The intermaterial area density generated by time- and spatially periodic 2D chaotic flows

*Corresponding author. Tel.: #1-732-445-3357; fax: #1-732-445-5313.

E-mail address: [email protected] (F.J. Muzzio)

Chemical Engineering Science 55 (2000) 1497}1508

The intermaterial area density generated by time- and spatiallyperiodic 2D chaotic #ows

Fernando J. Muzzio!,*, Mario M. Alvarez!, Stefano Cerbelli!, Massimiliano Giona",Alessandra Adrover"

!Department of Chemical and Biochemical Engineering, Rutgers University, PO Box 909, Piscataway, NJ 08855-0909, USA"Dipartmento Ingegneria Chimica, Universita& di Roma **La Sapienza++, via Eudossiana 18, 00184 Roma, Italy

Received 23 August 1998; received in revised form 29 December 1998; accepted 15 April 1999

Abstract

This paper explores in some detail the spatial structure and the statistical properties of partially mixed structures evolving under thee!ects of a time-periodic chaotic #ow. Numerical simulations are used to examine the evolution of the interface between two #uids,which grows exponentially with a rate equal to the topological entropy of the #ow. Such growth is much faster than predicted by theLyapunov exponent of the #ow. As time increases, the partially mixed system develops into a self-similar structure. Frequencydistributions of interface density corresponding to di!erent times collapse onto an invariant curve by a simple homogeneous scaling.This scaling behavior is a direct consequence of the generic asymptotic directionality property characteristic of 2D time-periodic#ows. Striation thickness distributions (STDs) also acquire a time-invariant shape after a few (&5}10) periods of the #ow and arecollapsed onto a single curve by standardization. It is also shown that STDs can be accurately predicted from distributions ofstretching values, thus providing an e!ective method for calculation of STDs in complex #ows. ( 1999 Elsevier Science Ltd. Allrights reserved.

1. Introduction

Mixing processes in chaotic #ows have been the focusof considerable attention in the past 15 years, startingwith Aref (1984), who "rst pointed out that passivetracers advected by relatively simple time-dependent#ows can exhibit complex Lagrangian behavior. Sincethen, a large number of studies has followed (severalreview articles are available in the literature; see forexample Aref, 1990; Ottino, 1990; Hobbs, Alvarez,& Muzzio, 1997a). The majority of chaotic systems dis-cussed in the early literature are two-dimensional, time-dependent #ows. More recently, attention has shifted to#ow systems of complexity similar to those encounteredin direct applications, such as cylindrical cavities (Miles,Nagarajan, & Zumbrunnen, 1995; Zhang & Zumbrun-nen, 1996; Zumbrunnen, Miles & Liu, 1996), simpli-"ed extruders (Lawal & Kalyon, 1995; Kim & Kwon,

1996), stirred tanks (Lamberto, Muzzio, Swanson,& Tonkovich, 1996; Harvey, Lee & Rogers, 1995; Harvey& Rogers, 1996), and static mixers (Hobbs & Muzzio,1997; Hobbs, Swanson & Muzzio, 1997b). These studieshave shown that many #ows of direct practical relevanceare indeed chaotic and that mixing processes in such#ows can be characterized, and perhaps even optimized,using concepts and tools from dynamical systems theory.However, in spite of such e!orts, the theoretical under-standing of mixing processes is relatively incomplete evenfor the simplest #ows. Practical mixing problems must beapproached by empirical means, including trial-and-er-ror procedures that are unlikely to generate e$cient orelegant solutions.

At short mixing times, and for viscous #uids, di!usionis negligible. Mixing occurs as a combination of stirring,stretching, and folding, which generates intricate lamellarstructures with wide distributions of length scales thatspan many orders of magnitude (Alvarez, Muzzio,Cerbelli, Adrover & Giona, 1998). For time-periodicchaotic #ows, the mixing process is controlled by astationary multiplicative operator that generates struc-tures that are self-similar with respect to time. The length

0009-2509/00/$ - see front matter ( 1999 Elsevier Science Ltd. All rights reserved.PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 3 5 9 - 0

Fig. 1. Sketch of the sine #ow, which takes place in a unit box withperiodic boundary conditions. The #ow is the combination of twoorthogonal motions, each with a sinusoidal velocity pro"le. Eachmotion acts alternatively for half a #ow period.

1 In fact, the initial location of the interface is entirely irrelevant,provided that it is contained within the chaotic region; after just a few#ow periods, the evolved material line is entirely independent of itsinitial location. The reason for this independence w.r.t. IC's is veryimportant and is discussed in detail in Section 5.

scale distributions characterizing such structures ulti-mately control the rates of di!usional homogenizationand the rate of reactions taking place at small scales ofthe #ow. Therefore, to understand reactive mixing pro-cesses, we need to be able to describe in detail the par-tially mixed structures created by the #ow. At the presenttime the genesis and evolution of partially mixed struc-tures in chaotic #ows remains poorly understood even inthe absence of di!usion. Essential questions remain un-answered; for example, (1) it is not known what controlsthe rate of the mixing process in a globally chaotic #ow(in fact, `mixing ratea does not currently have a preciseand objective meaning); (2) the length scale distributiongenerated by chaotic #ows remains to be determined; (3)the relationship between #uid mechanical stretching andmicromixing remains an open question.

Experimental characterization of mixture microstruc-ture is impractical because chaotic #ows create structureswith length scales much smaller than can be experi-mentally resolved. While computations can be used tosimulate in detail early stages of the mixing process, mostprevious studies have only simulated mixing of pointtracers or stretching of in"nitesimal vectors. Such simula-tions do not preserve the continuity of material lines, andtherefore are poorly suited to examine the length scaledistribution of a partially mixed structure. Evolution ofmaterial "laments in chaotic #ows was "rst consideredby Franjione and Ottino (1987), who concluded at thetime that the large computational resources required tosimulate the exponentially growing "lament made suchsimulations impractical. Fung and Vassilicos (1991) at-tempted a fractal analysis of convected lines, but theiralgorithm failed to preserve continuity along the lines,and did not examine the distribution of length scalescreated by the #ow. Beigie, Leonard and Wiggins (1993)did preserve continuity, but did not examine the topol-ogy of the structures generated by the #ow.

