Simulations with Dispersive Mixtures Model

P.J.A. Janssen, P.D. Anderson & G.W.M. PetersMT03.05

May 12, 2003

1 Introduction

The combination of two different materials in a blend can yield interesting results.The food industry, the polymer processing industry and more generally, all mixingprocesses are in some way connected to the phenomena occurring. If the volumefraction of one of the two components is relatively small, one can distinct dropletsin a matrix fase. The behavior of these droplets can be quite complex and includephenomena like stretching of droplets, breakup and coalescence. Varies modelsto describe droplet deformation have been suggested and further there is the cou-pling of the single droplet behavior to bulk parameters like the stress in the totaldispersion.

2 Modelling of drop behavior

The model used here is the one as suggested by Peters [5]. This is a phenomenolog-ical model, which describes the morphology changes of the dispersed fase duringflow and couples the state of the droplets to the bulk stress. The major drop statesare shown in figure 1. From top to bottom one can see in figure 1 respectivelystretching, filament break-up, necking and coalescence. When stretching, dropletscan be stretched into long, slender filaments. These filaments in turn can breakupinto many smaller droplets. Droplets can also break in two, in a proces callednecking. Finally two droplets can coalesce into one new, bigger droplet.

2.1 Parameters

To avoid any confusing, the first thing to do is to make a summary of the mostimportant parameters used in the model. The parameters are extracted from thedescription of the deformation of a droplet. R stands for the original radius of

1

Figure 1: Droplet deformation modes

the droplet, that is in un-deformed state, while L and B stand for the greatest andsmallest length parameters of the now stretched droplet. Furthermore there are thezero shear viscosities ηm and ηd of matrix and the droplet fase and the interfacialtension α between them. N denotes the number of droplets per unit volume, whileφ stands for volume fraction of the dispersed fase. When the mixture is under flow,γ̇ equals the local shear rate.

The capillary number Ca is used as a dimensionless number to characterize theflow. Simplified can be said that is represents the ratio of the viscous stresses tothe surface tension. The ratio between the viscosities is denoted by p. The stretchfactor β is used to indicate the stretch of the droplet in comparison with its initial,equilibrium diameter. A spherical drop thus yields a β of 1 and stretched dropletshave a β larger than 1. The last parameter used is the interfacial area Q betweenthe dispersed and the matrix fase. This is a measure for the degree to which thedrops are dispersed in the matrix.

2.2 Droplet deformation

A droplet can deform in many ways. These include stretching, necking, breakupand coalescence. Below are given the mathematical formulations for these different

2

Table 1: Physical parameters used to characterize a drop phase dispersed in a con-tinuous matrix.

Parameter Definitionφ Volume fraction occupied by the dropletsN Number of drops per unit volumeηd Zero shear viscosity of droplet phaseηm Zero shear viscosity of continuous matrixα Surface tensions between droplet and matrixγ̇ Local shear rate

Ca = ηm γ̇

α/R Capillary Numberp = ηd

ηmViscosity ratio

β = L2R Stretch ratio

Q = 2π N B Rβ Interfacial surface area

modes as included in the model used.

2.2.1 Stretching

If a droplet deforms perfectly with an imposed velocity field, the deformation istermed affine. The stretch ratio of a droplet for affine deformation can be writtenas:

d ln(β)

dt= Di j mim j . (1)

The orientation vector if the droplet, m j , can be calculated from the eigenvector ofthe maximum eigenvalue of the rate of deformation tensor D j j .

The deformation of droplets is, however not always affine. For non-affine de-formation of a droplet in an elongational flow, a relation was derived by Stegemanet al [8]. This results van be simplified to:

d ln(β)

dt= f (β)Di j mi m j . (2)

The current stretch ratio thus directly influences the rate of deformation of thestretch factor. Stegeman’s result is used in this study for general flows, not justelangational. This is done by calculating the elongational component of the generalflow substituted in Stegeman’s formula.

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2.2.2 Necking

Now that a method is provided to describe the drop deformation, the next step willbe to determine the deformations which will break the droplets up into smallerones.

