Analyzing Fluid Mixing in Periodic Flows byDistribution Matrices:The Mapping Method.

Peter G. M. Kruijt, Oleksiy S. Galaktionov1,Patrick D. Anderson,Gerrit W. M. Peters, and Han E. H. Meijer2

Materials Technology, Dutch Polymer Institute, Eindhoven University of Technology,P.O. Box 513, 5600 MB Eindhoven, The Netherlands

1visiting scientist from Institute of Hydromechanics, Kiev, UkraineFebruary 11, 2000

Topical Heading: 1. Fluid Mechanics and Transport Phenomena or 4. Process Systems Engineering.Keywords: Mixing, Process Optimization, Mapping Method

A method is presented to analyze the long term efficiency of distributive mixing inStokes flows. This mapping method, which is based on the determination of a distri-bution matrix, offers, for the first time, an efficient way to investigate the influenceon overall mixing quality of different mixing protocols in time-periodic prototypeflows or —finally— of different geometrical configurations and operating conditionsin real, generally spatial periodic, mixing devices as e.g. static mixers or twin screwextruders. The method is based upon and, moreover, evaluated with, the results of anaccurate front tracking method. The prototype flow investigated here is a two dimen-sional, lid driven, time periodic Stokes cavity flow.

IntroductionComputational tools available for the analysis ofmixing behavior usually give limited results or,when more detailed, are memory and time con-suming. They are therefore in general not flexibleenough to allow for optimization of mixing pro-cesses and are, moreover, mostly implicit.Examples are residence time distribution and straindistribution (Tadmor & Gogos, 1979; Hobbs &Muzzio, 1997), and local measures of the exten-sional or rotational character of the flow (Chella &Ottino, 1985; Bigio & Conner, 1995; Yao & Manas-Zloczower, 1996). Chaos theory provides tools thatelucidate the fundamentals of efficient mixing, butthese tools do not yield a direct description of the

2Corresponding author

final mixture. Examples include Poincare sections(Ottino, 1989; Ling, 1994; Jana et al., 1994b), man-ifold analysis (see e.g. Ottino (1989)), and peri-odic point analysis sometimes exploring symmetryconcepts Franjione et al. (1989); Franjione & Ot-tino (1992); Meleshko & Peters (1996); Andersonet al. (1999), interface tracking techniques (Noh &Woodward, 1976; Hirt & Nichols, 1981; Avalosse& Crochet, 1997a,b; Rudman, 1997; Galaktionovet al., 2000), and stretching distributions Muzzioet al. (1991); Liu et al. (1994). Problems of optim-ization of mixing processes can only be solved byexplicit models (Souvaliotis et al., 1995), that con-tain variables that represent the instantaneous stateof mixing, together with equations that describe howthese variables evolve during flow. Such models are

1

necessarily global.

At the macro scale, the smallest object of interest is acell and the largest is the entire mixture. The appro-priate variables are the cell-average concentrationsCi . Useful measures of distributive mixing can becalculated from the Ci ’s and their spatial arrange-ment, see e.g. Tucker & Peters (2000 submitted),which can be used in optimization and scale-up.

Mixing flows in general are repetitive, i.e. mixingis established by repeating a certain action. Theseflows are therefore periodic. A periodic flow gener-ally has some kth order periodic points, points thatreturn to their initial location after exactly k peri-ods. Three different types of periodic points exist:elliptic, hyperbolic and parabolic. Elliptic pointsform islands and no exchange of fluid is realizedfrom a region around an elliptic point, separated by aKAM (Kolmogorov–Arnold–Moser) boundary (Ot-tino, 1989; Ottino et al., 1992), to the rest of theflow-domain. Hyperbolic points are points aroundwhich fluid is stretched unidirectionally and is ingeneral spread over the entire flow-domain. In para-bolic points only simple shear occurs.

