Pattern Selection by a Granular Wave in a Rotating Drum for which the Angular Frequency Varies...

7/27/2019 Pattern Selection by a Granular Wave in a Rotating Drum for which the Angular Frequency Varies Sinusoidally.

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Pattern Selection by a Granular Wave in a

Rotating Drum for which the Angular Frequency

Varies Sinusoidally

R.D.P.East

August 9, 2013


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Contents

1 Introduction 1

2 Method 2

3 Results and Analysis 4

3.1 Linearity between the ratios . . . . . . . . . . . . . . . . . . . . . 43.1.1 Considerations . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2.1 Observations on the linearity between the ratios . . . . . 73.2.2 Petal splitting . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Further Experiments 11

5 References 13

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Abstract

Two types of granular particle were rotated in a shallow drum. This drumwas rotated at such a frequency petals formed [1]. The base angular velocitywas varied sinusoidally, and the subsequent effect on the petal pattern wasobserved. A linear relationship between the ratio of the sine wave frequencyover the granular wave frequency, and the ratio of the observed number ofpetals when the sinusoidal variation was applied over the original number. Asecond relationship was also observed in which there would be multiple petaltips per petal would form when the imposed sine-wave induced a petal numbergreater than the original. The number of split petals was also linear with theaforementioned ratio between the sine and granular wave frequency.


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Chapter 1

Introduction

The recent interest in granular medium has shown that the mixing of differ-ent granular particles is a complex and difficult problem and furthermore canprovide some surprising results. M.Mobius et al, demonstrated that smallerparticles would interact with a single larger particle by pushing the larger par-ticle to the top of the mix if the system was vibrated [2]. The partitioningof the mixtures of two granular materials was investigated by N. Burtally etal[3]. They demonstrated amongst other things that these particles separatedunder vertical vibration formed distinct layers. Most relevant for this experi-ment was the pattern formation caused by two granular materials in a rotatingdrum demonstrated by Zuriguel et al[1].This work is directly built on in thisexperiment, where an almost identical set-up was used with the addition of awaveform generator in order to impose a sinusoidal variation in the angularfrequency of the drumb.

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Chapter 2

Method

A drum of diameter of 24.5 0.2 cm and depth of 3 0.05 mm was half filled(3%) with two types of granular materials of similar density. The smallerwhite particles had a diameter of 120 30 m and made up a proportion of thevolume equating to a small particle volume fraction of = 0.35 . The largergreen grains had a diameter of 710 30 m. The drum was driven by a motorconnected to the disk via a belt drive attached to a gearbox in order to attainappropriate rotational speeds (see figure 3.2) . The voltage supplied to themotor was the sum of the steady voltage provided by a control box and thatof an arbitrary wave form generator. A camera was also used to document thepetal patterns formed.

Figure 2.1: A diagram of the experimental set-up.

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Initially a base petal pattern was found at some angular velocity such thatit displayed a small number of clear petals . The number of petals in the base

patterns considered in this experiment were 5,6,8 and 10. These patterns wereformed by the constant angular velocities of 0.139 rads1, 0.95 rads1,0.09rads1, and 0.065 rads1 respectively. The initial base pattern was formedusing only the steady input voltage from the control box in order to have acontrol pattern against which to compare the effects of sinusoidal variation inthe angular frequency.Once a stable pattern had been formed the time period of the granular wavewas found using a stopwatch to time the period of 10 wavelengths (ten petal for-mations) and from this deriving the its frequency. The reader should note thatthe granular wave is the mechanism by which patterns are formed in the drumas granular avalanches separate the particles (see Zuriguel for more detail[1]).The waveform generator was then used to vary the angular velocity of the drumsinusoidally, the rms amplitude of this sine variation was 0.06 rads1. Thefrequency of the imposed sine variations were fractions of the granular wave,leading to the use of the ratio ratio where

ratio =sine

wave.

Note sine and wave are the frequency of the imposed sine wave and the fre-quency of the granular wave associated with the base state respectively.

Figure 2.2: An image of the base petal pattern of 10 petals prior to the intro-duction af any angular frequency variation in the drum.