Recalling the definition of the capillary number as the ratio of the deformingforces to the restoring forces, or the strength of the flow, one can define a criticalcapillary number, which represents a flow that is strong enough to break or neckan initial spherical drop into two droplets.

The time required for an initial spherically drop to neck into two drop wasdetermined by Grace [4] by studying two dimensional flows:

tneck ≈ 85.3

(

R0ηm

α

)

p0.45. (3)

Peters determined, by assuming a linear necking rate for capillary numbersbetween Cacrit and 2Cacrit , the rate of change of the equivalent spherical radius ofa drop undergoing necking:

d R

dt= −3.91 × 10−3 γ̇ R

p0.45Cacrit. (4)

Additionally it is assumed that necking drops have a stretch ratio of β = 4.

2.2.3 Filament breakup

For stronger flows, Ca > 2Cacrit , it is necessary to consider how drops that havebeen stretched into long filaments break into many smaller drops. If this happens,the elongational processes occur more quickly than the breakup processes.

Janssen [6] found that when the cross sectional radius of a filament, B, droppedbelow a critical value determined by capillary wave instability growth, the filamentwould breakup into many droplets:

Bcrit ≈ 0.04ω0.10

(

ηm ε̇m

α

)−0.9

p−0.45 (5)

If (B < Bcrit) then:

Ravg ≈(

3π

2

)1/3

Rcrit . (6)

After a breakup occurs, the drops are assumed to have a stretch ratio of β = 1.5.

4

If a stretched droplet is placed into a static or stationary fluid, the breakupconditions are slightly different. For a static filament, Janssen [6] found:

Ravg ≈(

3λm(p)

4R

)1/3

R. (7)

Since such drops are breaking up under static conditions, the resulting drops areassumed to have a stretch ratio of β = 1.

Experiments show the appearance of so called satellite droplets. These aresmaller droplets emerging between the greater ones. In this analysis these areneglected.

2.2.4 Coalescence

The final item included in the model is coalescence. Based on the work of Chesters [2],Peters [5] derived a relationship describing the rate of change of the equivalentspherical radius of a drop undergoing coalescence. When the local capillary num-ber is below the critical capillary number, the drop radius changes according to:

d R

dt= exp

(

−√

3RpCa3/2

4hcrit

)

4γ̇ φR

3π. (8)

After coalescence, the drop stretch ratio can be determined from the work ofCox [3] as:

g (p, Ca) =5 (19p + 16)

4 (p + 1)

√

(

19p2)

+(

20Ca

)2. (9)

β =(

1 + g (p, Ca)

1 − g (p, Ca)

)2/3

. (10)

2.3 Constitutive model

Now that all modes for a single droplet have been covered, one must link the be-havior of a single droplet to the bulk stress. This can be done by modelling alldroplets or by modelling one, average droplet, which represents the mean prop-erties of all droplets. The choice has been made for the latter approach, but thisintroduces problems, like step wise behavior of certain parameters, like the stretchratio and the interfacial surface, especially after breakup and coalescence and isn’tvery realistic, since droplets vary in size. To overcome these problems an effec-tive measure parameter has been introduced. It links the average drop modelled

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to a more realistic and smoother value for the mean of all droplets, using a simpleexponential decay relation:

τQ Q̇ef f + Qef f = Q; (11)

τβ β̇ef f + βef f = β. (12)

The value of the time scale parameters associated with this effective measure ap-proach have to be estimated from experiments. Furthermore these values vary fordifferent states in which the drops/ filaments are, but these aren’t very large, i.e.the order of magnitude is the same.

To describe the stress for a dispersion of droplets in matrix, Peters [5] derivedthe following relationship:

⟨

Ti j

⟩

= −Pδi j +[

2 +10 (ηd − ηm)

2 (1 − φ) ηd + (3 + 2φ) ηmφ

]

ηm Di j + Si j . (13)

Essentially, the volume average bulk stress,⟨

Ti j

⟩

is equal to an isotropic pressureterm, P , a rate of deformation term, Di j which is modified by an effective Newto-nian viscosity and an extra stress due to the presence of the drops, Si j .

