Breakup of (non-) Newtonian droplets in atime-dependent elongational flow

Y.W. Stegeman A.K. Chesters F.N. vd Vosse H.E.H. Meijer

March 1999

Abstract

Within industry, blending is often used to obtain specific product properties. Besides the materialproperties of the phases, the rheology of the final product largely depends on the micro-structureformed. This micro-structure is the result of deformation, breakup and coalescence of the dis-persed phase during processing. In this paper, an analytical model is presented which predictsthe deformation of Newtonian droplets in an axisymmetric, time-dependent, elongational viscousflow. The predictions of this model, which is based on a slender, ellipsoidal droplet shape, iscompared with literature and experimental results. In these experiments, a droplet was subjectedto a hyperbolic flow with a time-dependent intensity (triangularCa− t profile). Qualitatively, thetime-dependent deformation is predicted well, although at low capillary numbers (almost spher-ical drops) the deformation is gravely under-predicted. In addition, preliminary results are presen-ted on the deformation of visco-elastic drops in hyperbolic Newtonian flow.

Keywords: drop deformation, time-dependent elongational flow, analytical model, experiments

1 Introduction

The morphology created by the blending of immiscible fluids in industrial processes is the productof droplet deformation, breakup and coalescence. While the continuous-phase flow in such sys-tems can often be predicted adequately by current CFD packages, the response of the droplets tothis flow cannot. This is particularly true of drop deformation and breakup, the main reason beingthe complexity of the situations involved. Perhaps the simplest case concerns the deformationand breakup of Newtonian drops in a second Newtonian phase. The continuous-phase flow in thevicinity of a drop, though linear, is then a mixture of extensional and rotational components.

Breakup of Newtonian droplets and threads has been subject to research for a long time. Infact, the breakup of a viscous thread in a inviscid medium was already described by Rayleigh [1]in the 19th century. In the thirties, this theory was extended to breakup of viscous thread inelongational flow of a surrounding viscous thread by Tomotika [2], while in the same decadeTaylor [3, 4] investigated the formation of such threads and published a breakup criterion in bothsimple shear and hyperbolic flow. In the last sixty year, these theories for the deformation andbreakup of viscous droplets and threads have been refined and extended to general linear flows.In 1984, Rallison [5] presented a review on the deformation of droplets, which was updated tenyears later by Stone [6], who also included drop breakup. A brief overview was also given byAcrivos [7] and Sadhal [8].

During industrial mixing processes, however, the flow conditions encountered by the blends,are usually not quasi-steady, and the phases are often visco-elastic. Although much theories

1

and experimental results are published concerning Newtonian systems, little is known about theinfluence of visco-elasticity on the breakup process. Various publications (partly) concerned thebreakup behavior of visco-elastic drops [9, 10, 11, 12, 13, 14, 15] and threads [10, 11, 16, 17, 18,19, 20], but little data is published and the results are not-consistent and at most qualitative only.

Thorough investigation of a time-dependent break-up process has been performed by Stone,Bentley and Leal [21, 22, 23], who subjected Newtonian drops to step-wise changes in flow con-ditions, while de Bruijn [9] applied a supercritical triangular strain rate - time profile in simpleshear. Torza, Cox and Mason [24] examined the influence of the rate of increase of shear rateon the mode of breakup. These studies indicate that in time-dependent flow, drop deformationand breakup can be radically different (larger time scales, more fragments etc.) from those inquasi-steady flow.

Despite this diversity, two parallel processes underly all deformation and breakup phenom-ena: drop stretching in response to the extensional component of the flow, and thread pinch-off(breakup) driven by interfacial tension (Rayleigh-type instability). The present paper presents anattempt to model the first of these processes in the simplest time-dependent case of viscous, New-tonian fluids and axisymmetric extensional flow. The theoretical analysis, presented in section 2,is an approximate one, making use of classical results in fluid mechanics. In contrast to existingmodels [5, 25], which describe the response of almost spherical droplets to time-dependent flow,this model assumes an elongated droplet shape. The result is an approximate analytical expres-sion for the rate of drop elongation as a function of the applied flow intensity and the physicalproperties. In order to verify the model, experiments have been performed subjecting droplets tosupercritical time-dependent hyperbolic flow. The experimental setup used is described in sec-tion 3. In section 4, model predictions are compared with these experiments and with availableresults in the literature. Because of its simplicity, the modeling approach can probably be exten-ded to non-Newtonian fluids and some preliminary experimental results are presented in section 5on deformation and breakup of visco-elastic liquids in a Newtonian continuous phase. Finally, theresults are discussed in section 6, leading to the conclusions in section 7.

