Chemical Engineering Science 73 (2012) 195–207

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Chemical Engineering Science

0009-25

doi:10.1

n Corr

E-m1 Pr

Mechan2 Pr

Princeto

journal homepage: www.elsevier.com/locate/ces

Mixing by cutting and shuffling 3D granular flow in spherical tumblers

Gabriel Juarez a,1, Ivan C. Christov b,c,2, Julio M. Ottino c,d,e, Richard M. Lueptow e,n

a Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USAb Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208, USAc Department of Chemical and Biological Engineering, Northwestern University, Evanston, IL 60208, USAd The Northwestern Institute on Complex Systems (NICO), Northwestern University, Evanston, IL 60208, USAe Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, USA

a r t i c l e i n f o

Article history:

Received 26 July 2011

Received in revised form

23 January 2012

Accepted 26 January 2012Available online 2 February 2012

Keywords:

Granular flows

Mixing

Cutting and shuffling

Piecewise isometries

Rotating tumbler

Simulations

09/$ - see front matter & 2012 Elsevier Ltd. A

016/j.ces.2012.01.044

esponding author.

ail address: r-lueptow@northwestern.edu (R.M

esent address: Department of Mechanica

ics, University of Pennsylvania, Philadelphia,

esent address: Department of Mechanical

n University, Princeton, NJ 08544, USA.

a b s t r a c t

Cutting and shuffling, a novel approach to mixing, can be used to predict the degree of mixing for three-

dimensional (3D) granular flows in rotating tumblers. An idealized prototype, a half-full ‘‘blinking’’

spherical tumbler alternating in rotation about each of two horizontal axes in the limit of a vanishingly

thin flowing layer, is studied using piecewise isometries (PWIs). Mixing is characterized by the

normalized difference between the center of mass of a collection of tracer particles and the centroid of

the filled portion of the container. The degree of mixing is highly dependent on the combination of

rotation angles about the two axes and the angle between the axes of rotation. Various rotation angle

combinations ðy1 ,y2Þ are investigated over 11ryir891 (in 11 increments). Combinations of rotation

angles that can lead to good mixing after only a few iterations of the blinking flow are identified.

Protocols with non-orthogonal axes are also studied by varying the angle f between rotation axes over

151rfr901 (in 151 increments). When f¼ 601 or 751, we observe the highest number of rotation

angle pairs that lead to good mixing.

& 2012 Elsevier Ltd. All rights reserved.

Fig. 1. Illustration of the 3D blinking flow in a spherical tumbler half filled with a

granular material. (a) Particles undergo solid body rotation with the tumbler upon

rotation about the z-axis until reaching the critical angle of repose, at which point

they begin to flow down the slope in a finite-thickness lens-like flowing layer at

the surface. After rotating the tumbler through the specified angle, yz , the tumbler

and particles are slowly rotated back as a solid body until the free surface is

horizontal. (b) Analogous rotation about the x-axis through the angle yx . The

rotation rates about the two axes are assumed equal. Used with permission from

Juarez et al. (2010). & 2010 IOP publishing.

1. Introduction

The problem of granular mixing in a tumbler is an old one(Lacey, 1954; Carley-Macauley and Donald, 1962, 1964), butthree-dimensional (3D) granular flow in biaxial tumblers has onlyrecently been modeled theoretically (Gilchrist, 2003; Gilchristand Ottino, 2003; Meier et al., 2007). Though mixing andsegregation in quasi-2D (Clement et al., 1995; Metcalfe et al.,1995; McCarthy et al., 1996; Boateng and Barr, 1996; Khakharet al., 1997; Hill et al., 1999; Gray, 2001; Mellmann et al., 2004;Cisar et al., 2006) and 3D (Wightman et al., 1998; Wightman andMuzzio, 1998a,b; Alexander et al., 2003; Gilchrist and Ottino,2003; Lemieux et al., 2007; Meier et al., 2007; Doucet et al.,2008a; Rycroft et al., 2009; Dubey et al., 2011) granular flowshave been explored through simulations and experiments, theoverall mixing properties of biaxial tumblers are not well under-stood (Mehrotra and Muzzio, 2009). One such biaxial granularflow is produced by the 3D blinking spherical tumbler illustratedin Fig. 1, in which a half-full spherical container rotates

ll rights reserved.

. Lueptow).

l Engineering and Applied

PA 19104, USA.

and Aerospace Engineering,

sequentially about each of two different horizontal axes. Thisblinking flow also presents an opportunity to study a novelmixing mechanism in granular systems: cutting and shuffling.The latter, which gives rise to complex dynamics without the‘‘usual symptoms’’ of chaos, can be observed in certain granularsystems due to the discrete nature of the materials (Juarez et al.,2010). Cutting and shuffling is fundamentally different (Christovet al., 2011) from the well-known stretching and folding mechan-ism of chaotic fluid mixing (see, e.g., Ottino, 1989), yet can stilllead to mixing (Juarez et al., 2010).

G. Juarez et al. / Chemical Engineering Science 73 (2012) 195–207196

At first glance, the spherical mixer may appear too idealizedfor practical applications. This view, however, may not be accu-rate, and a parallel with early work in chaotic mixing of fluids isinstructive. The partitioned-pipe mixer (PPM), published in 1987(Khakhar et al., 1987), was undoubtedly idealized, but this wasfollowed in 1992 with detailed experiments demonstrating theconcepts (Kusch and Ottino, 1992). The system was still not readyfor practical applications. But by 2006 the fundamental conceptsbehind the PPM gave rise to the Rotated Arc Mixer (RAM)(Metcalfe et al., 2006; Metcalfe and Lester, 2009). The RAM isnow protected by patents (Metcalfe and Rudman, 2006, 2010),has won an award for energy efficiency (Ziemlewski, 2010), and isbeing commercialised by an Australian manufacturer under alicensing agreement with the Australian Commonwealth Scien-tific and Industrial Research Organization (CSIRO) (Martin, 2011).The initial target is the food industry, Australia largest manufac-turing industry. The spherical granular mixer that we describehere has the potential to follow a similar trajectory.

