Chaotic Mixing Analyses by Distribution Matrices

Bilder 14.02.2001 9:34 Uhr Seite 119

CHAOTIC MIXING ANALYSES BY DISTRIBUTION MATRICES

Patrick D. Anderson and Han E. H. Meijer

Materials Technology, Dutch Polymer Institute, Eindhoven University of Technology P. O. Box 513, 5600 MB Eindhoven, The Netherlands Fax: x31.40.2447355 E-mail: [email protected] and [email protected]

Received: 31.5.2000, Final version: 7.6.2000

ABSTRACT Distributive fluid mixing in laminar flows is studied using the concept of concentration distribution mapping matrices, which is based on the original ideas of Spencer & Wiley [1], describing the evolution of the composition of two fluids of identical viscosity with no interfacial tension. The flow domain is divided into cells, and large-scale variations in composition are tracked by following the cell-average concentrations of one fluid using the mapping method of Kruijt et al. [2]. An overview of recent results is presented here where prototype two- and three-dimensional time- periodic mixing flows are considered. Efficiency of different mixing protocols are compared and for a particular example the (possible) influence of fluid rheology on mixing is studied. Moreover, an extension of the current method including the microstructure of the mixture is illustrated. Although here the method is illustrated making use of these simple flows, more practical, industrial mixers like twin screw extruders can be studied using the same approach.

ZUSAMMENFASSUNG Die Vermischung von Flüssigkeiten in laminarer Strömung wird anhand der ”Concentration Distribution Mapping Matrice“-Methode, basierend auf den Arbeiten von Spencer & Wiley, untersucht [1]. Durch diese Methode wird die zeitliche Entwicklung der Zusammensetzung zweier Flüssigkeiten gleicher Viskosität ohne Grenzflächenspannung beschrieben. Hierfür wird der Strömungsbereich zunächst in Zellen unterteilt und anschließend wird die Veränderung der Zusammensetzung, durch das Verfolgen der zellengemittelten Konzentrationen eines der beiden Fluide mittels der ”Mapping Method“ nach Kruijt et al., beobachtet [2]. Dieser Beitrag gibt ein Überblick über neuere Resultate zu typischen zwei- und dreidimensionalen, zeitperiodischen Mischungströmungen. Die Effizienz verschiedener Misch- methoden wird verglichen und für ein ausgewähltes Beispiel wird der (mögliche) Einfluss der rheologischen Eigen- schaften der Flüssigkeiten auf den Mischungsvorgang untersucht. Ausserdem wird eine Erweiterung der Methode vorgestellt, die die Mikrostruktur der Mischung miteinbezieht. Obwohl diese Methode hier an einfachen Strö- mungssituationen illustriert wird, kann sie auch zur Untersuchung von mehr anwendungsorientierten, industriellen Mischern, wie dem Doppelwellenextruder verwendet werden.

RÉSUMÉ Le mélange distributif de fluide dans les écoulements laminaires est étudié au moyen du concept de matrices de traçage de distributions de concentration, qui est basé sur les idées originales de Spencer & Wiley [1], décrivant l’évo- lution de la composition de deux fluides avec des viscosités identiques et sans tension interfaciale. Le domaine d’écoulement est divisé en cellules et les variations à grande échelle en composition sont suivies en estimant les concentrations moyennes d’un des deux fluides à l’aide de la méthode de tracé de Kruijt et al. [2]. Une revue des résul- tats récents est ici présentée, où des écoulements de mélange prototypes à deux et trois dimensions et avec une dépendance temporelle périodique sont considérés. Les efficacités des différents protocoles de mélange sont com- parées et pour un exemple particulier, l’influence (éventuelle) de la rhéologie du fluide sur l’action de mélange est étudiée. De plus, une extension de la méthode, où la microstructure du mélange est incorporée, est présentée. Mal- gré le fait que nous donnions ici des exemples où la méthode est appliquée dans le cas d’écoulements simples, d’un point de vue plus pratique, des mélangeurs industriels, comme les extrudeurs à vis jumelles, peuvent être étudiés en utilisant une approche similaire.

1 INTRODUCTION study these flows are overviewed in this paper. It is well known that even for laminar flow at very Fluid mixing processes receive a considerable low Reynolds numbers mixing can lead to com- amount of recognition because of their import- plex flow patterns, and that the exposure of such ance in nature and industry. Although in many patterns may be understood by imploring the cases mixing is associated with turbulent fluid theory of dynamical systems [3]. The motion of motions, mixing of very viscous fluids constitutes passive particles in such a flow is described by the an important class of mixing occurrences. These set of ordinary differential equations, are typical for polymer blending, compounding, food processing etc, and some new models to xu˙ = (,)x t 1