The extra stress is governed by the following differential equation where thefirst term, S(1)

i j represents the lower convective derivative of the extra stress:

S(1)

i j +[

1

τ− D

Dtln(

Qef f

)

]

Si j = L i j kl

(

Qef f , φ, S)

Dkl , (14)

with:

L i j kl =19µd + 16µc

[5(1 − φ)µd + (5 − 2φ)µc]Q

Q0φ

µc

τ0

4I +

2

3ISi j −

2

Eσ QSi j Si j . (15)

In this equation σ represents the interfacial tension and E is a function of the zero-shear viscosities and the volume fraction. Equation(14) couples the extra stressto changes in the interfacial surface area, Q e f f . Additionally, the relaxation timeτ , appearing in equation(14) is also a function of interfacial surface area and dropstretch ratio, βe f f as

τ =3φ[

(

βef f − 1)1.5 − 1

]

4αQef f

{

(19ηd + 16ηm) [2 (1 − φ) ηd + (3 + 2φ) ηm]

10 (1 − φ) ηd + (10 − 4φ) ηm

}

.(16)

With methods to model drop dynamics (i.e. deformation, necking, breakup andcoalescence) and a constitutive model to connect drop morphology to bulk stress,specific applications can be investigated to test the abilities of this work to modeldrop dispersion behavior.

6

Table 2: Physical parameters for a PID/PDMS dispersion studied by Vinckier [9].Parameter Value

matrix component poly-dimethylsiloxane (PDMS)ηm 195 Pa · sτm 2.7 × 10−2 sec.

droplet component poly-isobutene (PIB)ηd 86 Pa · sτd 2.7 × 10−4 sec.

dispersion parametersφ 0.10 (PIB)α 2.3 × 10−3 N/m

initial morphologyR0 10−5 mN 2.4 × 1013 m−3

2.4 Computational methods

The equations stated above were implemented and solved in the finite elementpackage SEPRAN using a discrete elastic-viscous stress splitting, discontinuousGalerkin (DEVSS/DG) approach with first-order time integration. See the Bo-gaerds [1] for more detailed information. Summarized one can say that this is acomputational method that solves all unknown variables concerning the macro-scopic behavior simultaneously. In other words: the deformation and the stressesare coupled. Each two-dimensional element consisted of nine nodes with second-order polynomials used as basis functions. For all of the simulations, physicalparameters for the dispersion were taken from Vinckier [9] and are reproduced intable 2.

7

3 Step Shear

In the step shear study, a 128 element, 561 node mesh was used with a time stepof 0.01 seconds. Following the results of Schiek [7], some more simulations havebeen done in a step shear flow. Since step shear is a relatively simple flow, not onlynumerical, but results can also be easily interpreted, it is very suited for analysis ofthe influence of parameters. Before variations are investigated, the original resultsare depicted in figure 2 with explanation of the different zones. As one can see, the

0 20 40 60 80 10010

13

1014

1015

1016

Time [s]

Num

ber

of d

rops

per

uni

t vol

ume

[m−

3 ]

← A

ffine

Str

etch

ing

← Filament Breakup

Necking

Necking balanced with Coalescence

Figure 2: Original result for step shear

initial droplets are stretched until the value of B drops below the critical one andthe drops breakup, resulting in a great increase in the number of droplets. After thebreakup, necking is the most important morphology state. This again leads to anincrease of droplets, but this time slower and more gradually. Finally the neckingis balanced with coalescence, which leads to no netto increase in the number ofdroplets.

8

3.1 Shear rate

First the influence of different values of the end shear rate is examined. Original astep was made from a shear rate of 0.3 s−1 to 3.0 s−1. The end value of the shearrate was made 30 s−1, 15 s−1 and 1.5 s−1. Initial morphology was as equilibrium ina shear rate of 0.3 s−1. In figure 3 the evolution of the number of droplets in timeis depicted. As can be seen, a higher shear rate will lead to more droplets. This

0 20 40 60 80 10010

13

1014

1015

1016

1017

1018

1019

Time [s]

Num

ber

of d

rops

per

uni

t vol

ume

[m−

3 ]