In the idealized 3D blinking spherical tumbler flow, cutting ofmaterial occurs when the axis of rotation changes, while shufflingoccurs during the subsequent rotation of the tumbler (see Fig. 2).Mixing by cutting and shuffling has theoretical foundations in arelatively new area of mathematics known as piecewise isome-tries (PWIs) (Goetz, 1998, 2000; Ashwin and Fu, 2002; Deane,2006). One way in which the mixing properties of PWIs arisingfrom the kinematics of granular flow are fundamentally differentfrom the stretching and folding mechanism characteristic ofchaotic fluid mixing is that all Lyapunov exponents of a PWI,when they exist (i.e., away from discontinuities), are equal to zero(Fu and Duan, 2008). A single isometry, i.e., a distance-preservingmapping, is not chaotic in the sense that the distance betweennearby particles stays constant. When different isometries are

Fig. 2. Cutting and shuffling of the two semi-circles that represent the curved edges of th

The two semi-circles are cut into smaller sections and shuffled about the lower hemisp

on the same spherical shell they start on (Meier et al., 2007; Sturman et al., 2008), the

pieced together discontinuously, however, the resulting dynamicsexhibits great complexity (Goetz, 2002; Ashwin and Goetz, 2005;Kahng, 2009).

It has been shown that cutting and shuffling in a 3D blinkingflow represents the skeleton, or underlying framework, for themixing that occurs, at least over a relatively small number ofiterations of the protocol (Juarez et al., 2010). Stretching due tothe shear experienced by the material passing through theflowing layer plays a role only over larger numbers of iterations,as does granular diffusion due to particle collisions, though to alesser degree. Thus, cutting and shuffling (as described by PWIs)can be used to examine mixing in a granular flow over a smallnumber of iterations under a 3D blinking protocol. This, in turn,provides an approach to determine the global dynamics andeffectiveness of mixing in this model 3D tumbler system.

Characterizing mixing experimentally in 3D granular flows inrotating tumblers is quite difficult due to the opaque nature ofthe materials, requiring techniques such as nuclear magneticresonance (Ehrichs et al., 1995; Hill et al., 1997). At the sametime, exploring a large parameter space by direct numericalsimulation using the discrete element method is prohibitivelytime consuming (Theuerkauf et al., 2003). On the other hand,while the complete theoretical description of the collectivebehavior of granular flow remains a topic of active research(Aranson and Tsimring, 2006; Khakhar, 2011), the simple kine-matic PWI model underlying cutting and shuffling can providesignificant understanding. In this paper, we show that goodmixing (without the stretching and folding associated withchaotic advection) can occur for a range of rotation angles. Weshow that the degree of mixing depends not only on thecombination of rotation angles, but also on the angle betweenthe two rotation axes.

e quarter-sphere of particles shown in Fig. 3, under the ðyz ,yxÞ ¼ ð901,451Þ protocol.

here of the tumbler. Though it can be shown that the segments of particles remain

intra-shell mixing by cutting and shuffling is substantial even after 25 iterations.

G. Juarez et al. / Chemical Engineering Science 73 (2012) 195–207 197

2. Mathematical model and simulation method

To simulate mixing by cutting and shuffling in a half-fullspherical tumbler of dimensionless radius R¼0.5 under a biaxialblinking protocol, we use the PWI derived by Sturman et al.(2008). An initial configuration of approximately 16,500 particles,representing a segregated batch of granular materials, is seeded inthe quarter-sphere

Q¼ fðx,y,zÞ9x2þy2þz2oR2,�Rox,yo0,�RozoRg, ð1Þ

as shown in Fig. 3. This is analogous to having monodisperseparticles of one color in one half of the lower hemisphere of thetumbler and transparent particles of the same size and density inthe other half. This seeding pattern was chosen to facilitate thevisualization of mixing and to ensure consistency with the initialconfiguration of the physical system in the experiments of Juarezet al. (2010).

An operational definition of the PWI (Sturman et al., 2008;Juarez et al., 2010) is as follows: as the tumbler is rotated by theangle yz about the z-axis, in each 2D slice perpendicular to theaxis of rotation, particles undergo solid body rotation until theyreach the infinitely thin flowing layer (y¼0), then they arereflected across its midpoint (x¼0 in this case) and continue insolid body rotation thereafter. If the tumbler’s cross-section isnon-circular, this definition can be successfully generalized as themixing mechanism of streamline jumping (Christov et al., 2010a,b).

This idealized model accurately captures the limiting behaviorof granular flow in a rotating tumbler for very thin flowing layers.In a physical system, the flowing layer has a nearly symmetriclenticular shape (Khakhar, 2011) and is, typically, only 5–20particles thick for millimeter-sized particles in laboratory equip-ment (Felix et al., 2007). Particles in solid body rotation at a radialposition close to the edge of the tumbler enter the flowing layernear its upstream end, staying near the surface as they travel theentire length of the flowing layer, and eventually re-enterthe fixed bed at the downstream end of the flowing layer nearthe tumbler wall. Particles near the center of the tumbler enterthe flowing layer at a deeper position, and, because of itslenticular shape, fall out of the flowing layer at a similar radiusto where they enter (after traveling only a short distance in theflowing layer). Thus, idealizing the flow of particles as solid bodyrotation in the bulk and reflection across the midpoint of the freesurface describes the actual flow quite well when the flowinglayer is very thin and the effects of diffusion due to inter-particlecollisions are negligible.

Fig. 3. Initial seeding of red particles in the quarter-sphere Q as seen from a 3D

view (left) of the tumbler and seeding of the outer shell as seen from the bottom of

the tumbler (right). The green sphere is a virtual surface used to visualize the

positions of the outermost red particles and to represent the location of green

colored particles that are not explicitly tracked. (For interpretation of the

references to color in this figure legend, the reader is referred to the web version

of this article.)

After the rotation of the tumbler about the z-axis is complete,the material undergoes the equivalent motion about the ortho-gonal, in this case, x-axis by an angle yx. As described, thisprotocol is identical to the one shown in Fig. 1, except that theflowing layer is assumed to be infinitely thin and coincident withthe free surface, so that particles traverse it in zero time (i.e., theyare instantaneously reflected across its midpoint). Once again, wenote that changing the axis of rotation leads to ‘‘cutting’’. In whatfollows, while seed particles can be visualized as finite sizeparticles for clarity, they cannot actually be identified as suchsince this is a continuum model. The motion of ‘‘particles’’ simplycorresponds to the kinematic model of solid body rotation withreflection across the midpoint of an infinitely thin flowing layer.

