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Droplet dynamics in mixed flow conditions: Effect of

shear/elongation balance and viscosity ratio

Elia Boonen, Peter Van Puyvelde, and Paula Moldenaers

Citation: Journal of Rheology (1978-present) 54, 1285 (2010); doi: 10.1122/1.3490661

View online: http://dx.doi.org/10.1122/1.3490661

View Table of Contents: http://scitation.aip.org/content/sor/journal/jor2/54/6?ver=pdfcov

Published by the The Society of Rheology

Articles you may be interested inDynamics of collapsed polymers under the simultaneous influence of elongational andshear flowsJ. Chem. Phys. 135, 014902 (2011); 10.1063/1.3606392

Effects of viscosity ratio and three dimensional positioning on hydrodynamic interactionsbetween two viscous drops in a shear flow at finite inertiaPhys. Fluids 21, 103303 (2009); 10.1063/1.3253351

Effect of confinement and viscosity ratio on the dynamics of single droplets duringtransient shear flow

J. Rheol. 52, 1459 (2008); 10.1122/1.2978956

Effects of shear flow on a polymeric bicontinuous microemulsion: Equilibrium and steadystate behaviorJ. Rheol. 46, 529 (2002); 10.1122/1.1446883

Three-dimensional shape of a drop under simple shear flowJ. Rheol. 42, 395 (1998); 10.1122/1.550942

Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP:

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Droplet dynamics in mixed flow conditions: Effect ofshearelongation balance and viscosity ratio

Elia Boonen, Peter Van Puyvelde,

a)

and Paula Moldenaers

Department of Chemical Engineering and Leuven MRC, Katholieke Universiteit

Leuven, Willem de Croylaan 46, 3001 Leuven, Belgium

(Received 14 July 2009; final revision received 12 August 2010;

published 5 October 2010

Synopsis

The dynamics of single droplets dispersed in a second, immiscible liquid undergoing a controlledmixture of shear and elongational flow has been studied using a home made eccentric cylinder

device. The model system consists of polydimethyl siloxane droplets in a polyisobutylenematrix, both Newtonian liquids at room temperature. In continuation of previous work Boonenet al., Rheol. Acta 48, 359371 2009, the effect of changing the balance of shearing andelongational flow components and varying viscosity ratio on the deformation and orientation of

the droplets has been systematically investigated under sub-critical flow conditions. The

experimental results obtained from optical microscopy are compared with theoretical predictions of

the phenomenological model by Maffettone and Minale J. Non-Newtonian Fluid Mech. 78,227241 1998, obtained using the transient form of the model and incorporating a flow typeparameter that accounts for the relative amount of extension in the flow. Overall, a fair agreement

was found between the model predictions and the experimental results for all sub-critical mixedflows applied and all viscosity ratios investigated here. This work provides an experimental

reference data set which can be used to guide future modeling efforts. 2010 The Society ofRheology. DOI: 10.1122/1.3490661

I. INTRODUCTION

The behavior of droplet dispersions is of interest for many industrial applications, e.g.,

food emulsions, cosmetics, pharmaceuticals, and polymer blending. An understanding of

the droplet dynamics in such systems helps explaining the rheological behavior of theflowing emulsions, such as viscoelasticityeven if the components are Newtonianand ashear-dependent viscosity. Conversely, from the rheology of the dispersion substantial

information about the morphology development can be obtained e.g., Vinckier et al.1996. Since the pioneering work ofTaylor1932,1934, a large number of experimen-tal and theoretical studies on droplet dispersions have been carried out. These are well

described in several recent reviewsGuido and Greco2004;Ottinoet al.1999;Stone1994; Tucker and Moldenaers 2002. Most studies have focused on the structuredevelopment in either purely elongational flow or simple shear flow. However, since most

aAuthor to whom correspondence should be addressed; electronic mail: [email protected]

2010 by The Society of Rheology, Inc.1285J. Rheol. 546, 1285-1306 November/December2010 0148-6055/2010/546/1285/22/$30.00


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industrial processing operations consist of complex mixtures of shear and elongation, asystematic investigation of the microstructure evolution in such mixed flow conditions

is considered to be very relevant.

In our previous workBoonen et al.2009, a summary has been given of the priorresearch conducted on the morphology development of droplet dispersions in controlled

complex flow fields. Important to recall here is the work of Leal and co-workersBentleyand Leal1986;Stoneet al.1986;Stone and Leal1989, who explored a whole rangeof two-dimensional2D flows using a computer controlled four-roll mill. They studieddroplet deformation and break-up in different types of steady flow, ranging from simple

shear to pure 2D-elongation, and provided a systematic set of data for the droplet dy-

namics, in these complex flow types.Godbille and Picot2000and Khayatet al.2000studied the influence of shear and elongation on drop deformation and break-up inconvergent-divergent channels. They found that the initial droplet diameter is of major

importance for the droplet deformation in this specific geometry, and different break-up

mechanisms, driven by shear and extensional flows, respectively, were identified. An

alternative set-up to study the morphology development in controlled mixed flow condi-

tions is the so called eccentric Couette system, as shown in Fig.1.This type of flow has

previously been studied extensively in fluid mechanics e.g., Ballal and Rivlin 1976;Diprima and Stuart 1972; Wannier 1950 and has received renewed interest fromWindhab and co-workers Feigl et al. 2003; Kaufmann et al. 2000; Windhab et al.

