6, 8]. Unfortunately, the results of these studies do not appear to be directly applicable to our work, because they do not discuss, in a quantitative manner, the low-convection acceleration regime addressed in this study. We should expect, from Eq. (12), that the total rise time of the disk is inversely proportional to the density of the disk. This behaviour arises, because the form of Eq. (1) is that of a simple harmonic oscillator, where – on small time scales – the potential energy of the bed increases as the square of the penetration distance of the disk into the bed. To determine the validity of this relationship, we made six disks of the same size (20 mm diameter), but with different densities. We were able to do this by drilling holes in five of the six disks (Fig. 9), thereby converting the disks to annuli. This method decreased the mass and average mass density of each annulus and produced disks with densities ranging from 1.95 to 7.46 g cm−3 . Each disk was placed at the bottom of the granular bed and the system was then shaken sinusoidally at a frequency of 4.45 Hz and with an amplitude of 1.5 cm until the disk reached the surface of the bed. At least 12 trials were run for each density and the results obtained are shown in Fig. 10. Here we plot the rise time of the disk as a function of the disk’s density, where the disk’s density is normalised relative to the bulk density, b , of the glass ballotini (1.52 g cm−3 ). The experimental results are
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 the disk density relative to the bulk density of the glass, b , (1.52 g cm−3 )
shown as filled-in squares and the theoretical result is given as an unbroken line. The error bars on the experimental results show the standard deviation of the mean. To obtain the theoretical line, we used Eq. (12) and set and d so that the line passed through the experimental value for the disk with the highest density (the right most experimental value in Fig. 10). As can be seen, all the other experimental values, except for the lowest density disk (1.95 g cm−3 ), lie on the line and so appear to be consistent with theory. 6.4 Total rise time versus shaking frequency For our final experimental result, we considered the total rise time of the disk as a function of the shaking frequency. Again, we took the 20 mm disk and measured the rise time in the manner discussed in §6.1. This time, however, we set the frequency of vibration to different frequencies in the range from 4.07 Hz to 4.99 Hz. The lower bound of the frequency range is the critical frequency, fc , for the movement of the disk, where the value for fc , is obtained by setting = 1. Thus fc is given by the formula g 1 , (17) fc = 2 r
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that has the value 4.07 Hz for r = 1.5 cm. The disk will not move when < 1 since the bed will not be dilated and the particles cannot move relative to one another. The upper bound for the frequency was chosen from observing the influence, or otherwise, of convection on the central regions of the granular bed. Nine separate frequencies were chosen and at least five separate experimental runs were conducted for each frequency. The experimental results are the black squares shown in Fig. 11, where we have plotted the average ascent speed, , against the non-dimensional acceleration . The error bars, representing the sample standard deviation of the results, are small relative to the absolute magnitude of the results and cannot be seen in Fig. 11. The average ascent speed is defined by the formula y0 = , (18) T where y0 is the initial distance between the top of the disk and the bed surface, and T is the total rise time of the disk. We have chosen to plot instead of T , because it provides a better indication of the disk’s behaviour for higher vibrational frequencies. From Eqs. (12), (13) and (18), our model gives a more detailed equation for : ad y0 d ap g √ = . (19) F (ad , , y0 ) 2 6b If we assume that is independent of the shaking frequency, then Eq. (19) predicts that should be directly proportional to . However, as can be seen from the experimental results shown in Fig. 11, there is a non-linear relationship between and and so Eq. (19) is not correct. One way to reconcile the data with our model is to suppose that is a function of frequency, since all the other quantities in Eq. (19) – besides – are independent of the shaking frequency. To determine the possible dependency of on f , we set
q f0 − fc = 0 , (20) fs − fc where f0 is the “normalisation” vibrational frequency (which we set to f0 = 4.45 Hz) such that when fs = f0 then = 0 . In this case, 0 is our standard value of (0 = 122.5), and q is a free parameter. By varying q, we were able to obtain the “best fit” line shown in Fig. 11. Via this process, it was found that q = 4 gave the best fit to the experimental data. At this stage of our analysis, it is not clear why q should have this value. One possibility is that is somewhat proportional to the number of contacts between particles in the granular bed. Due to atomic forces, (e.g., Van der Waal forces) each contact may have a small, attractive force component making the particles slightly sticky. For the disk to move through the granular bed, some of these force contacts would have to be broken to allow particles to move past one another. As the shaking frequency increases the number of contacts between the particles in the granular bed would decrease, thereby decreasing . Whether or not one could obtain q = 4 from such a process remains to be seen, although, it may have