To quantify mixing due to cutting and shuffling, the center ofmass (COM) is computed for the configuration of seed particles,whose initial locations are shown in Fig. 3, and its distance fromthe centroid of the lower hemisphere of the tumbler is calculatedfor each iteration of the protocol (Metcalfe et al., 1995; Prasad andKhakhar, 2010). The centroid of the lower hemisphere of radius R

is given by ðx,y,zÞcent¼ ð0,�3R=8;0Þ, while the COM of the initial

configuration of seed particles form Fig. 3 is ðx,y,zÞcom�

ð�3R=8,�3R=8;0Þ.We conjecture that if a protocol leads to mixing, then

ðx,y,zÞcomðnÞ-ðx,y,zÞ

centas n-1. Thus, the distance between

the COM and the centroid after n iterations of the protocol

rðnÞ ¼ Jðx,y,zÞcomðnÞ�ðx,y,zÞ

centJ ð2Þ

provides a global measure of mixing. Normalizing r by the initialdistance between the COM and the centroid, r0 ¼ rð0Þ, gives asegregation index between 0 and 1 for the initial configurationshown in Fig. 3. Large values of the segregation index represent apoorly mixed system, with r=r0 ¼ 1 corresponding to the initialcompletely segregated state, and r=r0 ¼ 0 corresponds to acompletely mixed state. The key point that allows this to be asuccessful measure for mixing is that any particle’s distance fromthe origin is fixed as long as the rotation rates about the z- and x-axes are equal (Meier et al., 2007; Sturman et al., 2008) and thecross-section of the tumbler is circular, meaning streamlinejumping does not occur (Christov et al., 2010a,b). Since, cuttingand shuffling in a 3D blinking spherical tumbler always redis-tributes particles across the same hemispherical shell (i.e., thePWI model does not account for the radial dispersion of particlesthat would occur in a physical granular system), the segregationindex is a promising measure of mixing by cutting and shuffling.

To implement this measure of mixing, the location of eachseed particle is recorded after every iteration so that the COM ofthe redistributed particles in the volume can be calculated to findr=r0. Plotting r=r0 as a function of the iteration provides arepresentation of how mixing proceeds under a given protocol.Meanwhile, a plot of the path that ðx,y,zÞ

comtraces out in space

over many iteration is a way to visualize the approach to awell-mixed state (or lack thereof). As conjectured earlier, whengood mixing is achieved, the COM of the seed particles converges tothe centroid of the lower hemisphere.

3. Mixing by cutting and shuffling

Fig. 4 shows an example of a protocol ðyz,yxÞ ¼ ð22:51,22:51Þ thatmixes poorly. From the 3D view, it is evident that particles areredistributed throughout the volume but they do not fill the entirehemisphere—pockets remain that the seed particles do not reach. Tovisualize the mixing pattern more clearly, a green sphere of radius0.97R is used to display only the seed particles on the outermostshell of the hemisphere. The pattern is identical on any shell becausethe PWI does not permit particles to move from one hemispherical

Fig. 4. Mixing due to the protocol ðyz ,yxÞ ¼ ð22:51,22:51Þ over 200 iterations. The 3D view gives a sense of how particles are redistributed from their initial configuration

after 200 iterations. From the bottom view, regions that are not well mixed even after 200 iterations are evident. Accordingly, r=r0 exhibits large-amplitude oscillations

and does not decay to 0. Visualizing the path traced out by the COM in space shows that it settles on a ‘‘limit cycle’’, orbiting around the centroid (blue dot), but never

converging to it. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 5. Mixing due to the protocol ðyz ,yxÞ ¼ ð451,22:51Þ over 200 iterations. From the 3D and bottom views, it appears that particles have been evenly redistributed

throughout the hemisphere, implying good mixing. Accordingly, the plot of the segregation index quickly approaches zero with only small amplitude fluctuations.

Visualizing the path of the COM over many iterations shows that it converges to the centroid (blue dot). (For interpretation of the references to color in this figure legend, the

reader is referred to the web version of this article.)

G. Juarez et al. / Chemical Engineering Science 73 (2012) 195–207198

shell to another. The particle-free ‘‘islands’’ of no mixing are clearlyevident. These persist even after 200 iterations of this protocol.

Tracking the r=r0 as a function of the iteration indicates poormixing under this protocol. In this case, the amplitude offluctuations in r=r0 is large and it does not decay significantly

over 200 iterations. Moreover, the average value for r=r0 does notapproach 0. The reason for this is evident in the path that the COM

of the seed particles traces out in 3D space. The COM is locked in anorbit (a ‘‘limit cycle’’) around the centroid of the hemisphere,indicating that the majority of the seed particles flip back and

G. Juarez et al. / Chemical Engineering Science 73 (2012) 195–207 199

forth from one side of the hemisphere to the other, neverbecoming evenly distributed throughout the volume.

Fig. 5, on the other hand, shows an example of well-developedmixing that occurs for the combination of rotation anglesðyz,yxÞ ¼ ð451,22:51Þ, again for orthogonal axes. The 3D viewindicates a uniform distribution of particles throughout thehemispherical volume. The view of the outermost shell confirmsthis largely homogeneous distribution of particles, showing noapparent regions of poor mixing. The time record of r=r0

supports the observation of good mixing as it falls under 0.2 invalue in 50 iterations; the amplitude of the fluctuations is small,unlike the protocol shown in Fig. 4. The path that the COM tracesout in 3D space over many iterations shows that the COM of theseed particles converges to the centroid of the filled portion of thetumbler, implying a uniform distribution of red particles throughthe entire volume of the hemisphere.

Fig. 6. Comparison over 200 iterations of the segregation index, r=r0, its running

average over 9 iterations, and the intensity of segregation, I , for (a) the protocol

ðyz ,yxÞ ¼ ð22:51,22:51Þ and (b) the protocol ðyz ,yxÞ ¼ ð451,22:51Þ.