Shear rate = 1.5 s−1

Shear rate = 3.0 s−1

Shear rate = 15.0 s−1

Shear rate = 30.0 s−1

Figure 3: The number of droplets in time for different shear rates

is to be expected, since a higher shear rate leads to a higher shear stress and canmore easily overcome the interfacial tension of the droplets. A shear rate of 30 s−1

and 15 s−1 leads to an intermediate platform in the number of drop after the firstbreakup. Apparently the shear rate is still high enough to continue to stretch thedroplets until a second breakup occurs, after which the the droplets are so smallthat the interfacial tension will prevent them from stretching too far and breakingup again. In fact equilibrium between necking and coalescence is found. Thisintermediate platform is notably smaller in time then the stretching zone prior tothe initial breakup. This has part to do with the modelling; the stretch factor afterbreakup is set to 1.5. So droplets have initially a higher stretch factor then in theoriginal configuration, in which the stretch factor was 1. Furthermore the droplets

9

will be smaller, and thus the critical cross section of the stretched filament as statedin equation 6, which is independent of the drop radius, will be reached earlier, andthus the filament will break up earlier.

After the breakup a necking regime is seen for a shear rate of 30 s−1, 15 s−1 and3 s−1. As can be derived from equation 4 a higher shear rate will lead to a fasterchange in radius and thus faster breakup and more droplets. This zone is followed

100

101

102

1014

1015

1016

1017

1018

1019

Shear rate [s−1]

Num

ber

of d

rops

per

uni

t vol

ume

[m−

3 ]

Figure 4: The end value for the number of droplets for different shear rates

by a area in which necking and coalescence are in equilibrium. If the number ofdrops in the end is plotted against the shear rate, a straight line is observed, as seenin figure 4 This is to be expected, as results in the past have clearly pointed out:Vinckier [9]. In here the radius of the droplets is plotted against the shear rate. Theradius of the drops is reversibly proportional to the number of drops. This behaviorwas inserted in the model, so this is actually nothing more then a validation of themodel.

3.2 Effective measure parameter

In the model the step behavior of certain parameters were smoothed. The introduc-tion of these effective measures lead to a time constant τ for the effective stretch

10

factor β and the interfacial area Q. The values of these constants were obtainedfrom experiments, but under different conditions. The assumption that the valuewould be the same for a different situation doesn’t hold automatically. To inves-tigate if these assumptions still hold, the time constants were varied slightly. Theobjective is not to make the best fit, but rather study the influence.

3.2.1 Interfacial Area

In the first series the parameter for the effective interfacial area Q is investigated.The results are shown in figure 5 together with the original. Besides the values forthe effective interfacial area, the only variables plotted here are the normal and theshear stresses, in respectively figure 6 and figure 7, since they can be measured andare directly influenced by the effective measures.

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3x 10

5

Time [s]

Inte

rfac

ial A

rea

[m2 ]

τQ

= 4 τQ

= 2

Original τQ

= 0.5

Unaffected value for area

Figure 5: Evolution of the effective value of the interfacial area for different timeconstants

In the filament stretching zone there is no difference between the different val-ues of τ . This is to be expected, since that no step wise changes in the interfacialarea have occurred. The effective value will follow the real value. After the breakup

11

0 20 40 60 80 1000

50

100

150

200

250

300

350

400

450

Time [s]

Nor

mal

Str

ess

τQ

= 4 τQ

= 2

Original τQ

= 0.5

Figure 6: Normal stress with different time constants for the effective value for theinterfacial area

things become more interesting. Increasing the value for τQ , which means a slowerdecay to the real value, leads to a lower ”peak” in the normal stress. This can beexplained by the step the interfacial area makes towards a greater value, due tothe smaller droplets, which have more area per volume than big droplets. A slowerresponding effective interfacial area will lead to a smaller value for the effective in-terfacial area. When the coalescence starts, the greater value for the time constantwill again lead to a slow response. According to equation( 14) the interfacial areahas a major influence in the relation describing the stress. If the effective value ofthe interfacial area doesn’t undergo major changes, it is likely that the stresses alsohave a more gradual course. Of course the interfacial area is not the only parameterdetermining the stress, but a greater value for the time constant for the interfacialarea will lead to a smoother course of the stress.