2 Drop stretching rate

In this section, an analytical expression for the drop stretching rate will be derived, for a Newto-nian droplet subjected to a (time-varying) Newtonian elongational flow. The shape of the dropletis assumed to be an ellipsoid of revolution, which, in first order approximation, is confirmed byexperimental results. In the first subsection, the governing equations, as well as the boundaryconditions are stated. To derive the expression for the drop stretching rate, the problem is dividedin four separate parts, describing the components of the stresses exerted on the interface betweenthe dispersed and continuous phase.

• the shear stress exerted on the interface by the dispersed phase [2.2]

• the shear stress exerted on the interface by the continuous phase [2.3]

• the normal stress exerted on the interface by the dispersed phase [2.5]

• the normal stress exerted on the interface by the continuous phase [2.6]

The different parts are coupled by the boundary conditions. In subsection 2.4, the interface velo-city is determined by matching the shear stresses on the interface. Balancing the normal stressesexerted on the interface with the jump in normal stress due to the interfacial tension, finally givesthe expression for the drop stretching rate, as will be shown in subsection 2.7. Because of theapproximation of an ellipsoidal drop, these interface conditions can be satisfied exactly in onlyone location, which is chosen here to be the central region of the drop.

2

2.1 Governing Equations

We consider a single drop of a Newtonian liquid in an elongational viscous flow. The velocityvc far from the droplet is given byEvc(r =∞) = ε(t) z Eez, whereε(t) is the elongation rate asfunction of timet andz is the distance from the center of the droplet in the directionEez, parallelto the axis of the droplet. If the droplet is small enough, inertial and gravitational forces may beneglected, and the flow is governed by the Stokes equations:

∇ · Evi = 0 Continuity equation (1)

∇ · (−pi I + τi ) = 0 Momentum equation (2)

whereEvi = ui Eer + vi Eez is the velocity vector,pi the pressure,τi the deviatoric stress tensor in thedispersed phase (i = d) or in the continuous phase (i = c). If surface effects, such as interfacialviscosity and interfacial tension gradients are absent, the boundary conditions for an ellipsoid ofrevolution, given byR(z) = R0

√1 − (z/ l )2, are:

1. Velocity is continuous on the interfacevd(R) = Vif = vc(R).

2. Tangential stress is continuous on the interfaceτd(R) = −τc(R).

3. Balance of normal stresses by interfacial tension(−p + τnn)c − (−p + τnn)d = σ R1+R2R1·R2

where 2R0 and 2l are the width and the length of the droplet. The interfacial tension is denotedwith σ andR1 andR2 are the principal radii of curvature.τxx,i indicates the (xx)-component ofthe deviatoric stress tensor, wheren is the direction normal to the interface. If the subscriptxx isomitted, the shear stressτnt is intended, wheret is the direction tangential to the interface.

r

r

z

u

v

v

l

R0

In an axisymmetric coordinate system, theStokes equations for Newtonian liquids read(η is the dynamic viscosity):

1

r

∂

∂r(ru) + ∂v

∂z= 0

η

[1

r

∂

∂r

(r∂u

∂r

)+ ∂2u

∂z2

]= ∂p

∂r

η

[1

r

∂

∂r

(r∂v

∂r

)+ ∂2v

∂z2

]= ∂p

∂z

2.2 Dispersed phase: Velocity field and shear stress

Assuming a slender body,(l/R0)2 � 1, the Stokes equations reduce to the Poisseulle equation:

ηd1

r

∂

∂r

(r

∂vd

∂r

)= ∂pd

∂z

Using the boundary conditions∂vd/∂r |r=0 = 0 andvd(R) = Vif , the velocityvd is given by:

vd(r, z) = Vif + τd R

2ηd

(1 − r 2

R2

)(3)

whereτd denotes the shear stress exerted by the dispersed phase on the interface. Due to theassumption that the droplet shape is an ellipsoid of revolution, the mean elongation rate in anycross-section,εd ≡ ∂Vd/∂z, proves to be constant and equal tol−1dl/dt for all −l < z < l . ThesymbolVd denotes the mean velocity at a cross-section:

Vd(z) ≡∫ R

0

∫ 2π

0vd(r ) r dφ dr = Vif + τd R

4ηd= z

l

dl

dt= εdz (4)

The shear stress is therefore given by:

τd(z) = 4ηd

R(z)

(εdz − Vif (z)

)(5)

3

2.3 Continuous phase: shear stress

In 1970, Batchelor published an expression for the shear force per unit of length on a rigid slenderbody in elongation flow [26, eq. 4.7]:

F(z) = 2π ηc V(z)

[ξ + ξ2

(3

2− ln

√1 − z2/ l 2

R(z)/R0

)+ o(ξ3)

], ξ =

(ln

2l

R0

)−1

(6)

whereV(z) is a linear function inz, denoting the velocity far from the body. In our case, thevelocity at the body-surface is non-zero and as a first approximation, equation 6 is generalized bytaking V(z) = εz − Vif . Assuming the pressure gradient along the interface of the ellipsoid ofrevolution is negligible, the shear stress becomes:

τc(z) = F(z)

2π R= ηc(εz − Vif )

R

[ξ + 3

2ξ2]

= ηc(εz − Vif )

Rf, f = ξ + 3

2ξ2 (7)

2.4 Interface velocity

From the third boundary conditionτc = −τd, an expression for the interface velocityVif can beobtained, involving the viscosity ratioλ = ηd/ηc:

Vif = f ε + 4λ εd

f + 4λz (8)

2.5 Dispersed phase: Normal stress

The normal stress exerted by the dispersed phase is given byTrr = −pd +2ηd∂ud/∂r . To obtaina first approximation of the normal stresses, the flow inside the droplet is treated as a plug flowwith the same mean elongation rate as the actual flow. In this case, the deviatoric normal stressesare given byτzz = 2ηdεd andτrr = −ηd εd. The quantity entering into the normal stress balanceproves to be∂Trr /∂z = −∂pd/∂z. To the lowest order inz/ l , an expression for∂Trr /∂z can bederived from a force balance:

∂Trr

∂z= −2τc

R+ 3 (∂ R/∂z)

Rτzz+o

(z2

l 2

)= −2ηc (ε − εd) f z

R20 (1 + f/4λ)

+ 6ηd εd z

l 2 +o

(z2

l 2

)(9)

2.6 Continuous phase: Normal stress

The normal stress exerted on the interface by the continuous phase is given byTrr = −pc +2ηc∂uc/∂r . From the mass balance at the interface in the central part of the droplet, one canderive that:

∂uc

∂r

∣∣∣∣R

= 1

R

∂(R Vif )

∂z+ 1

R

∂ R

∂ t(10)

The interface velocity is given by equation 8, while the droplet shape in the central part of thedroplet can be approximated byR = R0[1 − 1

2(z/ l )2]:∂uc

∂r= 1

R

∂(R Vif )

∂z+ 1

R

∂ R

∂l

∂l

∂ t= f ε + 4λ εd

f + 4λ[1 − R0 z2

R l2] − εd R0

2 R[1 − 5z2

2 l 2 ]Batchelor [27, p. 248] published the pressure distribution and the flow surrounding a fluid

droplet with a constant spherical shape embedded in a pure straining motion, from which one canderive a relation between the pressure generated by the presence of the sphere and the (rr )-componentof the deviatoric stress tensor:

p = − 2 + 5λ

6τrr = − 2 + 5λ

3η

∂uc

∂r

4

Assuming this relation is still approximately valid for a deforming, slender ellipsoid of revolution,the pressurepc is given by

pc = − 2 + 5λ

3ηc

∂uc

∂r= − 2 + 5λ

3

ηc

R

∂(R Vif )

∂z

The total equation for the normal stress exerted by the continuous phase becomes, to the secondorder inz/ l :

Trr = (8 + 5λ) ( f ε + 4λ εd)

3 ( f + 4λ)ηc[1 − z2

l 2 ] + ηcεd − 2ηcεdz2

l 2 + o

(z4

l 4

)(11)

2.7 Normal stress balance and dimensionless drop stretching rate Ed

Using the last boundary condition(−pc+τnn,c)−(−pd+τnn,d) = σ R1+R2R1·R2

, an implicit expressionfor the drop stretching rateεd is obtained.