The Poincare method is an asymptotic method thatis generally used to visualize islands (unmixed re-gions) around elliptic points in a chaotic flow andgives a useful indication of their size and location.Manifold analysis reveals the stable and unstable re-gions in a chaotic flow, which can be used to de-termine regions in the flow where high stretchingoccurs or where islands exist. Tracking reveals thedistribution and local stretching of a predeterminedboundary after a number of periods. After tracking,the amount of fluid that stretches is quantified andmixing properties can be studied by analyzing theirdistributions. If the description of the boundary ofa deformed sub-domain is accurate, mixing meas-ures as e.g. the intensity of segregation and scaleof segregation can be applied on a discretized flowdomain. These mixing measures can subsequentlybe used to compare different mixing protocols orchanges of geometry. Although the tracking methodis accurate, it is limited in its use, since the numberof points required to accurately describe the bound-ary of a tracked sub-domain, may increase exponen-tially every period.

In optimization of mixing flows, parameters of theflow have to be adapted and the analysis needs to berepeated. Since optimization generally requires nu-merous analyses, the methods listed above are notefficient, or even useless, and an improved, moreengineering, method to evaluate mixing behavior isrequired.

This paper presents such a new method, the so-called repeating mapping method, that is based onthe use of a distribution matrix. A similar, but ratherrestrictive approach applied to the most simple prob-lem, that of the bakers transformation, was usedby Spencer & Wiley (1951) and describes a dis-crete map of the flow-domain over some time-span.Methods like these have recently become attract-ive due to development of accurate adaptive track-ing methods (Galaktionov et al., 1997, 2000) andthe availability of computational resources. In orderto evaluate the mapping method, results are com-pared with those obtained from an accurate track-ing method. The lid driven, time-periodic, two-dimensional rectangular cavity flow serves as a testflow. Nevertheless, the algorithm can and is be-ing expanded to three dimensional, time periodic(Galaktionov et al., 2000) and space periodic flows.Intensity and scale of segregation are chosen as mix-ing measures in accordance with Danckwerts (1952)and Mohr et al. (1957). The coarse grain, or loc-ally averaged, density is mapped (see e.g. Tucker(1991)). Also the residence time distribution can bemapped, but this is only of interest in space peri-odic flows. Tracking and mapping techniques are,subsequently, compared. The mapping method canbe used to locate periodic points and can detect is-lands and their region of influence. The main ad-vantage, however, is that with this mapping tech-nique, mixing analysis and optimization can be per-formed at little extra costs with respect to computingthe mapping, while with boundary tracking or Poin-care mapping the entire analysis has to be repeatedat the same computational costs, if at all possible.

MappingThe method proposed subdivides an arbitrary do-main � into (a large number n of) sub-domains�i with boundaries �i . The boundaries of all sub-domains are tracked from t = t0 to t = t0 +�t . These computations are expensive, but needto be performed only once for every geometry andare highly parallellisable (Galaktionov et al., 1997)since there is no interdependence between particletracking computations. A distribution or mappingmatrix Φ is computed where Φi j contains the frac-tion of sub-domain � j at t = t0 that is tracked tot = t0 +�t and found in the domain �i as it existedat t = t0 (figure 1):

Φi j =∫

� j |t=t0+�t⋂

�i |t=t0

dA

/∫� j |t=t0

dA. (1)

The mapping matrix Φ has the following properties:

2

• The amount of fluid transported from a sub-domain j equals the area A j of the sub-domain(conservation of mass). This implies:

n∑i=1

Φi j = 1. (2)

• The amount of fluid transported to a sub-domain i equals the area Ai of the sub-domain(again: conservation of mass). Thus:

n∑j=1

Φi j A j = Ai . (3)

• Φ is sparse since, in a relatively limited timespan �t , fluid from one sub-domain at the be-ginning or entrance of the flow, is transportedto only a limited number of sub-domains at theexit or next time level (this allows for compactstorage in computations).

� j |t0� j |t0+�t

�i |t0

j

i

Φ i j�t

Figure 1: sub-domain advection; Φi j is the partof the area of �i that is advected to � j

In general, the interval �t does not span an entireperiod. If optimization of a mixing protocol is re-quired, and this includes the effect of the length ofthe period, then the minimum step size to computea mapping matrix should be equal to the minimumstep size that is used for varying the period time.Combination of maps with different time intervals�ti could be used to create the map ΦT for someperiod time T :

ΦT =k∏

i=1

Φ�ti where:k∑

i=1

�ti = T (4)