In this paper, the evolution of the continuous pe-rimeter delimiting the boundary between two #uids isexamined as such interface is stretched, stirred, andfolded by the time and spatially periodic sine #ow (Fig. 1).

The spatial density of the material line is strongly andpermanently non-uniform, and is characterized by a self-similar family of probability distributions that becomeinvariant upon simple re-scaling. Such scaling propertiescan be simply understood in terms of asymptotic direc-tionality, a fundamental property of the #ow that is alsothe cause of the invariant structure of the #ow manifolds.Moreover, the striation thickness distribution generatedby the sine #ow is also self-similar and can be accuratelyand e$ciently predicted from stretching calculations.

2. System

A two-dimensional time-periodic chaotic #ow, the sine#ow (Liu, Muzzio & Peskin, 1994; Alvarez et al., 1998), isadopted here as a case study (Fig. 1). The #ow is de"nedby two motions:

(<x, <

y)"(sin 2py, 0), n¹)t((n#1/2)¹, (1a)

(<x, <

y)"(0, sin 2px), (n#1/2)¹)t((n#1)¹, (1b)

where ¹ is the #ow period, n is the number of periods,and t is the time. The #ow is de"ned on the 2D torus;whenever a particle exits the unit square, it re-enters thebox through the opposite side. This simple #ow scheme isuseful for understanding chaotic mixing because a varietyof #ow behaviors can be obtained by suitably varying the#ow period ¹.

Four main types of calculations are performed in thesine #ow: (I) Evolution of interfaces, which in two dimen-sions are represented as continuous material lines; (II)calculations of the coarse-grained density o of the inter-face, which we call `intermaterial area densitya, (III)length scale distributions of the evolved interfaces, and(IV) distributions of stretching values (for details, seeAlvarez et al., 1998). As described below, these calcu-lations unveil important topological and statistical prop-erties of material lines evolved in chaotic #ows.

3. Evolution of material lines

As an illustration, let us consider the evolution ofa material line for ¹"1.6, which generates a `globallychaotica condition characterized by very small regularislands (for partially chaotic cases, see Alvarez et al.,1998). Figs. 2a, b shows the evolved "lament for n"4and 5 periods, respectively. The initial condition in thesimulation corresponds to a circle of radius equal to 0.01centered at (x, y)"(0.5, 0.5).1 Since this #ow condition is

1498 F.J. Muzzio et al. / Chemical Engineering Science 55 (2000) 1497}1508

Fig. 2. Evolution of a material "lament in the sine #ow, ¹"1.6, (a) n"4, (b) n"5.

devoid of large islands, the "lament quickly invades al-most all of the #ow domain. As it is convected, stretched,and folded by the #ow, it gives rise to a lamellar systemcomposed of thousands of striations with a wide distribu-tion of local length scales. After just a few #ow periods,a complex structure emerges, closely resembling the in-variant unstable manifold of the #ow. The structure con-tinues evolving as the number of periods increases; thelamellae become increasingly thinner as the result ofa complex iterative process. Fig. 2a, b illustrates that theevolution of the structure is governed by intrinsic self-similarity (as expected from previous experimental andnumerical studies (Swanson & Ottino, 1990; Leong& Ottino, 1989; Muzzio, Swanson & Ottino, 1991; Lam-berto et al., 1996; Hobbs & Muzzio, 1997; Hobbs et al.,1997a,b). As time increases, the chaotic #ow producesa partially mixed structure that, when recorded at peri-odic intervals, is essentially identical to the structurerecorded a period earlier, except that a larger number ofthinner striations is found in each region. The dynamicalself-similarity evidenced by Fig. 2a, b involves a complexevolution process wherein the entire structure evolves inconcert to generate time periodic properties. Interfacebranches that cover a given region A of the #ow domainat the end of period n move to a di!erent region B byperiod n#1. Such branches were present at a di!erentregion C a period earlier. Clearly, the overall evolution ofthe structure is controlled by some local property of the#ow, such that once a interface branch reaches a certainregion, it acquires the orientation corresponding to suchregion, so that the overall structure `always looks thesamea.

An important fact that is immediately apparent fromFig. 2 is that the elongated "lament populates the chaoticregion in an extremely non-uniform manner. Certain

regions of the #ow show extremely tight bundles ofstriations (such as, for example, the neighborhood of thepoint (0.5, 0.5)), while other regions are populatedmuch more sparsely. This observation, indicates that themixing process results in a wide distribution of `degreesof mixednessa.

An essential issue is to determine the overall rate ofmixing, which for a #uid-mechanical process is given bythe growth of the interface between the #uids. In oursimulations, this rate is computed by measuring the rateof growth of the length ¸

nof a closed material line as

a function of the number of periods. The fact that the #owis chaotic implies that any in"nitesimal vector dx

0is

stretched after n periods according to EdxnE+

Edx0E ) exp (nK) (where K is the Lyapunov exponent of

the #ow), regardless of its initial location and orientation.Then, by considering a generic material line as made ofin"nitely many di!erential elements of arc, one wouldexpect that an asymptotic growth rate ¸

n+¸

0) exp (nh)

(h is called the topological entropy of the #ow) holds,with h"K. However, computational studies on severalmodel and real #ows have shown that in general h'K,i.e. material lines stretch faster than predicted by theLyapunov exponent. While a complete understanding ofthis phenomenon is yet to be achieved, computer experi-ments seem to indicate that the inequality h'K is one-to-one with the unavoidable presence of periodic ellipticpoints within the #ow domain.