σ∂

∂z(R−1

1 + R−22 ) = ∂Trr,c

∂z− ∂Trr,d

∂z

σ z

R0 l 2

(1 + 2

R20

l 2− 3

R40

l 4

)= 2ηc(8 + 5λ)(εd + ε f/4λ) z

3 l 2 (1 + f/4λ)− 4ηcεd z

l 2+

+ 2ηc (ε − εd) f z

R20 (1 + f/4λ)

− 6ηd εd z

l 2

Using the following dimensionless quantities (a is the radius of the undeformed droplet):

L = l

a, Ca = ηcεa

σ, g =

√L(L6 + 2L3 − 3

)2L6

, ξ = 1

ln 2√

L3, f = ξ + 3

2ξ2

the explicit relation for the dimensionless drop stretching rateEd is given by:

Ed = εd

ε= dl / dt

ε l=

1 + L−3(

23λ

+ 512 − g

f Ca − g4λ Ca

)1 + L−3

(34 + 1

2λ+ 4λ

3 f − 23 f

) (12)

By way of illustration, figure 1 shows the response (rate of elongation / contraction) of a dropletof a certain dimensionless lengthL to the elongational flow applied at that moment, for variouselongational intensities. In steady, sub-critical flow, an initially undeformed spherical droplet

1 2 3 4 5 6 7 8 9 10

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

0.25 Cac

0.50 Cac

1.00 Cac

2.00 Cac

4.00 Cac

8.00 Cac

L

εdε

Figure 1: Dimensionless drop deformation rateEd as function of the dimension-less length of the dropletL = l

a and the capillaryCa: λ = 1.

5

(L = 1) will elongate with the flow, until the steady deformation (L ≈ 1.1, Ed = 0) is reached.If for any reason (for example if the flow was temporarily supercritical) the droplet length slightlyexceeds this steady deformation, it will retract (Ed < 0) until it reaches its steady deformationagain. Only if a very elongated drop is subject to a sub-critical flow (0.25Cac : L > 2.5, 0.5Cac :L > 3.5), the retracting force (due to surface tension) is smaller than the deforming force (due tothe flow) and the droplet will elongate ever faster, until it finally extends at the same rate as thecontinuous flow (Ed = 1).

3 Experimental Setup

Break-up experiments are performed in a cross-slot device, shown in figure 2, which generatesa plane hyperbolic flow. For studies using visco-elastic fluids, this device is preferred above afour-roll mill [28], since a pressure induced flow is less dependent on the non-Newtonian featuresof the continuous fluid. In a four-roll mill, the shear stresses generated by the rollers may be tooweak to overcome the large elongational stresses occurring at the inflow between the two rollerswhile using a visco-elastic fluid, whereas the rollers might not even get the bulk fluid flowing if thecontinuous fluid is shear thinning. Figure 3 shows the setup around the cross-slot. The continuous

40

10

12

80

20

290

145

Figure 2: Cross-section of the cross-slotdevice. Dimensions in millimeters

.γ

C

B

A

D

Microscopewith camera

Homogeneousillumination

Dropposition

controlPosition

Pump speed Pump

Reservoir

Figure 3: Position control system

phase fluid is pumped through the tubing and the flowcell by a gear pump (Liquiflo 37.FS.1133.Uattached to a Pacific Scientific SRF3752-4984 DC motor), of which the (time-dependent) speedis computer controlled. At low shear rates, a bypass is provided conveying a large part of the con-tinuous phase directly back into reservoir. Due to this bypass, higher rotational speeds can be used,which diminishes the pulsing effect of the gear pump on the flow. Downstream of this bypass, thematrix fluid is divided into two streams, which enter the flowcell at A and C. The flowcell con-sists of two parallel perspex plates, with four rounded solid blocks in between. The geometry anddimensions are chosen such, that the velocity field is approximately two-dimensional and the areaof uniform velocity gradient is about 1 cm2. Numerical simulations and Laser Doppler Anemo-metry measurements proved that the geometry produces an acceptable 2-D flow up toε = 10 s−1.The dispersed phase is injected between point A and the middle of the flow cell. The hyperbolicflow will force this drop toward the center, but the position of the dispersed drop at the stagnation