2005.Feiglet al.2003,for instance, investigated, through experiments and numericalsimulations, the drop deformation and break-up when only the inner cylinder rotates at aconstant speed. However, they neglected the wall effects in their study, which may be

questionable, in view of the ratio of droplet diameter to gap size usedseeVan Puyveldeet al.2008;Vananroyeet al.2006. Finally,Egholmet al.2008used a rather similargeometry to explore droplet dynamics in complex flow conditions. Their flow channel

consists of two concentric cylinders with toothed walls as a model for extruding flow.

They reported that for small deformations, the relation between the time-averaged drop

deformation and a time-averaged apparent shear rate can be described by Taylors small

deformation theory. Also, numerical simulations agreed fairly well with the experimental

results, although the calculations predict a somewhat higher deformation than experimen-

tally observed.

Here, we use a newly designed eccentric cylinder device ECD to study the defor-mation and orientation of single Newtonian droplets dispersed in a Newtonian matrix

undergoing controlled mixed flow conditions. Details of the cell can be found inBoonen

et al. 2009. For such a system with matching fluid densities and Newtonian compo-

FIG. 1. a Schematic of eccentric cylinder device. b Sketch of top view of eccentric cylinder geometry.

1286 BOONEN, VAN PUYVELDE, and MOLDENAERS


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nents, only three dimensionless parameters play a role for the dynamics in slow flows: theviscosity ratio p =d /m, where d is the viscosity of the drop fluid and m is the

viscosity of the matrix, the capillary numberCa which represents the ratio of viscous to

interfacial tension stresses, and the flow type which can be represented by the flow type

parameter see Sec. II.Cacan be defined asmR0E /, whereR0is the radius of theundistorted spherical droplet, is the interfacial tension, and E is the flow intensity

which equals42 + 2 for 2D flows. InBoonenet al.2009,the experimental results fora viscosity ratio of p = O1 have been compared with predictions obtained using thetransient form of the model ofMaffettone and Minale1998 adapted to complex flowssee also Sec. II. Under sub-critical flow conditions, good agreement was found betweenmodel predictions and experimental data, providing a quantitative assessment of dropshape predictions in controlled complex flows. In the present study, a systematic inves-

tigation is made of the effect of viscosity ratio on the droplet dynamics for a wide range

of viscosity ratiosranging from 0.1 to 10. In addition, we will explore more extremeflow conditions with an enhanced extensional contribution as compared to the previous

studyBoonen et al.2009.

II. EXPERIMENT

A. Materials

The materials used in this work are polyisobutylene PIB Glissopal from BASFand polydimethyl siloxane PDMS Rhodorsil from Rhodia. SinceGuidoet al.1999reported that PIB is slightly soluble in PDMS, PIB is chosen as the matrix phase. In order

to vary the viscosity ratio, three different grades of PDMS have been selected as the

droplet phase. TableI gives the zero shear viscosities0and the corresponding viscosity

ratios p at 23 C. All fluids are transparent at room temperature and exhibit Newtonian

behavior over the range of strain rates investigated here. The interfacial tension for the

PIB/PDMS system was measured by three methods small deformation theory, dropletretraction measurements, and pendant drop technique and was found to be2.80.1 mN/m. Moreover, this value was measured to be independent of the molecular

weight for the three grades of PDMS used here. All experiments were performed at

ambient temperature23 C; as the viscosity of PIB is very sensitive to temperature,the temperature of the sample was directly monitored by immersing a fine thermocouple

needle in the continuous phase. The viscositiesm and d, and the resulting viscosity

ratio p could be back-calculated using an Arrhenius equation.

TABLE I. Physical properties of component fluids.

Material

023 CPa s

p =PDMS/PIB23 C

Activation energy Ea

kJ/mole

PIB 51.6 Matrix 60.1

Glissopal 1300

PDMS 60.4 1.17 15.4

Rhodorsil 47V60.000

PDMS 501.6 9.72 15.3

Rhodorsil 47V500.000

PDMS 5.10 0.10 15.2

Rhodorsil 47V5000

1287DROP DYNAMICS IN MIXED FLOWS


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The density difference between both polymers is quite smallabout 0.08 g/cm3 sothat in combination with the high viscosity of the matrix, gravitational effects can be

neglected. Furthermore, the difference in refractive indices is high enough to have good

contrast for observations by optical microscopy.