It can also be noticed that the normal stress is influenced far more drasticallythan the shear stress. This is rather difficult to explain, but the 4th order tensor 4L,as stated in equation( 15), has a part in it with Q on the diagonal in 4th order 4I,which may dominate the normal stress, rather then the shear stress in the non-maindiagonal parts of the stress tensor.

12

0 20 40 60 80 100580

590

600

610

620

630

640

650

660

670

680

Time [s]

She

ar S

tres

s

τQ

= 4 τQ

= 2

Original τQ

= 0.5

Figure 7: Shear stress with different time constants for the effective value for theinterfacial area

13

3.2.2 Stretch factor

Besides the time constants for the interfacial area, the time constants for the effec-tive stretch ratio are varied also. The results are shown in figure 8. Besides thestretch ratio, the only variables plotted are the normal stress and the shear stress.

0 20 40 60 80 1000

5

10

15

20

25

30

35

Time [s]

Str

etch

rat

io

τλ = 4

τλ =2

Original τλ=0.5

Real value for stretch ratio

Figure 8: Evolution of effective stretch ratio with different time constants

If the time constant for the effective stretch ratio β is varied, one can again seea slower responding stress for higher values for the time constant. The explanationfor this is found in the value for the stress relaxation time as stated in equation( 16).A greater value for β leads to a higher value for the relaxation time. After thebreakup the drop stretch ratio is set to 1.5. The effective measure parameter willdecay from a value of approximately 30 just before the breakup to this end value. Ifthe decay is even more slowed, then the value for the relaxation time of the stresswill also be higher. A slower response of the stress is expected again and seenin figure 9 and figure 10. It is also seen that the relaxation has a major influenceon both the normal stress and the shear stress. The stress relaxation time affectsboth. But as seen in equation( 16) the effective value for the interfacial value isalso included in the stress relaxation time. Due to the power law coefficient of 1.5,which leads to a super-linear behavior for the effective stretch ratio, in comparison

14

0 20 40 60 80 1000

50

100

150

200

250

300

350

400

450

Time [s]

Nor

mal

Str

ess

τλ = 4

τλ =2

Original τλ=0.5

Figure 9: Normal stress with different time constants for the effective value for thestretch ratio

to a linear behavior, in the numerator that is, for the effective interfacial area, willthe stress relaxation time be influenced more by the stretch ratio then the interfacialarea. Furthermore ,the stretch ratio will have to make a greater decay, i.e. from 30to 1.5, in comparison with the interfacial area, which varies within 1.5×104 and2.5×104m2. All in all will parameters that affect β have a greater influence on thestress relaxation time then that parameters that affect the interfacial area do, andthus will have a greater influence on the general course of the stress.

If one would like to make new fits for the time constants, a good approachwould be to first fit the effective stretch ratio on the shear stress and then the inter-facial area on the normal stress, followed by fine tuning.

3.3 End value of β

As stated before, the model is not complete. For example: complex mechanismslike ”end-pinching” and ”tip-streaming” are not included. This has partly to dowith the absence of a decent description of the phenomena, but nevertheless onecan at least make an attempt. To multiply the end value of β with a constant, a very

15

0 20 40 60 80 100590

600

610

620

630

640

650

660

670

680

Time [s]

She

ar S

tres

s

τλ = 4

τλ =2

Original τλ=0.5

Figure 10: Shear stress with different time constants for the effective value for thestretch ratio

crude first step has been made to model phenomena influencing the stretch ratioof droplets, a parameter with great influence. Once again it is not the objective tomake the best fit, but to study influence. One of the most important tests for themodel is the end value of the stress. Therefore the multiplication parameter hasonly been implemented in the coalescence modelling, since this is the most impor-tant phenomena occurring after the breakup, and thus will be of major influenceon the end value for the stresses. The value of β, i.e the right hand side of equa-tion 10, has been multiplied with 2 and 0.5. In figure 11 the results are shown forthe normal stress. The shear stress results are seen in figure 12.

As seen the normal stress is affected far more then the shear stress. One shouldbe careful including these factors.