6

point is unstable to disturbances. In an uncontrolled system, a small disturbance would cause thedrop to move away from the center with a velocityεz. For break-up studies, the drop has to stayat the center of the flow cell. Therefore, the flow is adjusted by varying the ratio of the flow ratesexiting the flowcell at B and D. If the drop is moving, the control valve is rotated in such way, thatthe stagnation point shifts, forcing the drop back to the center of the flowcell. Since only the ratiobetween the exit flows is varied, the total flow rate, and therefore the elongation rateε, remainsconstant. To detect the position of the drop (and to record its shape) a CCD camera (JAI M30)is attached to a microscope (Olympus SZ4045), which limits the minimum drop radius to about250 µm (field of view at maximum magnification is 2 mm). The camera signal is transmittedto both a VCR and a frame-grabber. The frame grabbed is thresholded, after which the positionof the drop is determined. If the position of the drop is known, the computer calculates the newrequired position of the stagnation point and the corresponding angular position of the controlvalve. The rotation of this valve is realized via a DC motor (Maxon) with a gear wheel reduction.

4 Comparison of the model with literature and experiments

In this section, the predictions of the model will be compared to results available in literature andwith new experimental results on drop deformation in time-dependent elongational flow. First,the critical capillary numberCac = ηcaεc/σ is regarded in subsection 4.1. For any two-phasesystem (ηc, a), this determines the critical steady elongation rateεc of the continuous flow abovewhich a droplet of a given radiusa can no longer assume a stable shape, but extends exponen-tially until breakup occurs. Since, in general, breakup does not occur below this critical capillarynumber, it is important that the magnitude of the critical capillary number is predicted correctly.Because all droplets will initially be spherical (no flow), the model should, to some extend, also beable to predict the deformation of slightly deformed drops (l ≈ R0), even though this is outsideits regime of validity (l 2 � R2

0). Therefore, a comparison with existing linear small deforma-tion theories and experimental results is made in subsection 4.2. Although in quasi-steady flowbreakup will always occur if the critical capillary number is exceeded, in time-dependent flow,the droplet deformation (and consequently breakup) is also dependent on the time scales relatedto theCa − t profiles imposed. Since the main objective is to predict the time-dependent dropletresponse during processing, the deformation (and breakup) behavior in a realistic, although rathersimplified, flow is determined. To this end, a single droplet is subjected to a Newtonian hyper-bolic flow, applying a triangularCa − t profile, comparable to the profile applied by de Bruijn insimple shear [9]. As will be shown in (sub)sections 4.3 and 5, the deformation of the droplet andthe occurrence of breakup, depend on the time interval during which the critical capillary numberis exceeded, as well as the magnitude of this excess. The results shown in this section are allobtained with Newtonian two-phase systems. In section 5, some preliminary experimental resultswith visco-elastic droplets will be presented.

4.1 Prediction of critical capillary number

Under steady conditions, the dimensionless drop elongation rateEd is zero. Solving equation 12,one obtains an expression for the capillary numberCa necessary to sustain a given dimensionlessdrop lengthL, see figure 4.

Ca ≡ ηcaε

σ= g (12λ + 3 f )

12λ f L3 + 8 f + 5λ f(13)

On increasingCa slowly from zero,L will increase smoothly up to some critical valueLc, afterwhich no stable shape exist unless the capillary number is decreased. The capillary number atLc is called the critical capillary numberCac. Although the critical capillary number is under-predicted at higher viscosity ratio’s, the overall agreement with values found in literature is fairlygood, see figure 5.

7

1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

Ca

L

λ = 0.01

λ = 0.1

λ = 1

λ = 10

Figure 4: Necessary ca-pillary number Ca to sus-tain dimensionless dropletlength L for various viscos-ity ratiosλ (steady flow).