Applied Rheology May/June 2000 119 Bilder 14.02.2001 9:34 Uhr Seite 120

Figure 1: Geometries of the 2D y and 3D cavity. T In the upper figure 1 during the first half of the period the top (back) wall is moved from left to right, -h 0 h x and in the second half of the period the opposite (front) wall moved from left to right. where x = (x, y, z) denotes the position vector and -1 This protocol is denoted as u the local Eulerian velocity. The amount of B the TB protocol. stretching is often used to characterise the qual- The scheme of the flow domain for the cubic cavity ity of mixing: and notation for possible motions of front and back Bu walls is denoted in the lower 2 figure; see text for Br explanation of the where I and L are the present and initial length Bl notation used. Fu of a material filament, respectively [4]. The efficiency of stretching in a two-dimensional flow Bd Fr ranges from 1/t where the flow is steady and the domain is closed, to exponential for a hyperbolic, Fl unbounded, chaotic, flow. In chaotic flows the Fd reorientation prevents the material lines from full alignment with the streamlines, which would lead to linear stretching, and this general phenomena is known as chaotic advection. It is relatively straightforward to generate a flow that original ideas of Spencer and Wiley [1]. Instead of can generate chaos; a necessary condition is in trying to understand all details of mixing, by general the crossing of streamlines. For three- using abovementioned tools from chaos theory, dimensional flows, the efficiency of stretching we use the concept of concentration distribution can even be exponential for steady flow. matrices to study the advection of fluid. Once Computational analysis is an important tool such a distribution matrix is constructed it is very in the quantitative estimation of stretching in simple (fast) to study different mixing protocols practical mixers. Most flows are fully three or the influence of initial conditions. dimensional and time-dependent and numerical Although the method will be exemplified methods are required to obtain the velocity field making use of relatively straightforward two- and the resulting stretching. An example is the and three-dimensional time-periodic prototype Kenics static mixer, a spatially-periodic duct flow, flows, see Figure 1, in our laboratory at present studied by Avalosse [5] using a finite element also more practical, industrial, efficient mixers method. Hobbs and Muzzio [6] showed that the like 3D space-periodic static mixing devices, as stretching distributions in the Kenics mixer are e.g. the multiflux mixer, and 3D time- or space self-similar. Another example is the partitioned- periodic dynamic mixing devices, as e.g. the co- pipe mixer [7] consisting of a rotating pipe parti- rotating, closely intermeshing, self-wiping twin tioned with a sequence of orthogonally places screw extruder, are being studied using the same rectangular plates. The system is characterised approach. Results, that allow for fast optimis- by a parameter B, the ratio of cross-sectional ation processes, e.g. concerning screw design, fall twist to axial stretching, and for b = 0 the flow is outside of the scope of this paper and will be pre- regular and becomes chaotic with increasing sented elsewhere. values of b. Meleshko et al. [8] demonstrated, The following Section discusses some using this system, the importance of an accurate results based on the application of ”classical“ velocity field in studying chaotic mixing flows. dynamical methods applied to a three-dimen- Modern tools of chaos theory – Poincaré sec- sional prototype flow. Results will show that it is tions [4], periodic point analysis [9, 10], and possible to analyse this type of flow, but that it stretching distributions [11, 25] – provide a deep is hard to make a quantitative comparison understanding of the mechanisms that make between mixing protocols. In Sections 3 and 4 the chaotic mixing so effective, even though they do same flow is studied using a technique based on not give a direct description of the final mixture. distribution matrices and now actual compari- The goal of this paper is to present an overview sons are more straightforward. Finally, in Section of some recent results where mixing is studied 5 it is shown that microstructure of the mixture by using a more explicit approach based on the can be added to the computations.

Applied Rheology 120 May/June 2000 Bilder 14.02.2001 10:20 Uhr Seite 121

2 DISTRIBUTIVE MIXING IN TWO- AND The flow in cavity is described by the Stokes THREE-DIMENSIONAL FLOWS equations: Characteristics of chaotic mixing periodic flows are the determination of the location and nature of the periodic points [4]. By definition, periodic 6 points are points which return to their original position after one or more periods. They are clas- supplemented with boundary conditions. Here u T sified according to the nature of the stretching in = (u, v, w) is the velocity, p is the pressure and h their neighbourhood. Elliptic (stable) periodic is the dynamic viscosity of the fluid. For all sta- points are at the centre of non-mixing regions, tionary walls of the three-dimensional cubic cav- called islands, while hyperbolic (unstable) peri- ity, homogeneous Dirichlet boundary conditions odic points are centres of stretching and folding are imposed, while the condition u = 1, v = 0, w=0 in the flow. Around hyperbolic points the fluid is is applied for the moving front wall (the velocity compressed in one eigendirection and stretched field for Fr is obtained). The numerical solution of in the other eigendirection. The importance of Equations 6 for the two- and three-dimensional periodic points is clear; if a flow has many peri- flow problems is performed by a projection odic points of hyperbolic nature, and no elliptic scheme that decouples the pressure p and vel- periodic points (or only a few with very small sur- ocity u[12]. A spectral element approximation [13, rounding islands) we expect that the flow 14] is used for the spatial discretization and yields exhibits excellent mixing. an accurate representation of the velocity field. In general any motion can be represented, The cavity is subdivided into 15 * 15 * 15 spectral based on the solution of Equation 1 by elements, each of sixth order in all three directions, leading to 753,571 nodal points and a total 3 system with 2,868,477 degrees of freedom. The resulting system of equations is solved using a mapping particle X to x after a time t, e.g. [4], conjugate gradient solver with a finite element where Xrefers to the initial position at t = 0. Using preconditioner. this definition, a periodic point P of order n is In a Stokes approximation, the result of the defined as fluid deformation caused by the wall movement is completely defined by the dimensionless dis- 4 placement of the wall D, defined here as the wall displacement divided by the half of the cavity 5 edge length (which equals 1 in dimensionless coordinates). The relatively simple geometry T is the duration of one period of motion. forms a convenient model for testing the tech- Figure 1 shows the geometry of the time- niques designed to study three-dimensional periodic mixing devices discussed in this paper. laminar time-periodic distributive mixing. Due to On the bottom part of the figure a two-dimen- symmetry properties of the flow, the velocity sional cavity flow is depicted (studied in a later field has to be computed only for one type of the section). We first introduce the three-dimen- wall motion (in this case Fr). The velocity fields sional cubic cavity flow on the lower part of Fig- for other wall motions can be obtained using a ure 1 with fixed top, bottom, and side walls, and simple coordinate transformations. where the motion is generated by moving the In Anderson et al. [10] algorithms to locate front and back walls. In the notations the bold and classify periodic points in 3D mixing flows are letters Fand Bdenote a front or back wall motion, presented, and it is observed that for 3D flows while the subscript denotes the direction (r, l, u, these points form open or closed lines. The struc- and d stand for right, left, up and down respect- ture of the lines can be complex and their type ively). Thus, for example, Fr corresponds to the can change along a line. The four-step protocol front wall moving to the right. Compounded pro- FrBuFlBd was introduced to study three-dimen- tocols, like FrBuFlBd, should be read from left to sional effects in mixing, and the flow is induced right. by the successive motion of the two opposite

Applied Rheology May/June 2000 121 S.122122/2 14.02.2001 10:4210:50 Uhr Seite 122

Figure 2: Periodic lines in the four-step 3D flow: 1 thin – unstable (hyperbolic) lines; thick – stable (elliptic) lines.