B. Methods

The experiments are performed using a home built ECD which is equipped with a

microscope and camera system to enable visualization of droplet dynamics in controlled

complex flows. A detailed discussion of the experimental set-up has been given in

Boonen et al.2009. In Fig.1a, a schematic of the basic ECD is shown. In brief, theapparatus allows applying controlled mixed flow conditions, i.e., a combination of

shear and elongational components, in the narrowing and expanding areas of the gap

between the rotating, eccentrically positioned cylinders. Figure1bshows a 2D sketch of

the geometry, consisting of an inner and an outer cylinder with radii Ri and Ru andangular velocities iand u, respectively. The axes of these cylinders can be displaced by

a certain distancee, the eccentricity. The eccentricity ratioXof the system is then defined

as e / Ru Ri. This is an important, adjustable parameter that influences the relative, aswell as the absolute magnitudes of shear and elongational strain rates along the stream-

lines of the flow field. In addition, the flow configuration, i.e., inner cylinder rotating,

outer cylinder rotating, counter-rotating, or co-rotating, also has a profound effect on the

type of mixed flow obtainedsee further. For the experiments conducted in this study,the ECD has been used in two configurations: configuration A in which only the outer

cylinder is rotating with a constant speed, and configurationB where both cylinders are

co-rotating with the ratio of outer to inner cylinder velocity

u/

i equal to 1.5. For bothconfigurations the eccentricity ratioXis set to 0.2. In configurationA, a shear dominated

type of flow is obtained, while configurationB produces a flow with an enhanced exten-

sional contribution, which allows exploring more extreme flow conditions.

Single PDMS droplets with initial radiiR0 in the range of 300700 m were intro-

duced in the matrix material by using a home made injection system. The spherical drop

is injected in the widest part of the gap and positioned so that its center is at a radial

distance around 30 mm half the clearance from the axis of the inner cylinder. Uponstart-up of the flow, the droplet deformation and orientation is visualized during several

revolutions using optical microscopyOlympus SZ61-TR stereo microscope. The micro-

scope, equipped with a CCD cameraBasler A301f, is mounted on top of the flow cellso that images are captured in the velocity-velocity gradient planesee Fig.2 using theStreampix Digital Video Recording SoftwareNORPIX. Due to the limited field of viewof the microscope and difficulties in illumination, images of the deforming drop could, in

a single experiment, only be captured in the first twoIand IIor the last twoIIIand IVquadrants of the Cartesian coordinate system, defined in Fig.1b.In order to obtain theshape of the droplet, an ellipse is fitted to the recorded images after performing basic

image processing, including a convolution operation and threshold using IMAGEJsoftware

for WindowsRasband 1997. Throughout the experiments, the flow intensity is keptbelow the critical conditions for breakup, so that no significant deviations from the

ellipsoidal shape are present. Using this approach, it is then possible to obtain the defor-

mation and orientation of the droplets, i.e., the lengthL of the long axis, the lengthB of

the short axis, and the anglebetween the long axis and the flow direction, as shown in

Fig.2.

The experimental results are compared with the predictions of the MaffetoneMinale

modelMaffettone and Minale1998,1999, which has been applied to complex flows



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Boonenet al.2009. This phenomenological model assumes that during flow the dropshape is ellipsoidal at all times and that its volume is preserved. The drop shape can

therefore be described by a symmetric, positive definite second order tensor S, which

eigenvalues represent the square semi-axes of the ellipsoid. The evolution equation ofS ,

resulting from the competing actions of the hydrodynamic forces and the interfacial

tension, is given byMaffettone and Minale1998

dS

dtWSSW=

f1MM

pSgMMSI+ f2

MMp,CaDS+SD, 1

where the characteristic time equals mR0 /, I is the second rank unit tensor, andD

and W are the deformation rate and vorticity tensor of the flow field. The characteristic

timewill be used to define a dimensionless time t= t /in which t is the time since the

start-up of the flow. The scalar functiongMM is required to preserve drop volume, while

the functions f1MM and f2

MM have been determined to recover the asymptotic analytic

limitsMaffettone and Minale1998;Taylor1934:

f1MM = 40p+ 13 + 2p16 + 19p

f20 =

5

3 + 2p, f2

c =3Ca2

2 + 6Ca2+1

1 + p2,

f2MM =f2

0 +f2c ,

gMM =3IIIS

IIS. 2

Here, IISis the second scalar invariant ofS, and and are the small positive numbers

which, for all practical purposes whenCa is not too large andp is far from infinity, are

set to zero. The flow field generated in the ECD can be represented by a deformation rate

and vorticity tensor of the form

FIG. 2. Schematic view of a deformed droplet.



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D=G

1

20

1

2

0

0 0 0

, W=G

0 1

20

1 +

2

0 0

0 0 0

. 3

Here, the flow type parameter expresses the relative amount of elongation present in

the flow and is defined asFeigl et al.2003

=

+=

G , 4

with G the sum of absolute shear and elongation rates and 11. Mixed flow

conditions between eccentric cylinders can thus be represented by the flow strengthG

tand the flow parametertwhich in general vary in time. In this work, we will then referto the MM model as the model predictions according to the evolution equation, Eq.1,with the parameters given by Eq.2 and with the flow field represented by Eq.3.

III. RESULTS AND DISCUSSION

In Sec. III A and III B, the experimental results on droplet deformation obtained in

two types of sub-critical complex flows with different relative contributions for shear and

elongation, and for three viscosity ratios, are presented and discussed. These results will

be compared to the model predictions of the MM model as described above to help

elucidate the effect of flow type and viscosity ratio in these mixed flow conditions.