4. Spatial and temporal structure of the intermaterial areadensity

An issue that is as important as the rate of growth of¸n

is the spatial distribution of the intermaterial area

F.J. Muzzio et al. / Chemical Engineering Science 55 (2000) 1497}1508 1499

Fig. 3. Schematic representation of a dye advection simulation. (a) initial condition, (b) macroscopic structure of the interface, (c) close-ups showing thedi!erent microstructure corresponding to two locations of the #ow domain.

density. As is apparent from Fig. 2, the #ow generatesa structure with a very non-uniform distribution of ma-terial line densities. The importance of such density #uc-tuations is easily appreciated by means of a simple`thought experimenta, in which two miscible substancesA and B are mixed, starting from an initial condition(depicted in Fig. 3a) in which a `bloba of A is surroundedby a `seaa of B. Let the material "lament be initiallyde"ned by the perimeter of the region containing A. AsA and B are mixed by the #ow, the material line is onceagain evolved into an elongated "lament (Fig. 3b) that fora given value of ¹ would have essentially the sameproperties as the structures displayed in Fig. 2. Let ussubdivide the chaotic region of the #ow into identicallysized boxes, and measure the amount of A in each box.The ergodic principle ensures that the same amount ofA will eventually be present in every box, the larger theboxes, the shorter the time required to approach sucha `macroscopically uniforma condition. The simpli"edinterpretation of this property is that a globally chaotic

#ow should generate a completely homogeneous mixturein just a few #ow periods. However, actual mixing behav-ior is more complex. `Macroscopic uniformitya does notprovide a complete description of the process, becausethe intimacy of the mixture in each box depends on thenumber of striations present in each box. In regions thatcontain many layers (region X, Fig. 3b, intimate micro-mixing), substance A will populate the smaller boxesuniformly, but in regions that contain only a few layers(region Y, Fig. 3b, coarse micromixing), the concentra-tion of A will exhibit strong #uctuations from small boxto small box.

Such di!erences in degree of micromixing can bequantitatively described in terms of the intermaterialarea density o, de"ned as the length of the interfacecontained in each box, divided by the size of the box.Fig. 4 displays the spatial distribution of o/SoT (whereSoT is the average of all boxes) for ¹"1.6, n"5 periods.Regions of the #ow are color-coded ad hoc to facilitategraphical illustration; density decreases in the sequence

1500 F.J. Muzzio et al. / Chemical Engineering Science 55 (2000) 1497}1508

Fig. 4. (a) Material line density, ¹"1.6, n"6, (b) Stretching "eld, ¹"1.6. Regions of the #ow are color-coded ad hoc to facilitate graphicalillustration; both density (a) and stretching (b) decrease in the sequence given by red, yellow, green, and blue.

given by red, yellow, green, and blue. As shown in the"gure, o in some regions is several orders of magnitudehigher than in other regions. This observation meansthat, even if the system is homogeneous from a macro-mixing standpoint, in some regions the #ow achievesmuch more intimate mixing (much more intense micro-mixing) than in other regions.

An important result is illustrated in Fig. 4b, whichshows the spatial distribution of the coarse-grainedstretching "eld for ¹"1.6, which is computed by calcu-lating the stretching values of 1 000 000 tracer vectorsand then averaging the stretching values of all vectors ina box. Comparison between Figs. 4a and b con"rms thato in each box is strongly correlated to the coarse-grainedstretching "eld in the box. As expected from previouswork, regions of high intermaterial area density corres-pond closely to regions of fast stretching (Hobbs et al.,1997a,b; Muzzio et al., 1991; Liu et al., 1994; Hobbs& Muzzio, 1997; Liu et al., 1994). This result plays a keyrole in the prediction of the striation thickness distribu-tion for the #ow.

Moreover, such non-uniformities in "lament densityare a permanent feature of the chaotic mixing process. Ifthe spatial distribution of o is computed after additionalperiods of the #ow (not shown in the interest of brevity),although SoT increases by several orders of magnitude,its spatial distribution remains essentially unchanged.The time-invariance in the spatial distribution of o,which is another manifestation of the self-similarity of thestructures generated by time-periodic chaotic #ows, can

be demonstrated by considering the probability distribu-tion F

n(o), which is given by

Fn(o)"(1/No) dN(o)/do, (2)

where No is the total number of boxes and dN(o) is thenumber of boxes with densities between o and o#do. Inthe numerical simulations F

n(o) was computed by divid-

ing the #ow into (1024]1024) boxes, computing theamount of material line length in each box, and thendetermining the frequency of each value of o.

If o asymptotically approaches a self-similar spatialstructure, then distributions F

n(o) corresponding to dif-

ferent #ow periods should be given by a self-similarfamily of curves that should asymptotically collapse ontoan invariant (time-independent) curve by an appropriatescaling. The simplest form of such a scaling is a homo-geneous stretching/contraction of both the vertical andhorizontal axis (see Fig. 5),

F(u)"SoTFn(o), u"o/SoT, (3)

which means that the density distribution can be collaps-ed onto an invariant curve simply by dividing the densityo by the mean density SoT and multiplying the frequencyFn(o) by the mean density. The e!ective collapse of F(u)

over several orders of magnitude in Fn(o) and o indicates

that the above-mentioned non-uniformities in o are per-manent features of time-periodic chaotic #ows. As thenumber of periods increases, the intermaterial area den-sity preserves the same distribution, meaning that thenumber of low density and high density locations, and

F.J. Muzzio et al. / Chemical Engineering Science 55 (2000) 1497}1508 1501

Fig. 5. F(u) vs. u for ¹"1.6, n"4}7.

the relative magnitude of the density on such regions,remains unchanged.