However, necessarily ΦT is less sparse than Φ�ti .When Φ is computed, a quantity that is related tothe sub-domains can be mapped. The chosen quant-ity should not influence the flow field, or its influ-ence should be negligible, since otherwise the map

itself would change. The method assumes a uniformdistribution of the chosen quantity over every initialsub-domain. This introduces a systematic error inthe method: separate contributions of different sub-domains at t = t0 to a sub-domain at t = t0 +�t areaveraged in that sub-domain. Since the state afterone map defines the new state for the next map, thiserror propagates. This implies that the time step �tshould be chosen large, however, this will result ina denser matrix Φ . We will show that if the sub-domains are small enough, the method still providesvaluable results.To illustrate the mapping method, the mixing prop-erties of the well studied cavity flow are examined.The locally averaged concentration (or coarse graindensity) of a marker fluid in �i , Ci , is regarded asthe quantity that will be mapped. Thus, for every �i

in � its initial concentration Ci is stored in a columnC(0). Now, the concentration distribution C(n�t) atn × �t could be computed by C(n�t) = Φ�t

nC(0),provided that the map Φ�t is the same for every con-secutive �t (repetitive mixing). Spencer & Wiley(1951) suggested to analyze mixing behavior bystudying the properties of Φn for very simple trans-formations. However, where Φ will generally bea sparse matrix, less sparse, Φn will generally bedense, since fluid from an initial sub-domain will fi-nally be advected throughout the flow-domain. Thismakes studying the properties of Φn both inattract-ive and even impossible for efficient three dimen-sional exponential mixing flows. Therefore, insteadof investigating Φn , e.g. C(n�t) could be investig-ated. C(n�t) is computed by the sequence for i = 1to n:

C (i�t) = Φ�t · C((i−1)�t) (5)

This procedure does not change the matrix Φ�t andis, therefore, much cheaper, both in number of op-erations as well as in the computer memory needed,than computing Φ�t

n . Disadvantage is, of course,the (artificial) diffusion introduced by sub domain-averaging, which occurs in every mapping step andcan only be decreased by decreasing the number ofmapping steps, thus increasing the time-span of themap. This requires a balance between the length ofthe elementary time step �t and the number of stepsn to span the total mixing time n�t .An example of how the boundaries �i deform for atime-periodic prototype flow like the lid-driven cav-ity as they are advected during a time step �t ofthe flow is shown in figure 2. The figure shows thedeformation of the sub-domains due to fluid flow in-duced by a displacement of the top wall of twice itslength. Note that this figure serves only as an illus-

3

(a) (b)

Figure 2: Sub-domain advection in a lid driven cavity flow (height to width ratio is 1 : 1.67). (a) Initialsub-domain distribution (25 × 15 grid) and (b) deformed grid after displacement of the topwall by two times its length (D = 2).

tration. Actual analyses are computed with a twoto five orders of magnitude finer grids (200 × 120,400 × 240 or 800 × 480 sub-domains).

Adaptive Boundary TrackingThe adaptive tracking method (Galaktionov et al.,1997) was developed to accurately track the bound-ary � of an arbitrary domain � in a chaotic flow.It starts with using a small number of nodes to de-scribe the boundary of a domain. Since it is difficultto determine beforehand what part of the boundaryundergoes high stretching and which part will barelychange, it is not optimal to start with a large num-ber of equally distributed nodes describing the initialboundary of the sub-domain. Therefore, the adapt-ive boundary tracking method inserts nodes locallyon the boundary only where preset criteria on accur-acy are exceeded. First, an initial sub-domain �|t=t0is defined. The boundary �|t=t0 of this domain issubdivided in a small number of vertices. The nodesconnected by these vertices are stored. Each nodeis tracked by integrating the differential equationsof particle advection (x = u(x, t)) for a number ofsmall time intervals dt . At the end of each interval,the following conditions are checked (Galaktionovet al., 1997, 2000):

l < l1c if αi � αc ∧ αi−1 � αc (6)

l < l2c if αi < αc ∨ αi−1 < αc (l2c < l1c)(7)

with:

l = ||xi−1 − xi ||αi = arccos

((xi−1−x i )·(xi+1−x i )

||x i−1−x i || ||xi+1−x i ||)

where l is the distance between two adjacent nodes,α and αc the angle and minimum angle between twoadjacent vertices and l1c and l2c the preset maximum

lengths in straight and curved regions of the bound-ary respectively. If conditions (6) and (7) are notsatisfied, the vertex between xi−1 and xi is split intotwo parts and a new node is inserted at the initialconfiguration (at t = t0) in order to reduce posi-tion errors due to interpolation between neighbour-ing points. This procedure is repeated for the newlycreated vertices.