.1

1se

10

00

ss

10 10 10 10 10 10 1010-1-2-3-4-5

10 1032

Shear flow

Hyperbolic flowtay

0.1

1.0

10

100

1e-5 1e-4 1e-3 1e-2 1e-1 1e+0 1e+1 1e+2 1e+3

Experiments simple shear(Grace 1982)Experiments elongation(Grace 1982)Prediction Ed = 0(this paper)

Slender body theory: shear(Hinch and Acrivos 1979)Slender body theory: hyperbolic(Hinch and Acrivos 1980)First order theory(Taylor 1932)

Ca

λ

Figure 5: Comparison of critical capillary num-ber as a function of viscosity ratioλ predicted bythis method (Ed = 0) with experimental results byGrace [29], and analytical results by Taylor [3] andHinch and Acrivos [30, 31]

4.2 Prediction of small deformations

Steady flow at low capillary numbersThe steady deformation at low capillary numbers can be described by the first order deformationtheory of Taylor [3], based on hydrodynamical equations of a spherical drop in a shearing fluid:

D ≡ L − B

L + B= 19λ + 16

16λ + 16Ca (14)

In figure 6, the response of a Newtonian droplet (a = 0.733 mm,ηd = 8 Pa·s) to a step-profilein the elongation rate (ε = 0.05 s−1 if 0 ≤ t ≤ 30 seconds, elseε = 0 s−1) is shown. Theviscosity ratio of the system (λ = 0.8) was measured using a rheometer, but the surface tension(σ = 5 · 10−3 N/m) was determined using the Taylor equation (D = 0.08). It is clear that themodel seriously under-predicts the steady deformation of the droplet.

Start-up flowThe time evolution of the shape of a drop which is suddenly subjected to a hyperbolic flow is

0 5 10 15 20 25 300

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

D

time [s]

Figure 6: Droplet response to a step-profile in elongational rate: experimentalresults and predictions of the model (solid line) and first order equations fromliterature (dotted line).λ = 0.8, ηc = 10 Pa·s, ε = 0.05 s−1, σ = 5 · 10−3 N/m.

8

given by Cox [25]:

D(t) = [1 − e−βt] (19λ + 16)Ca

16λ + 16, β = 20ε

19λCa(15)

whereL andB denote the dimensionless half-length and -width of the droplet respectively. Thedroplet response predicted by this equation is given in figure 6 as a dotted line, time< 10 s. Asexpected, the final deformation is under-predicted by the model, since the steady deformationpredicted by equation 15 is identical to the one predicted by equation 14. The experimentallyobserved time interval which is necessary to reach the steady deformation is slightly larger thanthe time interval predicted by both equation 15 and the model.

Retraction rateThe retraction rate of a slightly deformed droplet in a quiescent matrix, can be deduced [32] fromthe analysis of Rallison [5]:

D(t) = D0 exp

( −40(λ + 1)

(2λ + 3)(19λ + 16)

tσ

ηca

)(16)

whereD0 is the deformationD at the beginning of retraction. Figure 6 shows that the retractiontime of the droplet which was initially subject to the steady elongational flow described before, iswell predicted.

4.3 Triangular strain rate profile

Although in quasi-steady flow any Newtonian droplet will, by definition, break up if the elongationrate exceeds the critical elongation rateεc = Cac σ/ηca, in time-dependent flow this is not neces-sarily the case. In fact, if the elongation rate is increased rather fast, the droplet deformation willlag behind and might even be less than the critical deformation, even if at that moment the criticalelongation rate is exceeded (for a limited amount of time). In this research the droplet response toa triangular strain rate profile is examined. We performed two kinds of experiments to obtain theinfluence of the parameters1T (ramp time) andCam/Cac, as is shown in figures 7 and 8. With a

0 5 10 15 20 25 30 350

0.05

0.1

0.15

0.2

0.25

Dimensionless time

Ca

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

Dimensionless time

D

Cam

1T

Figure 7: Determination of the influenceof 1T on the deformation of the droplet.Results given are model predictions:λ = 1,Cam = 1.5Cac. Ramp time1T is increasedfrom 2.5 to 17.5.

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

Dimensionless time

D

0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Dimensionless time

Ca

Cam

1T

Figure 8: Determination of the influence ofCam/Cac on the deformation of the droplet.Results given are model predictions:λ = 1,1T = 10. Cam is increased fromCac to2Cac.

9

0 10 20 30 40 50 60

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

la

Time [s]

Figure 9: Droplet response to a triangulart − Ca profile: experimental results.1T =27 s,Cam = 0.25, 0.28, 0.31 and 0.35. Inthis last experiment, breakup occurs.