4. Comparison of two mixing measures

Broadly speaking, there are two classes of mixing measuresand diagnostics: Eulerian and Lagrangian. Some of these (Doucetet al., 2008b) are motivated by mathematical definitions ofmixing from the ergodic theory of dynamical systems (see, e.g.,Sturman et al., 2006, Section 3), while others (Sturman andWiggins, 2009) arise from the study of linked twist maps (see,e.g., Sturman et al., 2006, Section 2). The segregation indexdescribed above is a computationally efficient Lagrangianmeasure. However, any new mixing indicator should be com-pared to established ones. The most common measure of thequality of mixing is an Eulerian measure based on the variance ofconcentration, i.e., Danckwerts’s intensity of segregation I(Danckwerts, 1953). In this method, the local concentration isestimated by dividing the hemispherical volume into bins andcounting the number of particle of each color inside of each bin.Then, the concentration c of red particles in the ith bin is

ci ¼nr,i

nr,iþng,i, ð3Þ

where nr,i and ng,i are the number of red and the number of greenparticles, respectively, in this bin. (Note that, in addition to thepositions of the red particles, the positions of the green particlesinitially filling the quarter-sphere opposite the red particles inFig. 3 must be known to compute the local concentration.)

A local sample of a well-mixed material should have an equalnumber of red and green particles, and therefore, a concentrationclose to c ¼ 1=2. Hence, the variance of the concentration over allbins within the volume, i.e., the measure of how close to the meanthe concentration is on average, rescaled by the initial valueð1�cÞc ¼ 1=4, results in a quantity that can be compared to thesegregation index r=r0. This is the so-called intensity of segrega-tion given by

I ðnÞ ¼ 41

N

XN

i ¼ 1

½ciðnÞ�0:5�2( )

, ð4Þ

where N is the total number of bins. Similar to r=r0, I ¼ 0 for aperfect mixture when the local concentration is equal to the meaneverywhere, while I ¼ 1 for a completely segregated mixture;values between 0 and 1 correspond to cases in which there arepatches of material where the concentration has not been homo-genized to the mean. In general, this calculation requires moreinformation than the calculation of the segregation index and,hence, is more computationally intensive.

Starting with the same initial condition of a segregated mixtureshown in Fig. 3, the location of both red and green particles aftereach iteration for the cutting and shuffling protocols with

ðyz,yxÞ ¼ ð22:51,22:51Þ and ðyz,yxÞ ¼ ð451,22:51Þ was computed byadvecting 64,000 particles using the PWI mapping. The volume ofthe lower hemisphere was then divided into ‘‘wedge-shaped’’ binsand the intensity of segregation was calculated after every iteration,up to 200 iterations. Specifically, 125 tracers were uniformlydistributed in each wedge-shaped bin. An equal number of binswas used in each of the coordinate (i.e., radial, azimuthal andlatitudinal) directions with N¼512. The bins were constructed in amodified spherical coordinate system to ensure that they have equalvolumes; details are provided elsewhere (Christov, 2011, AppendixB). Since we study purely advective transport, the system is time-reversible and backwards tracking of particles was used to ensure auniform distribution of tracers within the hemisphere for thecalculation of I .

Results are shown in Fig. 6. The segregation index as well as itsrunning average over nine (i.e., the current, four before and fourafter) iterations are plotted for comparison. (A running average isnecessary to remove the large-amplitude fluctuations for certainprotocols, such as in Fig. 6(a).) The intensity of segregation for theprotocol ðyz,yxÞ ¼ ð22:51,22:51Þ does not exhibit the fluctuationsthat are present in the segregation index, but there is very goodagreement between the intensity of segregation and the runningaverage of the segregation index, as shown in Fig. 6(a). Similarly,in Fig. 6(b), the intensity of segregation agrees well with both thesegregation index and its running average for the protocol

G. Juarez et al. / Chemical Engineering Science 73 (2012) 195–207200

ðyz,yxÞ ¼ ð451,22:51Þ, since the fluctuations in r=r0 are small inthis case.

Thus, the segregation index r=r0 is a reliable measure ofglobal mixing under PWI mappings in tumblers. The aboveexamples suggest that protocols that do not lead to a well-mixedstate display large amplitude fluctuations r=r0, while protocolsthat do lead to a well-mixed state display only small amplitudefluctuations. Therefore, in what follows, we use a moving averageof the segregation index instead of its raw value.

Fig. 7. Color density plots of the segregation index r=r0 for non-orthogonal axes

1 and 2 and orthogonal axes z and x for different values of f¼ 151, 301, 451, 601,

751 and 901. The left column shows the segregation index after 10 iterations, while

the right column shows the segregation index after 25 iterations. The color scale is

normalized so that black is 0 and white is 1 with darker colors indicating better

mixing. (For interpretation of the references to color in this figure legend, the

reader is referred to the web version of this article.)

5. Exploring the parameter space of mixing protocols

The approach to quantifying the quality of mixing described inSection 3 can be applied to any combination of rotation anglesabout two given horizontal axes. In this section, we studyprotocols with rotation angles limited to 11ryir891, in incre-ments of 11, and applied for up to 40 iterations. It is sufficient toconsider angles only up to 901 because the PWI is periodic in eachrotation angle, meaning rotation by yi ¼ 901þW identical torotation by yi ¼ 901�W (Sturman et al., 2008). It is not necessaryto consider a large number of iterations because, in a physicalsystem, stretching in the flowing layer and granular diffusioneventually overtake cutting and shuffling as the dominant mixingmechanisms. We also consider a half-full spherical tumblerrotating about horizontal non-orthogonal axes now labelled ‘‘1’’and ‘‘2’’. The angle between the axes of rotation is represented byf, with f¼ 151, 301, 451, 601, and 751, as well as f¼ 901 fororthogonal axes. The results are presented in the form of colordensity plots of the running average of the segregation index r=r0

(over nine iterations, as in the previous section).Fig. 7 shows such color density plots of the running average of

r=r0 as a function of y1 and y2 (or yz and yx for orthogonal axes,i.e., f¼ 901) after 10 (left panels) and 25 (right panels) iterations.It is evident that the degree of mixing is dependent on thecombination of rotation angles, ðy1,y2Þ, and the angle betweenthe axes, f. Protocols that mix well or mix poorly result fromspecific combinations of angles, and these are easily discerned inthe color density plots: good mixing occurs for parameterscorresponding to the dark regions, while poor mixing occurs forparameters corresponding to the light regions.

