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( 1 cm 2 31293 01691 4586

This is to certify that the
dissertation entitled

THE MOVEMENT AND BREAKUP OF VISCOUS AND
VISCOELASTIC DROPS IN PERIODIC ARRAYS OF CYLINDERS

presented by

DOUGLAS JAMES BACKES

has been accepted towards fulfillment
of the requirements for

Ph.D. Chemical Engineering

degree in

1W
Ial/M/i 7 MW

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THE MOVEMENT AND BREAKUP OF VISCOUS AND VISCOELASTIC DROPS
IN PERIODIC ARRAYS OF CYLINDERS

Douglas James Backes

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Chemical Engineering

1 997

ABSTRACT

THE MOVEMENT AND BREAKUP OF VISCOUS AND
VISCOELASTIC DROPS IN PERIODIC ARRAYS OF
CYLINDERS

By

Douglas James Backes

The removal of air pockets and binder droplets from fiber bundles is important for
producing high quality composite parts. One method of removal is to mobilize them and
push them out of the fiber preform during mold filling (in most commercial processes Ca >
1). To further understand this phenomenon, mobilization was modeled with drops moving
through an array of cylinders aligned transverse to the flow of suspending fluid (resin).

Drops of a Newtonian water/ glycerol mixture or a viscoelastic water/glyceroll-
polyacrylamide mixture were suspended in an immiscible chloroparaffin oil. The drops were
neutrally buoyant in the oil whose refractive index was matched with that of the solid
PMMA rods for easy viewing of the drops. Three rod arrays were examined here: a square
and a hexagonal array with a fiber volume fraction (V f) of 0.40 and a square array with Vf
= 0.50. Drops of different lengths were placed in the array and were mobilized with different
flow rates of the oil (resulting in different Ca).

V The shape of the drop as it moves through an array is determined by a combination

of the initial drop shape, the array geometry, and the drop fluid rheology. This drop shape

will determine the mobility, the critical capillary number (Ca"'), and the breakup behavior
of the drop as it moves through the cylindrical array.

The critical capillary number (Ca*) is defined as the point at which the drop first
becomes mobilized. For both types of drops, Ca“ increased with decreasing drop length and
Vi, and was higher for square arrays. For both arrays, Ca" was higher for the viscoelastic
drops. A test with drops composed of a much more viscous Newtonian fluid (pure glycerol)
with a higher k resulted in lower Ca“ indicating that the higher Ca“ for the viscoelastic drops
resulted from their elasticity.

The mobility of a Newtonian drop increased with increasing Ca, drop length, and Vf;
and with decreasing viscosity ratio (k). Mobility was also higher for square arrays. For
viscoelastic drops, there was a Ca above which mobility increased with decreasing drop
length (all other trends remained the same). The Deborah number (the ratio of the relaxation
time vs the residence time in the throat) for the drop was low at this inversion point (De <
0.05). Despite this, the elastic nature of the drop caused the inversion due to stiffening of
viscoelastic filaments as they are stretched.

Above a certain capillary number (Cab) the drop would break into smaller droplets.
If these droplets were too small, they would become immobilized in the array. For the
square array there was also an upper Cab above which breakage did not occur. Longer drops

were more unstable, breaking more frequently and at a lower Cab.

To Everett 1.]. Paulson (1910-1997)

iv

ACKNOWLEDGMENTS

I would like to thank Dr. Krishnamurthy Jayaraman for his guidance and assistance
throughout the course of this research. I would also like to thank the other members of my
committee for their time and assistance: Dr. Lawrence Drzal, Dr. Karen Klomparens, and Dr.
Alec Scranton. My appreciation also extends to Mike Rich for providing much needed
technical guidance throughout this project. Further thanks go to Terry Casey and Mike
McClean for their assistance in making the equipment used here. I would finally like to
thank Dr. James Steffi for use of his liquid Rheometrics equipment and Dr. Stanley Flegler
for his assistance in the use of the electron microscopes.

Finally, I wouldn’t have gotten through this without the comradarie of my fellow
students and post-docs, in particular Sanjay Mishra, Himanshu Asthana, and Ashim Dutta
whose coffee chats were enlightening to say the least. I also had the pleasure of knowing a
bizarre group of people including: John, Rob, Nancy, Min, James, Julie, Ajay, Rik, Mark,

Marketta, Prasad, Sunil, and the gang at the MSU Bridge Club.

TABLE OF CONTENTS

Page
LIST OF TABLES ........................................................................................................... x
LIST OF FIGURES ......................................................................................................... xi
NOMENCLATURE ....................................................................................................... xvi
1. INTRODUCTION ...................................................................................................... 1
1.1. Voids ............................................................................................................ 1
1.1.1. Void formation .............................................................................. 1
1.1.2. Effect of voids on part properties ................................................... 7
1.1.3. Void mobilization .......................................................................... 10
1.2. Binders .......................................................................................................... 11
1.2.1. Function of binder in preforrning ................................................... 11
1.2.2. Effect of binder on part properties ................................................. 15
1.2.3. Binder mobilization ........................................................................ 21
2. EQUIPMENT AND MATERIALS .............................................................................. 23
2.1. Equipment ...................................................................................................... 23
2.1.1. Flow cell .......................................................................................... 23
2.1.2. Cylinder arrays ................................................................................ 26
2.1.3. Experimental configuration and procedure ..................................... 29
2.2. Materials ........................................................................................................ 32

vi

3. DEF ORMATION AND BREAKUP BEHAVIOR OF CYLINDRICAL DROPS IN

FLOW THROUGH ARRAYS OF CYLINDERS ............................................................ 39
3.1 . Introduction .................................................................................................... 39
3.2 Background .................................................................................................... 39

3.2.1. Motion and breakup through capillary tubes and channels ............ 40
3.2.2. Motion and breakup through a constriction .................................... 42
3.2.3. Motion and breakup through a packed bed ..................................... 45
3.3. Materials and procedure ................................................................................. 48
3.4. Objectives ...................................................................................................... 49
3.5. Results and discussion ................................................................................... 50
3.5.1. Drop motion through the arrays ...................................................... 50
3.5.1.a. Square arrays .................................................................... 50

3.5.1 .b. Hexagonal array ............................................................... 64

3.5.2. Drop breakup .................................................................................. 68
3.5.2.a. Square arrays .................................................................... 69

3.5.2.b. Hexagonal array ............................................................... 73

3.5.3. Consequences of breakup for drop mobilization ............................ 81

3.6. Conclusions .................................................................................................... 89

4. MOBILIZATION OF CYLINDRICAL DROPS IN FLOW ACROSS AN ARRAY

OF CYLINDERS ............................................................................................................... 91
4.1 . Introduction .................................................................................................... 91

4.2. Background .................................................................................................... 91

4.2.1. Flow through capillary tubes and channels ..................................... 93

vii

4.2.2.Flow through a constriction ................................................................... 96

4.2.3. Flow through a packed bed ............................................................... 97

4.2.4. Theory ............................................................................................... 99

4.3. Objectives ....................................................................................................... 101
4.4. Materials and procedures ............................................................................... 102
4.5. Results and discussion ................................................................................... 104
4.5.1. Newtonian drops ................................................................................ 104

4.5.1 .a. Vr=0.40 square array ............................................................ 106

4.5.1 .b. V,«=O.50 square array ........................................................... 109

4.5.1.0. Vf=0.40 hexagonal array ................................................... 116

4.5.1.d Comparison of Newtonian drops in the arrays .................. 120

4.5.2. Viscoelastic drops ........................................................................... 124
4.5.2.a. Vr=0.40 square array ........................................................ 128

4.5.2.b. V,=O.40 hexagonal array .................................................. 133

4.5.2.c. Comparison of viscoelastic drops in the arrays ................ 139

4.5.3. Comparison of Newtonian and viscoelastic drops .......................... 139
4.5.3.a. Square array ...................................................................... 139

4.5.3.b. Hexagonal array ................................................................. 147

4.6. Drop mobilization model ................................................................................. 151
4.7. Conclusions .................................................................................................... 153
5. CONCLUSIONS .......................................................................................................... 155
6. RECOMMENDATION ................................................................................................ 159

viii

APPENDIX A ...................................................................................................................... 160

APPENDIX B ..................................................................................................................... 166

REFERENCES ................................................................................................................... 169

ix

LIST OF TABLES

Page

Table 2.1: Array specifications ...................................................................................... 29
Table 2.2: The interfacial tensions for the liquid-liquid systems ................................... 38
Table 3.1: Array specifications ........................................................................................ 48
Table 3.2: Properties of the fluids ................................................................................... 48
Table 4.1: Parameters for different bubble shapes in a Hele-Shaw cell

[from Kopf-Sill and Homsy] ........................................................................... 95
Table 4.2: Array specifications ............................................................................................ 102
Table 4.3: Properties of the fluids ..................................................................................... 103
Table A]: The surface tension for the solid surfaces ....................................................... 164
Table A2: The surface tension for the liquid materials .................................................... 165
Table A3: The interfacial tensions for the liquid-liquid systems ..................................... 165

Figure 1.1:
Figure 1.2:
Figure 1.3:
Figure 1.4:

Figure 1.5:

Figure 1.6:

Figure 1.7:
Figure 1.8:
Figure 2.1:
Figure 2.2:
Figure 2.3:
Figure 2.4:
Figure 2.5:
Figure 2.6:
Figure 2.7:

Figure 2.8:

Figure 2.9:

LIST OF FIGURES

Page
Methods of void entrapment ....................................................................... 3
Two levels of resin impregnation ............................................................... 5
Consolidated nonwoven mat ...................................................................... 13
Consolidated bidirectional mat ................................................................... 14
Effect of binder concentration on the conversion profile
of a vinyl ester resin ..................................................................................... 17
Effect of binder concentration on the reaction rate of a
vinyl ester resin ............................................................................................. 18
Effect of binder concentration on the cure of polyurethane ......................... 19
Mobilization of binder in a fiber bundle ....................................................... 22
Aluminum flow cell ...................................................................................... 24
Schematic of window .................................................................................... 25
Rod array ....................................................................................................... 27
Packing geometries ........................................................................................ 28
Rod array ........................................................................................................ 30
Experimental setup ......................................................................................... 31
Calibration curve for rotameter with oil ......................................................... 33
Dynamic moduli for viscoelastic drop ............................................................ 35
Relaxation time vs. Frequency for the viscoelastic drop ................................ 36

xi

Figure 3.1: Trajectory of the drop through an array of cylinders ....................................... 48
Figure 3.2: Motion of a Newtonian drop through a square array (cross section view) ...... 51

Figure 3.3: Newtonian drop movement through an array of transversely aligned
cylinders with a square arrangement (top view) ............................................. 53