Upper plot for D = 3; z 0 lower plot for D = 5. (Reproduced from Anderson et al.[10] with

permission of Cambridge 1 of 258 nodes. The first blob is positioned around 1 University Press). 1 0 an elliptic point, the second one around a hyper- 0 1 1 bolic point. The results for the case D = 5 are pre- y x sented in Figure 3. The difference in deformation between the two blobs is large and the blob 1 around the hyperbolic point is advected through the whole domain, while the stretching of the blob around the elliptic point is very small. For

z 0 the blob around the elliptic part of the line, the interfacial area remains nearly constant; however, for the hyperbolic blob a substantial

1 increase in area is observed. 9689 nodes are 1 1 0 required to describe the final deformation after 0 1 1 three periods of flow. For both material blobs the y x volume is preserved within a percent after three periods of mixing. Using the definition of the deformation gra- T walls in crossed directions as depicted in Figure 1. dient FX = (—XFT(X)) with F as defined in Equa- The net resulting displacement of each wall, after tion 3 a useful definition of a stretching coeffi- one complete period, is zero. cient at a periodic point can be defined: The first-order periodic lines found are plotted in Figure 2. Hyperbolic parts of the line are plotted thin, elliptic lines are displayed thick. For 7 both dimensionless displacement D-parameters three periodic lines are found. In the first case, where s(FX) is the eigenvalue spectrum of FX. D = 3, two completely elliptic lines are revealed Such an analysis is in particular of import- and one periodic line consisting of one elliptic ance when we are interested in the local behav- part and two hyperbolic tips. In the second case, iour of periodic flows. The presence of low-order D = 5, the periodic structure consists of three lines hyperbolic periodic points is a clear indication with mixed types of periodic points. In Galak- that a mixing flow generates an exponential tionov et al. [15] symmetry concepts of this flow increase of interfacial area (albeit in a limited are discussed and rotational symmetry of the part of the flow domain). The stretching coeffi- periodic lines is proven. cient at such a point is a measure for the rate of The presence of hyperbolic periodic points in mixing at the periodic point. If one has to favour the flow is important since it provides informa- one of the two mixing protocols, D = 3 or D = 5, tion of stretching. To emphasise the importance the periodic point analysis may help to compare of the periodic structure analysis, the motion of these flows. However, guideness how to com- two blobs is calculated for a few periods. An pare these protocols is unavailable. One might adaptive front tracking model is used to follow compare the arclength of the periodic lines con- the deformation of the material volumes in an sisting of hyperbolic points, and try to use the Figure 3 a, b, c, d efficient way [16]. The initial sphere is covered stretching coefficient at these points, but a sim- (fromf the two material with an unstructured triangular mesh consisting ple quantitative comparison based on the first- volumes after one, two, and th left to right): 3a shows two material blob placed around a hyperbolic and an elliptic periodic point for the FrBuFlBd protocol with D = 5. Sub-figures (b), (c), and (d) show the deformation oree periods, respectively.

Applied Rheology 122 May/June 2000 AR 3 14.02.2001 10:24 Uhr Seite 123

Figure 4 (upper): Initial sub-domain discretization 10 * 6 grid; and deformed grid (below) after displacement of the top wall by two times its length. During the actual computations a finer 200 * 120 was used.

ution of, for example, fluid concentration in the original grid is then mapped to obtain a new distribution. The area of the intersections of the deformed sub-domains with the original ones, determine the elements of the mapping (or distribution) matrix y, where yij equals the fraction of the deformed sub-domain Wj at time t = t0 + ∆t that is found in the original (t = t0) sub- domain Ωi

8 The polygonal descriptions of the sub-domains order results is still lacking. The maximum are used to determine the matrix elements yij, stretching coefficient equals 3.37 at the point and the accuracy of these elements is deter- (0.264, 0.469, 0.068) for the protocols D = 3, and mined by the accuracy of the velocity field and for the flow with D = 5 the coefficient equals 10.1 the error tolerances defined in the adaptive front at the point (–0.394, –0.458, 0.377). Based on this tracking procedure. For details on the validation information the higher dimensionless displace- and accuracy of the mapping method we refer to ment seems to be favourable. A more extensive Kruijt et al.[2]. The matrix y is essentially sparse comparison of protocols becomes a cumbersome if ∆t is not too large [2]. Where y will generally repetition and the need arises for a more flexible be a sparse matrix, yn will generally be dense, method to distinguish a large numbers of mix- since fluid from an initial subdomain will finally ing protocols, the topic of the remainder of this be advected throughout the flow-domain. This paper. makes studying the properties of yn, as originally proposed by Spencer and Wiley [1], both unat- tractive and even impossible for exponential 3 MAPPING METHOD mixing flows. Therefore, instead of investigating n (n) (n) The ”mapping“ method is proposed based on the y , e.g. C could be investigated. C , the con- original ideas of Spencer & Wiley [1], and the centration distribution after n mapping steps, is main idea is not to track each material volume in computed by the sequence for i = 1 to n the flow domain separately, but to create a discretized mapping from a reference grid to a 9 deformed grid. Instead of tracking the boundaries of material volumes a set of distribution C(0) is the initial concentration distribution. This matrices are used to advect concentration in the procedure does not change the matrix y and is, flow. Within the mapping method a flow domain therefore, much cheaper, both in number of Ω is subdivided into N non-overlapping sub- operations as well as in the computer memory n domains Ωi with boundaries ∂Wi, see Figure 4 needed, than computing y (top). The boundaries ∂Wi of these sub-domains The computation of the matrix elements yij are represented by polygons and tracked from t is time consuming ; more than 100 hours using a = t0 to t = t0 + Dt using an adaptive front track- 12 CPU multiprocessor 225 Mhz R10K Silicon ing model [16], and, as a result, deformed poly- Graphics™ for a 200 * 120 grid. However, once the gons are obtained, see Figure 4 (bottom). Note matrices are obtained matrix–vector multiplica- that the subdivision of Ω into Ωi’s is not related tions are extremly fast, less than 1 sec on a single to any finite differences, or finite element dis- CPU 100 Mhz Pentium™, and a large number of cretization used to solve the velocity field in Ω. mixing protocols are easily compared.To differ- Next, a discretized mapping from the initial grid entiate between the quality of mixing protocols, to the deformed grid is constructed. The distrib- mixing measures are applied.