A. Effect of shear/elongation balance

First, the dynamics of droplets with a constant viscosity ratio of about 1.2 in two

different types of mixed flow are considered: shear dominated flow in configurationA of

the ECD Figs. 3 and 4 and a flow field with an enhanced extensional componentobtained in configurationB Figs.5 and6. In both cases, the transient droplet deforma-tion is observed in the first two quadrants of the flow fieldFig.1b for a number ofrevolutions during start-up of the flow. Furthermore, by varying the velocities of the

cylinders and changing the initial droplet radiusR0, a range of capillary numbers can be

explored.

1. Configuration A: Shear dominated flow

Due to the motion through the ECD, the drop is subjected to a periodic flow field.

Hence a new characteristic time-scale, the oscillation period of the imposed flow, comes

into play. The droplet dynamics as reported in this paper are not influenced by the

presence of this additional time-scale. For instance, in the experiments presented in Fig.

3, the droplet relaxation time is about 0.7 s, whereas the ECD rotation period is 250 s.

These time-scales are well separated indicating that the droplet can rapidly respond to the

applied deformation and that the droplet shape is fully determined by the deformation at

that instant.

For the shear dominated flow of configurationA , the results at a viscosity ratio of 1.2

have been discussed in detail elsewhereBoonen et al. 2009. Typical profiles for thestrain rates and the corresponding flow parametersCat and t for a PDMS drop witha representative initial radius of 375 m are shown in Fig.3.The shear rate is seen to



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reach a sharp maximum when the gap is at its narrowestpointC and displays a broadminimum in the widest part of the gap point A. These features also show up in theCa-profile shown in Fig. 3b. Here, the flow conditions are mostly shear dominated,which is also evident from the values of the flow type parameter in Fig. 3c. Theaverage amount of elongationavg,III over the first two quadrantsIII in Fig.1b is

FIG. 3. Example of flow parameters for configuration A with i =0, u = 0.025 rad/s and eccentricity ratioX=0.2, along streamline starting at x= 30 mm from the axis of the inner cylinder:astrain rate profiles, shearand elongation;band cevolution ofCa and for a PDMS drop with initial radius of 375 m;dexample of a deforming droplet at different points along the trajectory.



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about 10% of the sum of strain ratesG. The flow type parameter logically follows thesame profile as the elongational strain rate, shown in Fig.3a:both exhibit a symmetricprofile with a positivestretching maximum and an equal but opposite, negativecom-pression minimum in the converging and diverging parts of the gap. The results fordroplet deformation under these flow conditions are summarized in Fig.4together with

the model predictions according to the MM model for complex flowsEqs.13.It is observed that the experimental deformation parametersL /2R0and B /2R0, and the

orientation angle show a steady oscillation in response to the applied, time periodic

capillary numbersCat see Fig.3ii. Furthermore, good quantitative agreement is foundbetween the experimental results and the model predictions. Quantitative deviations only

start to occur at Ca-numbers that are expected to be near the critical conditions for

break-up. It should be noticed that the experiments are characterized by the average

capillary number Caavg over one rotation rather than by the maximum and minimum

values Camax and Camin, as inBoonenet al. 2009.

2. Configuration B: Enhanced extensional contribution

For configurationB, in which both cylinders are co-rotating with the ratiou /iset to

a value of 1.5, the flow field comprises an enhanced extensional contribution. Here, the

flow conditions are drastically different from the one previously discussed, as shown in

Fig.5. In this case, the shear ratesee Fig.5a displays a minimum peak at the pointwhere the gap is the narrowestpointCand a broad maximum in the widest part of thegappointA. For this specific flow configuration, in some parts of the gap, the velocityprofiles in the radial direction appear the show a nonmonotic behavior reaching an ex-

tremum value. This could explain this counterintuitive finding that the shear rate is

FIG. 4. Droplet dynamics as a function of dimensionless time t for the flow conditions of Fig. 3 and for a

viscosity ratio ofp 1.2. The symbols represent the experimental results, whereas the lines are the predictionsaccording to the MaffettoneMinale model.



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FIG. 5. Example of flow parameters for configuration B with i =0.02, u = 0.03 rad/s and eccentricity ratioX=0.2, along streamline starting at x= 35 mm from the axis of the inner cylinder:astrain rate profiles, shear and elongation;b and c evolution ofCa and for a PDMS drop with initial radius of 500 m.

FIG. 6. Droplet dynamics as a function of dimensionless time t for the flow conditions of Fig. 5 and for a

viscosity ratio ofp 1.2. The symbols represent the experimental results, whereas the lines are the predictionsaccording to the MaffettoneMinale model.



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maximum in the widest part of the gap. The elongation rate on the other hand shows the

same type of symmetric profile as in Fig.3a,being positive in the converging part andnegative in the diverging part of the gap. The main difference as compared to the previ-

ous configuration is that the extensional strain rate has the same order of magnitude as the

shear rate during a substantial part of the profile. This results in relatively large values forthe flow type parameter shown in Fig.5c. This parameter follows the same sym-metric profile as in Fig. 3c, but with significantly higher maximum values of about0.50 in quadrants II and III. As a consequence, the average amount of elongation

avg,III in the first two quadrants is now increased to approximately 30% of the sum of

strain rates, as compared to only 10% for the previous configuration. The profile for the

capillary numberCa also differs substantially from that in Fig.3,as can be seen in Fig.