Consider first the case of orthogonal axes, f¼ 901, shown in thebottom row of Fig. 7. The segregation index after 10 iterations (to beprecise, its average from the 6th to the 14th iteration) is shown onthe left of Fig. 7. Protocols with angle combinations of 301tyit851provide the best mixing. Poorly mixed systems lie along the diagonaland at the outer edges of the plot. The poor mixing at yz or yx � 901is due to the fact that for such angles of rotation, the material simply‘‘flips’’ from one side of the hemisphere to the other about the givenaxis of rotation. In the case of yz � 901 (rotation about the z-axis inFig. 3), the initial condition is symmetric about rotation axis z (seeFig. 3), so rotation about the z-axis is an identity transformation ofthe domain. For the case of yx � 901, the overall mixing in Fig. 7(bottom row) is improved as compared to yz � 901 because theinitial condition is now anti-symmetric about this axis. However,there is not enough shuffling overall in the tumbler for such anglesof rotation to result in good mixing.

To better illustrate this ‘‘lack of shuffling’’, Fig. 8 shows theevolution of the mixing pattern for two pairs of rotation anglesslightly less than 901. For yz ¼ 851 in Fig. 8(a), the first rotationjust flips the position of the red and green particles, and thesecond rotation subsequently has little effect. For yx ¼ 851 inFig. 8(b), the first rotation brings a wedge of green particles overthe red particles, which ends up within the red particles upon thesecond rotation. In general, for both orthogonal and non-ortho-gonal axes, the protocols with ðy1,y2Þ and ðy2,y1Þ are not

Fig. 8. Evolution of the theoretical mixing pattern, as seen from below, for the protocols (a) ðyz ,yxÞ ¼ ð851,51Þ and (b) ðyz ,yxÞ ¼ ð51,851Þ, showing why poor mixing is

observed for these angle combinations.

Fig. 9. PWI mixing pattern viewed from below the tumbler after 10 iterations

(starting from the initial condition in Fig. 3), corresponding to example (a) good

(yz,yx)¼(641,421) and (b) poor (yz,yx)¼(731,181) mixing protocols as identified

from the segregation index plot in the left panel of the bottom row of Fig. 7.

G. Juarez et al. / Chemical Engineering Science 73 (2012) 195–207 201

equivalent. A mathematical proof of this non-commutativity isoutlined in Appendix A. This explains why all of the plots in Fig. 7are asymmetric about the line y1 ¼ y2 (or yz ¼ yx when f¼ 901),especially as f approaches 901.

Poor mixing along the diagonal of the plots in Fig. 7, y1 ¼ y2 (oryz ¼ yx for orthogonal cases), results from similar rotation anglesproducing a different type of inherent symmetry, especially for fclose to 901. This leads to the formation of unmixed regions andthe trajectory of the COM being trapped in an orbit around thecentroid as shown in Fig. 4.

It is helpful to compare the actual mixing pattern produced bythe PWI for protocol parameters chosen in (a) a region of goodmixing and (b) a region of poor mixing, as shown in Fig. 9 for 10iterations of the orthogonal-axes (i.e., f¼ 901Þ protocols with(a) ðyz,yxÞ ¼ ð641,421Þ and (b) ðyz,yxÞ ¼ ð731,181Þ. While these twopoints in the parameter space are relatively close when plotted onthe mixing map in Fig. 7 (bottom row, marked with � symbol),the degree of mixing under each of the two protocols is substan-tially different. For ðyz,yxÞ ¼ ð641,421Þ, sections of both colors ofmaterial are cut and shuffled effectively throughout the hemi-sphere. On the other hand, for ðyz,yxÞ ¼ ð731,181Þ, the dark mate-rial remains mostly on the left side of the hemisphere, where itwas initially placed, while the light material is mostly on theopposite side. After 10 iterations, only small patches of material ofthe opposite color make it to the side opposite the one thematerial was initially placed on. This comes about for the samereason that mixing is poor in Fig. 8(a).

Returning to the plots in Fig. 7, the segregation index after25 iterations (to precise, its average from the 21st to the

29th iteration) is shown in the right panels. Again consideringf¼ 901, poorly mixed regions remain along the diagonal andnear the edges of the contour plot, for the same reasons notedearlier. Away from these extremes, the regions of good mixing(dark) have grown in size, lying above and below the y1 ¼ y2

(or yz ¼ yx for f¼ 901) diagonal.The case of two horizontal orthogonal axes (the mixing maps

for which are shown in the bottom row of Fig. 7) is the simplestsystem to study since it is easy to implement experimentally.However, it is also worth determining whether by deviating fromorthogonal axes protocols it is possible to achieve better(or worse) mixing (the mixing maps for which are shown in theremaining rows of Fig. 7). For example, the mixing profile fornon-orthogonal axes with f¼ 151, shown in the top row of Fig. 7,is quite different from that corresponding to orthogonal axes.Here, few protocols mix well, with the exception of a very smallregion (less than 5% of protocols) that mixes better along thediagonal y1 ¼ 901�y2. In this case, the small value of f results inalmost no difference from the case of a tumbler rotating about asingle axis. Indeed, by considering f¼ 301 (second row of Fig. 7)we find that more protocols mix to a higher degree. However, it isnot until f\451 that there is significant mixing for a wide rangeof rotation angles ðy1,y2Þ. It appears that mixing for f¼ 601 andf¼ 751 is better than mixing for f¼ 901. This can be interpretedas a result of breaking inherent symmetries in the orthogonal-axes protocols.

By changing the angle between the axes of rotation, the initialcondition shown in Fig. 3 is no longer exactly anti-symmetricabout axis 1 and exactly symmetric about axis 2. This, in turn,prevents the degenerate situation illustrated in Fig. 8(a), whereinthe rotation about the z-axis with yz � 901 flips the two sides ofthe hemisphere so that the mixing state at the beginning ofrotation about x-axis is almost symmetric about it, meaning thesecond rotation leaves the mixing pattern almost unchanged.Clearly, non-orthogonal axes make such a situation impossible. Ofcourse, the angles still cannot commute (as discussed in AppendixA), so the plots in Fig. 7 (even for fa901Þ cannot be expected tobe symmetric about y1 ¼ y2.

A summary of the percentage of protocols that mix to aparticular level after a given number of iterations is shown inFig. 10 for different values of f. Non-orthogonal-axes protocolswith f¼ 151 (top left panel) result in poor overall mixing for themajority of choices of rotation angles (recall also the top row ofFig. 7). For f¼ 301, there is a clear dependence of the degree ofmixing on the number of iterations. For few iterations, the number

Fig. 10. Percentage of protocols that mix to a particular value of the segregation index r=r0 after different numbers of iterations (5, 10, 25 and 40), as the angle f between

the two axes of rotation is varied from 151 to 901 in 151 increments.