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Chapter 3

Results and Analysis

3.1 Linearity between the ratios

In analysing the pattern formed in varying the imposed sine variations frequencya linear relationship was discovered in the frequency region considered. Therelationship was between the ratio of the imposed sine wave frequency over thegranular wave frequency in the base state ratio, and the ratio of the observednumber of petals over the original number as seen in the base state, nratio. Thisrelationship is represented in figure 3.1. This relationship demonstrates thatan imposed sine wave of a frequency that is some fraction of the granular waveresults in the formation of that very fraction of the original number of petals inthe base state. This is provided that said fraction results in an integer numberof petal tips. If this was not the case more complex patterns formed which

followed no discernible regimen and as such are not considered here.

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0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

4

ratio

nratio

Base petal tip number = 8

linear

Base petal tip number =5



Figure 3.1: Plot ofnratio against ratio, note the linear fit which is described byy=x.

To illustrate this phenomena, consider figure 3.2 where the image furthest leftis the base state of the five petals, and moving right we see the effect of theimposition of sinusoidal variation rotation at frequencies that are fractions ofthat of the granular wave. It is clear that the ratio ratio directly dictates thenumber of petals that will form.

Figure 3.2: The base 5 petal pattern with (from left to right): No imposed sinefrequency; ratio =

15 ; ratio =

35 ; ratio =

45 .

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3.1.1 Considerations

It should be noted that for the smaller fractions equating to one or two petals,small formations would occur such as those seen in figure 3.3. These howeverdid not occur in patterns with greater number of petals. When considering thenumber of petals these formations did not contribute to the result previouslydiscussed. The justification for this has been that these formations are notbelieved to be natural products of the rotations.In the patterns with higher peal number the petals do not come in contact withthe edge of the drum; In contrast the conservation of the volume of the particlesmeans patterns of lower petal number which form wider and longer petals would.The petals that reach the drum edge as they form are presumably disruptedhence leading to said formations. Though this explanation does hold well withthe patterns observed, in which we see no extra formations when the petals arenot of sufficient length to reach the drum edge. It would still be advisable to

use larger drums in future or alternatively smaller volume fractions.

Figure 3.3: Circled in red are the anomalous formations for the patterns whosebase petal number was 5,8,and 10 from left to right. The ratio for each imageare 25 ,

38 , and

210 respectively.

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3.2 Observations

3.2.1 Observations on the linearity between the ratios

It was noted that the edge of the petal facing in the direction of rotation wasmore distinct than its other edge. Unfortunately the camera glare makes thisdifficult to analyse with the current experimental set-up. A second observationwas that the greater the number of petals in the base state the slower the petalformation (and the splitting which is mentioned in section 3.2.2).A final observation is that there was an interesting variation in petal widthwhile the imposed frequency equated to ratio 1 (but was still leading to aninteger number of petals).As seen in figure 3.4 there is a subtle difference in thepetal widths. Unfortunately a proper characterisation of this was not obtainedduring this experiment.

Figure 3.4: The petal pattern for the base petal number of 8 with an imposedfrequency ratio ratio equal to

68 . Note the subtle variation in petal widths.

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3.2.2 Petal splitting

A second relationship was observed during the the experiment, namely that forimposed ratios ratio > 1 (but again still leading to an integer number of petaltips) the patterns became stable homogeneous and separate for a period beforeslowly transforming (see figure 3.5) into an entirely different pattern. Thissecond pattern presented the same number of petals as in the base pattern onwhich the sine variation was imposed. That said the number of petal tips wasthe same as in the patterns pre-transformation. This can be understood as oneor more petals would have multiple tips, or had split. This effect was only fullyanalysed for the five petal base state, as the results from this seemed to formfastest and clearest. Though all results discussed from here on pertain to the 5petal base state, every other base number tested showed evidence of splitting.The manner in which the petals split is also of interest.Further observation leadto the conclusion that each petal would split as little as possible viz. all petals

would split twice (corresponding to a frequency, , where sine = 2wave ) beforeany would split to have 3 tips. The number of times any petal split, that is tosay a single tip petal became a double, or a double split became a triple and soon, was plotted against ratio. This revealed a linear relationship which can beseen in figure 3.6.It indicates that after the number of petal tips exceeds thebase number, every extra induced petal tip is produced via a split. Each splitoccurs in one of the petals of the pattern of which the number of tips is theminimum for the observed pattern.