As only incompressible fluids are considered, andthus the domain �i should be area conserving, theerror ε can be estimated by:

ε =∣∣∣∫�|t=t0

dA − ∫�|t=t0+�t

dA∣∣∣∫

�|t=t0dA

(8)

If the area conservation is not within certain limits,the values for l1c and l2c have to be adapted.

In chaotic flows, the length of the boundary can in-crease exponentially (Ottino, 1989; Muzzio et al.,1992), so the increase in number of nodes willalso be exponential if an accurate description of theboundary has to be maintained. Therefore, althoughaccurate, this method is computationally expensivefor analysis of even a relatively small number ofperiods, or flows where stretching of the boundaryis extremely non-uniform, and becomes infeasiblefor large numbers of periods.

Quantification of MixingTo quantitatively compare mixing protocols, mixingmeasures need to be defined. Relevant mixing meas-ures are functions of moments of the concentrationdistribution. Here, the intensity of segregation I andthe scale of segregation S are chosen.

4

Intensity of SegregationThe intensity of segregation is a second momentstatistic (variance) of the concentration distribution.It can be calculated as follows: first define the aver-aging operator 〈 f (x)〉� over the domain � as:

〈 f (x)〉� =∫�

f (x)dA∫�

dA(9)

Intensity of segregation is a measure for the devi-ation of the local concentration from the ideal situ-ation (i.e. when the mixture is homogeneous). Theintensity of segregation is defined as, Danckwerts(1952):

I = σ 2c

c(1 − c)with c = 〈c(x)〉� (10)

where σ 2c is the variance in concentration over the

entire domain �:

σ 2c = 〈(c(x) − c)2〉� (11)

and c(x) is the concentration in a point x in the do-main.Since only passive interfaces and no diffusion is as-sumed, c(x) will either be 1 or 0. Therefore I willalways be equal to 1 independent of the distribu-tion. To avoid this situation, the coarse grain dens-ity Ci (Welander, 1955) on a finite sub-domain �i isdefined:

Ci = 〈c(x)〉�i (12)

Coarse grain density can take values between, andincluding, 0 and 1. The domains �i are chosenidentical to the domains in the discretization for themapping method. Equation (10) can be rewritten ina discrete form using equation (12) as:

Id = 1

A�

n∑i=1

(Ci − C)2

C(1 − C)A�i with C = c (13)

in which A� and A�i represent the area of theflow-domain and a sub-domain, respectively. Usingthis discretization, the intensity of segregation (orsample variance) changes when a mixing protocolis applied. The top limiting value (worst case) forId is 1 when all �i have a coarse grain density ofeither 0 or 1 (the boundary of the dyed area is coin-ciding with the boundaries of the sub-domains). Inthe ideal case Id = 0 since Ci = c in all �i . Notethat Id depends on the size of the domains �i . Thismeans that the sub-domain size should be chosenaccording to the level of interest; if striations in themixture of a certain size are critical for the mixtureand thus must be visualized, the sub-domain sizeshould be chosen accordingly.

Scale of SegregationThe scale of segregation is another second mo-ment statistic of the concentration distribution. Thismeasure indicates whether large unmixed regions inthe mixture are present and whether a periodicity inthe concentration distribution exists. The scale ofsegregation was defined by Danckwerts (1952) as afunctional of the correlation function:

S =∫ ∞

0

∫ ∞

0ρ(r)dr (14)

with the correlation function ρ(r):

ρ(r) = 〈(c(x) − c) (c(x + r) − c)〉�σ 2

cwith: r = �x

(15)This indicates that ρ(0) = 1 since numerator anddenominator then are equal. The correlation func-tion can be regarded as the auto convolution of(c(x) − c). The correlation function ρ(r) can haveany value from 0 to 1, unless some form of longrange segregation (periodicity) in the mixture ispresent. In that case ρ(r) can range from −1 to1. This indicates that S according to (14) can besmall, although the mixture is bad. In the paperof Danckwerts, it was assumed that this long rangeperiodicity is not present. When diffusive mixtures(0 ≤ c ≤ 1) are studied, the ideal situation isreached when C(x) = C ∀x∈�. Then the correl-ation function will be equal to 0 in its entire do-main. Consequently, the best possible diffuse mix-ture in the sense of scale of segregation occurs whenS = 1 ∧ I = 0.When the concentration distribution is discretizedsimilar as for the determination of the discrete in-tensity of segregation, the correlation function iseasiest computed via a two dimensional fast Four-ier transform (Tucker, 1991).