Figure 3.4: Movement of a 3.82 cm Newtonian drop through the Vf = 0.40

square array at different Ca .............................................................................. 55
Figure 3.4 (continued): ....................................................................................................... 56
Figure 3.5: Movement of a 0.60 cm Newtonian drop through the Vf = 0.40
square array at Ca = 0.0266 ............................................................................ 58
Figure 3.6: Movement of a 2.87 cm viscoelastic drop through the Vf = 0.40
square array at different Ca ............................................................................ 59
Figure 3.6 (continued) ........................................................................................................ 60
Figure 3.7: Movement of a 1.08 cm viscoelastic drop through the V, = 0.40
square array at Ca = 0.0907 ........................................................................... 62
Figure 3.8: Movement of viscoelastic drops of different lengths through the
Vf = 0.40 square array at Ca = 0.524 ............................................................. 63
Figure 3.9: Path of a drop in a hexagonal array ................................................................. 65
Figure 3.10: Fluid flow around a cylinder in a hexagonal array ........................................ 66

Figure 3.11: Motion of a Newtonian drop through a hexagonal array (cross section
view) ............................................................................................................. 67

Figure 3.12: Breakup of a Newtonian drop in an array of transversely aligned
cylinders with a square arrangement (top view) ............................................ 70

Figure 3.13: Cab for a Newtonian drop in a Vf = 0.50 square array ................................... 71

Figure 3.14: Breakup of a longer Newtonian drop in a hexagonal array (cross section
view) ............................................................................................................. 74

Figure 3.15: Breakup of a 4.66 cm Newtonian drop in a V, = 0.40 hexagonal
array at Ca = 0.015 ........................................................................................ 76

xii

Figure 3.16: Breakup of a longer viscoelastic drop in a hexagonal array (cross section
view) ............................................................................................................. 77

Figure 3.17: Breakup of a 1.89 cm viscoelastic drop in a V, = 0.40 hexagonal

array at Ca = 0.052 ........................................................................................ 79
Figure 3.18: Cab for Newtonian and viscoelastic drops in a V, = 0.40

hexagonal array ............................................................................................. 80
Figure 3.19: Drop path through a square array ................................................................. 82
Figure 3.20: Path through a hexagonal array of a 3.50 cm Newtonian drop at

Ca=0.0064 ..................................................................................................... 84
Figure 3.21: Path through a hexagonal array of a 3.50 cm Newtonian drop at

Ca=0.012 ....................................................................................................... 85
Figure 3.22: Path through a hexagonal array of a 3.50 cm Newtonian drop at

Ca=0.019 ....................................................................................................... 87
Figure 3.23: Path through a hexagonal array of a 4.66 cm viscoelastic drop at

Ca=0.012 ....................................................................................................... 88
Figure 4.1: Drop moving between two ........................................................................... 100
Figure 4.2: Surface area/volume ratio vs drop length for drops in various arrays ........... 105

Figure 4.3: Relative drop velocity vs Ca for Newtonian drops in

a square array (V 50.40) ................................................................................. 107
Figure 4.4: Determination of V* using 1/U vs V curve (drop length of 3.82 cm) ........... 110
Figure 4.5: 1/U vs V for Newtonian drops in a square array (V 50.40) .......................... 111
Figure 4.6: Ca“ vs drop length for Newtonian drops in a square array (V (-0.40) ........... 112
Figure 4.7: Relative drop velocity vs Ca for Newtonian drops in a square

array (V 50.50) ............................................................................................... 114
Figure 4.8: Ca“ vs drop length for Newtonian drops in a square array (V 50.50) ........... 115
Figure 4.9: Path of a drop in a hexagonal array ............................................................... 117

xiii

Figure 4.10: Relative drop velocity vs Ca for Newtonian drops in a hexagonal array
(V5040) ........................................................................................................ 119

Figure 4.11: Retraction of a drop branch in a hexagonal array ........................................ 121
Figure 4.12: Ca“ vs drop length for Newtonian drops in a hexagonal array (V 5040).. 122
Figure 4.13: Relative drop velocity vs Ca for long Newtonian drops in the three

arrays ............................................................................................................ 123
Figure 4.14: Relative drop velocity vs Ca for short Newtonian drops in the three

arrays ............................................................................................................ 125
Figure 4.15: Ca" vs drop length for Newtonian drops in the three arrays ....................... 126
Figure 4.16: Ca“ vs dimensionless drop length (A) for Newtonian drops in the three

arrays ........................................................................................................... 127
Figure 4.17: Movement of viscoelastic drops through a V, = 0.40 square array at

Ca = 0.0907 .................................................................................................. 129
Figure 4.18: Relative drop velocity vs Ca for a viscoelastic drop in a square array

(V5040) ..................................................................................................... 130
Figure 4.19: Relative drop velocity vs Ca for a viscoelastic drop in a square array

(V5040) [Redrawn] .................................................................................... 131
Figure 4.20: Ca" vs drop length for viscoelastic drops in a square array (V5040) ....... 134
Figure 4.21: Relative drop velocity vs Ca for a viscoelastic drop in a hexagonal array

(V5040) ...................................................................................................... 135
Figure 4.22: Relative drop velocity vs Ca for a viscoelastic drop in a hexagonal array

(V 504.40) [Redrawn] .................................................................................. 137
Figure 4.23: Ca” vs drop length for viscoelastic drops in a hexagonal array (V 50.40).138
Figure 4.24: Relative drop velocity vs Ca for viscoelastic drops in the V5040 arrays..140
Figure 4.25: Ca“ vs drop length for viscoelastic drops in the V5040 arrays ................. 141
Figure 4.26: Relative drop velocity vs Ca for Newtonian and viscoelastic drops in a

square array (V5040) ................................................................................. 143

xiv

Figure 4.27: Relative drop velocity vs Ca for the three drops of length 1.55 cm and

different k in a square array (V5040) ......................................................... 145
Figure 4.28: Ca" vs drop length for Newtonian and viscoelastic drops of different k in

a square array (V 50.40) .............................................................................. 146
Figure 4.29: Relative drop velocity vs Ca for Newtonian and viscoelastic drops in a

Hexagonal array (V 50.40) ........................................................................... 149
Figure 4.30: Ca* vs drop length for Newtonian and viscoelastic drops in a hexagonal

array (V 50.40) ............................................................................................ 150
Figure 4.31: Model results vs experimental results for Ca" ............................................ 152
Figure A. 1: Sessilc drop method for determination of contact angle .............................. 162
Figure B]: Cylindrical drop with round ends ................................................................. 167
Figure B.2: Dimensions of a drop moving through a rod constriction ............................ 167

XV

Ca

Ca“

fl, f2
6’

G”

NOMENCLATURE

Capillary number

Critical capillary number for mobilization
Critical capillary number for breakup
Bubble capillary number

Pore capillary number

Curvature of leading and trailing feet
Deborah number

Throat diameter

Numerical functions for sessile drop method
Dynamic storage shear modulus
Dynamic loss shear modulus

Film thickness

Viscosity ratio

Effective permeability

Length of drop

Length of main body of the drop

Length of downstream edge of the drop
Length of leading foot of the drop

Length of upstream edge of the drop

xvi

Oh

pcapi

pvisc

Ohnesorge number

Capillary pressure

Viscous (hydrodynamic) pressure

Sessilc drop factor

Radius of a constriction

Radius of drop

Body radius of the array

Radius of downstream edge of the drop
Throat radius of the array

Radius of upstream edge of the drop

Surface area of drop

Surface area/volume ratio of the drop
Distance between leading and trailing drop feet
Drop velocity

Relative velocity of drop

Fiber volume fraction

Interstitial velocity

Factor of how much U exceeds fluid velocity
Work of adhesion

Work of adhesion for a liquid-liquid interface
Work of adhesion for a solid-liquid interface
Sessilc drop factor

xvii

X1, X2

1’1

75

Y1

Y2

712

Location of feet along the flow direction
Angle between fluid direction and drop direction
Geometric factor of the array

Surface tension

Dispersion component of the surface tension
Surface tension of a liquid

Polar component of the surface tension
Surface tension of a solid

Surface tension of suspending fluid

Surface tension of drop

Interfacial tension

Resin viscosity

Contact angle

Time of deformation

Dimensionless size of the drop

Relaxation time

Density

Snap-off time

Frequency

xviii

1. INTRODUCTION

Resin Transfer Molding (RTM) and Structural Reaction Injection Molding (S-RIM)
are processes widely used in industry for the production of composite parts. In both
processes a preform made of fiber mats is placed in a mold which is then heated. The mold
is then filled with a polymer resin and heated for curing and post-curing after which the part
is removed. Inhomogeneities and flaws in the part can result from several processing factors
such as channeling and resin shrinkage. Two factors that can have a great effect on part
properties and will be examined in this study are trapped pockets of air (voids) and the
melting of a thermoplastic binder used to hold the preform together. The movement of both
will be modeled with an immiscible drop moving through an array of transversely aligned

cylinders. This alignment will match that of a bidirectional mat.

1.1 Voids

1.1.1 Void formation
During the mold filling process, when the resin penetrates the fiber preform, voids
will often form at the fiber/resin interface. This is the result of the resin trapping pockets of

air in regions of the fiber bed that are difficult for the resin to penetrate. Areas of the fiber

1

2

surface are left unwetted with no fiber/resin adhesion. The degree to which voids are created
depends on a variety of factors: the viscosity of the resin; the contact angle of the resin on
the fiber; the number of defects (such as nicks and grooves) on the fiber surface; and the
structure of the preform [1]. An early study of voids was conducted by Bascom and Romans
[2] in a fiber wound system. They found that small micro voids would form between
individual filaments in a fiber tow when the tow was drawn through the resin reservoir.
Tang [3] saw void formation in compression molding using a resin and woven fabrics. The
void content decreased exponentially with increasing cure pressure from 15% at 10 psig to
under 5% at 90 psig.

A variety of studies on RTM have been conducted to determine the degree to which
void formation occurs in that process. Peterson and Robertson [4] worked with a tube packed
with glass fiber rovings in the direction of resin flow. At constant injection pressure, they
found that with increasing fiber volume, the total volume of voids increased while the size
of the individual voids decreased. The relatively closed structure made it more difficult for
the incoming resin to push the voids out of the mold. At fiber volumes of >50%, there was
a uniform dispersion of voids and they had no mobility. Increasing injection pressure
decreased the total void volume and the size of the individual voids. In their study, the voids
formed at the fiber/resin interface as the result of three conditions: surface defects,
constriction in the pathway, and constrainment between two or more fibers (Figure 1.1).