From a practical standpoint, Fig. 5 indicates that onceFn(o) has reached the asymptotic regime (which for the

sine #ow takes 5}10 #ow periods), all of its time depend-ence is contained in the homogeneous scale SoT. Thetime evolution of SoT is explicitly known, i.e.,

SoT+¸n+Sj(n)T+enh, (4)

which can be used to express SoT with respect to its valueat the end of a given period n

1, i.e.,

SoTn/SoT

n1"eh(n~n1).

Therefore, once F(u) is determined, Fn(o) can be predicted

for any period simply by inverting Eq. (3),

Fn(o)"(SoT

n)~1F(u)"(SoT

n1)~1e~(n~n1)hF(u), (5a)

o"SoTnu"SoT

n11e(n~n1)hu. (5b)

This scaling is not valid for arbitrarily long times. Event-ually, as the short-time Lyapunov exponents collapseonto a delta function around the main Lyapunov expo-nent, Sj(n)T+enK and under such conditions Eq. (5b)over-predicts SoT

n. However, for almost all cases, such

a crossover occurs after at least 100 periods of the #ow(and in many cases it occurs much later); for most systemsof interest to mixing applications, a much smaller num-ber of #ow iterations is actually used. In cases wherelong-time behavior is relevant (as is perhaps the case instirred tanks that use hundreds of impeller rotations) thevalidity of Eq. (5b) is restored simply by substitutingenh by Sj

nT.

Fig. 5 demonstrates that the spatial and statisticaldistributions of o are self-similar with respect to time, andEq. (5a) and (5b)) provides an explicit scaling equation. Inthe next section, we examine why the intermaterial areadensity is self-similar, and why it exhibits the speci"c typeof self-similarity demonstrated above.

5. Asymptotic directionality: The source of self-similarityin time-periodic chaotic 6ows

A number of previous studies have qualitatively re-lated the topology of the partially mixed structures to thestructure of manifolds of hyperbolic periodic points. Ingeneral, such studies have used continuation principles to"nd the invariant sets (manifolds) that spawn from thestable and unstable eigendirections associated with hy-perbolic "xed or low-order periodic points. After quali-tatively observing partially mixed structures that closelyresembled such manifolds, some authors suggested thatmanifolds play a role on the development and evolutionof self-similar partially mixed structures (Beigie, Leonard& Wiggins, 1991; Rom-Kedar, Leonard & Wiggins, 1990;Swanson & Ottino, 1990).

A somewhat di!erent approach can be proposed byfocusing on a local property of chaotic #ows, which wename `Asymptotic Directionalitya (AD). This property isthe source of the invariant manifold structure. Compari-son of the orientation of material line branches thatpopulate the same region of the #ow at di!erent timesdemonstrate that once a branch of the material linereaches a certain location in the #ow, it adopts an

1502 F.J. Muzzio et al. / Chemical Engineering Science 55 (2000) 1497}1508

orientation that is characteristic of that position. In otherwords, orientation of a material element in a time-peri-odic #ow is determined by instantaneous location, not bytime. This time-dependent local orientation of materiallines is a completely general property of time- or spatiallyperiodic chaotic #ows that has been reported for manyother systems (Swanson & Ottino, 1990; Leong & Ottino,1989; Lamberto et al., 1996; Hobbs et al., 1997a,b; Hobbs& Muzzio, 1997).

In order to re-state the above observation in math-ematical terms, let us consider the map x

n"U(x

n~1) of

a generic time-periodic continuous #ow "eld. This map isa vector function, which for the sine #ow is de"ned by

xn"U

x(x

n~1, y

n~1)"x

n~1#(¹/2) sin 2py

n~1, (6a)

yn"U

y(x

n~1, y

n~1)"y

n~1#(¹/2) sin 2px

n

"yn~1

#(¹/2) sin [2p(xn~1

#(¹/2) sin 2pyn~1

)]. (6b)

Consider a vector z0(x

0) tangent to the material line. As

the #ow iterates, the vector is advected, stretched, andre-oriented by the #ow. This evolution is given instan-taneously by

zn(x

n)"UH(x

n~1) ) z

n~1(x

n~1), (7)

where UH(xn~1

) is the Jacobian of U(xn~1

). Moreover,zn~1

(xn~1

)"UH(xn~2

) ) zn~1

(xn~2

), and therefore,

zn(x

n)"UH(x

n~1) )UH(x

n~2)2UH(x

n~m) ) z

n~m(x

n~m)

"UHm(xn~m

) ) zn~m

(xn~m

). (8)

The evolution of the vector z has been expressed interms of its past history while by keeping its "nal positionconstant. Taken literally, Eq. (8) can be used to examinehow the orientation and length of a vector at a givenposition and time depends on its past evolution. This canbe done by choosing a given position x

nand "nding all

the past locations corresponding to such a "nal position,which are given by the sequence

xn~1

"U~1(xn), x

n~2"U~1(x

n~1), 2. (9)

Subsequently, an arbitrary vector, labeled zn~m

, can beplaced at location x

n~mand evolved forward in time to

end at the desired position and time. After just a few #owperiods (typically 2}3 #ow periods or so), z

n(the "nal

vector) approaches a "xed orientation. This orientation,which we call eu

=, depended exclusively on the "nal posi-

tion and is independent of the initial orientation of zn~m

.A similar calculation can be performed by evolvingpoints forward in time. Random vectors placed at theevolved positions and evolved back to the initial positionby using the inverse map approach a (di!erent) asymp-totic local orientation, which we call es

=. Similar results

were obtained for the #ow between eccentric cylinders,the cavity #ow, the standard map, the Du$ng oscillator,

and Anosov's map. These results provide strong numer-ically justi"cation for the existence of local orientation ofthe map as a generic property of time-periodic chaotic#ows.