Comparison of Tracking and Map-ping TechniquesThe mapping technique is, by its nature, less ac-curate than the adaptive contour tracking technique.This does not imply any serious limitations, as willbe shown by comparing both methods for a limitednumber of periods (due to limitations of the contourtracking technique).In the test case of the lid-driven cavity flow theStokes approximation is valid since inertia can beneglected with respect to viscous contributions. Thelength-to-width ratio of the cavity is 1.67 in accord-ance with Leong & Ottino (1989). The flow is in-duced by successive motion of the top and bottomwalls of the cavity; the side walls are fixed (figure

5

x

y

O

fixed

wal

l

fixed

wal

l

Moving wall (T)

Moving wall (B)

2L = 3.34

2W=

2.0

(a) (b)

Figure 3: (a) Dimensions and coordinates of the cavity; (b) the deformed blob after 5 periods ofmotion, the gray area around (x, y) = (1.15, 0) indicates the initial blob.

25 × 15

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

50 × 30

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

100 × 60

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

200 × 120 400 × 240 800 × 480

Figure 4: Concentration distribution after 5 periods, obtained by mapping from 1-period distribution(computed using adaptive front tracking and discretized) with different resolutions

3a). The dimensionless displacement D is definedas:

D = 1

L

∫ Tp

0(vT (t) + vB(t)) dt (16)

where vT (t) and vB(t) are the velocity of top andbottom wall, Tp is the period time and L is the half-length of the cavity. First, the value of D is set to6.24.The top wall (T) and bottom wall (B) move alternat-ingly; the top wall from left to right and the bottomwall from right to left. This particular flow protocolis well examined, both experimentally and numeric-ally, by the aforementioned authors. Periodic points,which are a key item to evaluate the mixing abilit-ies of the flow have been found (Meleshko & Peters,1996). To describe the Newtonian velocity field, an

accurate semi-analytical solution (Meleshko, 1994)is used.

The mapping method is compared to the trackingmethod for three to five periods. The initial andfinal situations as obtained by the tracking methodare shown in figure 3b: a dyed area with a ra-dius of 0.10 around the hyperbolic periodic point(x, y) = (1.15, 0) in a 3.34 by 2 cavity with thelocation of the origin in the center of the cavity. Theerror in area conservation after five periods is 0.4%(equation 8); the boundary is stretched approxim-ately 1.5 × 103 times (figure 3b). To compare track-ing and mapping, the coarse grain intensity of se-gregation is applied to both techniques. The resultsof the tracking method are discretized on the samegrid as the mapping method (see figure 5).

In case of the tracking method, Id is computed after

6

trac

king

3 periods

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

4 periods

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

5 periods

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

map

ping

Figure 5: Comparison of the discretized concentration distribution for the tracking method and themapping method (initial location of the blob is shown in Figure 3).

five periods by discretizing the tracked boundary ofthe dyed area, as shown in figure 3, on the grid onwhich the corresponding mapping matrix was com-puted (figure 5). For the mapping method, the con-tour of the dyed area was tracked for one period,discretized, and then mapped to five periods. As in-dicated in table 1 the accuracy is apparently onlymoderate, and worse for the larger domain sizes,which was to be expected given the artificially in-duced diffusion. The difference between mappingand tracking is also explained by this numerical dif-fusion. The contribution of tracking to one cell isalways accurate; the contribution of mapping to acell is determined by the average concentrations ofthe contributing cells at the previous step.