Mahale [5] investigated void formation using an RTM system with continuous strand
nonwoven preforrns. In this studies, they determined how void formation varied with the

capillary number:

Resin
flow

 

 

/

Figure 1.1: Methods of void entrapment: 1. Attach at
surface defect, 2. Constriction of pathway, 3.
Constrainment between two or more fibers (Robertson,
1991)

Ca = u (1.1)
Y

where Ca is the capillary number, n is the resin viscosity, v is the interstitial velocity of the
resin, and y is the surface tension. They found that void entrapment is negligible above a
certain critical capillary number and that below it, the void volume increased exponentially
with decreasing capillary number. A variety of void sizes occur inside the mold and with
increasing velocity, the number of large voids decreases and the number of small voids
increases. These results indicate that increasing either velocity or viscosity (while keeping
the other constant) will result in a decrease in the void content. Since increasing velocity or
viscosity results in an increase in resin pressure, this study shows that an increase in pressure
causes a decrease in void volmne which matches the results of Robertson [4]. Hayward and
Harris [6] found that void formation in RTM can be greatly reduced with the use of a vacuum
to assist in mold filling. This can reduce void volume from over 2% to less than 0.2%. They
also observed that the number of voids increased further away from the mold inlet.

Chen [7] performed visualization experiments on void formation in continuous strand
mats to elaborate and Mahale’s results. A mechanism for air trapping and void formation
was described in three phases (Figure 1.2): initial liquid bypassing and air entrapment; later
capillary penetration of fiber bundles; and finally air bubble mobilization as the result of flow
pressures. If the capillary velocity in the fiber bundle was greater than the mold filling
velocity, then meso voids formed by the converging of flow fronts predominate. In fast
filling, micro voids trapped within fiber bundles predominate. With sufficient viscous force

(greater than the capillary forces), the micro voids become mobilized and are forced out of

Fiber bundle

 

 

 

 

 

 

 

 

fl,

 

 

 

Fluid
injection

 

> Fiber bundle

 

Figure 1.2: Two levels of resin impregnation (Macosko, 1995)

6

the bundles where they converge to form large spherical macro voids and are pushed toward
the mold outlet. Increased fiber volume fraction will also reduce the amount of voids
entrapped by increasing the mold filling pressure.

Chen also illuminated an apparent anomaly of Mahale’s work: that increased resin
flow rate and viscosity decreased the amount of voids formed. One would expect that
increased resin flow rate through the bed would create greater inhomogeneities in the flow
front resulting in more voids. Zisman [1] and others also determined that increased resin
viscosity would make it harder for the resin to penetrate surface defects in the fiber and that
would also result in a greater amount of voids. Chen determined that what happened during
mold filling was that when the resin initially wetted the fiber, increased velocity and
viscosity caused more voids to be formed. However, because of the increased filling
pressure that results, the voids are more likely to be mobilized and pushed out of the fiber
bed resulting in a smaller number of voids in the final part.

Void formation can also be affected by the overall mat geometry. Patel and Lee [8,9]
studied the mold filling for a unidirectional double stitched mat aligned parallel with the
flow. At low flow rates, the resin front within the fiber tows is ahead of the front in the gaps
between the tows. This is the result of capillary forces within the tows being greater than the
hydrodynamic force. At high flow rates, the flow front in the gaps is ahead of that within the
tows which results fiom greater hydrodynamic than capillary forces. The result is that at low
flow rates, macro voids are formed inside the gaps while at high flow rates, micro voids are
formed within the tows. Stitching, which is ofien used to hold a nonwoven mat together,

may also cause voids by acting as a barrier to resin flow.

7

Void formation is a significant problem in S-RIM because the high rate of resin flow
allows little time for fiber wetting. McGeehan [10] did extensive work on the effect of S-
RIM processing conditions on void formation involving radial flow. Two different types of
prefonns were studied, one consisting of continuous nonwoven mat and a hybrid composed
of two layers of tightly woven mat with a layer of chopped random mat sandwiched between
them. Twice as many voids were formed in the hybrid mat compared with the continuous
strand mat. This was caused by channeling through the woven mat (where resistance was
less) which resulted in non uniform flow through the thickness of the mold at the flow fiont
and entrapment of large air pockets. This did not occur in the nonwoven mat since the
resistance to fluid flow was uniform. The use of a vacuum for S-RIM reduced the void
volume from 4.4% to 2.6% but increased the size of the voids to increase by 27%. The
number of voids varied with the location fi'om the mold inlet, staying constant until two-
thirds of the way in the mold when it increased and reached a maximum at the mold edge.
The size of the voids did not change along the length of the mold and the void content did

not depend on the injection rate.

1.1.2 Effect of voids on part properties

The formation of voids at the resin/fiber interface where adhesion occurs can have
detrimental effects on the properties of the composite. Early studies by Mylonos [11] found
that a predominant cause of stress failure was the formation of stress concentrations at the
adhesion interface. He determined that this was the result of poor wetting and that as the

wetting angle increased, the stress concentration increased by a factor of two. Griffith [12]

8

concluded that the degradation of adhesion is the result of inhomogeneities such as voids,
air bubbles, or surface defects occurring at the interface. He also ascertained that premature
failure will occur when the resulting stress concentrations are significantly higher than the
mean stress applied to the sample.

This phenomenon was readily explained for voids by Zisman [1]. In an ideal
situation where the wetting angle is zero (spontaneous spreading of the liquid) and any
defects in the fiber surface are filled by the resin, the theoretical joint strength of the
fiber/resin interface will be much greater than the tensile strength of the matrix. However
this joint strength is never reached in practice because of the stress concentrations and the
voids that result at the interface. In fact, if the wetting angle is zero it may not fill all the
fiber defects leaving micro voids at the interface. This will only intensify when the wetting
angle is greater than zero. Voids are especially detrimental if they occur close together in
the same plane because it is likely that a crack will propagate from one void to the next in
a similar fashion to a zipper and will result in joint breakage. Although very rough surfaces
may prevent planar voids from occurring, it is much more desirable to smooth the surface
with a chemical sizing.

In their initial study of voids in fiber wound systems, Bascom and Romans [2] found
that voids affect the strength of composites in an adverse way. When the void volume was
reduced from 5% to 0.2% in test ring samples made by fiber winding, the interlaminar shear
strength would increase form 40 to 100% depending on the resin system used. Kohn [13],
working with the same system used by Bascom and Romans, determined an empirical

relationship between void content and interlaminar shear strength. These results indicate that

9

reducing the void content below 0.5% will not give sufficient gains in strength. Tang [3]
found in compression molding that the macro voids formed between laminates will sharply
reduce three properties of the part: longitudinal compressive strength, short beam shear
(SBS) strength and SBS modulus. These properties were constant below 3-4% void content,
the properties remained relatively constant indicating that to maintain good mechanical
properties, it is necessary only to keep the void content below a certain limit.

Hayward and Harris [6] compared the SBS and flexural strengths of RTM samples
made with and without vacuum assistance with vacuum reducing the void content from 1%
to 0.15%. They found that the SBS strength decreased 7% for every 1% increase in void
content. They then compared how different production methods affected the void content
and the physical properties RTM samples with fiber volume fiactions of 0.40. Hand lay up,
compression molding, RTM, and vacuum assisted RTM produced void contents of 3.5%,
2.8%, 1%, and 0.15% respectively. SBS strength, flexural strength and flexural modulus
increased with decreasing void content and hence vacuum assisted RTM is the best
production method based on its ability to reduce void content.

McGeehan [10] studied the relation between void volume and mechanical properties
in an S-RIM system. In fatigue testing on samples with increasing void content, there was
no apparent decrease in fatigue strength which resulted from fiber support of the load in
tension. The fatigue modulus decreased by 20% as the void content increased from 1% to
7%. In the study of impact damage, the damage length (cracks) decreased with increasing
void content. The voids apparently act to arrest the growth of cracks and this was most

significant at high void content (>2%). In terms of impact energy the voids are detrimental

10

with increasing void content resulting in decreasing maximum load at high impact energies.
A 2.1% void content would result in a 13% reduction in the maximum load withstood an this
may result from a decrease in the bulk density of the sample.

Voids can also affect the mechanical properties of a part by promoting moisture
absorption. Moisture content will reduce properties by plasticizing the resin and weakening
the fiber/resin bond. Harper [14] studied the moisture absorption in compression molded
specimens by exposing them in humid environment and found that specimens with a 5%
void content absorbed 50% more moisture than samples with a 1% void content. McGeehan
[10] studied moisture absorption in their S-RIM samples by immersing them in water and
found that when void content increased from 1.0% to 3.2%, the moisture content increased
by 50%. In comparing samples with 1.1% void content, the tensile strength decreased by up
to 30% for the samples with the high moisture content.

One way voids can have a positive impact on the part is when resin shrinkage occurs.
During polymerization, the a resin will compress as the result of cross linking. When this
occurs, the matrix may pull away from the fiber causing micro cracks to form at the interface
which will result in a decrease in part strength. Void may be heated and expand during the

cure and compensate for the resin shrinkage, reducing the number of cracks that appear [15].

1.1.3 Void mobilization
Chen, et al [7] determined a mobilization mechanism for micro void mobilization
after the initial wetting of the fiber bundle. Resin penetration of the bundle results in

interfacial tension and viscous forces being placed on the microvoid and the void will try to

11

balance the forces by moving along the axial direction. During this movement, several
situations may occur: the voids may merge with other micro voids, they could find a gap or
larger pore in the bundle and escape, or they could remain trapped inside the pore. If the
interfacial tension cannot hold the microvoid, the pressure gradient of the resin will force the
microvoid to move transversely in the bundle and eventually are forced out. Rohatgi [9]
found that for void mobilization, the pressure of the incoming fluid must be greater than the
capillary and air pressures of the void and the resistance due to stearic hindrances. For micro
voids as well as macro voids, they observed that at high flow rates, the hydrodynamic
pressure caused the voids to elongate and break apart into smaller voids. This decreased the
stearic hindrance and forced the voids out of the bundle.

Lundstrom [16] saw that air pockets could also be removed from the void as a result
of collapse and dissolution into the resin. He found that three factors increased the degree
of dissolution of the binder: high pressure, high local flow rate, and the presence of a
vacuum. He also determined that the type of resin affected the degree of dissolution with

vinyl ester being much more favorable than epoxy.

1.2 Binder

1.2.1 Binder function in preforming
Continuous strand mats are used in making preforms for RTM and S-RIM because
they are less bulky and rigid than the fiber directed mats and allows more flexibility in mold

design. They also offer greater flexural strength and modulus, creep resistance, fatigue

12

endurance, and impact strength [17]. Woven mats have to be placed precisely one on top of
the other to give the final part the optimum properties of flexibility and strength. However,
the large amount of time needed to cut, overlap, and stitch the mats will result in decreased
production and higher labor costs [1 8]. Nonwoven continuous strand mats were the solution
to this problem since they didn't need to be precisely stacked. Because these mats would fall
apart while being handled during stacking and mold loading, the mats are sprayed with a
chemical binder that holds the individual strands together. This binder is usually a
thermoplastic powdered polymer and is applied in quantities up to 10% of the total mat
weight.

To manufacture a nonwoven preform, mats are stacked with the number determined
by desired final porosity. The stack is then placed in a hydraulic press and is compressed to
the desired thickness with heated press platens to melt the binder. As a result of the
compression forces, the binder will flow and reconfigure among the fibers so that it holds the
plies together upon cooling. The mats do not spread tangentially during compression so that
the preform can be transferred directly form the press to the mold.