Applied Rheology May/June 2000 123 Bilder 14.02.2001 10:11 Uhr Seite 124

Figure 5: D = 5 Comparison of different 0 10 mixing protocols. TTTT TBTB TBTBBTBT 1 10 TBTB

2 10

3 10 Here, the intensity of segregation I is used, Intensity of segregation 4 which is a property of the mixture that is effect- 10 ed by molecular diffusion. The intensity of seg-

5 regation is a second-moment statistic (variance) 10 0 10 20 30 40 50 60 of the concentration distribution and is defined Number of wall displacements as [17]:

D = 7 0 10 10 TTTT TBTB TBTBBTBT 1 2 10 TBTB where sc is the variance in concentration over an entire domain Ω, defined as:

2 10 11

3 10 and c(x) is the concentration in a point x ŒWin

Intensity of segregation the domain. The brackets < > denote an aver- 4 Ω 10 aging over the quantity in the brackets in Ω. Intensity of segregation is a measure of the devi- 5 10 0 10 20 30 40 50 60 ation of the local concentration from the ideal Number of wall displacements situation (i.e. when the mixture is homogeneous). A value of I equal to zero means no intensity, where a value of unity means a maxi- D = 9 0 10 mum intensity (only black and white fluids and TTTT TBTB no gray fluid). TBTBBTBT 1 10 TBTB 3.1 2D MAPPING METHOD

2 10 To demonstrate the strength of the mapping method, we first consider the two-dimensional

3 10 cavity flow as depicted in Figure 1 (top). The mapping method is a flexible tool for studying and Intensity of segregation 4 10 optimising of mixing processes, since it makes it possible to incorporate variations on an existing

5 mixing protocol without having to recompute 10 0 10 20 30 40 50 60 Number of wall displacements the entire protocol. As a first illustration, two parameters of the cavity flow are varied: the dimensionless displacement D and the number of wall movements n; the product of which is pro- D = 12 0 10 portional to energy input. Different protocols are TTTT TBTB compared by changing the order of consecutive TBTBBTBT 1 10 TBTB mappings. Four different protocols are investigated. In

2 10 protocol TTTT only the top wall moves, avoiding periodicity in the flow, and a linear mixing flow

3 results. In protocol the top wall ( ) and bot- 10 TBTB T tom wall (B) move alternatingly, in the directions Intensity of segregation 4 as depicted in Figure 1 (originally proposed by 10 Franjione et al. [18]). Protocols TBTBBTBT and TB- are variations on the second protocol pro- 5 T-B 10 0 10 20 30 40 50 60 posed [4, 19, 20] to reduce the regularity of the Number of wall displacements protocol, thus decreasing the size of regions

Applied Rheology 124 May/June 2000 Bilder 14.02.2001 10:11 Uhr Seite 125

Figure 6: Concentration distributions for the TB-T-B protocol with D = 9. Initially, the left half of the cavity is filled with white fluid and the right half with black fluid. After already four periods the fluids are reasonable mixed except for t=1T t=2T a relatively large zone in the core of the flow. The fluid in this weak zone is only slowly interchanged with the fluid outside this zone.

t=4T t=6T

t=8T t=9T

t=10T t=15T

around elliptic points (symmetry breaking pro- for the top wall, but rotated (x and y values mul- tocols). The minus-signs in the last protocol tiplied by –1 in the coordinate system). Wall denote a negative wall movement direction. movement in the opposite direction and move- For these four protocols the concentration ment of the opposite wall, can be computed by distribution is studied using the mapping matri- mirroring (multiplying the x and/or y-coordinate ces: yD=0.25, yD=1, yD=2 and yD=4 which are con- by –1 in the coordinate system of Figure 1), re- structed for the first protocol TTTT (considering ducing the number of required matrices. For the a movement of the top wall only). Combinations mapping calculations, a grid of 120 * 200 rec- of these mapping matrices are used to obtain tangular cells covers the interior of the cavity, deformations for larger displacements. For and extends from –0.995h < x < 0.995h and example, to obtain the concentration distribu- –0.999 < y < 0.999, leaving a single, very thin cell tion after 7.5 wall displacements the matrix multi- around the boundary. This boundary cell reduces (0) computational expenses, by avoiding the neces- plications yD=4yD=2yD=1yD=0.25yD=0.25C are applied. Since the geometry is symmetric (with sity to track points that pass very close to the cor- the origin in the centre of the cavity), the map for ner singularities. a movement of the bottom wall is the same as

Applied Rheology May/June 2000 125 Bilder 14.02.2001 10:11 Uhr Seite 126

Figure 7: D = 5 Shear-thinning results for 0 the TBTB flow. 10

1 10

2 The four protocols are compared and results 10 for four different dimensionless displacement Intensity of segregation Newtonian are presented, i.e. D=5, 7, 9, and 12. Figure 5 shows Carreau, n = 0.6 Carreau, n = 0.2

the decrease in intensity of segregation versus 3 10 the number of wall displacements for each proto- 0 10 20 30 40 50 60 Number of wall displacements col. Except for the protocol with D = 7 it is observed that the linear flow TTTT performs worst. From earlier studies [13, 4] it is known that D = 7 0 the protocol TBTBhas a large island for Dapproxi- 10 Newtonian mately between 5 and 7. The fluid within the Carreau, n = 0.6 island is isolated and cannot reach the chaotic Carreau, n = 0.2 mixing zone. As a result the intensity of segre- 1 gation remains about 10-1 for D = 7. The other pro- 10 tocols provide better mixing and a faster rate at which these protocol reach the minimum level of