Fig. 11. The average ascent speed of a 20 mm diameter disk as a function of . The squares represent the experimental results and the solid line shows the line from Eq. (19), where – as is explained in the text – it is assumed that the granular friction coefficient, , is dependent on the shaking frequency
something to do with the quasi two-dimensional structure of our experiment. We hope to investigate this, and other hypotheses, in later studies. For now, however, we simply note the result. 7 Conclusions In this paper we document our observations of the ascent of a large intruder disk in a 2D vibrating bed of granular material, where the granular bed is subject to relatively low vibrational accelerations, typically at = 1.2. For this value of , the bulk convection of the granular bed would appear to have little or no influence on the motion of the large intruder disks. Our experiments showed that the ascending disk appears to push through the granular bed, entraining granular material in the its wake. In an attempt to understand this behaviour, we have developed a “penetration” model for the disk’s motion. We suggest that the disk’s upward motion is driven by its momentum and that for it to move through the bed, the disk has to interact with a wedge of material directly above the disk. This insight allows us to construct an analytic model for the disk’s upward motion. In this model the disk can move in any direction within the granular bed, i.e. either up or down. The granular material simply acts as a source of frictional drag on the movement of the disk. For the disk to move upwards, it has to displace particles above it. This means it has to displace the top surface since, in a granular medium, particles are interconnected via rigid, long-range contact chains. If the disk moves downwards, it must displace particles below it – but this is much more difficult, since the stress chains will terminate at the fixed base of the granular bed. So the granular medium, coupled with the free top and fixed bottom surfaces, imposes a bias that favours the upward movement of the disk. The disk’s upward motion is subject to drag from particles in the granular bed, where the drag force is proportional to the weight of the material displaced by the disk. To determine the weight of displaced material, we use the
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fact that particles within the granular bed are subject to long range interactions with other particles via long contact chains. Thus, to move through the bed, the disk has to interact with a wedge of material directly above the disk. This insight allows us to construct an analytic model for the disk’s upward motion. The model predicts that the total time for a disk to reach the surface of the bed should be approximately inversely proportional to disk size, and inversely proportional to the square root of the disk density. Both predictions are consistent with our experimental results. The model can also provide an accurate description of the depth of the disk as a function of time, but the model requires an extra parameter to account for a deceleration in the disk’s ascent near the surface of the bed. Finally, our experiments indicate that the friction coefficient between the disk and the granular bed is a fourth order polynomial of the shaking frequency. We suggest that this is due to the number of static, “sticky” contacts between particles in the granular bed, which may be a function of the shaking frequency. We hope to study this behaviour in future work. 8 Appendix: Rise time and depth of the disk as a function of time To determine the depth of the disk as a function of time, we integrate Eq. (11) over a time t: t
a + by +
0
cy 2
dy dt = dt
t ad
0
d ap g √ dt . 2 6b
(A1)
The left hand side of Eq. (A1) is t 0
dy a + b|y| + cy 2 dt = dt
−y a + b|y| + cy 2 dy , (A2) −y0
where y is the depth of the particle at a time t (y is taken to be a positive quantity) and y0 is the initial depth of the particle. Eqs. (A1) and (A2) imply (2cy0 + b) a + by0 + cy02 4c
2 c(a + by0 + cy02 ) + 2cy0 + b √ + √ ln 8c c 2 ca + b (2cy + b) a + by + cy 2 − 4c
2 c(a + by + cy 2 ) + 2cy + b √ − √ ln 2 ca + b 8c c d ap g √ = ad t , 2 6b
(A3)
where
= 4ac − b2 = −4a2d 1 + tan . 2
(A4)
The disk reaches the top of the bed at the rise time t = T , and y = 0. Substituting these values into Eq. (A3) gives the rise time as √ F (ad , , y0 ) 2 6b , (A5) T = ad d ap g where F (ad , , y0 ) =
√ (2cy + b) a + by0 + cy02 − b a 4c
2 c(a+by0 +cy02 )+2cy0 +b √ . + √ ln 8c c 2 ca + b (A6)
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