16

0 20 40 60 80 1000

50

100

150

200

250

300

350

400

450

Time [s]

Nor

mal

Str

ess

[Pa]

Original β=2x org β=0.5x orgData Vinckier

Figure 11: The influence of a multiplication factor for β on the normal stress

0 20 40 60 80 100580

600

620

640

660

680

700

Time [s]

She

ar S

tres

s [P

a]

Original β=2x org β=0.5x orgData Vinckier

Figure 12: The influence of a multiplication factor for β on the shear stress

17

x

y

scalex: 3.000

scaley: 3.000

MESH

Figure 13: Mesh for 4-to-1 contraction

4 Contraction

An first approach has been taken to run the model in a 4-to-1 contraction. In ad-vance one can say that there are a lot of problems to be expected. The combinationof a corner singularity and a time-dependent problem is usually very problematicfor numerical routines. The DEVSS/DG method used can handle singularities, buthas stability problems with time dependent problems. But since Schiek [7] hasmanaged to run his step shear and journal bearing flow, it is to be expected that thecontraction will also work, although there were also problems reported. Further-more there is a combination of shear flows and elongational flows. Some param-eters are flow type dependent, especially the ones concerning critically capillarynumber. There are made no adjustments for this.

4.1 Mesh

Initially, the mesh had a straight corner, but after some divergence problems a smallcorner radius of 0.1 was introduced in the mesh. The mesh included further 711elements with 4898 free degrees of freedom. A picture of the mesh is seen infigure 13. The height from the symmetry line to the top wall is 1.6 and the thetotal length is 5. A parabolic velocity profile, with a maximum fluid velocity of1.5 m/s at the center line, was prescribed at the inlet on the left side of the mesh.The top walls all had impenetrable, no-slip boundary conditions, i.e. the x- andy-velocity were set to zero. At the outlet the y-velocity was set to zero, as in afully developed outflow. Simulations without this boundary condition justified thisassumption. The bottom side of the mesh had the y-velocity set to zero, as in asymmetric flow.

18

scalex: 3.000

scaley: 3.000

time t: 0.000

LEVELS

-18.704

-17.122

-15.539

-13.957

-12.374

-10.792

-9.209

-7.627

-6.044

-4.462

-2.879

-1.297

0.286

1.868

3.451

Contour levels of Stress 11 (TIME sec)

Figure 14: Normal stress

4.2 Results

Due to numerical problems it was not possible to complete a full simulation andreach an equilibrium state. The results represented below are after 0.5 seconds,with a maximum fluid velocity of 1.5 m/s in the symmetry line. Although this isdefinitely not equilibrium, it represents the general trends. The numbers for thestate of the droplets mean:

• 0: Static filament

• 100: Filament stretching

• 200: Start filament stretching

• 300: Filament breakup

• 400: Static filament breakup

• 500: Coalescence

• 600: Coalescence after static breakup

• 700: Necking

As one can see, the greatest values for the shear stress and for the stretch ratioare found in the contraction at the top wall of the outlet. This has lead to the greateststretch ratios and hence at these zones the greatest concentration of droplets isexpected.

The major problems were identified, but no origin or satisfactory solution werefound. The major problem was the value of the stretch factor dropping below 1,which gives for example problems in equation 16, were the power of 1.5 applied

19

scalex: 3.000

scaley: 3.000

time t: 0.000

LEVELS

-22.095

-19.234

-16.372

-13.511

-10.650

-7.788

-4.927

-2.066

0.795

3.657

6.518

9.379

12.240

15.102

17.963

Contour levels of Shear Stress ?? (TIME sec)

Figure 15: Shear stress

scalex: 3.000

scaley: 3.000

time t: 0.000

LEVELS

1.137

1.960

2.782

3.604

4.427

5.249

6.071

6.894

7.716

8.538

9.360

10.183

11.005

11.827

12.650

Contour levels of DMM Beta eff. (TIME sec)

Figure 16: Effective stretch ratio

scalex: 3.000

scaley: 3.000

time t: 0.000

LEVELS

100.000

200.000

300.000

400.000

500.000

600.000

700.000

Contour levels of DMM DPFLAG (TIME sec)

Figure 17: State of the droplets

20

to a negative number will lead to problems. In most of these points the trace ofthe rate of deformation tensor D was not zero anymore, which is in violation ofthe incompressibility constraint. Artificial solutions, like building in constraintsto prevent the stretch factor dropping below 1, have been inserted. This lead to aslightly longer simulated time, but there was an ever increasing number of pointsin which the stretch factor dropped below 1, resulting eventually in divergence ofstress to infinity.