2 intensity of segregation (a noise level resulting 10

from small numerical errors). For most systems Intensity of segregation in Figure 5 it is noted that after some initial transient effects a constant decrease of intensity of 3 10 segregation is observed until the minimum level 0 10 20 30 40 50 60 is reached. Except for the TB-T-B protocol with Number of wall displacements D = 9 and we examine this system in more detail in Figure 6. D = 9 It shows the fluid concentrations after some 0 10 integer number of periods. After one period of Newtonian Carreau, n = 0.6 flow many striations are already observed, but Carreau, n = 0.2 also large areas of white and black fluid are seen. 1 A weak zone appears to be present in this flow 10 which only slowly interchanges fluid outside this zone, explaining the behaviour of the intensity

of segregation as observed in Figure 5. 2 10

Figure 7 shows some effects of the rheology Intensity of segregation of the fluid on mixing. Here the Carreau model is used with zero infinite-shear-rate viscosity: 3 − 10 ηη=+()λγ˙ 2 ()n 12 0 10 20 30 40 50 60 0 1 ( ) 12 Number of wall displacements

where n is the power-law parameter, l a time h constant and 0 is the zero shear-rate viscosity. D = 12 0 10 The following dimensionless numbers charac- Newtonian terise the shear-thinning nature of the flow: Carreau, n = 0.6 Carreau, n = 0.2 • The shear-thinning index or power-law 1 parameter n, 10 • The Carreau number Cr = (l U)/L, which can be regarded as a dimensionless shear rate, 2 where the transition from the Newtonian 10 plateau to the shear-thinning region takes Intensity of segregation place, and is fixed Cr = 5, meaning that shear-

thinning behaviour is clearly present in the 3 10 flow, and the fluid behaves like a power-law 0 10 20 30 40 50 60 Number of wall displacements fluid.

Applied Rheology 126 May/June 2000 Bilder 14.02.2001 10:11 Uhr Seite 127

Figure 8: The concentration distribution for D = 3 after a total of 100 dimensionless wall displacements is plotted on two cross-sections of the cubic cavity. Below it the result for D = 5 is presented.

The results presented in Figure 7 are for the TBTB flow and the power-law parameter is varied, i.e. n = 1, 0.6, 0.2. Here the grid of interior cells extends from –0.985h < x < 0.985h and –0.985 < y < 0.985, reducing computational expenses, by avoiding the tracking of points that pass very close to the corner singularities. For D = 5 the Newtonian flow (n = 1) provides the highest rate of mixing, but for the other dimensionless displacements the shear-thinning fluids seem to reach the minimum level of intensity of segregation faster. For D= 7 the rate of mixing is about equal for n = 0.6 and n = 0.2, where larger differences are observed for D= 9 and D=12. These results can be partially explained by studying the velocity field in the cavity flow [21]. It is observed that as n decreases shear-thinning reduces the influence of the moving walls in the core of the cavity. This may lead to better or worse mixing as shown in Figure 7.

3.2 3D MAPPING METHOD In Galaktionov et al. [22] a full three-dimensional extension of the mapping method is introduced. Basically the method is similar to the two- dimensional version, except that now a three-dimensional velocity field is required, and that the mapping matrices are constructed using for the FrBuFlBd flow for D = 3 and D = 5 that the a subdivision of the domain in bricks. These sub- latter one is probably a better mixing flow. To val- domains are advected for a time interval ∆t and idate or question this conclusion, half of the cav- the mapping matrix coefficients are found by ity is filled with black fluid, and the other half evaluating the intersections of deformed bricks with white fluid. For both dimensionless dis- with the original ones. Details and validation of placements the concentration distribution after the 3D model, and implementation of it, are avail- a total of 100 wall displacements (so 8 1/3 periods able in [22]. for the D = 3 system, and 5 periods for D = 5) the The strength of the model is that again once concentration is plotted in Figure 8. In order to the mapping matrices are computed (which is visualise the mixing patterns, gray-scale plots of now really time consuming), concentration dis- the concentration distribution at two selected tributions are found very quickly. Several proto- cross-sections of the cavity are presented. The cols can be easily compared this way and the volume (and not just in the two cross sections) cubic cavity flow is studied again. Only the mapp- averaged intensity of segregation equals 0.0664 ing matrices corresponding to the movement Fr for D = 5 and 0.146 for D = 3 as expected from the are computed. Mappings for all other motions periodic point analysis. Figure 8 shows a rather are easily obtained by using the symmetry of the large light gray-scaled unmixed region for D = 3, velocity field and flow domain. For example the where the fluid concentration seems to be bet- mapping Fl can be presented as SxFlSx, where Sx ter distributed for D = 5. is the symmetry operator Sx(x, y, z) = (-x, y, z). An interesting question that arises is Similar transformations yield all other types of whether a larger D will always provide better wall motion considered here. mixing. A small dimensionless displacement In Section 2 we concluded from the stretch- makes it impossible to have effective stretching ing values computed on the periodic lines found and folding in the flow, whereas a very large

Applied Rheology May/June 2000 127 Bilder 14.02.2001 10:11 Uhr Seite 128

Figure 9:

0 Comparison of different 10 dimensionless wall-displacements for the

four-step mixing protocol. 1 10

2 10

D = 3 D = 5 Intensity of segregation D = 7 tion could also be this large. But in most cases we 3 10 D = 9 D = 12 D = 15 desire a mixture whose final scale of segregation D = 20 -6 D = 25 is of the order of 10 m.