Each individual tow (strand) in the fiber mat consists of many individual filaments
held together with a sizing. When the stack of mats is compressed, individual filaments of
each tow are forced to spread out resulting in the flattening of each tow and the formation
of gaps between the individual filaments. The binder, which is initially in the form of small
globules on the tow surface, will melt and smear out along the length of the fibers and fill
in the filament [19]. Figure 1.3 shows a consolidated nonwoven mat and Figure 1.4 shows

a consolidated bidirectional mat. In both cases, the dark smeared areas are the binder which

 

Figure 1.3: Consolidated nonwoven mat (Knight, 1995)

 

. l- I. 1
" A .l. glut- ILL... .13.;

Figure 1.4: Consolidated bidirectional mat

15

has penetrated the filament gaps. Along sections of the tows where no binder was initially
located, the filament gaps will remain unfilled (dry). Squeeze flow is the primary cause of
binder reconsolidation during compression with the capillary effect in the filament gaps
playing only a minor role (although it is more significant at higher temperatures) [20]. When
compression is complete, the tows are wide and thin with filaments gaps present and sections

of the tow smeared with binder.

1.2.2 Effect of binder on part properties

The presence of a binder may adversely affect the molding and cure processes and
the physical properties of the resulting composite part. During mold filling with a soluble
binder, washout will occur resulting in a binder concentration gradient along the length of
the mold with binder concentration lowest at the mold inlet and highest at the outlet. This
is caused by the fibers nearest the inlet being exposed to the resin for a longer period of time.
The binder is more thoroughly dissolved and is flushed towards the outlet. Increased binder
concentration results in increased resin viscosity resulting in a viscosity gradient along the
length of the mold. This causes viscous fingering and other inhomogeneities in the part that
affects the part's physical characteristics [21].

This binder may also precipitate during the cure process and alter the resin cure
kinetics. Ruffier [22] and Lee [23] showed that high concentrations (~15%) of a polymeric
low profile additive (LPA) will result in the suppression of the cure reaction. Lee's group
also found that the degree of cure suppression depended on the LPA compatibility.

However, these studies were conducted with high concentrations of LPAs and these effects

16

may be the result of reactant dilution.

A series of experiments were performed using low concentrations (055%) of binder
to examine the effect of low LPA concentrations on the cure with minimal dilution. This
would help determine the cure behavior of actual molded parts since the binder concentration
is usually less than 5% except at the mold outlet [21]. Figure 1.5 shows the effect of binder
on the conversion profile for the resin at a cure temperature of 50°C. The binder acts as an
inhibitor in the resin cure and the degree of inhibition increases with increased binder
concentration. The percentage time increase to reach 50% conversion is much larger than
the percentage increase in the binder concentration, indicating more than just reactant
dilution was at work. Figure 1.6 shows that the maximum cure rate decreases with increased
binder concentration.

A possible mechanism for binder inhibition of cure when it is soluble in the resin is
that the binder decreases the free volume present in the resin. At the molecular scale, the
density of the resin is not constant, instead there are pockets that occur where no resin
molecule is situated and this is known as the free volume. Free volume is important because
molecules move by filling an adjacent free volume thereby leaving a new one in the space
they once occupied. Increased free volume results in greater resin mobility. In a reactive
system, the reaction rate increases because reactive sites become more mobile and this allows
for more collisions. Binder may suppress cure simply by occupying free volume resulting
in decreased reactive site mobility and number of collisions.

Time/temperature curves in Figure 1.7 show the effect of insoluble binder

concentration on a resin (polyurethane). The time needed for the curing resin to reach the

l7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.7
I 1 u .
0.6 ..
0.5
f; 0.4
“:5 0%
C:
,3 0.3 1/2%
1%
0.2 2%
5%
0.1
OJ

 

 

0 500 1000 1500 2000 2500 3000 3500 4000
Cure time (seconds)

Figure 1.5: Effect of binder concentration on the conversion profile
of a vinyl ester resin

18

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.0009 J I
1 I 0%
0.0008‘ c ”2%
I
g A 1%
‘ I
0.0007
. + 2%
, i
. I _ i 0 5%
0.0006 '
. I 0
. '1‘ o
£00005 '5 5] +
g . n’ U I
E 1 s”
500004 +3
o . 4‘
Cl
, .e‘cr I
0.0003 ‘ I.ACI *’
. 0
0.0002 *0
* a
4 V0
0.0001 ‘ L‘T
. A0
. O ' U
4 o
0 'r l l I *2”; ‘ 0 o a
0 500 1000 1 500 2000 2500 3000 3500 4000 4500

Cure time (seconds)

Figure 1.6: Effect of binder concentration on the reaction rate of a
vinyl ester resin

19

BK. _ . coo. . . com. . . bow. _ . .oom. . . .ocm. _ _ 9:.

2552322. Co 2:0 :0 coach—80:00 BEE fie acetm "5F 2:2“.

Amucooomv can.

_ _ _

 

 

 

 

 

 

 

"r*'r
O O
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mow

 

 

3) armeraduiol

20

maximum temperature decreases with increasing concentration. This indicates that the
presence of binder increases the rate of reaction for an insoluble resin. A possible
explanation for this result is that the binder forms an interpenetrating polymer network (IPN)
within the resin matrix. When this happens, the binder penetrates the resin network and
forms its own separate network interwoven inside the matrix. With increasing
concentrations of binder, the binder network may penetrate the resin matrix to a greater
extent. If the binder acts to facilitate the curing reaction, then a larger IPN will result in a
greater amount of the resin matrix coming in contact with the binder, resulting in a faster
cure.

The addition of LPA is may also cause a reduction in the amount of resin shrinkage
that occurs during cure. Studies by Lee [23], Kroekel [24], Montarnal [25], and Kiaee [26]
that have shown and quantified this effect. They demonstrated that increased compatibility
of the LPA resulted in less resin shrinkage. There are many explanations for this behavior,
but generally it is agreed that the LPA forms a second phase inside the resin and that micro
cracks form at the interface surface to offset the shrinkage. For compatible LPA's, the
morphology is more homogenous with smaller particles which results in larger smface areas
available for micro cracks to form. Less compatible LPA's will form a larger, more distinct
phase with less area for micro cracks.

These binder droplets may also adversely affect the part by forcing the resin to flow
around it (if it isn’t melted) and form additional voids. The binder can also physically
prevent void mobilization. During postcure of the part, the binder may melt and may migrate

to the surface of the part through the voids in the part. This can significantly reduce the

21

appearance of the resulting part.

1.2.3 Binder mobilization

If the binder is immiscible (incompatible) in the resin, the binder may be mobilized
during mold filling in a similar process as would occur with the voids. As the mold is
injected with the resin, the mold is usually heated to help propagate cure. In order to be
thermoformable, the binder usually has a low melting temperature (~5 5°C for polyester
binder) and often the binder will melt in the mold. If the binder is immiscible in the
incoming resin, it will form small droplets within the fiber tows that are viscoelastic. Once
again, if the hydrodynamic pressure of the incoming resin is greater than the capillary forces
of the drop, it will be pushed through the fiber bundle and out into the matrix.

Figure 1.8 shows the mechanism for mobilization which is similar in many respects
to void mobilization. First the binder melts into a drop when the mold is preheated (Fig.
1.8.a). When mold filling begins, the resin will first flow around the fiber tows and then
penetrate the tow, forcing the drop to squeeze through the filament gap (Fig.1 .8.b). If the
drop reaches a stagnation point, it may wrap around the filament (Fig. 1.8.c). Because the
binder droplets are viscoelastic, a small thread may form connecting the two droplets (see
chap. 3). Eventually, this thread may break forming very small droplets that could affect the
morphology of the part. When the binder leaves the bundle, it forms small spherical droplets

(Fig. 1.8.d) and is carried toward the mold outlet.

e::::_ .3... e a 5...: he 535:2: "5 9...»:—

 

 

2. EQUIPMENT AND MATERIALS

2.1. Equipment

2.1.1. Flow Cell

The flow cell which was used to house the arrays is shown in Figure 2.1 and is based
on a design conceived and implemented by Chmielewski [27]. The rectangular cell was
constructed of six aluminum plates held together with Allan screws and sealed with a 100%
clear silicone sealant (Dow Coming). The front plate (Fig. 2.1.A) has two pressure
transducer ports for measuring the pressure drop. Two PX102-ZOOSV pressure transducers
(Omega Corporation) were used for permeability testing. They have flat, circular surface
that are flush with the inside surface of the flow cell. The inlet and outlet ports are located
at the end plates. Aluminum triangular wedges filled the flow cell comers at the inlet and
outlet to block out stagnation flow in these areas.

The bottom and top cell plates contained windows made of PMMA. These widows
fit into the plate via slots (Fig. 2.2) and were sealed in place with silicone. These windows
allowed observation of the drop as it moved through the array. The front plate had a window
which also served as a bracket for the array (see sec. 2.1.2).

The back plate (Fig. 2.1.B) does not have a window. Instead, there is a hole through

which a port can be connected for a syringe. A ball valve is attached to the port on one end

23

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26

and on the other is a connector with a thick rubber membrane covering its open end. A
syringe needle is pushed through the membrane and the open valve and through a hole in the
array between cylinders (see sec. 2.1.2) and a drop is placed in the array. The syringe is then

withdrawn and the ball valve is closed so as not to put pressure on the membrane.

2.1.2. Q'linder arrays

A diagram of a cylindrical rod array is shown in Figure 2.3 and like the flow cell is
based on a design used by Chmielewski [27]. For all the arrays discussed here, both the side
brackets and rods are made of polymethylrnethacrylate (PMMA). The array consists of two
side brackets that fit into the side plates of the flow cell and are sealed in place with the
silicone sealant. One of the brackets may consist of a raised surface which constitutes the
side window. Each bracket has a series of holes drilled into them in a certain arrangement.
The cylindrical rods are fit into these holes tightly (no adhesive is used to hold them in
place), this aligns the rods perpendicular to the fluid flow. The cylindrical rods have a cross
section diameter of 0.476 cm. In the flat (non-raised) bracket, a small hole is drilled in which
the syringe needle passes through for drop impregnation.

For this study, two array geometry factors were varied: the packing arrangement and
the fiber volume fraction (V f). The two packing arrangements were square and hexagonal
which are shown in Figure 2.4. The throat radius (r,) is defined for both packing
arrangements as the radius of the constriction between two rods. The drop is impregnated
in a large cavity and this cavity is defined as the body. The body is a circle that is

circumscribed by the nearest adjoining rods (the body radius rb is the radius of this circle).