Table 1: Intensity of segregation after five peri-ods for the mapped and tracked method

Grid size tracked mapped ratio25 × 15 5.906 · 10−3 1.413 · 10−3 4.1850 × 30 1.089 · 10−2 3.413 · 10−3 3.19

100 × 60 1.992 · 10−2 7.195 · 10−3 2.77200 × 120 3.823 · 10−2 1.347 · 10−2 2.84300 × 180 5.538 · 10−2 2.011 · 10−2 2.75400 × 240 7.065 · 10−2 2.740 · 10−2 2.58600 × 360 9.883 · 10−2 3.823 · 10−2 2.59800 × 480 1.257 · 10−1 5.055 · 10−2 2.49

Despite this diffusion, figure 5 nevertheless showsthat the overall structure of the mixing patterns that

are predicted by the mapping method, are in reas-onable agreement with the results from the track-ing method. So, despite of the obvious differencesin computed values for Id , it can still be concludedthat results obtained with the mapping method couldserve as a valuable starting point for the analysis andoptimization of mixing flows. This statement will befurther explored in the next section.

Application of the Mapping MethodThe results of conventional analyses as e.g. Poincaremaps and, to some extent, manifold analysis, canbe obtained by applying the mapping method. Fig-ure 6 compares the resulting concentration distribu-tion after twenty periods, obtained by using the map-ping method, with the Poincare section for a situ-ation known from literature, D = 8, L/W = 1.67(Meleshko & Peters, 1996). The size of the largecentral island is readily recognized. Also, the threesmaller islands (a,b,c) around the first island (A) arefound as well. In contrast to Poincare sections, map-ping allows for the determination of the order of theperiodic points: island A is of order 1; islands a, b,and c are of order three and rotate counterclockwise.Notice that the structure (striations) in the concen-tration distribution is still distinguishable after 20mapped periods.The mapping method is a flexible tool for optim-izing mixing, since it allows the incorporation ofvariations on an existing mixing protocol, withouthaving to recompute the entire protocol adapted re-peatedly. As an illustration, two parameters of thecavity flow are varied: the dimensionless displace-

7

a

b

cA

(a)

(b)

Figure 6: (a) Concentration distribution after 20periods (alternatingly moving the topwall T and bottom wall B) with D = 8and n = 40; initially the right half ofthe cavity was black, the left half white.(b) Poincare map to show the corres-pondence in the location of the island.The three third order islands are visiblewith both techniques.

ment D and the number of wall movements n; theproduct of which is proportional to energy input.Different protocols are compared by changing theorder of consecutive mappings.Five different protocols are investigated. In protocolA only the top wall moves, avoiding periodicity inthe flow. In protocol B the top wall (T) and bottomwall (B) move alternatingly. Protocols C and D arevariations on protocol B proposed by Ottino (1989);Franjione et al. (1989); Jana et al. (1994a); Aref &El Naschie (1995) to reduce the regularity of theprotocol thus decreasing the size of regions aroundelliptic points (symmetry breaking protocols). Pro-tocol E is a variation of protocol B where a finitewall length is taken into account and the directionof the wall motion is changed.In first instance, for all these protocols the concen-tration distribution is computed with only two map-ping matrices: Φ D=2 and Φ D=4 (movement of thetop wall only). Bottom wall movement and wallmovement in the opposite direction are realized by

coordinate transformation, thus preventing that extramatrices need to be computed and stored. Resultsare shown in figures 7a and 7b; for all concentrationdistribution plots the energy input (n ·D) is constant.Figure 7 clearly shows different mixing behavior forthe different protocols, i.e. different values for Dand n. From figure 7a it is suggested that simplyincreasing D at the cost of n is beneficial for themixing performance. This is not the case, however,since when the parameter D increases further (fig-ure 7b), all protocols become similar to protocol A(only the top wall moving, no periodicity) and theintensity of segregation increases. Apparently (andnaturally), there is an optimum value for D and n,since both large D as well as large n yield laminarmixing, as already demonstrated for this flow by anumber of authors.The mixing properties for the five different protocolsare now investigated in much more detail, with Dvaried between 0.25 and 64.0, in steps of 0.25, andn between 1 and 64 using just five maps for differ-ent values of D. The first four maps (for D = 0.25,D = 0.5, D = 1.0 and D = 2.0) were computedwhile integrating to D = 4.0, thus computing thefive maps in a single run. This run was computedusing 50 personal computers in parallel during 16hours. The mapping matrices were then extractedfrom the tracked grids (see figures 1 and 2) whichcosted 2 hours on the same set of computers. Sincethe geometry is symmetric, the map for a movementof the bottom wall is the same as for the top wall,but rotated (x and y values multiplied by −1 in thecoordinate system of figure 3). Wall movement inthe opposite direction can be computed by mirror-ing (multiplying the x or y-coordinate by −1 in thecoordinate system of figure 3).The resulting maps are applied to the concentrationcolumn C to determine the concentration distribu-tion. If e.g. the concentration distribution C(5.25) forD = 5.25 is required, it is computed by:

C(5.25) = Φ4(Φ1(Φ0.25C(0))) (17)

Computing the concentration distribution by matrixcolumn multiplication requires less than a second ona single machine for each mapping operation.A two dimensional, a-periodic stationary Stokesflow will not be chaotic. Therefore mixing will belinear, and only depend on the amount of energyused. The efficiency plot for protocol A (see fig-ure 8) shows that in this case the lines where in-tensity of segregation is constant coincide with thelines where energy input is constant. This figurealso shows the influence of consecutive mappings:

8

64 steps, D=2 32 steps, D=4 16 steps, D=8

Protocol A: TTTT TTTT TTTT TTTT

I = 0.203, S = 1.70 I = 0.254, S = 1.12 I = 0.254∗, S = 1.12

Protocol B: TBTB TBTB TBTB TBTB

I = 0.455, S = 0.51 I = 0.349, S = 0.43 I = 0.133, S = 1.08

Protocol C: TBTB BTBT TBTB BTBT

I = 0.326, S = 0.91 I = 0.067, S = 3.36 I = 0.047, S = 4.32

Protocol D: TBBT BTTB BTTB TBBT

I = 0.202, S = 1.72 I = 0.046, S = 4.21 I = 0.013, S = 15.50

Protocol E: TB–T–B TB–T–B TB–T–B TB–T–B

I = 0.356, S = 0.40 I = 0.066, S = 2.16 I = 0.008, S = 26.50

Figure 7: (a) Results of dye advection using different protocols with the same energy input. Allcomputations were done using pre-computed mappings for D = 2 and D = 4. Initiallydyed fluid completely fills the right half of the cavity. Computed intensity of segregation isstated under each image. ∗) As for D = 4, the mapping with D = 8 was performed as twoconsecutive mappings with D = 4. (S = S · 104)

9

8 steps, D=16 4 steps, D=32 2 steps, D=64

Protocol A: TTTT TTTT TTTT TTTT

I = 0.254∗, S = 1.12 I = 0.254∗, S = 1.12 I = 0.254∗, S = 1.12

Protocol B: TBTB TBTB TBTB TBTB

I = 0.008, S = 24.80 I = 0.041, S = 6.20 I = 0.137, S = 1.92

Protocol C: TBTB BTBT TBTB BTBT

I = 0.019, S = 7.23 I = 0.041, S = 6.20 I = 0.137, S = 1.92

Protocol D: TBBT BTTB BTTB TBBT

I = 0.028, S = 8.20 I = 0.092, S = 2.30 I = 0.137, S = 1.92

Protocol E: TB–T–B TB–T–B TB–T–B TB–T–B

I = 0.009, S = 26.70 I = 0.041, S = 6.47 I = 0.137, S = 1.92

Figure 7: (b) Figure 7a continued.

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Protocol A (TT)

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Figure 8: Intensity of segregation Id as a func-tion of number of wall movementsn and dimensionless displacement D.Protocol A shows the error due tonumerical diffusion of the mappingmethod. The only independent para-meter is the product n · D (energy). Theiso-energy lines (· · ·) should thereforecoincide with the iso-intensity of se-gregation lines (—).