The next step is to determine what causes such localorientations to exist. Let us denote by eu

mand es

mthe

eigenvectors of the m-order Jacobian UHm(xn~m

) havingassociated eigenvalues ju

mand js

m, ju

m*js

m, ju

mPR;

jsmP0; ju

m) js

m"1. For any position x

nbelonging to

a chaotic region of an area-preserving #ow, as m in-creases, both eu

mand es

mapproach well-de"ned limits, i.e.,

limm?=

eum"eu

=; limm?=

esm"es

=, (10)

where eu=

and es=

are the asymptotic orientations de"nedabove. An important direct consequence of the existenceof the vector "elds eu

=and es

=is that these vector "elds are

self-preserving with respect to UH (Giona, Androver& Muzzio, 1998):

eu=(x

n)"(j

n~1,n)~1UH(x

n~1) ) eu

=(x

n~1), (11a)

es=(x

n)"j

n~1,nUH(x

n~1) ) es

=(x

n~1), (11b)

where jn~1,n

(xn~1

) as de"ned by Eqs. (11a) and (11b), isthe step elongation corresponding to position x

n~1. Eqs.

(11a) and (11b) means that a vector parallel to the asymp-totic direction eu

=(x

n~1) at location x

n~1becomes parallel

to the asymptotic direction eu=(x

n) when the #ow advects

it to the location xn"UHn(x

n~1), and its length increases

by a factor jn~1,n

. Similarly, a vector parallel to theasymptotic direction es

=(x

n~1) at location x

n~1becomes

parallel to the asymptotic direction es=(x

n) when the #ow

advects it to the location xn"UHn(x

n~1), and its length

decreases by a factor jn~1,n

. This property is the reasonwhy an arbitrary vector z(x

0) becomes asymptotically

parallel to eu=

(xn) when iterated forward in time for n #ow

periods, and to es=(x

~n) when iterated backward in time

for n #ow periods. This can be readily understoodby taking the linearly independent vectors eu

=(x

0) and

es=(x

0) as a basis for z

0, i.e., decomposing it into

z0"aeu

=(x

0)#bes

=(x

0), where a and b are coordinates

of z0

with respect to eu=

(x0) and es

=(x

0). After 1 period,

z0

becomes

z1(x

1)"UH(x

0) ) z

0(x

0)"UH(x

0) ) [aeu

=(x

0)#bes

=(x

0)]

"aj0,1

eu=

(x1)#b(j

0,1)~1es

=(x

1), (12)

and after n periods,

zn(x

n)"UHn(x

0) ) z

0(x

0)"UHn(x

0)*[aeu

=(x

0)#bes

=(x

0)]

"aj0,n

eu=

(xn)#b(j

0,n)~1es

=(x

n). (13)

Since for every chaotic location j0,n

"j0,1

) j1,2

2jn~1,n

diverges exponentially fast as n increases, the vector zn(x

n)

becomes parallel to eu=

(xn) as the number of periods

F.J. Muzzio et al. / Chemical Engineering Science 55 (2000) 1497}1508 1503

increases. The only exception is a vector that is initiallyparallel to es

=(x

0), i.e., a vector for which a"0. As n in-

creases, such a vector would remain parallel to es=(x

n),

but its norm decreases by a factor j0,n

. However, sincethere is only one such direction at each point, a randomlyoriented vector would never exactly coincide with it, andfor a long enough number of periods every vector in thechaotic region would grow exponentially.

Therefore, as the #ow deforms a material line, thein"nitesimal vector z

n~1(representing an element of the

material line) is evolved via an iterative process; in eachperiod of the #ow, the vector is displaced from x

n~1to

xn"U(x

n~1), and is stretched and reoriented as deter-

mined by zn"UH ) z

n~1, so that its component parallel to

eu=(x

n) grows by a factor j

n~1,nand its component paral-

lel to es=

(xn) shrinks by a factor (j

n~1,n)~1. For all chaotic

#ows studied so far, any arbitrary vector becomes e!ec-tively parallel to eu

=(x

n) in just 2 or 3 #ow periods, and

remains parallel to it ever since. This orientation processis extremely important to both the structure and thedynamical evolution of a partially mixed structure. Sincethe orientation process takes place everywhere along thematerial line, after just two of the three periods, thematerial line is everywhere parallel to eu

=(x). This means

that the entire structure of a material line can be predictedat a given time by integrating a continuous trajectory that iseverywhere parallel to the unstable asymptotic directioneu=(x) and that has the same length as the material line.

Therefore, since the material line is everywhere parallel tothe vector "eld eu

=(x), nearby elements of the material line

become parallel to one another, and the very fact that the"eld eu

=(x) is invariant causes material lines correspond-

ing to di!erent periods to `look alikea. In other words,AD is the direct cause of the self-similarity of partiallymixed structures and their independence of initial condi-tions discussed above.

In fact, AD can be used to discuss rigorously the role of`manifolds of hyperbolic periodic pointsa on the evolu-tion of partially mixed structures. Such manifolds areusually examined by using perturbation methods andcontinuation ideas to "nd the trajectories in phase space(or the streaklines in a #ow) that emanate from hyper-bolic points in the stable and unstable directions. Manyauthors have qualitatively remarked that partially mixedstructures closely resemble such manifolds. In fact, theAD can be used to explain and generalize this relation-ship. It is natural to look for a global geometric de"nitionof the unstable manifold that is representative of thegeometrical and statistical properties evidenced in theevolution of material lines. To this end, let us de"ne forall wandering points x

0belonging to the chaotic region

the dynamical system

dxu(u)/du"eu

=(x

u(u)), x

u(u"0)"x

0, (14)

where u is a real parameter. Let =ux0

be the continuouscurve (integral manifold) generated by Eq. (14) starting

from x0. The collection of all the integral manifolds

emanating from points x0

within the chaotic regionforms what is called a foliation Fu"M=u

xNx|Cc

and eachintegral manifold is referred to as a leaf of the foliation. Ifx0"x

Pis a periodic point, then =u

xPis just the local

unstable manifold at xP.