27

 

Front and
eeeeeeeeebmm

 

 

\ Acrylic
rods

 

Top and
bottom
view

 

Side
window

Side
view

 

 

Figure 2.3: Rod array

28

 

Figure 2.4.a: Square array

  

\||"'a
‘0‘ '0

 

I I ‘

 

Figure 2.4.b: Hexagonal array

Figure 2.4: Packing geometries

29

The body is circumscribed by four rods for the square array and by three rods for the
hexagonal. The two values for Vf examined here were 0.40 and 0.50 (porosities of 60% and
50% respectively). A total of three arrays were examined in this study: a V5040 square
array, a V5050 square array and a V5040 hexagonal array. The position of the drop in the
array and the specific geometries are illustrated in Figure 2.5. The specifications of the

arrays are listed in the following table:

Table 2.1: Array specifications

 

 

 

 

 

 

 

 

 

 

 

Rod Fiber volume rt rb rb/r, Permeability
alignment fraction (V f) (cm) (cm) (cm’)

Square 0.40 0.095 0.23 2.42 1.37x10'3

Square 0.50 0.050 0.17 3.40 3.25x10‘4
Hexagonal 0.40 0.12 0.21 1.75 1.25x10'3

 

The throat to body radius (rt/n) is important because it gives an indication of the degree of
distortion that a drop will experience as it moves through the constriction. The permeability
values were determined experimentally and are in good agreement with the theoretical values

found using Gebart’s model [28].

2.1.3. Experimental configuration and procedure
Figure 2.6 shows the overall experimental setup for the observation of drop
movement through the cylindrical arrays. A pressure pot was filled with the suspending fluid

(oil) and was pressurized with a nitrogen at 40 psi. The flow cell was then connected to the

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32

tank and was filled with the oil while standing vertically. This flushed the flow cell of air
and ensured that no large bubbles were trapped. The cell was then placed horizontally on a
stand and the outlet was connected to a needle valve and a rotameter. The needle valve
controlled the fluid flow rate which is measured by the rotameter (Key Instruments). The
rotameter was calibrated by the supplier for water flow and was rated for up to 2.5 gallons
per minute. For this study, a nrler was attached to the rotameter and a calibration curve was
constructed for oil flow (Figure 2.7).

For a run, the drop was injected into the array using a syringe (see sec. 2.1.1). The
inlet ball valve was opened (while the outlet valve was closed) to pressurize the cell. The
outlet valve was then opened and the drop began its flow through the array which was timed
with a stopwatch. During this period, the drop movement was often recorded with a video
camera focused on either the top or side windows. After the run was finished, the drop was
pushed out of the array and the valve for the drop purge was opened. The purge was just a

transparent hose and once the drop entered the hose, it was sucked out of the oil with another

syringe.

2.2. Materials

For this study, a chloroparaffin oil (Paroil 50, Dover Chemical Co.) was used as the
suspending fluid. The oil had a refractive index of 1.49 which matched the refractive index
of the acrylic rods. This matching caused the rods to fade into the background and allowed
easier viewing of the drops (especially when viewing from the top window of the flow cell).

The oil had a viscosity of 109 mPa-s at 23°C and a specific gravity of 1.19.

Volumetric flow rate (cc/s)

 

33

 

 

I I I I [fl I I I I I I I I I I T r j l'

20 40 60 80 100 120‘ 140
Rotameter reading

Figure 2.7: Calibration curve for rotameter with oil

34

The Newtonian drops used in most of the flow tests consisted of an immiscible
water/glycerol mixture (18.3% water). The specific gravity of this mixture was 1.19 which
matched that of the oil and negated the influence of gravity. The viscosity of the mixture

was 60 mPa-s at 23°C. The viscosity ratio (k) is defined as:

k : viscosity of drop (2 1)
viscosity of suspending fluid

 

and for the drop mixture, k = 0.55. To make the drops stand out during runs, the mixture
was dyed with Crystal Violet (Sigma Chemical). A special run was done with a pure
glycerol drop for comparison with the water mixture. The glycerol had a specific gravity of
1.26, its viscosity is 970 mPa-s at 23°C, and k=8.9.

The viscoelastic drop consisted of the same water/glycerol mixture and dye as was
used for the Newtonian drop but with 0.3% polyacrylamide added. The rheology of this
solution was determined with a Haake RS-lOO liquid Rheometrics instrument. Figure 2.8
shows the dynamic moduli curve of the viscoelastic drop solution for two runs. The
relaxation time (it) describes the degree of elasticity of the fluid (for Newtonian, A=O

seconds). For the viscoelastic drop, 1. is found using the following equation:

 

/
Am =[ G ] (2.2)
w - Y

Figure 2.9 shows the resulting relaxation time vs. fiequency graph. The viscoelastic solution

had a viscosity of 1430 mPa-s at 00~0 at 23°C which gave k=13.0.

G'(Pa). G"(Pa)

 

 

 

 

 

 

 

100,
101
11
0.1 —.
;
. _._ 0'
G"
0.01 g +
0.001 1, .,,
0.1 1 10 100

Figure 2.8: Dynamic moduli for viscoelastic drop

frequency (rad/s)

 

36

 

relaxation time (s)
o
.0 N
10 01
I I I l I l I

.0
_s
01
l

 

 

,,,,-4,
O 2 4 6 8 10 12 14 16 18 20
frequency (rad/s)

 

Figure 2.9: Relaxation time (M vs. frequency (00) for the
viscoelastic drop solution

37

The Deborah number (De) is the ratio of the relaxation time of the fluid to the time

of deformation imposed by the fluid flow:

De = (2.3)

i
6d

where 0d is the time of deformation and for flow through a cylindrical ray is determined by:

d
0,1 = ii (2.4)

where d, is the throat diameter and U is the drop velocity. De measures the significance of
the elastic behavior of the drop. For De < 0.5, the drop is considered to be behaving in a
Newtonian fashion. Figure 2.9 was used to calculate the Deborah number for each flow
condition and determine whether the elasticity of the drop will significantly affect the flow.

There are three interfaces that are being examined here: oil/glycerol, oil/glycerol-
water (Newtonian), oil/glycerol-water-PAC (viscoelastic). The surface tension of the fluids
and the interfacial tensions of the interfaces were determined using the Sessilc drop method.

This method and the corresponding calculations are described in detail in Appendix A. The

interfacial tensions are listed in the table below:

38

Table 2.2: The interfacial tensions for the liquid-liquid systems

 

 

 

 

 

 

 

 

 

Suspending fluid/drop y, (mN/m) 72 (mN/m) W, (lem) yl2 (mN/m)
system
oil/glycerol 33.2 64 83.8 13.4
oil/Newtonian 33.2 68.6 84.1 17.7
oil/viscoelastic 33 .2 53.5 78.6 8.1

 

Where 7, is the surface tension of the oil, y2 is the surface tension of the drop solution, W,

is the work of adhesion for the interface, and ylz is the interfacial tension.

 

3. DEFORMATION AND BREAKUP BEHAVIOR OF

CYLINDRICAL DROPS IN FLOW THROUGH

ARRAYS OF CYLINDERS

3.1 Introduction

This chapter will focus on two aspects of a drop as it travels through an array of
transversely aligned cylinders: deformation and breakup. The spotlight will be on the path
of the drop and on the mechanism of breakup, as well as the conditions necessary for its
occurrence, and the significance of this phenomenon for mold filling. Newtonian and
viscoelastic drops of different lengths will be compared and contrasted. Also, there are three
types of array geometries that will be examined: two square arrays with V5 0.40 and

V5050, and a hexagonal array with V5040.

3.2 Background

During mold filling with a fiber preform, pockets of air (voids) are often tapped
inside the fiber bundle. Binder, which is used to hold the preform together during handling,
is smeared into the fiber tow during consolidation. At higher filling temperatures, the binder

may melt forming small droplets. The presence of voids and binder droplets can be

39

40

detrimental to both the mechanical properties and appearance of the resulting part. During
mold filling, these inhomogeneities can be mobilized and pushed out of the preform by the
viscous forces of the incoming resin and break apart as they move through the fibers.
There has been a great deal of work over the years into the motion and breakup of
bubbles and drops through various constrictions such as capillary tubes, tube constrictions,
wavy walled tubes, and packed beds. These studies have improved our understanding of
how bubbles and drops move, distort, and break apart as they move through different

constraints.

3.2.1 Motion and breakup through capillary tubes and channels

The distortion and breakup of a drop in a capillary tube was studied by Olbricht and
Kung [29]. For 10'2 < Ca < 10", a drop will experience only slight distortion with the degree
of elongation increasing with increased Ca, increased size, or decreased k (viscosity ratio).
Elongation also results in a more tapered leading interface of the drop and a flatter trailing
interface. For 10“ < Ca < 1, the trends continue as before except at low k. There, a small
indentation appears in the backside of a drop and this indentation grows with increasing Ca.
At Ca > 1, the dependence of drop behavior on k increases. For low k, the indentation will
continue to grow until the drop breaks into separate droplets. For high k, any indentation
will eventually collapse and the drop will keep elongating until it becomes unstable and
shatters. Finally, the relationship of the critical capillary number for drop breakup (Cab) to
k and drop size was determined: Cab increased for both increasing k and decreasing drop

size.

41

More insight into the flow of bubbles came fi'om work on the Hele-Shaw cell. Taylor
and Saffrnan [30] first developed a model to determine the shape and velocity of a rising
bubble in a cell. Unfortunately there was a degeneracy in their work which Tanveer [31,32]
solved by including the effect of surface tension. Maxworthy [33] performed experimental
ms with a Hele-Shaw cell inclined vertically which consisted of two plates with a very thin
gap between them. In this study, he varied the incline angle of the cell and the size of the
bubble. He found that with increasing bubble width, the velocity increased sharply then
leveled off at a plateau before rising sharply again. This second sharp increase was the point
at which bubble instability occurred. Maxworthy found that both the Taylor and Tanveer
models were adequate only for large bubble widths and as the incline angle approached zero.
The degree of bubble elongation also increased with both bubble size and incline angle and
both were underestimated by the two models. He reasoned that at high velocity (high angle)
the bubbles experienced instabilities such as fingering and tip-splitting.

Kopf-Sill and Homsy [34] studied air bubble motion as a result of fluid flow in a
horizontal Hele-Shaw cell. They found that the bubble would take on seven distinct shapes
(near circles, flattened, elongated, long-tail, short-tail, “Tanveer” [flattened drop with a
concave leading edge], and inverse “Tanveer”) and which shape would occur depended upon
both Ca and the initial bubble radius. As an example, near circular bubbles would occur at
6x10" < Ca < 3x103 and a wide range of radii while a flattened bubble was seen at 3x103
< Ca < 2x10’2 and small radii. The most interesting shape observed was a circular bubble
with a long trailing tail which occurred over a wide range of Ca and bubble radii. Once the

flow and bubble reached a steady state, all of these shapes remained stable.