There have been made attempts by refining the mesh to run for longer time, butthese effects were not satisfactory enough. Mesh refinement did lead to slightlylonger simulated times, but the computational time was out of proportion and thusthere have been made no attempts to refine the mesh to extreme. Reducing the timestep also didn’t lead to major improvements. The best results were obtained witha reduction of the mean fluid velocity with a factor of ten, in combination with areduced time step and some mesh refining. The simulation ran longer, and moreimportantly the total fluid flux, velocity times total simulated time, was greater byabout a factor 2. However since the velocity was smaller, and thus local shear rateswere smaller, there were no phenomena observed that didn’t show in the original.

21

5 Conclusions and Recommendations

The dispersive mixtures model as suggested by Peters [5] shows promising results.The implementation in a step shear simulation has been successful and results haveproven to be realistic. A higher shear rate leads to more and smaller droplets. Ad-justing parameters for the effective measures for the interfacial area and stretchfactor can greatly influence the stress and thus need to be chosen with care. Sim-ple adjustments to the stretch ratio also influence the stress. All these parametersadjustments should be handled with care.

An attempt to run the model in a 4-to-1 contraction has proven to be unsuccess-ful. Numerical problems lead to divergence of stress and no satisfactory solutionhas been found. But trends found indicate generally in the right direction, of manysmall droplets found along the walls.

To run the contraction problem successful, one could decouple the deforma-tion, the stresses and the drop morphology. For this, the used DEVSS/DG methodwould have to be replaced with a decoupled routine. Furthermore the deformationof the droplets should be study in a more controlled way, with lots of checks dur-ing the simulation. Since the stretch ratio tends to drop below 1 at some points,there are probably faults in the calculation of the deformation of the droplets. So athorough check of these can perhaps yield some progress.

References

[1] A.C.B. Bogaerds, 3D Viscoelastic Analysis, Masters Thesis, Department ofMechanical Engineering, Eindhoven University of Technology, 1999.

[2] A.K. Chesters, The modeling of coalescence processes in fluid-liquid dis-persions: A review of current understanding Transactions of the Institute ofChemical Engineers, 69A pp. 259–270, 1991.

[3] R.G. Cox, The deformation of a drop in a general time-dependent fluid flowJournal of Fluid Mechanics, 37 pp. 601–623, 1969.

[4] H.P. Grace, Dispersion phenomena in high viscosity immiscible fluid systemsand application of static mixers as dispersion devices in such systems. 3rd En-gineering Foundation Conference on Mixing, 1971. Republished in ChemicalEngineering Communications, 14 pp. 225–227, 1982.

[5] G.W.M. Peters, S. Hansen & H.E.H. Meijer, Constitutive modeling of disper-sive mixtures Journal of Rheology, 37(3) pp. 659–689, 2001.

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[6] J. Janssen & H.E.H. Meijer, Droplet break-up mechanisms: Stepwise equi-librium versus transient dispersion Journal of Rheology, 37 597, 1993.

[7] R.Schiek, P.D. Anderson & G.W.M. Peters, Simulating Polymer Blends witha Phenomenologically Derived Constitutive Model, Internal Report, SectionMaterials Technology, Department of Mechanical Engineering, EindhovenUniversity of Technology, 2001.

[8] Y. Stegeman, F.N. van de Vosse, A.K. Chesters & H.E.H. Meijer, “Break-upof (non-) Newtonian droplets in a time-dependent elongational flow” Pro-ceedings of the Polymer Processing Society ’s Hertogenbosch, The Nether-lands (CDrom), 15, 1999.

[9] I. Vinckier, Microstructural Analysis and Rheology of Immiscible PolymerBlends, Ph.D thesis, Katholieke Universiteit Leuven, Faculteit ToegepasteWetenschappen, Departement Chemische Ingenieurstechnieken, de Croylaan46, 3001 Leuven, Belgium, 1998.

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