0 10 20 30 40 50 60

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

la

Time [s]

Figure 10: Droplet response to a trian-gular t − Ca profile: model prediction.1T σ/ηc a = 18, Cam = 1.01, 1.15, 1.29and 1.44Cac.

single drop, a number of experiments are performed, steadily increasing either the ramp time1Tor the maximum capillary numberCam with each experiment until breakup occurs. In figure 8,anyCam ≥ 2Cac will lead to breakup, since the deformation at 21T will then exceed the criticaldeformation for drop breakup in a quiescent matrix [21]:λ = 1 → (l/b)c = 4, Dc = 0.6.

With the fluid system of figure 6, the second type of experiment was performed, increasing themaximum capillary number from 0.25 to 0.35. The critical capillary number was not measured.The Grace-curve predicts a critical capillary number between 0.15 and 0.25, the model predictsCac = 0.167. Fitting the model to the experiment withCam = 0.35 givesCac = 0.243, whichwas used for the prediction of the droplet response in the other experiments. The ramp time1Tis 28 seconds, corresponding to a dimensionless ramp time of 18. Stone et al [21] predict that forthis viscosity ratio breakup occurs in a quiescent matrix as the dimensionless lengthL exceeds2.5. Indeed, during the last experiment (Cam = 0.35), breakup occurred.

5 Preliminary results with visco-elastic droplets

Critical length for breakup in a quiescent matrixFirst, experiments similar to those performed by Stone,Bentley and Leal [21] were carried out

with visco-elastic drops in a Newtonian matrix. Although quantitative results were not obtained,the experiments clearly showed that the critical length for breakup of visco-elastic droplet exceedsthose for Newtonian droplets, confirming observations made by de Bruijn [9], Elmendorp [11],and Milliken and Leal [13].

Response to a triangulart − Ca profileAn experiment of the first type has been made with a non-Newtonian droplet of 2.5 mm diameter.The dispersed phase is a visco-elastic, shear-thinning solution of 1.65 wt% PEO 5e6 in water,figure 11, while the continuous Newtonian phase is silicon oil with a viscosity of 10 Pa·s. Thecritical elongation rate was determined in quasi steady flow and proved to be 0.29 s−1 for a dropletof 2.5 mm diameter. The droplet is subjected to triangular profiles withCam = 1.25Cac and ramptimes of respectively 2.7, 5.6, 11, 17, 22, 28, 33, 39, 44, 50 and 56 seconds, see figure 12.

Although the model is strictly Newtonian, we tried to fit the model to the experimental data,in order to see how well it performs. In order to do so, the viscosity ratio and the surface tensionneed to be determined. Since the droplet phase is shear-thinning, an equivalent shear rate has

10

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100

101

102

100

101

102

20

30

40

50

60

70

+V

isco

sity

[Pa·s

]

oL

oss

angl

e[o]

Frequency [rad/s]

Figure 11: Viscosity (+) and loss angle (o) of 1.65 wt% PEO 5e6 in water.

to be estimated, in order to obtain the viscosity ratio. As the droplet is more viscous than itssurroundings, the shear rate inside the droplet is probably less than the elongation rate, whichwould increase the viscosity ratio. On the other hand, the flow field inside the droplet will not bepure shear flow, which increases the effective viscosity. Since for these higher viscosity ratios theeffect onEd is relatively low, we assume the equivalent shear rate is equal to the maximum strainrate of the continuous flow, estimating the viscosity ratio to be 3.5 (ηc =35 Pa·s).

Due to the visco-elastic nature of the droplet, the surface tension could not be easily determ-ined from the experiments. However, since the time constant is the only unknown left, it canbe estimated by fitting the predictions of the model to one of the experimental triangular profiles.Since the model becomes more accurate as the deformation increases, the time constant is determ-ined by fitting the triangular profile where the largest deformation occurs without breakup:1T =50 s. This results in a time constantηca/σ of 2.2 seconds. This value is also used to predict thedroplet response to the othert − Ca profiles, leading to figure 13.

0 10 20 30 40 50 60 70 80 90 1001

1.5

2

L

Time [s]

Figure 12: Response of a shear-thinningvisco-elastic droplet (1.65 wt% PEO in wa-ter) to a triangulart − Ca profile: Cam =1.25Cac, 1T = 2.7, 5.6, 11, 17, 22, 28, 33,39, 44, 50, and 56 seconds.