42

Armour etal [3 5] reported on how the shape of a drop would evolve when it was
placed in an air jet. The drops examined had an equivalent spherical diameter of 3.85 mm.
For Newtonian drops, they found that it would oscillate and after 2-3 seconds would
disintegrate. For viscoelastic drops, they observed that a filament was formed when the drop
was injected into the air jet. This filament would either retract into the drop or it would
elongate in the direction of the air flow. This process continues until the entire drop is
elongated and breaks into a fine mist of droplets. These experiments were carried out for
several different materials and the main property of the drop that would determine the extent
of tail formation was relaxation time (A). At low A (less than 10 milliseconds) no tail would
occur, while at high it (above 25 milliseconds) a tail would always occur but grow very

slowly.

3.2.2 Motion and breakup through a constriction

The movement of a void through a fiber bundle is analogous to a bubble moving
through a constricted capillary tube. A study of gas bubbles by Gauglitz et al [36]
determined the conditions at which the bubble will break as the result of sudden expansion
as it passes through the constriction (Haines Jump). Whether the bubble breaks is dictated

by the Ohnesorge nmnber:

ch = —"— (3.1)
(0011,)"2

where RT is the radius of the constriction (throat radius). They found both numerically and

43

experimentally that breakup occurs at low Oh where the bubble motion is impulsive and
dominated by inertia. At high Oh, viscous forces dominate and the motion is a viscous ooze.
The breakup of the drop usually occurs at the narrowest section of the constriction (the
throat). There the film becomes unstable and forms small peaks that eventually snaps off the
leading section fi'om the rest of the bubble. The time it takes for the filth to grow and break
the bubble is known as the snap-off time (1').

Lundstrom [37] examined the flow of a bubble through two tubes that contracted in
different ways, one sudden and one gradual. He found that the pressure required for a bubble
to move through a constriction depends on the ratio of the tube and constriction radii, the
surface energy, and the length scale of the geometry. He then related his findings to actual
molding situations and found that for macro voids, a sufficient pressure gradient can be
found that will force the bubble out while for micro voids it is difficult to force the bubbles
out and other means such as diffusion of the bubble into the resin may be needed.

A numerical study of a drop moving through a constriction was conducted by Tsai
and Miksis [38]. For a given radius and viscosity ratio, the value of Ca will determine
whether the bubble breaks. At Ca = 0.05, the film in the constriction is sufficiently thin so
that no instability in film grth occurs. At Ca = 0.1, the film thickness in the constriction
increases forming small peaks as were seen by Gauglitz and eventually resulting in the fluid
pinching off the leading portion of the bubble. At Ca = 0.2, the instability of the film in the
constriction continues to grow but the velocity of the bubble is so fast that its residence time
in the constriction is smaller than the snap-off time. The bubble moves through the

constriction without pinch off occurring although a trailing finger does result. This results

44

in a range of Ca values (between 0.05 and 0.2) in which the bubble will break as it passes
through the constriction. The pressure of a bubble increases slightly as it first penetrates the
constriction then drops off sharply as the bubble passes through and finally there is a sharp
pressure rise if pinch off occurs. The viscosity ratio has a large effect on the bubble
dynamics and breakup. At k = 0.0005 the drop will breakup at Ca = 0.1 which is much lower
than would be seen by k = 0.01. The Ca at which breakup occurs decreases with increasing
drop length. This results from the higher residence time of the larger bubbles which often
is larger than the snap-off time.

Another idealization of to the movement of a drop through a porous medium is its
movement through a wavy walled axisymmetric tube as was studied by Olbricht and Leal
[39] for a Newtonian drop in both Newtonian and viscoelastic fluids. In studying the excess
pressure drop as the drop moves through the constriction for the Newtonian fluid, they found
that the maximum value occurs when the drop is midway through the neck. This pressure
peak becomes larger and more defined as the drop length increases. For the lowest Ca values
studied here (0.004 < Ca < 0.088), the excess pressure difference is positive for small drops,
reaches a maximum and then falls ultimately reaching negative values for the longest drops.
In the range of 0.090 < Ca < 1.00 and k < 0.75 excess pressure drop is decreased by several
factors: flow interference by the drop, deformation, drop/wall interactions and replacement
of the suspending fluid with drop fluid with the last one being particularly significant for
long drops. For Ca > 1.00, the excess pressure drop increases with increasing viscosity ratio
and Ca has little effect. This is the result of the drop being deformed into a long strand that

doesn’t distort as it moves through the constriction.

45

Olbricht and Leal then performed similar experiments using a Newtonian drop in a
viscoelastic fluid. The maximum pressure occurred when the drop is midway through the
neck as before however, it is not as distinct as in the Newtonian case. This is the result of
the fluid not closely following the tube contours and a thicker film at the constriction.
Viscosity and density ratios also have less effect on the pressure and the shape of the drop
and the viscoelastic properties of the fluid become dominant. The effect of Ca is also
minimized since drop deformation is not as strongly linked with velocity in this case. For
Ca = 0.097, there is a pressure peak that is more distinct than that for lower Ca’s. This peak
also increases with increasing drop length despite the fact that the bubble is more elongated.
A tail forms on the upstream side of the drop. For smaller drops, this tail will retract after
the drop passes the constriction while longer drops or higher flow rates will result in a
permanent tail. At a critical Ca the tail will break off fi'om the parent drop. If the leading
edge of a drop invades the next constriction before the tail fully evacuates the preceding one

this can cause the peak pressure to level off at long drop lengths.

3.2.3 Motion and breakup through a packed beds

Ng [40] investigated the motion of an oil drop displaced by water through a spherical
porous medium. It was found that for high viscosity suspending fluid (~1000 mPa-s), the
drop moved in an ameboid fashion with an advancing “foot” protruding into an adjacent
pore. This caused a retraction of the drop body and any feet that moved laterally. For low
viscosity suspending fluids (~5 mPa-s), the drop moves rapidly as a result of interface

instabilities in the manner of a Haines jump. Drop breakage results in smaller droplets that

46

often become stagnant in the bed. Other experiments by Ng found that coalescence of the
droplets will occur at relatively low drop concentrations.

Liu [41] studied the deformation and breakup of an initially spherical drop entering
and moving through an array of cylinders. In this study, a spherical drop was injected into
a flow cell upstream from the array. The entry point of the drops was varied from the
stagnation region of the rod to the constriction between the rods. The drop came in contact
with the array at which point it began to deform and/or break apart as it moved through the
array. The study focused on two array geometries: square and hexagonal. Liu varied four
parameters: bubble size, Ca, k, and trajectory (where the drop comes in contact with the
array, [Figure 3.1]). For all cases in the square array, the critical Ca (Cab) at which breakup
occurs decreased for increasing drop size and Cab increases as the initial entry point of the
bubble gets closer to the centerline of the gap between cylinders. Drops with diameters
smaller than the gap width will not experience breakup. At high k ( > 1.00), three regions
of deformation exist: periodic, progressive elongation and breakup. These regions will
occur at Ca < 1.0 for increased size and for entry points close to the stagnation point of the
rod. For the hexagonal array, the trajectory had little effect on whether the drop would break
apart since a cylinder would always be in its path. It was discovered that only small bubbles
at Ca < 1.0 would remain intact while for other conditions it would break in two ways:
collision with the first cylinder or the formation of strands that eventually break. Collision
occurs at Ca < 1.0 and smaller drop size than elongation. Increasing viscosity ratio results

in the elongation occurring at Ca < 1.0 and smaller drop size.

47

00000
"If. ::C:>_§>_§>:Q§:z
cr=-l "ca-0009‘”

Square array

no 000. O__

cr=0 —_@:e___e.-_G__.

era-l _ e—Ej—e-a-G-H
O O O

Hexagonal array

Figure 3.1: Trajectory of the drop through an array of
cylinders (Liu, 1989)

48

3.3 Materials and procedure
The aluminum flow cell and experimental setup used in this study are described in
Chapter 2. The three rod arrays used here are described in detail in Section 2.1.2, with their

geometric specifications being as follows:

Table 3.1: Array specifications

 

 

 

 

 

Rod Fiber volume rt rb rb/r, Permeability
alignment fraction (V ,) (cm) (cm) (cmz)
Square 0.40 0.095 0.23 2.42 1.37x10'3
Square 0.50 0.050 0.17 3.40 3.25x10"
Hexagonal 0.40 O. 12 0.21 1.75 1 .25x10‘3

 

 

 

 

 

 

 

The suspending fluid is a chloroparaffin oil and drop materials are glycerol, glycerol/water
(Newtonian) and glycerol/water/polyacrylamide (viscoelastic) which are described in Section

2.2. The key factors of these materials are listed below:

Table 3.2: Properties of the fluids

 

 

 

 

 

 

Material viscosity viscosity surface tension interfacial tension
(mPa-s) at ratio, k (mN/m) with oil (mN/m)
23°C
oil 109 - 33.2 -
glycerol 970 0.55 64.0 13.4
glycerol/water 60 8.9 68.6 17.7
glycerol/water/PAC 1430 13 78.6 8.1

 

 

 

 

 

 

49

The drop is first injected into the fluid-filled array with a syringe. The flow is then
started by opening a pressure pot and is regulated with a needle valve and rotameter.
Movement of the drop is recorded with a video camera and can be analyzed with a VCR.
The critical capillary number (Cab) for breakup is defined as the number at which the onset

of drop breakup is first observed.

3.4 Objectives

This study will examine the mobilization of a drop moving through an array of
cylinders transversely aligned to the fluid flow. Particular attention will be given here to the
specific shape and path of the drop as it moves through the array. The process and
conditions necessary for drop breakup will also be examined. The effect of capillary
number, drop length, array geometry, and drop rheology on the motion and breakup of the
drop will be studied.

This study departs from previous studies in several respects. The first is that the
rheology of the drop, as opposed to the suspending fluid, will be varied between Newtonian
and viscoelastic behavior. The range of capillary numbers focused on here (0.005 < Ca <
0.1) is lower than what was examined in previous studies. Using an array of cylinders is a
departure from most studies which have focused on channels and capillary tubes with or
without constrictions. While Liu’s study is similar in certain regards, she did not examine
the differences between Newtonian and viscoelastic drops. Another key difference is with

the procedure: in Liu’s study, a spherical drop was placed in the flow cell upstream fi'om the

50
array; in this study, a cylindrical drop is impregnated directly into the array. The drop will
be initially at rest in each flow run and will start with a cylindrical shape. The drop behavior

will then be more comparable to that of voids and binder droplets.

3.5 Results and discussion

3.5.1 Drop motion through the arrays

This section will investigate the precise motion of a drop as it moves through an array
of transversely aligned cylinders. The effect of drop composition (Newtonian or visco-
elastic) and drop length on the motion will be investigated. Finally, the drop will move

through two different array geometries: square and hexagonal.