Consider a period-1 periodic point xp, such that

xp"U(x

p). For such a point, Eqs. (11a) and (11b) simply

becomes

eu=(x

p)"(j

0,1)~1UH(x

p) ) eu

=(x

p),

es=(x

p)"j

0,1UH(x

p) ) es

=(x

p), (15)

i.e., the asymptotic directions eu=

and es=

are eigenvectorsof UH(x

p) and therefore coincide with the stable and

unstable directions that emanate from the periodic point.We use induction to extend this idea to hyperbolic

periodic points xpof higher order. We start by consider-

ing all the period-2 hyperbolic points of U(xn), denoted

generically as Mx2N. De"ne the composed period-2 map

C(xm)"U(x

n) o U(x

n`1)"U(U(x

n)), such that n"2m. By

construction, since C and U generate the same mixingprocess, they have the same characteristic directionseverywhere (and the same invariant manifolds, and alsothe same period-2, period-4, etc, periodic points). There-fore, Eq. (15) can be applied (with minor modi"cations) toC(x

2) to demonstrate that the asymptotic directions at

such points coincide with the eigendirections of theperiod-2 points, and that therefore the manifolds emanat-ing from such period-2 points are locally parallel toeu=

and es=

at such points. The same argument can bemade for periodic points of any order. Since periodicpoints are dense in a chaotic region, and since unstablemanifolds from di!erent periodic points cannot intersecttransversally, unstable manifolds from periodic pointsmust be everywhere parallel to eu

=, and stable manifolds

must be everywhere parallel to es=

. The direct conse-quence is that unstable manifolds from all periodic pointsare in fact identical to one another, and they are allidentical to a global unstable manifold that is everywheretangent to eu

=. Similarly, stable manifolds from all peri-

odic points are identical to one another and to a globalstable manifold that is everywhere tangent to es

=. There-

fore, Eq. (14) represents a simple and non-perturbativeway to reconstruct the unstable manifold of the #ow.

As a consequence of these observations, the corre-spondence between the evolved material line and thestructure of global unstable manifold is a foregone con-clusion. Since material lines are continuous and asymp-totically tangent to eu

=, and since manifolds are also

continuous and everywhere tangent to eu=, as the number

of #ow periods increases, the structure of evolved mater-ial lines becomes asymptotically identical in both densityand orientation to the global manifold of the #ow.

The self-similarity of Fn(o), and its scaling properties,

immediately follow from this observation. Consider, for

1504 F.J. Muzzio et al. / Chemical Engineering Science 55 (2000) 1497}1508

Fig. 6. Hn(log s), the probability frequency of log s, for ¹"1.6, n"5}8. As shown in the inset, a simple scaling makes H

n(log s) collapse onto a single curve.

example, a material line that has been evolved for n peri-ods until it acquires a length ¸

n. Divide the #ow domain

into N equal-sized boxes, and compute ¸in, which is the

fraction of the material line ¸npresent in box i. At such

point, the density in box i after n periods is oin"N¸i

n/A,

and the average density is SoTn"¸

n/A, where A is the

area of the #ow domain. Recall that, since the materialline is everywhere parallel to the manifold, it can also bereconstructed as a trajectory along the manifold usingEq. (14). Denote with k(i) the manifold measure in regioni, (equal to the probability of sampling region i whilefollowing an arbitrary trajectory along the manifold).For a su$ciently long line, ¸i

n/¸

n"oi

n/(NSoT

n)"

un/N+k(i)/N. Consider now the material line at a later

period m'n. At this point, the line has acquired a length¸m'¸

n. Since the line at period m can also be recon-

structed as a trajectory along the manifold using Eq. (14),the same reasoning can be used to state that ¸i

m/¸

m"

oi/(NSoT

m)"u

m/N+k(i)/N. We can now de"ne the

probability distribution of u in terms of k(i);

un"u

m"uPk(i), and

F(u)"(1/N) dN(u)/du"(1/N) dN(k)/dk. (16)

Since the manifold is invariant, then for a given box, k(i)is invariant. For long enough n, this generates an invari-ant re-scaled density u and an invariant scaled densitydistribution F(u). Moreover, since d N(o)"d N(u) anddu"d(o/SoT)"(1/SoT) do, it follows that

Fn(o)"(1/N) dN(o)/do

"SoT~1(1/N) dN(u)/du

"SoT~1 F(u), (17)

i.e., as observed above, F(u)"SoTFn(o) is invariant. In

other words, the normalized PDF F(u) of the densityo converges towards an invariant distribution as a directconsequence of the fact that the spatial density o isa permanent feature of the #ow.

6. Computation of the striation thickness distribution

As mentioned in the introduction, in many cases ofpractical interest it is important to determine the distri-bution of length scales generated by the mixing processes.Such distribution is typically quanti"ed by means ofthe striation thickness distribution (STD). The STDgenerated by the sine #ow was computed following theprocedure described in Alvarez et al. (1999). Practicalimplementation of the procedure is quite di$cult, bothalgorithmically and computationally, obeying primarilythe fact that after just a few periods, the #ow generatesa structure with an enormous number of striations char-acterized by an extremely wide distribution of lengthscales. For example, for ¹"1.6 after just 5 #ow periods,computation of striation thickness based on intersectionsof the "lament with a single horizontal line yields morethan one million striations, some of them smaller than10~10. Under such conditions, extreme care is necessaryin order to prevent numerical error from overwhelmingthe results.