Fig. 11. Bottom view of a spherical tumbler rotated through the orthogonal-axis

protocol ðyz ,yxÞ ¼ ð451,451Þ. The effects of the finite-thickness flowing layer and

diffusion in the experiment become evident quickly. Nevertheless, mixing by

cutting and shuffling (i.e., under the PWI resulting from letting d0=R-0) is

observed in the initial stage of the process.

G. Juarez et al. / Chemical Engineering Science 73 (2012) 195–207202

of protocols that mix well is small, but at higher numbers ofiterations many protocols result in substantial mixing. For f¼ 451,the overall degree of mixing is further improved, particularly for alarge number of iterations. Overall, it appears that the quality ofmixing is best for f¼ 601 and 751. As noted earlier, protocols withf¼ 601 and 751 result in better mixing that orthogonal-axes

(f¼ 901Þ protocols because of symmetry-breaking for the choseninitial condition in Fig. 3. It would appear that modifying othersystem parameters such as the fill level or tumbler geometry shouldalso break such symmetries leading to better mixing.

6. Effects due to a finite-thickness flowing layer

Of course, the PWI model discussed above does not account forthe effects due to a finite-thickness flowing layer (Juarez et al.,2010). Through a comparison of the PWI dynamics with anexperiment and simulations with a finite-thickness flowing layer,Fig. 11 shows how the stretching and folding that is created bythe flowing layer eventually dominates over cutting and shuffling.Details of such a comparison are presented elsewhere (Juarezet al., 2010); here, it suffices to say that the experiments are basedon starting with a segregated mixture half of which consist ofblack particles 1 mm in diameter, while the other half consists ofwhite particles 1 mm particles in diameter. The spherical tumblerhas a diameter of 14 cm. The simulations of the positions of theblack particles with each iteration for a finite-thickness flowinglayer are performed by numerically integrating the kinematicequations of motion (Meier et al., 2007). The flowing layer’sthickness is chosen to be comparable to that observed in experi-ments, i.e., d0 ¼ 7d, where d0 is the maximal flowing layerthickness and d is the particle diameter. This corresponds tod0=R¼ 0:1 in the experimental system. A very thin flowing layerwith d0 ¼ d (corresponding to d0=R� 0:014Þ is also considered,and the PWI corresponds to the limiting case of case d0=R-0 forwhich only cutting and shuffling (no stretching and folding)occurs.

It is quite clear that the experiments and simulations withd0=R¼ 0:1 match each other quite well, confirming the accuracyof the simulations. Further, note that the details of the mixingpattern in the two simulations with a finite-thickness flowing

Fig. 12. A simulation of the initial clockwise rotation in a half-filled spherical tumbler for d0=R¼ 0:1 demonstrates the origin of stripes and the deviation of the angle of the

interface between the two colors of particles under the model with a finite-thickness flowing layer and the PWI model. Slices perpendicular to the axis of rotation show

how regions (a) A and B fill the space that becomes the fixed bed after rotation by (b) 301 and (c) 901. The layer C is the origin of stripes evident after completing the out-of-

plane rotation for this iteration of the protocol as in, e.g., the middle row of Fig. 11. (For interpretation of the references to color in this figure legend, the reader is referred

to the web version of this article.)

Fig. 13. Comparison of the evolution of the intensity of segregation, I , of the PWI

model (vanishingly thin flowing layer, d0=R¼ 0) and the model with a finite-

thickness flowing layer (d0=R¼ 0:1) for (a) the protocol ðyz ,yxÞ ¼ ð22:51,22:51Þ and

(b) the protocol ðyz ,yxÞ ¼ ð451,22:51Þ. The axes of rotation are orthogonal (f¼ 901).

G. Juarez et al. / Chemical Engineering Science 73 (2012) 195–207 203

layer match each other, except that the stripes and bands arethinner for d0=R� 0:014 than for d0=R¼ 0:1, which indicates thatthese details are a result of the finite-thickness flowing layer.Finally, note that these details disappear altogether when theflowing layer is infinitely thin in the case of pure cutting andshuffling (PWI), hence the conclusion that cutting and shuffling(the underlying framework of mixing in a blinking sphericaltumbler (Juarez et al., 2010)) is distinct from stretching andfolding, which creates these stripes and bands. For larger numbersof iterations, mixing is further enhanced in the experiments bythe effects of granular diffusion due to inter-particle collisions inthe flowing layer. Since the flowing layer thickness scales with theparticles’ size and the length of the free surface, sharper experi-mental images should be possible using smaller particles or alarger tumbler.

Fig. 12 shows, for a simulation with d0=R¼ 0:1, a slice througha half-filled spherical tumbler undergoing its initial rotationabout the z-axis. This serves to explain how the stripes of darkand light particles in the first three columns of Fig. 11 appear dueto the presence of a finite-thickness flowing layer. At the start ofthe first rotation, the downstream green half of the finite-thickness flowing layer, denoted as ‘‘B’’, flows down the incline tofill a wedge-shaped volume with its base at the wall of the tumbler,evident when the tumbler is rotated by 301 in Fig. 12(b). Theupstream red portion of the initial flowing layer, denoted by ‘‘A’’,similarly fills in the space above it. (The white space betweenparticles is an artifact of the initial particle seeding in the simula-tion; particles actually fill the entire space.) As a consequence of thewedge-shaped space that is filled in by the green particles (B), thearc of green particles touching the tumbler wall after the initialrotation of the tumbler is greater than the initial arc of 901. Thiscomes about because the wedge of green particles does not extendto the center of the tumbler in Fig. 12(b), since the red particles fillthe flowing layer. This increased green-particle arc length at thetumbler wall remains ‘‘frozen’’ in the fixed bed until the tumbler isrotated by a total angle of 901 (Fig. 12(c)), so the red–green interfaceis not vertical when the rotation ends. Upon the next rotation in theprotocol, which would be a rotation about a horizontal axis in theplane of the page, the deviation of the interface from the verticalpersists. In Fig. 11, this results in the angle of the diagonal black-grayinterfaces at n¼1 for the simulation with d0 ¼ 7d (i.e., d0=R¼ 0:1) aswell as the experiment being slightly different from those for thesimulation with d0 ¼ d (i.e., d0=R� 0:014Þ and the PWI simulation.Similar results are evident in Figs. 3 and 4 of Juarez et al. (2010).