3.5.l.a Square arrays

Figure 3.2 shows a cross sectional view of a drop moving through a square array.
The body gap is defined here as the empty space circumscribed by four adjacent rods.
Initially, the drop occupies a body gap between two rows of rods (Figure 3.2.a). As can be
seen, the cross section of the drop is not a perfect circle but rather conforms around the
adjacent rods. When the flow begins, the leading edge of the drop slowly penetrates the rod
gap, reaching the throat (Figure 3.2.b). At this point, the capillary forces are greatest and if
the drop is to become mobilized, its leading edge must move beyond the throat. When it
does, the drop quickly fills the adjacent body gap downstream of the fluid flow (Figure

3.2.0). Once the adjacent gap is completely filled, the result is two adjacent drops connected

51

 

Figure 3.2: Motion ofa Newtonian drop through a square array
(cross section view)

52

by a bridge (Figure 3.2.d) with the drop fluid flowing from the upstream drop into the
downstream drop. In this phase, the drop flows transverse to the flow and along the rod
direction. When the upstream body is nearly empty of the drop, the remainder forms a small
trailing foot (Figure 3.2.e) that quickly retracts into the main drop body. Finally, the drop
is completely in the adjacent gap (Figure 3.2.0. This entire process when a drop moves from
one body to the next is referred to as the drop moving one step.

Figure 3.3 is a top view representation of the drop motion. The elliptical drop
initially fills the body gap (Figure 3.3.a). When the resin flow is started, the leading edge
of the drop begins to bulge into the rod gap (Figure 3.3.b). Eventually, a small foot is
formed near the center of the drop that extends into the adjacent gap (Figure 3.3.c). For
every step the drop moves, these first stages of motion are the most time intensive. The
amount of time needed to form the bulge and the leading foot is defined here as the bulge
time. Once the leading foot is made and penetrates into the adjacent body, the drop quickly
drains into that body. When the drop fills in the cavity, it will essentially form two drops
connected by a bridge (Figure 3.3.d). The drop fluid will continue to drain from the
upstream drop into the downstream drop. Since resistance to flow is greater in the direction
of the bulk flow (as a result of the rod gap constriction), the drop fluid will flow in the
transverse direction, parallel to the rods. This will give the impression of two interconnected
drops with one contracting and the other expanding (Figure 3.3.d). This type of motion
matches the observations by Ng et a1 [45] where the drop tended to spread in all directions
of little resistance before it finally moved into the next gap. During this process, the

transverse length of the connecting bridge will match that of the shortest drop. When the

53

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drop is almost completely transferred to the downstream gap, a small trailing foot is left
(Figure 3.3.e) that slowly retracts into the leading drop. This process ends with the drop
completely transferred into the next gap (Figure 3.3.0. Figures 3.2 and 3.3 give a complete
overview on the motion of an immiscible drop through an array of transversely aligned rods
in a square arrangement.

Videotapes of the drops were made and photographs were taken off of them (dark
bands on the photos are a result of this process). Figure 3.4 shows the effect of Ca on the
motion of a Newtonian drop of length 3.82 cm moving through the Vf= 0.40 square array.
In this case, Ca is varied by varying the flow rate of the suspending fluid. Although it cannot
be seen in the still pictures, increased Ca will result in a higher drop velocity (U). This is
mostly the result of reduction in the time for the leading edge of the drop to bulge and form
a leading foot in the next gap. The bulge time becomes greater with decreasing length. At
Ca = 0.0266, on average it took 34.50 seconds for a 0.60 cm drop to move one step and 98%
of this time involved forming a lead foot (33.73 seconds). For the 3.82 cm drop at the same
Ca, the time to move one step was 2.80 seconds with the bulge time accounting for 50% of
this period (1.42 seconds). This is the result of two factors: decreased SV (dr0p surface
area/volume ratio) for the longer drops which results in quicker penetration of the foot and
increased residence time of the drop which reduces the relative amount of time for
penetration. It is this bulge time that gives the drops (particularly shorter ones) a
discontinuous motion through the array.

Figure 3.4.a shows the drop moving at Ca = 0.0136 and Figure 3.4.b shows it at Ca

= 0.0266. The main effect of increased Ca is that it will extend the drop across multiple

55

~. 1.42,

‘-v‘ummlmmywt “1

 

Figure 3.4.a: Ca = 0.0136

Figure 3.4: Movement of 3.82 cm Newtonian drop through the
Vr=0.40 square array at different Ca

56

 

Figure 3.4.b: Ca = 0.0266

Figure 3.4 (continued): Movement of 3.82 cm Newtonian drop
through the Vr=0.40 square array at different Ca

57

numbers of gaps (three or more). Increased Ca will decrease the amount of time for the drop
to form a lead foot through the downstream so that it is less than the residence time in the
throat (the amount of time the drop actually occupies the throat). The result is that the drop
will have both a leading and trailing foot occurring at the same time and the drop becomes
spread over three rod gaps. This effect is more apparent for longer drops than it is for shorter
drops.

Figure 3.5 presents a series of photographs of the movement of a short Newtonian
drop (0.60 cm) through the Vf= 0.40 square array. In theses pictures, Ca = 0.0266 and they
are compared with Figure 3.4.b. Shorter drops are harder to mobilize because a longer time
is needed to form a leading foot that penetrates into the next gap. Lower drop volume gives
results in a lower residence time for the drop to move through the gap. These two trends
combine to make the movement of a short drop very uneven with quick motion through a gap
followed by a relatively long period of stagnation while the drop forms a leading foot. Even
at high Ca, shorter drops will span no more than two steps while it moves through a throat.
Conversely, a long drop needs less time to form a leading foot and has a longer residence
time through the throat causing a more continuous flow through the array. The result is that
even at relatively low Ca, the drop will form a leading foot and a trailing foot at the same
time.

The motion of a viscoelastic drop is significantly different from that of a Newtonian
drop with comparable interstitial velocity (V) although no breakup of the viscoelastic drops
was seen in this square array as well. Figure 3.6 shows the motion of a 2.87 cm viscoelastic

drop through the Vf = 0.40 square at Ca = 0.0368 (Figure 3.6.3) and Ca = 0.0907 (Figure

58

 

Figure 3.5: Movement of a 0.60 cm Newtonian drop through the
Vr=0.40 square array at Ca = 0.0266

59

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. -.

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Figure 3.6.a: Ca = 0.0368

Figure 3.6: Movement of a 2.87 cm viscoelastic drop through the
V,=0.40 square array at different Ca

60

‘EWII’FI‘

 

Figure 3.6.b: Ca = 0.0907

Figure 3.6 (continued): Movement of a 2.87 cm viscoelastic drop
through the V,=0.40 square array at different Ca

61

3.6.b). Ca" is larger for the viscoelastic drops as the result of viscoelastic stresses opposing
flow. At low Ca, the viscoelastic drop would form leading and trailing feet as before but
here they are much wider transversely and not as distinct as they were in the Newtonian
drops. For high Ca, the feet tended to disappear altogether and the drop formed an elliptical
shape that had a continuous motion through the array. The bulge time here is essentially zero
since the drop is moving continuously forward. With this continuous motion, shorter drops
are faster than longer drops.

Figure 3.7 shows the motion of a short viscoelastic drop (1.08 cm) at Ca = 0.0907.
As with the longer drop, the feet were not as apparent and the drop formed a relatively
constant shape. The drop motion through the array tended to be uneven with pauses as the
leading edge moved through the throat. The bulge time was 60% of the total step time at Ca
= 0.0368. However these pauses were much shorter than those seen for Newtonian drops and
the motion was much more continuous in comparison.

The motion of a viscoelastic drop was even more unusual for very high Ca. Figure
3.8 shows the motion of 1.08 cm and 2.87 cm drops at Ca = 0.524. For both drop lengths,
the drop ends formed feet leading into the next gap. This resulted in the leading edge of the
drop becoming concave and the trailing edge becoming convex. For the 1.08 cm drop
(Figure 3.8.a), the final shape closely resembled the “Tanveer” bubbles seen by Kopf-Sill
and Homsy [34] in their Hele-Shaw studies. The 2.87 cm drop (Figure 3.8.b) formed a
similar shape that was more elongated across the array spanning five gaps when the shape

was fiilly developed.

62

 

Figure 3.7: Movement of a 1.08 cm viscoelastic drop through the
Vr=0.40 square array at Ca = 0.0907

W

A «J 'Igé- g; ‘IMTV‘ Wu 417,1} T‘v“ 1'

 

Figure 3.8.b: Drop length = 2.87 cm

Figure 3.8: Movement of viscoelastic drops of different lengths
through the Vr=0.40 square array at Ca = 0.524

64

3.5.1.1) Hexagonal array

In the hexagonal array, the body gap is formed by three rods. The drop moves in the
same uneven fashion as a drop in the square arrays with some key distinctions. The most
important distinction is that the drop does not move in a straight path parallel with the resin
flow as it would in the square array. The direction of the drop’s movement alternates
between parallel with the resin flow moving at a 60° angle from parallel (either upward or
downward) (Figure 3.9). This results in a tortuosity of 1.50 for the drop path through the
array.

Since the rods are in the path of the fluid flow, the rod will separate the fluid into two
streams. This results in a stagnation point at the lead edge of the rod (Figure 3.10). At this
point, extensional forces are realized. These forces can retard drop movement by causing
the drop wrap around the rod and either stall or break. However, if the drop doesn’t wrap
around or one of the branches retracts into the main drop body, these forces will serve to
elongate the drop over several steps and may increase the drop velocity as a result. Also,
as a consequence of the lower rb/r, ratio in the hexagonal array, resistance to drop flow
through the throat is smaller and the drop is more likely to spread out over several steps. For
the square array this extension of the drop only happens at high lengths and flow rates, but
with the hexagonal array, this can happen at any flow rate for drops with sufficient volume.

Figure 3.11 shows the cross sectional view of a Newtonian drop moving through a
hexagonal array. The drop initially fills the body radius (Figure 3.11.a). When fluid flow
is begun, the drop first moves horizontally into the next gap in a manner similar to that for

the square array (Figure 3.1 lb). When the drop hits the stagnation point of the adjacent rod,

65

Figure 3.9: Path ofa drop in a hexagonal array

66

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Figure 3.11: Motion ofa Newtonian drop through a
hexagonal array (cross section view)

68

the extensional forces at this point will serve to force the drop to flow in both directions
around the rod, forming two branches (Figure 3.11.c). As a result of slight inhomogeneities
in the array, the drop will prefer one direction over the other and it will start flowing in that
direction (Figure 3.11.d). If the leading edge of the secondary branch has not reached the
throat, the drop will remain intact (See sec. 3.5.2.b). This branch then begins to retract into
the main body of the drop (Figure 3.11.e) resulting in the complete drop moving completely
into the third body gap (Figure 3.11.0. The complete process is defined as the drop moving
one complete step. If breakage does not occur, the viscoelastic drop has similar flow
characteristics as the Newtonian drop except that it has a greater tendency to wrap around
rods and its movement is more continuous. A top view of the hexagonal array drop is similar
to one in a square array with the formation of a bulge and then a leading foot as it moves into

the next body gap.