5b. The Ca-number is more or less constant in the first quadrant of the flow field,subsequently drops rapidly to reach a sharp minimum at the end of quadrantIIpointC,and symmetrically continues in quadrantIIIand IV. It will be verified now to what extent

the drastically changed flow field will affect the droplet dynamics.Figure6shows the results for droplet deformation observed in quadrants IIIduring

start-up, as well as the model predictions according to the MM model. It can be seen that

the deformation parametersL /2R0and B /2R0, and the orientation angleagain seem to

follow an oscillating pattern for subsequent revolutions, in nice agreement with the model

predictions. In this case, however, the deformation strongly relaxes near the end of the

second quadrant as shown by the decrease and increase especially for the lower Cavaluesof the major axisL and the minor axisB of the droplet, respectively. Furthermore,the orientation angle starts to increase at the end of the second quadrant, indicating the

droplet rotates away from the velocity direction. This decreased deformation and orien-

tation corresponds to the steep decline of the capillary numberCa at the end of quadrant

IIsee Fig.5b. As the Ca-number drops, the driving force for deformation and orien-tation of the droplets decreases, and due to the action of the interfacial tension the droplet

relaxes to a less deformed and less oriented state. For the orientation angle, the model

predictions even indicatefor this range ofCa-numbersthat the droplet rotates past 45,which is the asymptotic limit for small Ca in simple shear flow Taylor 1934. Thispeculiar effect cannot be explained entirely by the evolution of Ca. To investigate this

phenomenon more thoroughly, experiments were also performed in the third and fourth

quadrants of the flow field, the droplet still being injected at pointA.

The results are shown in Fig.7;the relaxation of the droplet is clearly confirmed in the

evolutions ofL /2R0 and B /2R0 depicted in Figs.7a and7b. In addition, the orienta-tion angle quickly rises from the velocity direction toward the perpendicular direction to

a maximum value above 45, as predicted by the MM model. This effect can be attributed

to the local change in flow type the droplet experiences in this flow field. At the end of

the second quadrant the elongation rate changes sign and becomes negativecompressionflow, while the shear rate remains quite low for some timesee Fig.5a. As a conse-quence a local flow field with a large, negative-value exists for some time. This tends

to rotate the drop toward the 90 limit, resulting in a maximum orientation angle above

45. In summary, the balance of elongational and shear components in the flow field

clearly affects droplet dynamics in transient mixed flow conditions. Furthermore, it has

been shown that this effect can be correctly predicted by the MM model.

B. Effect of viscosity ratio

It is known that the viscosity ratio is a crucial parameter for droplet dynamics in

simple flow conditions. Here, we want to investigate the effect of the viscosity ratiop on



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droplet dynamics in mixed flow conditions. Hereto, droplets with viscosity ratios of

O10 and O0.1 have been observed during the type of complex flow obtained inconfigurationB of the ECD with enhanced extensional contributionsee Fig.5.

1. High viscosity ratio drops

Figure8shows the droplet dynamics for drops with a high viscosity ratio of p 10.Note that observations have only been made in the first two quadrants of the flow field,

and results are for low to intermediateCa higher Ca-numbers will be discussed further

on. Qualitatively, the same behavior is observed as for a viscosity ratio around 1.2seeFig. 6. The deformation and orientation increase in the first part of the flow field andstart to relax again near the end of the second quadrant, when the capillary number

quickly drops to a lower value see Fig. 5. It seems though that the relaxation of thedroplet starts somewhat later for this high viscosity ratio case as compared to p 1.2.This is more clearly illustrated in Fig.9 where the droplet dynamics for p 1.2 and p 10 are compared for about the same average Ca-number. As expected, the high vis-cosity ratio drops are less deformed lower L, higher B. They are also slightly moreoriented toward to the flow directionlower . Furthermore, nice agreement is obtainedbetween the model predictions and the experimental results for the low to intermediate

Ca-range for these high viscosity ratio drops.

For higherCa, as depicted in Fig.10,deviations with respect to the predictions of the

MM model start to appear for the major and minor axes of the drop, while the predictions

for the angles are still reasonable. Furthermore, for these high capillary numbers the

effect of p on the orientation angle becomes clearer, as shown in Fig. 11. Here, the

experimental results for the orientation of droplets with p around 10 and 1.2 are com-

FIG. 7. Droplet dynamics as a function of dimensionless time t for the flow conditions of Fig. 5 and aviscosity ratio of p 1.2, with observations made in quadrants III and IV of the flow field. The symbolsrepresent the experimental results, whereas the lines are the predictions according to the MaffettoneMinalemodel.



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pared for an average capillary number of about 0.33. A high viscosity ratio obviously

promotes orientation toward the flow direction, something that is also seen in simpleshear flowe.g.,Rumscheidt and Mason1961. If we apply even higherCa-numbers, asthe ones shown in Fig.12, quantitative deviations between model predictions and experi-

mental results grow even larger for L /2R0 and B /2R0, while the angle predictions still

coincide with the experimental data very well. Qualitatively, the predictions for the major

and minor axes still describe the experimental results. In addition, Fig.12illustrates that

at this high viscosity ratio, the amplitude of the oscillating deformation parameters varies

in time, as also seen in the model predictions. This peculiar transient oscillatory behavior

for high p in mixed flow conditions is similar to the damped oscillatory behavior ob-

served during start-up of simple shear flow for high viscosity ratios e.g., Torza et al.