the lines of constant intensity of segregation arejagged, which is explained by the fact that e.g. a Dof 8.0 is reached with two mappings (Φ4(Φ4C(0))),whereas a D of 7.75 is computed with five mappings(Φ4(Φ2(Φ1(Φ0.5(Φ0.25C(0)))))). Since every map-ping step introduces diffusion into the system, thecalculated intensity of segregation decreases if moremappings are needed. Computation of the efficiencyplot as shown in figures 8, and 10 requires approx-imately 150.000 mappings and takes about 3 hourson one personal computer.Figure 9 shows the mixing efficiency plot for pro-tocol B. To demonstrate the principle of this plot,some concentration distributions are plotted along-side. The total energy input for every concentrationdistribution is the same (n × D = 500). It is evid-ent that the homogeneity of the mixture varies alongthe equal energy line. Figure 10 shows plots of theintensity of segregation (eqn. 10) versus D and n.Black indicates a bad mixture (i.e. Id = 1), whiteindicates a good mixture (i.e. Id = 0). All plotsclearly show values for D where the intensity of se-gregation of the mixture only slowly changes withchanging D or n (the gray streaks), while for othervalues of D just a slight change in its value results ina stepwise improved mixture, often with less energyinput (the white areas).

The different protocols can also be compared mu-tually and the more sophisticated protocols imple-ment that symmetry breaking (like protocols C andD) yield more efficient mixing. Protocol B in partic-ular shows the worst overall mixing efficiency of allperiodic flows: a relatively high value for D (largerthan 15) is needed in order to reach a good mixturequality.

Similar plots can be drawn for the scale of segreg-ation (figure 11). This plot appears similar to theplots of Id versus n and D. However, there are somenoticeable differences. From the wider (in the direc-tion of the D axis) and longer (in the direction of then axis) dark streaks in this figure it follows that theperiodicity in the concentration distribution of themixtures exists for a longer time span (the numberof wall movements) and is less sensitive to the dis-placement D. This also indicates that the mappingmethod is able to preserve the structure (striations)of the mixture.

ConclusionsThe mapping method is an efficient, engineering,method to analyze mixing protocols and geometries.Most computation time is consumed by tracking theboundaries of the sub-domains and is mainly de-termined by the size of the grid chosen, the accuracyrequired of the description of the boundaries of thesub-domains and the time for which these boundar-ies have to be tracked. However, once these map-ping matrices are determined, analysis of arbitrarymixing protocols that can be constructed from thesematrices can be computed within seconds.

The efficiency of the mapping method lies in thefeature that parameter variations can be made andinvestigated easily. It thus leads to a better under-standing of the mixing properties of a flow and theinfluence of variation of its mixing parameters. Thedecrease in accuracy caused by diffusion in the map-ping method is clearly compensated for by its flexib-ility: on one hand the mapping method keeps trackof information concerning the actual distribution ofdyed fluid, in contrast to e.g. the Poincare methodin which points have no physical size. On the otherhand, in contrast to application of boundary track-ing methods the investigation of higher numbers ofperiods (> 10) and different initial dye shapes andposition is only possible by applying the mappingmethod. Finally, as clearly demonstrated, once aset of distribution matrices is computed, they can beused to compute a range of different protocols withdifferent parameters, allowing for an optimization ofthe mixing protocol.

11

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Figure 9: Mixing efficiency plot with corresponding concentration distributions where energy (n× D)is constant (500)

Protocol B (TB)

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Protocol E (TB−T−B)

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Figure 10: Intensity of segregation plots for four different protocols. The dotted lines (· · ·) indic-ate lines of constant energy, the drawn lines (—) indicate lines of constant intensity ofsegregation.

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Figure 11: Plot of the scale of segregation as function of D and n for protocol B (left) and protocol D(right)

Although here only a two dimensional prototypemixing flow with time periodicity is discussed, this,however, is by far not a limitation of the mappingmethod. Currently, three dimensional time periodicflows, as the flow in a closed cube, and space peri-odic flows as e.g. the partitioned pipe mixer, theKenicsTM mixer and the multiflux static mixer arebeing investigated by this method. In these casesthe z coordinate (z-axis in the general direction ofthe flow) takes the place of time. In case of e.g.the KenicsTM mixer an optimal length and angle ofa divider wall can be computed. Further extensionwill be in the investigation of optimization of mix-ing in corotating twin screw extruders, by apply-ing the mapping method. To be able to study andoptimize mixing in extruders a library of mappingmatrices will be created (on super computers us-ing parallel computing), and by changing order andnumber of the matrices, different mixing processescan be simulated and optimized. This last optimiz-ation step can be performed on a personal computerwith moderate memory requirements. Comparisonswith experiments focusing on concentration distri-butions will be made. Results will be reported indue time.

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