At the end of the rotation by 901 (Fig. 12(c)), a layer of redparticles, denoted by ‘‘C’’, remains at the surface of the upstreamportion of the flowing layer as a result of the way in which redparticles near the tumbler wall enter the flowing layer. That is,particles enter the flowing layer from solid body rotation whenthey reach the interface between the flowing layer and the fixedbed. This occurs last for the particles in the fixed bed that are

adjacent to the wall of the tumbler due to the small thickness ofthe most upstream portion of the flowing layer. The layer of redparticles remaining at the top of the left side of the flowing layer(C) forms the horizontal stripe during the next rotation by, e.g.,901 about a horizontal axis in the plane of the page. Thiscorresponds to the black stripe visible at n¼2 in Fig. 11 for thesimulations with d0=R¼ 0:1 and d0=R� 0:014 and the experiment.Subsequent iterations of the biaxial protocol result in similarstripes (for n¼3 and for n¼4) in Fig. 11, each also includingremnants of the previous stripes.

Fig. 14. Comparison of the color density plots of (a) the segregation index r=r0 of the PWI model (vanishingly thin flowing layer) and (b) the intensity of segregation I of

mixing for a finite-thickness flowing layer (d0=R¼ 0:1Þ as the angles of rotation ðyz ,yxÞ are varied (f¼ 901). White arrows indicate consistent features. Note the color scale

difference between (a) and (b). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

G. Juarez et al. / Chemical Engineering Science 73 (2012) 195–207204

As the thickness of the flowing layer decreases, the thicknessof the stripes and the deviation (between the case of the vanish-ingly thin and finite-thickness flowing layer) of the angle of theinterface between the particles of different colors decreases. Thisis evident when comparing the simulations with d0=R� 0:014 andd0=R¼ 0:1 in Fig. 11. In the limit of a vanishingly thin flowinglayer, there are no stripes, as seen in the fourth column of Fig. 11for the PWI mapping. Again, similar effects of the flowing layerare evident in Figs. 3 and 4 of Juarez et al. (2010).

Another way to quantify and understand the effects due to afinite-thickness flowing layer is to compare and contrast themixing produced by the PWI (d0=R¼ 0Þ and the model withd0=R¼ 0:1 for orthogonal rotation axes (f¼ 901Þ. This is shownin Fig. 13(a) and (b) for the same rotation angle combinations asin Figs. 4 and 5, respectively. For both protocols, the intensity ofsegregation decays faster in the case of a finite-thickness flowinglayer, which indicates that the additional effect of chaotic mixingvia stretching and folding is present. However, for the first � 10iterations, both curves track each other closely, which shows thatcutting and shuffling is present. Fig. 14 presents the completemixing maps for these two cases, i.e., the PWI (d0=R¼ 0Þ and themodel with d0=R¼ 0:1, after 10 (left panels) and 25 (right panels)iterations of the mixing protocols. Though, overall, mixing ismuch improved for the case of a finite-thickness flowing layerdue to the stretching and folding chaotic dynamics superposedonto cutting and shuffling, there are some common featuresbetween Fig. 14(a) and (b). Specifically, over 10 iterations,the choices of ðyz,yxÞ that lead to (some of) the best mixing bycutting and shuffling in (a) remain such even after the effects of a

finite-thickness flowing layer are accounted for in (b). Then, over25 iterations, the central region of angles that leads to poormixing for the PWI (d0=R¼ 0Þ in (a) remains a region of poormixing even when d0=R¼ 0:1 in (b). Thus, we can conclude thatmaximizing mixing by cutting and shuffling in the short term isbeneficial, as this leads to good mixing in the more realistic caseof a finite-thickness flowing layer. On the other hand, over largernumber of protocol iterations, if the underlying cutting andshuffling mechanism is not mixing well, the enhancement dueto stretching and folding may be diminished as well.

Here, it is important to note that the white region near yz � 11in Fig. 14(b) is an artifact of the differing maximum value in thecolor scale between Fig. 14(a) and (b). To better enhance theregions pointed out by the white arrows in (b), it is necessary tomake the maximum color value � 0:4, this roughly the samevalue observed in (a) in the region under question. Therefore,mixing has not worsened (for these choices of rotation angles)upon adding a flowing layer of maximal depth d0 ¼ 7d.

7. Conclusions

In the absence of stretching (in the sense of shear strain),segregation and diffusion, mixing in spherical tumblers undergoingrotation about two different horizontal axes is due to cutting andshuffling. This mechanism, which can be framed in terms of a classof discontinuous maps called piecewise isometries (PWIs), can leadto highly developed mixing for properly chosen combinationsof rotation angles. In particular, efficient mixing is achieved

G. Juarez et al. / Chemical Engineering Science 73 (2012) 195–207 205

under protocols with rotation angles ðy1,y2Þ each between 301 and851, although mixing is not as good for equal rotation angles. Inaddition, significant mixing results when the rotation axes are at anangle of 60–901 with respect to each other, with the angles of 601and 751 between the axes being slightly better than 901. Finally, afinite-thickness flowing layer improves mixing substantiallybecause it allows for chaotic mixing via stretching and folding(Hill et al., 1999).

The result that good mixing can occur solely due to cutting andshuffling has important implications for physical granular sys-tems, which also have chaotic advection due to stretching in thefinite-thickness flowing layer and diffusion due to collisionsbetween particles. Our study suggests that mixing at short timesis dominated by cutting and shuffling, while at longer times it issubstantially enhanced by advection and diffusion. It may even bepossible that a granular system in which mixing by cutting andshuffling can occur (e.g., a spherical tumbler subject to blinkingflows about two different axes), could mix better than a systemwith chaotic advection and diffusion only. Optimizing biaxialtumbler mixers for cutting and shuffling is desirable.

Acknowledgments

We thank Stephen Wiggins and Rob Sturman for providinghelpful comments and suggestions. I.C.C., J.M.O. and R.M.L. weresupported, in part, by NSF Grant CMMI-1000469.

Appendix A. Non-commutativity of the angles of rotation inthe blinking spherical tumbler

To show that the orthogonal-axes protocol with angles ofrotation ðy1,y2Þ is not equivalent to the protocol with angles ofrotation ðy2,y1Þ, it suffices to show that the locations of any oneparticle after one iteration of each protocol differ. The easiest wayto do so is to consider the case or orthogonal rotation axes,f¼ 901, and also assume that the particle never reaches theinfinitely thin flowing layer interface at y¼0.