FIG. 8. Droplet dynamics as a function of dimensionless time t for the flow conditions of Fig. 5for a viscosityratio ofp 10 and intermediate capillary numbers Ca . The symbols represent the experimental results, whereasthe lines are the predictions according to the MaffettoneMinale model.

FIG. 9. Comparison of droplet deformation and orientation for the flow conditions of Fig. 5, for a viscosityratio ofp 10 and p 1.2, respectively, for the same average capillary number Caavg. The symbols representthe experimental results, whereas the lines are the predictions according to the MaffettoneMinale model.



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1972and is not fully understood at present. Note that in Fig.12in addition to the MMmodel predictionsrepresented by the solid linesalso model predictions according to theso called Minale model corresponding to the dashed lines are shown. The Minalemodel Minale 2004 is an extension of the MaffettoneMinale model to account forelastic effects in the matrix and/or droplet phase. It has an extra term in the evolution

equation of the droplet shape tensorS with the coefficient f3. This extra term allows for

oblate droplet configurations, and for instance, enables this model to predict the widen-

ing behavior seen forp1 in transient simple shear flowCristini et al.2002. In the

FIG. 10. Droplet dynamics as a function of dimensionless time t for the flow conditions of Fig. 5 for aviscosity ratio of p 10 and high capillary numbers Ca. The symbols represent the experimental results,whereas the lines are the predictions according to the MaffettoneMinale model.

FIG. 11. Comparison of droplet orientation for the flow conditions of Fig. 5,for viscosity ratios ofp 10 andp 1.2, respectively, for high capillary numbers Ca.



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case of Newtonian components studied here,Minale2004 showed that the predictionsfor steady simple shear flow almost coincide with the MM model over a large range of

Ca and only differ at very high capillary numbers. The evolution equation for S for

Newtonian components is given as

dS

dtWSSW=

f1MM

pSgMinaleSI+ f2

Minalep,CaDS+SD

+f3

Minale

R0p,CaDSS+SSDDS+SD S:I

3,

where the extra coefficient f3 depends on the appliedCa and the viscosity ratio p notethat gMinale is again a term to preserve the volume and f2

Minale is a slightly adapted form

of f2MM using a weighting factor for f2

c, as defined in Eq. 2. In our study at highviscosity ratio and high capillary number, the extra term in the Minale model seems to

give different predictions as compared to the MM model, as shown in Fig.12.From this

figure, it can be concluded that the Minale model predictions show somewhat better

agreement with the experimental profiles ofL /2R0

and B /2R0, although not quite satis-

factorily yet. For the orientation angle on the other hand, the predictions of the Minale

model and the MM model are practically on top of each other and agree very well with

the experimental values.

Considering both models in more detail, one might suspect that the discrepancy is

related to the behavior of another almost mutual coefficient of both models: f2. The

FIG. 12. Droplet dynamics as a function of dimensionless time t for the flow conditions of Fig. 5 for aviscosity ratio of p 10 and very high capillary numbers. The symbols represent the experimental results,

whereas the lines are the predictions according to the MaffettoneMinale model solid line and the Minalemodeldashed line.



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parameters f2MM and f2

Minale depend on p as well as Ca and consists of two termsseeEqs.2. One term, f2

0, is determined to recover the asymptotic limits ofTaylor1934.The other term, f2

c, is an empirical factor added by Maffettone and Minale to improve the

predictions in simple shear flow and used in the Minale model with a weighting factor of

0.75. The factor f2c contains two empirical constants and , which for all practical

purposes, whenCa and p are far from infinity, have been set to zero by Maffettone and

Minale. In theory though, these parameters have to be small, positive numbers to be able

to recover the asymptotic limits of p, and of affine behavior for p =1 and Ca,

respectively. This could mean that, in our case of highp, a small but nonzero value of the

parameter

has to be chosen in order to reflect the behavior of the high viscosity ratiodroplets. In Fig.13,we have verified the effect of the parameter on the predictions of

droplet dynamics by the Minale model for the high viscosity ratio dropsp 10 for alarge Ca. By assigning a small positive value to 103 and 102 in Fig. 13, thepredictions forL /2R0and B /2R0are indeed improved as compared to the normal MM

and Minale model predictions. For the predictions of the orientation angle, on the other

hand, hardly any effect of is observed. This indicates that it is the high viscosity ratio

effect that causes deviations from the MM model predictions because we are reaching the

asymptotic limit for highpfor which a small, nonzero value of the parameter has to be

set in the coefficient f2MM of the model.