4 10 Galaktionov et al. [26] presented a global, 0 50 100 150 200 250 300 350 400 Total dimensionless wall displacement multi-scale model for liquid-liquid mixing in laminar flow, called the extended mapping method. The model is limited to passive mixing, where the two fluids have identical rheology and zero inter- dimensionless displacement will finally result in facial tension, and where diffusion is negligible. pure linear mixing. So if we expect bad mixing It is build on the original mapping method and for small and large D’s there should be a (local) treats the macroscopic problem, and to this cal- optimum in between. The mapping method is culation microstructure is added. Classical con- applied to validate this statement. For D = 3, 5, 7, tinuum mechanics provides the smallest length 9, 12, 15, 20, and 25 the decrease in intensity of scale: a ”material point“. In fact this is an aver- segregation is studied as a function of total age over a region that is larger than the molecu- dimensionless wall displacement. The results are lar scale but much smaller than the finest stri- presented in Figure 9. It is found that for an ation. To this we add a second division, at a length increasing D a higher rate of mixing is observed scale that we will call the cell size. We call fea- until D = 15. For D = 20 and D = 25 higher values tures of the mixture above the cell sizethe macro- of intensity of segregation are found indicating scopic aspects of the mixture, while features that a local optimum is found around D = 15. smaller than the cell size are the microscopic In a similar way it is easy to compute mixing aspects. Variables at both the macroscopic and measures for different mixing protocols with varying dimensionless wall-displacements. In microscopic scales are chosen to represent the Galaktionov et al. [22] two-step, four-step, and state of mixing, and develop evolution equations eight-step mixing protocols are compared, and for those variables. some indications for optimal D are given. In the At the macroscale the smallest object of next section a proposed extension of the algo- interest is a cell, and the largest is the entire mix- rithm is described that adds microstructure of ture. The major goal at this level is distributive the mixture to the macroscopic mapping mixing – achieving an even distribution of the method. components. At the micro-scale, the smallest object of interest is a point and the largest is a cell. The goal at this level is to create in each cell 4 EXTENDED MAPPING METHOD: a microstructure that imparts desirable proper- COUPLING OF LENGTH SCALES ties to the mixture. Here we need variables that describe the average microstructure within a cell. Few models have treated mixing in a global, The classical measures of scale and intensity of multi-scale manner. Chella & Viñals [23] used a segregation [17] are micro-scale descriptors, as Navier-Stokes/Cahn-Hilliard formulation to track c(x) globally (as used in Equation 11), includ- are droplet size distributions and striation thick- ing interfacial tension effects, but this calcula- ness distributions. A key issue in choosing micro- tion must stop when the striation thickness scale variables is evolution: the variables must approaches the size of the numerical mesh. Cal- contain sufficient information to allow their new culations of the distribution of stretching values values to be calculated after any given deform- [11, 24, 25] represent some features of the local ation of the mixture. Scale and intensity of seg- microstructure and treat the entire mixture, but regation do not offer this capability, nor does stri- do not describe distributive mixing. ation thickness alone, so these are not suitable Constructing an explicit, global model is dif- variables for a mixing model. ficult, because of the enormous range of length A useful generalisation of the interfacial scales involved. The length scale of the chamber area measure, which includes orientation, is the in a typical mixer for polymer melts is on the area tensor [27]. The second-order area tensor A order of 10-1m, and the initial scale of segrega- is defined as

Applied Rheology 128 May/June 2000 Bilder 14.02.2001 10:11 Uhr Seite 129

Table 1: Three microstructures and Geometry A their corresponding area tensors. Spherical 130 0   SV  0130  0013

For passive mixing, each part of the interface 1200 Cylindrical   dG deforms affinely with the material. This can S 0120 V   be used to derive a differential evolution equa-  000 tion for the area tensor [27]:

 100 Lamellar   In this equation the dot indicates a material SV 000 000 derivative, and L is the velocity gradient tensor. ¿ is the fourth-order area tensor, defined as

To integrate Equation 15 for any type of deformation a closure approximation for Equation 16 is required and an accurate closure called the RE 13 (rational ellipsoidal) closure is proposed by Wet- zel and Tucker [27] and uses a rational polynomi- The components of the area tensor have units of al approximation for the area tensor compon- interfacial area per unit volume, or (length)-1, and ents associated with ellipsoidal droplets. A major the trace of the tensor equals the total interfa- advantage of the area tensor is that it provides a cial area per unit volume compact description of complex microstructures capturing information on the length scale, the 14 geometric nature and the spatial orientation of the microstructure. In the extended mapping so trA is a useful scalar measure of microstruc- method microstructure is added to the mapping tural mixing. The averaging volume V should be method using the area tensor. The cell volume W large enough to provide a representative sample i serves as the averaging volume for the area ten- of the microstructure, but smaller than the scale sor. For cell i the associated area tensor is over which the microstructure varies. The area tensor captures information about the size, shape, and orientation of the micro- 17 structure. Table 1 shows the area tensors for three particular microstructures. These three ex- Here G is the interfacial surface lying within Ω . amples are limiting cases, and the area tensor can i i Because Ωi is explicitly chosen and has finite vol- represent any intermediate structure as well, ume, the cell area tensorA is a coarse-grain vari- such as a group of identical ellipsoidal droplets. i able, like C. For each non-zero entry yij in the The interfacial area of a non-spherical droplet is mapping matrix, we compute an inverse deform- distributed non-uniformly in space, and this is ation gradient tensor F -1. This is evaluated at the captured by the area tensor. The principal axes of ij centroid of Ωj|tk+1 » Ωi|tk, the intersection the tensor will also be co-axial with the principal between the deformed cell j and the undeformed axes of the droplet, so the area tensor describes cell i. This computation is carried out off-line, at the droplet orientation. Any other microstruc- the same time the mapping matrix y is being ture that possesses a distinct interface also pos- computed, and the results are stored for later sesses an area tensor, so even complex structures use. The extended mapping method now like co-continuous morphologies can be repre- updates the area tensor at each time step accord- sented. The area tensor determines the contri- ing to bution of interfacial tension to bulk stress [28], and has been used as the state variable in rheo- logical theories of polymer blends [29, 30, 31]. 18

Applied Rheology May/June 2000 129 Bilder 14.02.2001 10:11 Uhr Seite 130

Figure 10 a-f: Evolution of concentration (left) and trace of the area tensor (right) distributions in the flow described by protocol TB-T-B with dimensionless displacement D = 8. Marker fluid initially fills the left half of the cavity. The results are shown after 2, 4, and 8 periods of the flow. (g, h): Similar to (c) and (d), but marker fluid initially fills the lower half of the cavity. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 3 3.5 4 10 a 10 b

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2.5 3 3.5 4 4.5 5 5.5 6 6.5 10 c 10 d

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 8 8.5 9 9.5 10 10.5 11 11.5 12 10 e 10 f

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2.5 3 3.5 4 4.5 5 5.5 6 6.5 10 g 10 h