First, we set up the coordinate system to be used in thisanalysis. Fig. A1 shows the relation between the coordinatesneeded for the rotation about each axis and the Cartesiancoordinates common to both. For rotation yz about the z-axis,we use the cylindrical coordinates ðrz,Wz,zÞ, where x¼ rz cos Wz

and y¼�rz sin Wz. Meanwhile, for rotation yx about the x-axis, weuse the cylindrical coordinates ðx,rx,WxÞ, where z¼ rx cos Wx andy¼�rx sin Wx.

Fig. A1. Cylindrical coordinates, in the z and x cross-sections of the spherical

tumbler, as needed for rotation about the (a) z- and (b) x-axes. The opposite sense

of rotation (in relation to a right-handed coordinate system) is taken for

convenience and for consistency with Sturman et al. (2008).

Let us denote by ðx0,y0,z0Þ the particle’s starting location. It canbe re-written in the coordinates from Fig. A1(a) as

ðrz,0,Wz,0,z0Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

0þy20

q,tan�1 �

y0

x0

� �,z0

� �: ðA:1Þ

First, we perform the rotation about the z-axis, which takes theparticle to the new location ðrz,0,Wz,0þyz,z0Þ. Converting the latterback to Cartesian coordinates, we obtain

ðx1,y1,z1Þ ¼ ðrz,0 cosðWz,0þyzÞ,�rz,0 sinðWz,0þyzÞ,z0Þ: ðA:2Þ

Now, we must switch to the coordinates in Fig. A1(b)

ðx1,rx,1,Wx,1Þ ¼ x1,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiy2

1þz21

q,tan�1 �

y1

z1

� �� �: ðA:3Þ

Then, rotation about the x-axis takes the particle to the newlocation ðx1,rx,1,Wx,1þyxÞ. Returning to the Cartesian coordinatesone last time, we obtain the final position of the particle after oneiteration of the blinking protocol

ðx2,y2,z2Þ ¼ ðx1,�rx,1 sinðWx,1þyxÞ,rx,1 cosðWx,1þyxÞÞ: ðA:4Þ

Note that the restriction (of convenience) that the particle neverreaches y¼0 under either protocol is, thus, equivalent to thefour simultaneous conditions: (i) Wz,0þyzo901, (ii) Wz,0þyxo901,(iii) Wx,1þyzo901, and (iv) Wx,1þyxo901.

Next, we substitute the results from (A.1), (A.2) and (A.3) into(A.4) to obtain

x2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

0þy20

qcos tan�1 �

y0

x0

� �þyz

� �, ðA:5aÞ

y2 ¼�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx2

0þy20Þ sin2 tan�1 �

y0

x0

� �þyz

� �þz2

0

s

�sin tan�1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

0þy20

qsin tan�1 �

y0

x0

� �þyz

� �� �z0

� �þyx

�,

ðA:5bÞ

z2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx2

0þy20Þ sin2 tan�1 �

y0

x0

� �þyz

� �þz2

0

s

�cos tan�1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

0þy20

qsin tan�1 �

y0

x0

� �þyz

� �� �z0

� �þyx

�:

ðA:5cÞ

At this point we need the following easily derived trigono-metric identities:

cosðtan�1ðaÞþbÞ ¼ cos b�a sin bffiffiffiffiffiffiffiffiffiffiffiffiffi1þa2p , ðA:6aÞ

sinðtan�1ðaÞþbÞ ¼ a cos bþsin bffiffiffiffiffiffiffiffiffiffiffiffiffi1þa2p : ðA:6bÞ

One application of (A.6) to (A.5) gives

x2 ¼ x0 cos yzþy0 sin yz, ðA:7aÞ

y2 ¼�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx0 sin yz�y0 cos yzÞ

2þz2

0

q�sinðtan�1ððx0 sin yz�y0 cos yzÞ=z0ÞþyxÞ, ðA:7bÞ

z2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx0 sin yz�y0 cos yzÞ

2þz2

0

q�cosðtan�1ððx0 sin yz�y0 cos yzÞ=z0ÞþyxÞ: ðA:7cÞ

A second application of the identities in (A.6) to (A.7) gives

x2 ¼ x0 cos yzþy0 sin yz, ðA:8aÞ

y2 ¼�ðx0 sin yz�y0 cos yzÞ cos yx�z0 sin yx, ðA:8bÞ

z2 ¼ z0 cos yx�ðx0 sin yz�y0 cos yzÞ sin yx: ðA:8cÞ

G. Juarez et al. / Chemical Engineering Science 73 (2012) 195–207206

If we consider now the symmetric protocol ðyz,yxÞ/ðyx,yzÞ, wewould have instead

~x2 ¼ x0 cos yxþy0 sin yx, ðA:9aÞ

~y2 ¼�ðx0 sin yx�y0 cos yxÞ cos yz�z0 sin yz, ðA:9bÞ

~z2 ¼ z0 cos yz�ðx0 sin yx�y0 cos yxÞ sin yz: ðA:9cÞ

Clearly, the expressions in (A.8) and (A.9) are not identical ingeneral, therefore, the angles of rotation in the protocol do notcommute. Setting the expressions in (A.8) and (A.9) equal to eachother gives a set of three simultaneous algebraic equations forðx0,y0,z0Þ as a function of yz and yx, which identifies a special setof initial particle positions that reach the same location underboth the ðyz,yxÞ and the ðyx,yzÞ protocol. It remains to bedetermined if these play any special role in the development ofmixing in such blinking flows.

To extend this argument to the case in which particles mayreach the infinitely thin flowing layer requires that the angles betaken modulo 901 when they are incremented by yz and yx. Thisleads to several cases that need to be considered separately.However, for the present purposes the above argument is suffi-cient to disprove the claim that the blinking flow remainsunchanged under the rotation angle transformation ðyz,yxÞ/

ðyx,yzÞ. A similar analysis for non-orthogonal axes, while requiringa more complicated transformation when switching from rotationabout one axis to rotation about the other axis, should yield thesame result of non-commutativity.

This argument shows that the rotation angles about each oftwo given (fixed) axes cannot be interchanged (do not commute)in general. An analogous argument shows that the two rotationaxes with given (fixed) rotation angles about each also cannot beinterchanged (do not commute) in general.

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