Furthermore, it is knowne.g., Grace1982 that in simple shear flow when p4 asteady deformation limit, independent ofCa, exists for largeCa. In that case, the defor-

mation parameter D of the droplet, defined as L B / L +B, tends toward a constantvalue equal to 5/ 4pand independent ofCaTaylor1934. To check if similar behav-ior can occur in transient mixed flows, the deformation results for several large, but

differentCa-number experiments forp = O10, are compared in Fig.14.Comparing the

FIG. 13. Effect of parameter on droplet dynamics for the flow conditions of Fig. 5, a viscosity ratio of p 10 and large capillary number Ca. The symbols represent the experimental results, whereas the lines are thepredictions according to the MaffettoneMinale model solid line, the Minale model dashed line, and theMinale model with =0.001dotted line and =0.01dash-dotted line.



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results for two experiments for subsequent revolutions, the deformation, in terms ofL and

B see Fig.14a, appears to level off to the same value, independent ofCa, similar tothe limiting behavior seen in steady simple shear flow forp4. In the case of mixed flow

conditions, Fig.14bdemonstrates the evolution of the deformation parameterD, for thesame two experiments shown in Fig.14a. Here, two additional experiments have beenincluded with about the same average capillary numbers as the first two ones, to illustrate

the reproducibility. Inspecting Fig.14b, the deformation indeed seems to level off to asteady value. To show this more clearly, the average values of the experimental defor-

mation parameter D

avg

were calculated using 0t

Dtdt/

t for each revolution afterstart-up of the flow. In Fig.14c, Davg is plotted as a function of the number of cycles.Obviously, apart from the initial start-up transients, the average deformation for all four

experiments appears to go to a constant value, although the observations were only made

for a limited number of cycles. To confirm this steady limiting behavior for high p, the

experiments were repeated for a larger number of revolutions, the results of which are

shown in Fig.15. In this figure, clear evidence is given for the limiting behavior for high

viscosity ratiosp = O10 in mixed flow conditions. Note that for steady mixed flowswith a constant value of the flow type parameter, we can also derive a steady defor-

mation limit forp from the analytical solution of the MaffettoneMinale modelseeAppendix. This is represented by the solid line in Fig.15for a steady 2D flow with aconstant value of

equal to the average value

avg present in the first two quadrants of

the flow field. Clearly, the experimental deformation limit is close to the limit for steady

2D flows obtained from the MM model. The existence of a steady deformation limit for

high viscosity ratios in mixed flow conditions was already reported byBentley and Leal

1986 for steady flows with a constant -value. For instance, they found that for

FIG. 14. Limiting behavior of droplet deformation for the flow conditions of Fig. 5, a viscosity ratio of p 10 and large capillary numbers Ca:aDimensionless droplet axes L /2R0 andB /2R0.b Droplet deforma-tion parameter D= L B / L +B. c Average deformation parameter Davg as a function of the number ofcycles.



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0.36, a limiting deformation exists for p-values larger than about 27. In the presentcase, however, this steady limiting behavior is already observed forp 10. Probably, thisdifference is related to the transient nature of the mixed flow applied with the eccentric

cylinder device. Nevertheless, the average deformation observed for high p in the tran-

sient complex flow of the ECD is approximately the same as for a steady mixed flow with

the sameaverage value of the flow type parameter.

2. Low viscosity ratio drops

In the last part of this study, the effect of a low viscosity ratio on the droplet dynamics

in complex flows was investigated, withp-values of about 0.1 at 23 Csee TableI. Theresulting dynamics for these low viscosity ratio drops are shown in Fig. 16for low to

intermediate capillary numbers Ca and in Fig.17for high Ca. Again, qualitatively the

same profiles as for p 1.2 are obtained in Fig.16compare with Fig.6; the deforma-tion and orientation increase during the first quadrant of the flow field and subsequently

relax near the end of the second quadrant. As before, good quantitative agreement is

found between experimental results and predictions according to the MM model. Devia-tions only start to appear for higherCa, as illustrated in Fig.17.In thisCa-range we are

near the critical conditions for droplet break-up where the MM model obviously fails. In

fact, for the highestCa explored here, it was found experimentally that the droplet indeed

breaks up after many revolutionsoutside the time scale of Fig.17, as indicated by thearrows in theL /2R0-profile of Fig.17.

To quantify the effect of viscosity ratio in more detail, the results for p 0.1 and p 1.2 are compared in Fig.18.Note the difference in time-scales which is due not onlyto differences in revolution period but also to differences inR0 and m for the different

experiments. These differences only become visibleand pronounced at larger tmorerevolutions. First, Fig. 18a shows the comparison for intermediate Ca Caavg 0.2;here, no significant effect of the low viscosity ratio is visible, and the deformation and

orientation are approximately at the same level for both viscosity ratios. In Fig. 18bresults are given for a higher Ca Caavg 0.4, where some differences start to appear.The low viscosity ratio drops exhibit a somewhat larger deformatione.g., visible in theevolution of the major axisLand slightly less orientatione.g., a larger maximum value

FIG. 15. Average droplet deformation for a viscosity ratio ofp 10 and very high Ca for the flow conditionsof Fig.5 as function of the number of cycles. The solid line is the deformation limit for p for steady mixedflows with a constant, positive value ofaccording to the MaffettoneMinale model.



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FIG. 16. Droplet dynamics as a function of dimensionless time t for the flow conditions of Fig. 5, for aviscosity ratio of p 0.1 and intermediate capillary numbers Ca. The symbols represent the experimentalresults, whereas the lines are the predictions according to the MaffettoneMinale model.