Applied Rheology 130 May/June 2000 Bilder 14.02.2001 10:11 Uhr Seite 131

That is, the area tensor in any cell at time k f are identical in appearance, though the two fig- + 1 is the sum of contributions from all donor cells, ures use different scales for their gray level maps. after the donor tensors from time k have been The pattern after only two periods, Figure 10b, is transformed by the appropriate deformation only slightly different from the self-similar pat- gradients. The symbol ƒ denotes the transform- tern in Figure 10d and f. ing of the area tensor under finite strain and is For a globally chaotic flow, this self-similar used to convert an initial area tensor A0 to an pattern of the interface distribution is also inde- equivalent droplet shape tensor G0, find the pendent on the initial configuration of the mix- droplet shape tensor Gin the deformed state, and ture. Figures 10g and h show the distributions of then transform G back to find the deformed- concentration and of trA after four periods of the state area tensor A. Equations 9 and 18 constitute same flow, but for a different initial condition. one step of the extended mapping method. All Here the dark fluid initially occupies the lower details of the conversion between A and G and half of the cavity, and the initial interface is hori- on the validation of the extended mapping zontal. Although the concentration pattern in method can be found in Galaktionov et al. [26] Figure 10gis slightly different from Figure 10c,the interface distribution in Figure 10h is identical to 4.1 APPLICATION TO THE 2D CAVITY FLOW: Figure 10d. The average value of trA in Figure 10h SELF-SIMILARITY OF THE MICROSTRUCTURE is slightly higher than in the previous case, because the initial interface was longer. In Galak- For the globally chaotic flow TB-T-B with D = 8 tionov [26] the evolution of total interface is concentration distribution results are shown in studied in more detail. Figure 10. For this protocol, the striation thickness of the emerging lamellar mixture pattern quickly become too fine to resolve with the basic 5 CONCLUSIONS mapping technique using a 200 * 120 grid. Intrin- sic numerical errors, caused by averaging on the An overview of recent work studying liquid-liquid cell scale at every mapping step, also tend to chaotic mixing by distribution matrices is pre- erase the fine structure of the mixture. As a sented in this paper. The computation of the result, after only a few periods the computed mapping matrix is time consuming and its com- concentration distribution is nearly uniform (Fig- puting can be significantly reduced by parallel- ure 10e). This is a desirable mixing result on the isation of the numerical code. Once the matrix is macroscale, but in Figure 10e the pattern of cell computed the evolution of the composition of concentrations no longer reveals anything use- fluid concentration can be studied very fast com- ful about the state of the mixture. pared to the computation of the mapping matrix. The extended mapping method compen- It is shown that the mapping method can be used sates for this loss of information by tracking the to study a large variety of 2D and 3D mixing microstructure within each cell. These results are protocols, and that protocol parameters can be shown in Figure 10, parts b, d, e, and f. Here we optimised with respect to the mixing measure show the values of log(trA) within each cell, for chosen. The results are presented for simplified different times during the mixing process. A geometries, but the method is general and can logarithmic scale is used because the area tensor be applied to a wide range of mixing problems. distribution is very non-uniform. Although the con- The strength of the method is that it models mix- centration distribution quickly becomes nearly ing directly. One can specify the initial config- uniform, the microstructure continues to evolve uration of the two fluids, subject them to a pre- as mixing proceeds. In addition, the mixture scribed amount of mixing, and predict the remains highly structured at the microscale, and concentration distribution at every point in the the interface distribution is highly non-uniform. resulting mixture. Moreover, it is possible to A self-similar pattern of interface distribu- extend to algorithm not only to map concentration is established after a few periods. This pat- tion of fluid, but also to include microstructure tern is maintained for all subsequent mixing, to the mapping: the extended mapping method. while the average value of the trace of the area The extended mapping method not only tensor grows exponentially. Thus, Figure 10d and provides explicit prediction of mixing perform-

Applied Rheology May/June 2000 131 Bilder 14.02.2001 10:11 Uhr Seite 132

ance, but also important fundamental infor- BIOGRAPHY mation about the flow itself. The calculation Han Meijer received his Ph.D. degree from the reveals the self-replicating pattern of interface University of Twente in 1980 with Prof. J.F. Ingen orientation that arises in periodic flows. When Housz as his supervisor. He joined DSM research, the flow is globally chaotic, this pattern displays and was active in the area of Polymer Processing an exponential growth rate for interfacial area. Modelling and Explorative Research. In 1985 he Spatial analyses of the one-period dynamics for became part-time professor at the department interface stretching can be performed, to reveal of Polymer Chemistry and Technology in the area which parts of the flow field are most effective of applied rheology. In 1989 he became full pro- at small-scale mixing. fessor in Polymer Technology in the Division of One disadvantage of the present calculation Computational and Experimental Mechanics of is that it is subject to numerical diffusion. This the Department of Mechanical Engineering. He restricts its quantitative accuracy, especially has been chairman of the Dutch Society of Rhe- when studying the long-time behaviour of a ology since 1995 and is currently president-elect flow. However, the short-time behaviour is accur- of the Polymer Processing Society. Patrick Ander- ately predicted, and we expect that the relative son graduated in Applied Mathematics at the performance of different mixing protocols will be University of Eindhoven in 1994 under the super- predicted quite accurately, provided one uses the vision of Prof. A.A. Reusken. He received his Ph.D. same size mapping steps for both protocols. This degree in the research group Materials Technol- makes the extended mapping method a useful ogy lead by Prof. H.E.H. Meijer and Prof. F.P.T. engineering tool. Baaijens where he studied distributive mixing A topic of the current research in our la- processes. After a six month leave at Océ Tech- boratory, is the extension of the mapping nologies he joined the Materials Technology method technique to more complex, industrial, group as an Assistant Professor. mixers. Examples include the multiflux static mixer, described by Sluijters [32], an example of a three-dimensional space-periodic flow, and the closely intermeshing, corotating twin screw extruder, which can be regarded as either time- or space-periodic. Moreover, an experimental set-up of the cubic cavity flow will be used to validate the computational results for e.g. the four-step mixing protocol, and to study the influence of visco-elastic effects on mixing.

ACKNOWLEDGEMENTS Financial support for this work was provided by the Dutch Polymer Institute. Apart from the research staff in our group, the authors want to thank Prof. Charles Tucker of the University of Illi- nois at Urbana-Champaign for helpful discus- sions concerning the area and shape tensors.