Figure 3.5: Here the homogeneous separate petals can be seen on the left, theserelate to ratio =

53 . On the right is the eventual pattern depicting 5 base petals

each with 3 tips. Note there are still 15 induced petal tips but there is a returnto a 5 petal pattern.

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Figure 3.6: Graph of the number of splits against ratio, the partitions show themanner in which the petals split.

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As an example one can consider the region where single split petals, beginto break into three tips (a double split). In figure 3.7 where we see the single

split petals formed at ratio = 2 form double splits as the sine frequency isincreased to produce an extra tip. Here we clearly see with each increment ofthe frequency ratio to produce an extra tip, another single tip petal splits tobecome a double tip petal.

Figure 3.7: For a base state of five petals we see from right to left where:ratio = 2 and so we fine 10 tips on five single split petals; ratio =

115 ; ratio =

115

and as such we have 11 tips formed by 4 single split and 1 double; ratio =125

and as such we have 12 tips formed by 3 single split and 2 double.

Considerations

Though promising, this result only confirms the relationship for a limited rangeand further analysis must be done in future. The issues to over come are firstlythat it becomes problematic attempting to distinguish separate petals and theirrespective tips as the frequency is increased. Secondly there is an issue with the

time over which these patterns form on which they seem to have a non-lineardependence (It is important to note this was not quantified and is only a casualobservation).The result of these two issues were that It was not possible to test the petalsplitting above ratio=3 , and also it meant that obtaining results at lower sinefrequencies was difficult, assuming that they do in fact exist. Specifically whileit was possible to obtain a pattern for the frequency pertaining to ratio =

95 ,

which did produce the expected result of 4 single splits and an un-split petal (4double tips and a single), frequencies below this did not produce the expectedpatterns after forming the initial homogeneous petals even after several hours.

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Chapter 4

Further Experiments

Based on the findings and observations contained herein it seems appropriateto offer an outline for future experiments.

The key requirement for further analysis of the number of petals and the tipsper petal at higher frequencies must be the computer analysis of the imagestaken by the camera. It is obvious to an observer at lower petal numbers wherethe petal tips are split or are separate. This is judged based on the depth ofthe region with low small particle concentration between petal tips. If large,the next tip is part of a separate petal.If small, it is a tip attached to a petal onwhich there is at least one other petal tip. This is difficult for the human ob-server as the petals become more densely packed,but would not be a problem forcomputer image analysis. Especially when one considers the human observerspropensity to seek patterns in images. Furthermore when a human observerin some parts anticipates a certain pattern the likelihood of confirmation biasrenders objective computer analysis a must[4].

Secondly computer analysis is then the only reliable way one could hope toquantify the more distinct edges of the petals as mentioned in section 3.2.1as one would most likely need to set some intensity bound on what one wouldconsider the more distinct edge in order to asses its width perhaps, or otherpossible properties.

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The obstacle to reliable computer analysis of the images lies in the unsuitable

lighting conditions. The sporadic phenomena of camera glare and the reflectionsof the light in the room off of the glass leads to bright patches in photos, thesewould frustrate attempts to provide reliable analysis.

A future experimenter should also look into the different widths that were ob-served in the petals that formed at an imposed sine wave frequency equatingto values of ratio=3 below 1 as discussed previously in 3.2.1. It may also beworthwhile to make a study of the time scales over which petal formation varies,be it the time to form the homogeneous separate petals or the split petal stateimplied by section 3.2.2.

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Chapter 5

References

[1] Zuriguel, I., Gray, J. M. N. T. Peixinho, J. Mullin, T. Pattern selection by agranular wave in a rotating drum. Phys. Rev. E 73 (2006).

[2] N. Burtally, P. J. King, M. R. Swift, Science, 295, 1877 (2002).

[3] Mbius ME, Lauderdale BE, Nagel SR, Jaeger HM (2001) Brazil-nut effect:Size separation of granular particles. Nature 414: 270.

[4] Sternberg, Robert J. (2007), Critical Thinking in Psychology: It really iscritical, in Sternberg, Robert J.; Roediger III, Henry L.; Halpern, Diane F.,Critical Thinking in Psychology, Cambridge University Press, p. 292

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