Fig. 6 (main panel) shows the frequency distributionH

n(log s)"(1/N

s) dN(log s)/d(log s), where s is a value

of striation thickness, Ns

is the total number ofmeasurements, and dN(log s) is the number of values ofstriation thickness between log s and log s#dlog s. The

F.J. Muzzio et al. / Chemical Engineering Science 55 (2000) 1497}1508 1505

Fig. 7. Hn(log j), the probability frequency of log j, for ¹"1.6, n"5}8. As shown in the inset, the same scaling used in Fig. 13 for log s also makes

Hn(log j) collapse onto a single curve.

computation corresponds to ¹"1.6. For n*5, thecurves all have the same shape. The same behavior isobserved for ¹"0.8, n*6, and for ¹"1.2, n*8 (notshown). In all three cases, curves become almost identicalto one another after just a few periods as H

n(log s) asymp-

totically approaches a self-similar distribution.The self-similarity of H

n(log s) is made explicit by

re-scaling the curves via standardization, which isperformed by computing H(v)"dN(v)/dv wherev"(log s!Slog sT)/p

-0' s. Normalization then requires

Hn(log s)"p

-0' sH (v). The only parameters required in

this scaling are Slog sT and p-0' s

. For an area-preserving#ow such as the one considered here, the mean of logsevolves as Slog sT"!nh. The standard deviation alsoincreases linearly in time, i.e., p

-0' s+nK, where K is

a constant particular for each ¹ value. Therefore, v canbe expressed in terms of n, i.e. v+(log s#nh)/p

-0' s. As

shown in Fig. 6 (inset), when re-scaled in this manner,H

n(log s) asymptotically collapses onto a time-invariant

curve, highlighting the importance of properly predictingthe exponent h. As mentioned for o, this observation haspractical implications: once the STD achieves the self-similar curve, it retains it for all times thereafter, up to thepoint in which di!usion smears striations. Hence, STDsdetermined at early stages of the process can be used topredict STDs at later times, provided that early STDshave already achieved the self-similar regime. This isquite useful, since for long times and in complex #owsystems, striations cannot be examined either by experi-ments or by computer simulations.

Fortunately, striation thickness distributions corre-sponding to large numbers of periods can also be pre-

dicted directly from stretching calculations. Clearly, thesame iterative stretching-and-folding process that gener-ates self-similar density distributions drives the evolutionof striations, and therefore controls the dynamics ofSTDs. Both the shape and the scaling properties of STDsstrongly resemble those exhibited by stretching distribu-tions. An example is presented in Fig. 7, which showsvalues of stretching j computed for 100 000 vectors forthe sine #ow, ¹"1.6, n"5}8. Subsequently, thesevalues were used to compute the probability densityfunction of log H

n(log j)"(1/Nj) dN(log j)/d(log j),

where j is a value of the stretching, Nj is the total numberof vectors, and dN(log j) is the number of values betweenlog j and log j#dlog j. Similarly, to the STDs, distribu-tions of stretching values become wider as n increases,but they retain their shape. The inset shows that suchdistributions are also self-similar and collapse ontoa master curve by exactly the same type of scaling asthe STDs. The variable log j is re-scaled as w"

(log j!Slog jT)/p-0' j. Its scaled distribution H(w)"

dN(w)/dw is then computed, where normalization onceagain requires H

n(log j)"p

-0' jH (w). As shown in theinset, Fig. 7, this simple scaling collapses the distributionsof stretching values corresponding to di!erent times ontoa master curve.

Muzzio et al. (1991) argued that STDs could be com-puted directly from distributions of stretching valuessimply by realizing that as a portion of #uid is stretched,it generates striations with a thickness inversely propor-tional to the amount of stretching applied to the #uid.Moreover, the number of such striations in a portion of#uid, which is proportional to the density o, is also

1506 F.J. Muzzio et al. / Chemical Engineering Science 55 (2000) 1497}1508

Fig. 8. Comparison of the striation thickness distribution calculated directly from simulated "laments (continuous curves) and predicted fromstretching distributions using Eq. (39) (circles) for ¹"1.6, n"5}8.

proportional to the stretching. In other words,

s&1/j, (18a)

ds&!j2 dN(j), (18b)

and

dN(s)&j dN(j), (18c)

where the minus sign in Eq. (18b) simply re#ects the factthat a list ordered in increasing magnitude of s resultsinto a list ordered in decreasing order of j. Eqs.(18a)}(18c) can be simply re-stated in terms of the logar-ithms of s and j as

d(log s)&!d(log j), (19a)

and

dN(log s)&!j dN(log j). (19b)

Finally, one needs to take into account the fact that whilethe number of stretching values remains constant with n,the number of striations, N

s, increases as

NsSoT&SjT. (20)

Eqs. (19a), (19b) and (20) are everything one needs topredict the striation thickness distribution:

Hn(log s)"(1/N

s) dN(log s)/d(log s)

&(SjT~1)jdN(log j)/d(log j)

"(j/SjT)Hn(log j), (21a)

or, in the same format as Muzzio et al. (1991),

Fn(s)"(1/N

s) dN(s)/d(s)&(SjT~1)j3 dN(j)/d(j)

"(j3/SjT)Fn(j), (22)

Fig. 8 compares the STDs with the prediction of Eq. (22).Clearly, an extremely accurate prediction of the striationthickness distribution is obtained. The importance of thisprediction is that direct computations of the STD for assimple a case as the sine #ow require hundreds of hoursof CPU in state-of-the-art computers, and are altogetherimpossible for realistic #ows. Calculations of distribu-tions of stretching values, on the other hand, take onlya few seconds for the sine #ow, and just a few hours forrealistic three-dimensional #ows such as stirred tanks(Lamberto, 1997) or static mixers (Hobbs & Muzzio,1998), thus providing a convenient approach for predic-ting STDs in industrially relevant systems.