0 10 20 30 40 50 60 70 80 90 1001

1.5

2

L

Time [s]

Figure 13: Droplet response predicted byOth order Newtonian model: Cam =1.25Cac, 1Tσ/ηca = 1.2, 2.5, 5.0, 7.7,10.0, 12.7, 15.0, 17.7, 20.0, 22.7, and 25.4

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6 Discussion

The comparison in the previous section show that the overall, qualitative agreement of the modelpredictions with literature and Newtonian experimental results are good, the quantitative agree-ment improving with droplet deformation. This is due to the model assumption that the dropletis a highly elongated (slender) body. This explains why the prediction of the critical capillarynumber becomes poorer with increasing viscosity ratio (aboveλ = 0.1, l < 2 R0), and why thedeformation is seriously under-predicted in figure 6 (D ≤ 0.08 → l < 1.2 R0).

Although the deformation is under-predicted at these low capillary numbers, the time scaleinvolved with start-up flow and retraction rate, are well predicted. Also the prediction of thedroplet response to the triangulart − Ca profiles becomes better with increasing deformation.

Comparing figure 9 and 10, one should keep in mind that the exact height of the curve corres-ponding toCam =1.44Cac is obtained by fitting the exact value of the critical capillary number.However, the exact value ofCac only influences the maximum value of the deformation and notthe shape of the curve. Although the maximum deformation of the other three curves is slightlyunder-predicted, the (qualitative) shape of all predicted curves match the experimental results.

A more detailed comparison of the model prediction with the experimental results shows thatthe deviation for less deformed droplets is not only due to the assumption of a slender body,but also due to the fact that the model is developed for axisymmetric elongational flows, while,experimentally, the flow imposed upon the droplet is hyperbolic. Although at higher deformationsthe surface tension causes the droplet to become (almost) axisymmetric, despite of the hyperbolicflow, this is not the case for slightly deformed drops as is shown in figures 14 and 15. This causesthe model to predict a smaller deformation than experimentally observed in the direction parallelto the inflow.

Concerning the preliminary visco-elastic results, a number of observations are made. If thesurface tension is determined by fitting the height of the curve with1T = 50 s, the following(1T = 56 s) and previous (1T = 44) curve are predicted well. All other curves show a verypoor correspondence. Although this might be due to the visco-elasticity, the experimental cir-cumstances which produced these curves are outside the validity regime of the model (as has beenshown above), due to the low deformations and the discrepancy in flow type. The comparison istherefore only made to illustrate the difficulties in determining the parameters1T andCam / Cac.

Figure 14: Experimentally observed de-formation: cross sections of droplet par-allel to the inflow, corresponding tot =0, 10, 20 and 30 seconds (figure 9,1T = 27 s,Cam = 0.22).

Figure 15: Deformation at cross sectionperpendicular to the inflow. Calculatedfrom experimentally observed deforma-tion parallel to inflow (fig 14), assumingan ellipsoidal cross section.

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7 Conclusions

In this paper, an analytical model is presented which predicts the deformation of Newtoniandroplets in an axisymmetric, time-dependent elongational flow. The predictions of this model,which is based on a slender, ellipsoidal droplet shape, are compared with literature and experi-mental results obtained in hyperbolic flow. Although the circumstances under which these lastresults were obtained were technically outside the range of validity of the model, it proved topredict the droplet response to supercritical triangulart − Ca profiles well, although it gravelyunder-predicted the magnitude of the deformation in a sub-critical step profile. Overall, the modelpredictions have to be considered good, even if the deformations are too small to justify the as-sumption of a slender body. For very small deformations, the model could be extended with thelinear theories which have been published in literature [3, 4, 5, 25]. Due to its simplicity, wesuspect that the modeling approach can also be extended to non-Newtonian fluids. The prelim-inary experiments performed with non-Newtonian droplets are not conclusive, but indicate thenecessity of more thorough investigations concerning the determination of the surface tension forvisco-elastic materials, and the influence of elasticity and shear-thinning behavior on deformation(especially in time-dependent flows) and breakup.

Acknowledgments

This research is part of the FAIR project BioMix (CT 97 3022), which is funded by the EuropeanCommunity.

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