FIG. 17. Droplet dynamics as a function of dimensionless time t for the flow conditions of Fig. 5, for a

viscosity ratio of p 0.1 and high capillary numbers Ca. The symbols represent the experimental results,whereas the lines are the predictions according to the MaffettoneMinale model.



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for at the beginning of first quadrant. These observations are also in agreement withthe model predictions according to the MM modelnot shown in Fig.18. Finally, Fig.18c compares the data at p 0.1 and p 1.2 for a large capillary number Caavg

0.5. Clearly, the low viscosity ratio drops are more deformed, especially the majoraxis of the dropletL. In addition, they show significantly less orientation toward the flow

direction as evident from the average level of the orientation angle. In summary, theeffect of low viscosity ratio p = O0.1 in transient mixed flow conditions as appliedwith the ECD only becomes apparent for highCa, leading to increased deformation and

decreasing orientation, as compared to the case for p 1.2.

IV. CONCLUSIONS

In this work, we have studied the dynamics of single Newtonian droplets suspended in

an immiscible Newtonian matrix, which is subjected to transient mixed flow conditions.

The controlled complex flows are applied using a home built ECD, as described in

Boonen et al. 2009. By exploring different operation modes for the ECD, and varyingthe viscosity ratio p over a range of 2 decades from 0.1 to 10, the effect of shear/elongation balance and viscosity ratio has been systematically explored. In addition to the

experimental results on droplet dynamics obtained from optical microscopy, model pre-

dictions have been used for comparison. To this end, the model ofMaffettone and Minale

1998 was used, adapted to complex flows by incorporating a flow type parameter thataccounts for the relative amount of elongation in the flow field.

For the different types of sub-critical flow applied and all viscosity ratios explored in

this work, good agreement is found between the experimental results and model predic-

tions. The effect of varying the balance of shear and elongational components is pre-

dicted, as evident from the comparison between two different flows obtained in the ECD,

using two different configurations shear dominated versus enhanced extension. Next,the experimentally observed effect of a high p = O10 and low p = O0.1 viscosityratio, respectively, as compared to the base case ofp 1.2, is also in agreement with themodel predictions. Quantitative deviations only start to appear for higher capillary num-

bers Ca, for p 0.1 and p 1.2 because we are near the critical conditions for droplet

FIG. 18. Comparison of droplet dynamics for the flow conditions of Fig. 5,for viscosity ratios ofp 0.1 andp 1.2, respectively, foraintermediate capillary number Ca ,bhigh capillary numberCa , andcvery highcapillary number Ca.



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break-up, where the MM model obviously is not valid anymore. Forp 10, on the otherhand, a deformation limit exists, independent of the appliedCa. This limiting behavior

can only be incorporated into the MM model by assigning a nonzero value to the em-

pirical factor in the definition of the coefficient f2MM in the model.

All these findings provide evidence that the rather simple phenomenological of Maf-fettone and Minale performs quite well in predicting single droplet dynamics for arbitrary

sub-critical mixed flow fields. Hence, it can be regarded as a useful tool when predicting

the morphology development in more complex systems e.g., incorporating elastic andconcentration effects, and real processing flows. The extensive data set provided herecan serve as a reference to guide and evaluate future modeling efforts.

ACKNOWLEDGMENTS

The authors would like to thank Ir. Bart Caerts for his help with the design and

construction of the ECD. This work has been financially supported by the Onderzoeks-fonds KULeuvenGrant Nos. GOA 03/06 and GOA 09/002.

APPENDIX

The non-dimensional form of the MM modelMaffettone and Minale1998is givenby

dS

dt CaW S S W= f1

MMpS gMMSI+Ca f2MMp,CaD S

+S D, A1

where S= S /R02, t= t/, and D and W have been made dimensionless with the flow

intensity E:

D =1

42 +1 2

1

20

1

2 0

0 0 0

,

W =1

42 +1 2 0

1

20

1 +

20 0

0 0 0

. A2The steady state values for the dimensionless axesL /2R0 and B /2R0 can then be calcu-lated from Eqs.A1 andA2 to be

L2R0

2 = f12 +Ca21 2 +Ca f2f12 +Ca21 21 2 + 42f1

2 +Ca21 21/31 f221 2 42f2

2Ca2 + f122/3

,



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B2R0

2 = f12 +Ca21 2 Ca f2f12 +Ca21 21 2 + 42f1

2 +Ca21 21/31 f221 2 42f2

2Ca2 + f122/3

. A3

Thus, the deformation parameterD = L B / L +B equals

D=f12 +Ca21 2 f12 +Ca21 2 Ca2f221 2 + 42

Ca f21 2 + 42 . A4

Taking the limit for 1/p0 where f1goes to 20/ 19pand f2to 5 / 2p see Eqs.2for, 0, finally we obtain

Dlim =5

4p

1 2 + 42

1 . A5

Hence, Dlim

represents the steady deformation limit for steady 2D mixed flows with a

constant

. For

= 0 Dlim equals 5 / 4p, the Taylor limit for simple shear flowTaylor1934; for=1, on the other hand, no steady limit exists as a pure extensional flow canbreak-up a drop of any p.

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