Applied Rheology 132 May/June 2000 Bilder 14.02.2001 10:11 Uhr Seite 133

REFERENCES [21] P.D. Anderson, O.S. Galaktionov, G.W.M. Peters, F.N. van de Vosse, and H.E.H. Meijer: Mixing of non-Newtonian [1] R.S. Spencer and R.H. Wiley: The mixing of very viscous fluids in time-periodic cavity flows, J. Non-Newt. Fluid liquids, J. Colloid Sci. 6 (1951) 133-145. Mech. (2000) Accepted. [2] P.G.M. Kruijt, O.S. Galaktionov, P.D. Anderson, G.W.M. [22] O.S. Galaktionov, P.D. Anderson, P.G.M. Kruijt, G.W.M. Peters, and H.E.H Meijer: Analyzing fluid mixing in Peters, and H.E.H. Meijer: A mapping approach for 3D periodic flows by distribution matrices, AIChE J. (2000) distributive mixing analysis, Computers & Fluids (2000) submitted. Accepted. [3] H. Aref: Stirring by chaotic advection, J. Fluid Mech. 143 [23] R.Chella and J.Viñals: Mixing of a two-phase fluid by cav- (1984) 1-21. ity flow, Phys. Rev. E 53 (1996) 3832-3840. [4] J.M. Ottino: The kinematics of mixing: stretching, chaos [24] F.J. Muzzio, C.Meneveau, P.D. Swanson, and J.M. Ottino: and transport, Cambridge University Press (1989). Scaling and multifractal properties of mixing in chaot- [5] Th. Avalosse: Simulation numerique du melange lami- ic flows, Phys. Fluids 4 (1991) 1439-1456. naire par elements finis, PhD thesis, Université [25] M.Liu, R.L. Peshkin, F.J. Muzzio, and C.W. Leong: Struc- Catholique de Louvain, Belgium (1993). ture of the stretching field in chaotic cavity flows, AIChE [6] D.M. Hobbs and F.J. Muzzio: Effects of injection location, J. 40 (1994) 1273-1286. flow ratio and geometry on Kenics mixer performance, [26] O.S. Galaktionov, P.D. Anderson, G.W.M. Peters, and AIChE Journal 43 (1997) 3121-3132. Charles L.Tucker III: A global, multi-scale simulation of [7] H.A. Kusch and J.M. Ottino: Experiments on mixing in laminar fluid mixing: The extended mapping method, continuous chaotic flows, J. Fluid Mech. 236 (1992) 319- Int. J. of Multiphase Flows (2000) Submitted. 348. [27] E.D. Wetzel and C.L. Tucker: Area tensors for modeling [8] V.V. Meleshko, O.S. Galaktionov, G.W.M. Peters, and microstructure during laminar liquid-liquid mixing, Int. H.E.H. Meijer: Three-dimensional mixing in Stokes flow: J. Multiphase Flow 25 (1999) 35-61. the partitioned pipe mixer problem revisited, Eur. J. of [28] C.E. Rosenkilde: Surface-energy tensors, J. Math. Phys. 8 Mech. / Part B - Fluids 18 (1999) 783-792. (1967) 84-97. [9] V.V. Meleshko: Steady Stokes flow in a rectangular cav- [29] M. Doi and T. Ohta: Dynamics and rheology of complex ity, Proc. Roy. Soc. London A 452 (1996) 1999-2002. interfaces I, J. Chem. Phys. 95 (1991) 1242-1248. [10] P.D. Anderson, O.S. Galaktionov, G.W.M. Peters, F.N. van [30] M. Grmela and A.Ait-Kadi: Rheology of inhomogeneous de Vosse and H.E.H. Meijer: Analysis of mixing in three- immiscible blends, J. Non-Newt. Fluid Mech. 77 (1998) dimensional time-periodic cavity flows, J. Fluid Mech. 191-199. 386 (1999) 149-166. [31] N.J. Wagner, H.C. Õttinger and B.J. Edwards: Generalized [11] F.J. Muzzio, P.D. Swanson, and J.M. Ottino: The statistics Doi-Ohta model for multiphase flow developed via of stretching and stirring in chaotic flows, Phys. Fluids 3 GENERIC, AIChE J. 45 (1999) 1169-1181. (1991) 822-834. [32] R.Sluijters: Het principe van de multiflux menger, Inge- [12] L.J.P. Timmermans, P.D. Minev, and F.N. van de Vosse: An nieur of Chemische Techniek 77 (1965) 33-36 (in Dutch) approximate projection scheme for incompressible flow using spectral elements, Int. J. Numer. Meth. Fluids 22 (1996) 673-688. [13] P.D. Anderson: Computational Analysis of Distributive Mixing, PhD thesis, Eindhoven University of Technology, The Netherlands (1999). [14] Y.Maday and A.T. Patera: Spectral element methods for the incompressible Navier-Stokes equations, In A.Noor, editor, State-of-the-Art surveys on computational mechanics, ASME, New York (1989). [15] O.S. Galaktionov, P.D. Anderson, and G.W.M. Peters: Symmetry of periodic structures in a 3D mixing cavity flow, Phys. Fluids 12 (2000) 469-471. [16] O.S. Galaktionov, P.D. Anderson, G.W.M. Peters, and F.N. van de Vosse: An adaptive front tracking technique for three-dimensional transient flows, Int. J. Numer. Meth. Fluids 32 (2000) 201-217. [17] P.V. Danckwerts: The definition and measurement of some characteristics of mixtures, Appl. Sci. Res. A 3 (1953) 279-296. [18] J.G. Franjione, C.Leong, and J.M. Ottino: Symmetries within chaos: A route to effective mixing, Phys. of Fluids A 1 (1989) 1772-1783. [19] S.C. Jana, G.Metcalfe, and J.M. Ottino: Experimental and computational studies of mixing in complex Stokes flows: The vortex mixing flow and multicellular cavity flows, J. Fluid Mech. 269 (1994) 199-246. [20] H.Aref and M.S. Naschie: Chaos Applied to Fluid Mixing, Elsevier Science Ltd., London (1995).

Applied Rheology May/June 2000 133