Now that the scaling behaviors of Fn(o) and H

n(log s)

are clear, and that the relationships between SoT, SsT,and SjT are apparent, we turn out attention to thediscussion of some of its consequences. In the "rst place,Eq. (16) and (17) have an important physical interpreta-tion: Once o approaches su$ciently the characteristicinvariant and statistical distributions generated by theglobal invariant manifold, it then evolves everywhere atthe same rate as the mean density. In other words, if themean intermaterial area density is doubled, then the localdensity is doubled everywhere. Intimacy of mixing im-proves everywhere by the same factor. This is importantto practitioners because it means that the time evolutionof SoT+enh determines the time evolution of o at alllocations of the chaotic #ow. Similarly, since SsT+SoT~1 both locally and globally, the local average stri-ation thickness decays everywhere at the same rate aspredicted by SsT+e~nh. Therefore, the mixing processcan indeed be characterized by a single quantity, thegrowth of the average intermaterial area density SoT.

F.J. Muzzio et al. / Chemical Engineering Science 55 (2000) 1497}1508 1507

Moreover, h, the topological entropy exponent, can beregarded as a mixing rate not only in a global sense, butalso in a local sense. While this observation is not true innon-chaotic regions, this distinction is immaterial inpractical applications because #ows with large non-cha-otic regions are likely to generate such processing prob-lems that in all likelihood such #ows will be avoided inwell-designed mixing applications.

7. Conclusions

In this paper, the evolution of a material line wascontinuously tracked in a chaotic #ow until it acquireda well-de"ned self-similar structure. Due to the Asymp-totic Directionality property, this structure is determinedby the "eld of asymptotic orientations characteristic ofthe #ow and is for all practical properties identical to thestructure of the global unstable manifold of the #ow.Hence, simulation of the evolution of a material line isakin to a non-perturbative iterative reconstruction of themanifold structure, and the invariance of the manifoldguarantees the self-similarity of the evolved material line.For the sine #ow considered here, and also for other 2D#ows, such evolution occurs very quickly; the materialline is stretched and oriented exponentially fast and ac-quires the properties of the self-similar structure in justa few (less than 10) iterations of the #ow.

As a result of the intrinsic invariance of the "eld oforientations, which generates an invariant manifoldstructure, the evolution of material lines is characterizedby a self-similar distribution of intermaterial area densit-ies that scale homogeneously and are determined solelyby the average intermaterial area density SoT. Since theevolution of SoT can be easily predicted from elementarycalculations of the stretching of point tracers, once theintermaterial area density distribution is known fora given period of the #ow, it can be predicted for allsubsequent periods.

Similarly, striation thickness distributions generatedby the #ow are self-similar and can be rendered invariantby standardization. Moreover, they can be accuratelyand e$ciently predicted from calculations of stretchingvalues, thus providing a convenient means for examiningthe length scale distribution generated by realistic #ows.

References

Alvarez, M. M., Muzzio, F. J., Cerbelli, S., Adrover, A., & Giona, M.(1998). Physics Review Letters, 81(16), 3395.

Aref, H. (1984). Journal of Fluid Mechanics, 143(1), 1.Aref, H. (1990). Philosophical Transactions of the Royal Society of London

A, 333, 273.Beigie, D., Leonard, A., & Wiggins, S. (1991). Physics of Fluids A, 3(5),

1039.Beigie, D., Leonard, A., & Wiggins, S. (1993). Physics Review Letters, 70,

275.Franjione, J. G., & Ottino, J. M. (1987). Physics of Fluids, 12, 3641.Fung, J. C. H., & Vassilicos, J. C. (1991). Physics of Fluids A, 3, 2725.Giona, M., Adrover, A., & Muzzio, F. J. (1998). Journal of Fluid

Mechanics. submitted for publication.Harvey, A. D., & Rogers, S. E. (1996). A.I.Ch.E. Journal, 42, 2701.Harvey, A. D., Lee, C. K., & Rogers, S. E. (1995). A.I.Ch.E. Journal, 41,

2177.Hobbs, D. M., Alvarez, M. M., & Muzzio, F. J. (1997a). Fractals, 5, 395.Hobbs, D. M., Swanson, P. D., & Muzzio, F. J. (1997). Chemical

Engineering Science, 53, 1565.Hobbs, D. M., & Muzzio, F. J. (1997). Chemical Engineering Journal, 67,

153.Hobbs, M. D., & Muzzio, F. J. (1998). Physics of Fluids, 10, 1942.Kim, S. J., & Kwon, T. H. (1996). Advances in Polymer Technology, 15(1),

41.Lamberto, D. J. (1997). Ph.D. Thesis. Rutgers University.Lamberto, D. J., Muzzio, F. J., Swanson, P. D., & Tonkovich, A. Y.

(1996). Chemical Engineering Science, 51, 733.Lawal, A., & Kalyon, D. M. (1995). Polymer Engineering Science, 35(17),

1325.Leong, C. W., & Ottino, J. M. (1989). Journal of Fluid Mechanics, 209,

463.Liu, M., Muzzio, F. J., & Peskin, R. L. (1994). Chaos, Solitons, and

Fractals, 4, 869.Miles, K. C., Nagarajan, B., & Zumbrunnen, D. A. (1995). Journal of

Fluids Engineering } Transactions ASME, 117, 583.Muzzio, F. J., Swanson, P. D., & Ottino, J. M. (1991). Physics of Fluids

A, 3(5), 822.Ottino, J. M. (1990). Annual Review of Fluid Mechanics, 22, 207.Rom-Kedar, V., Leonard, A., & Wiggins, S. (1990). Journal of Fluid

Mechanics, 214, 347.Swanson, P. D., & Ottino, J. M. (1990). Journal of Fluid Mechanics, 213,

227.Zhang, D. F., & Zumbrunnen, D. A. (1996). Journal of Fluids Engineer-

ing } Transactions of ASME, 118, 40.Zumbrunnen, D. A., Miles, K. C., & Liu, Y. H. (1996). Composites A,

27A(1), 37.

1508 F.J. Muzzio et al. / Chemical Engineering Science 55 (2000) 1497}1508