3.5.2 Drop breakup

The previous section examined the motion of a drop through an array no drop
breakup occurred and the drop remained intact. Drop breakup often occurs at high flow rates
and for long drops. Breakup is important because the result is smaller and more numerous
drops that can be harder to mobilize and remove from the array. This section will study the
breakup mechanisms of Newtonian and viscoelastic drops in both square and hexagonal

arrays.

69

3.5.2.1! Square arrays

Figure 3.12 shows the top view of the mechanism for drop breakup a the square array.
The drop penetrates into the adjoining body gap in the same manner shown in Figures 3.3.a-d
resulting in two drops connected by a bridge and this is represented in Figure 3.12.3. As the
leading drop continues to expand and the trailing drop contracts, the bridge starts to become
concave in the transverse directions (Figure 3.12.b). As the trailing drops becomes shorter,
the bridge also becomes narrower and starts to pinch off (Figures 3.12.0 and 3.12.d). This
pinching occurs at the throat of the rod gap. When the upstream drop forms a trailing foot,
it is connected to the main drop body by only a small thread (Figure 3.12.e). Eventually, the
trailing foot is entirely out off from the main drop resulting in a small droplet behind the
main drop (Figure 3.121). This pinching behavior is the result of increase in film thickness
at the throat [36,3 8] that eventually snaps off the trailing foot from the main body. The
resulting droplet, which will always be shorter then the main drop, will have a lower mobility
and will need a longer time to be removed fi'om the array. It may even become demobilized
and stop moving in the array. In a square array, the droplets never recombine into larger
droplets.

For the flow rates examined here, breakup did not occur in the Vf = 0.40 square array
for either Newtonian or viscoelastic drops. However, breakup was prevalent for Newtonian
drops in the V, = 0.50 square array. The critical capillary number for breakup (5.3a) is
defined as the Ca where the drop becomes unstable and begins to split apart. Figure 3.13
shows Cab vs length for drops in the 50% array. It is immediately apparent that there is a

range of Ca’s over which breakup occurs and the drop remains intact for Ca’s above and

70

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0.025
0.02;
0.015.:
Ca

0.01 L

0.005 I.

 

Region of drop
breakup

 
 

 

 

I upper limit

0 lower limit

 

 

  

 

Drop length (cm)

Figure 3.13: Cab for a Newtonian drop in a

Vf = 0.50 square array

I l r I I

1.5 ‘5 ,,.2,5,,,.é..,3.5 14.5

 

72

below this range. Below the lower limit, the fihn thickness between the drop and rod is too
thin and doesn’t grow fast enough to pinch off the trailing foot. Above the upper limit, even
though the rate at which the film grows increases, the drop moves so fast that the residence
time of the drop in the throat is greater than the snap-off time and the drop moves through
the rod gap without breaking. The breakup mechanism and Cat, range observed here are
similar to the results seen be Tsai and Miksis in their numerical study [3 8].

Another trend that can be discerned from Figure 3.13 is the relationship of Ca, to
drop length. Both lower and upper limits of Cab decrease with increasing drop length: as
drop length increases from 1.53 cm to 4.03 crrr, the lower limit goes from 0.018 to 0.012 and
the upper limit goes from 0.022 to 0.018. For the two shortest drops studied (0.57 and 1.05
cm) breakup did not occur. The decrease in the lower limit results from the increase in the
residence time of the longer drops. This indicates that a larger drop is more unstable than
a smaller drop. The decrease in the upper limit is the result of shorter snap off times for
larger drops (Tsai and Miksis [38]).

Another factor of breakup affected by the drop length is number of times the drop
breaks. As stated before, the shortest drops examined did not break at all. The shortest drops
that did experience breakup (1.53 cm and 2.00 cm) broke only once throughout the Ca,
range. The longest drops ( 2.96 cm and 4.03 em) both broke only once at or near the upper
and lower Cab limits. Near the middle of the Ca, range (0.017 and 0.015 respectively), the
drop will break on average 3.5 times for the 3.96 cm drop and 3.7 times for the 4.03 cm drop.
Because the resulting trailing droplet is so much smaller than the main drop (Figure 3.12.1),

the drop volume and hence the residence time will be reduced only slightly. This means that

73

the drop experience further breakup until it residence time is sufficiently reduced. These
results are averages because the number of time drops of the same length broke varied widely

fiom run to run.

3.5.2.b Hexagonal array

Drops are far more unstable and tend to break up more readily in the hexagonal array
as opposed to the square arrays of similar porosity. The drop break up is a direct result of the
tortuosity of the bubble path and the stagnation point of the rod. Figure 3.14 displays the
breakup of a longer Newtonian drop in a hexagonal array (short drops often don’t break at
all). At first, the leading edge of the drop is reaches the stagnation point of the adjacent rod
(Figure 3.14.a). As a result of the extensional forces at this point, the drop will begin to
move in both directions and wrap around the rod (Figure 3.14.b). The drop will continue to
wrap around the rod forming two distinct branches (Figure 3.14.e). These first stages are
identical to the initial steps for drop movement described in Section 3.5.1.b. If the leading
edges of these branches extend beyond the throat, then neither branch will retract into the
other. In this case, the branches will continue to extend and the bridge connecting them will
become thinner at the stagnation point (Figure 3.14.d). Eventually, this bridge will become
unstable and split apart (Figure 3.14.e). The final result is two smaller droplets that are
independent of one another (Figure 3.14.0. Although the final drawing shows the two
droplets as being equal in size, it is very common for one droplet to be much smaller than
the other. When this occurs, there can be difficulties in removing the droplets (see Section

3.5.3). Newtonian droplets in this array would never recombine into larger droplets.

74

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75

Figure 3.15 is a series of photographs that show the top view of the breakup of a 4.66
cm Newtonian drop at Ca = 0.015. In the first photograph, the drop has wrapped around a
rod and the leading edge of the main branch is moving into the next body gap. The next two
pictures show the main drop branch continuing to move forward while the smaller branch
and the bridge connecting it continue to narrow in the transverse direction. If Ca is low (<
0.01 for longer drops), this smaller branch will eventually retract into the main body of the
drop (Figure 3.11.e). However, in this case Ca is too high and the two branches eventually
spilt. A trailing droplet is formed behind the main drop that is much smaller.

Figure 3.16 shows the breakup behavior of a viscoelastic drop in a hexagonal array.
Figures 3.16.a and 3.16.b shows the drop wrapping around the rod at its stagnation point in
a similar manner to the Newtonian drop. As with the Newtonian drop, as the two branches
extend downstream, the bridge connecting them becomes thinner (Figure 3.16.c). At this
point, the mechanism for drop breakup of the viscoelastic drop diverges from that of the
Newtonian drop. The main difference between the two types of drops is that the bridge will
break at a much higher extension for the viscoelastic drop. Instead, the bridge continues to
get thinner and the branches coalesce into drops. The result is two droplets joined together
by a thread (Figure 3.16.d). As the droplets continue to move downstream, the thread is
extended (Figure 3.16.e) and may span over two or three rods. Eventually, the thread breaks
and the two droplets are separated (Figure 3.16.1). Another important difference is that
viscoelastic droplets will often recombine to form larger droplets. This may be the result of
the lower interfacial tension between the drop and the fluid which allows the droplets to

break any fluid film between them and combine.

76

   
 

.gvm'grwllfirllil'Mll‘fl r .

Figure 3.15: Breakup of a 4.66 cm Newtonian drop in a Vf = 0.40
hexagonal array at Ca = 0.015

77

0:0

elastic drop in a

view)

Figure 3.16: Breakup of a longer visco

ay (cross section

agonalarr

hex

78

A good example of the breakup of a viscoelastic drop is shown in Figure 3.17. Here
the drop length is 1.89 cm and Ca = 0.052. The first two photographs show the drop
wrapping around the rod and the narrowing of the connecting bridge as was seen in the
Newtonian case. As the droplets move forward, the bridge continues to narrow until it forms
a thread (third photograph). The fourth photograph shows that as the droplets move forward
along the array, this thread will continue to be stretched and lengthened until it reaches a
point at which it will break apart. The degree to which this thread is lengthened depends
most strongly on Ca: increased Ca results in greater extension of the thread. This would be
expected since at higher Ca (and U) the Deborah number (De) increases (De ~ 0.065 in
Figure 3.17). This results in the greater elastic behavior of the drop and therefore greater
stretching of the thread.

The main difference between breakup of drops in square and hexagonal arrays is that
the drops do not pinch off for hexagonal arrays but rather stretch around the stagnation point
and this results in a lack of an upper limit for Ca, in the hexagonal array. Figure 3.18 shows
Cab for both Newtonian and viscoelastic drops in the hexagonal array. Cab is lower for the
hexagonal array than it is in the 50% square array. Above Cab the drop will break and below
it the drop will remain intact. For both drop types, Cab decreases with increasing drop length
and this dependence is greater for shorter drops. For Newtonian drops, Cab goes from 0.018
at 1.32 cm to 0.0070 at 4.66 cm. Over this range of drop lengths, Cab goes from 0.015 to
0.0070 for the viscoelastic drop. For the shortest drops (0.73 cm) no breakage occurred in
either case. The small difference between Cab for the Newtonian and viscoelastic drops

indicate that drop length is the major factor in determining the drop’s stability. Increased

79

 

Figure 3.17 : Breakup of a 1.89 cm viscoelastic drop in a V,= 0.40
hexagonal array at Ca = 0.052

80

 

0.025

 

I Newtonian

 

 

 

O 02; . viscoelastic

0.015 S
Cab

0.01 _.

0.005 -

 

 

I I T T T I I T T r I T l T l I l I T I T l T T f 7

. . . T . . . . I . .
1 1.5 2 2.5 3 3.5 4 4.5
Drop length (cm)

Figure 3:18: Cab for Newtonian and viscoelastic
drops in a Vf=0.4 hexagonal array

 

81

drop length results in greater viscous forces pushing the drop forward. This will cause the
longer drop to wrap around the rod more frequently than the shorter drops. When the drop
wraps around a rod, the viscous forces will push the branches further forward along the array.
This will thin the connection between the branches to a greater extent and result in increased
breakage.

As with the square array, the drop can break several times as it moves through a
hexagonal array and the number of times depends on both Ca and the drop length. In the
Newtonian case, the drop will break only once near Cab. The shorter drops (1.32 cm) will
break only once or twice in the range of Ca’s studied. For the longest drop (4.66 cm), the
average number of breaks increases to 3.0 at Ca = 0.019. Similar behavior is seen with
viscoelastic drops. The largest average number of breaks that occurs is 5.5 for a 4.66 cm
drop at Ca = 0.063. Unlike the square array where a small trailing droplet (about 1/1/10 the
size of the drop) is consistently formed, breakage in the hexagonal array can result in a wide
variety of droplet sizes ranging from droplets of equal size to droplets where one is much
smaller than the other. However, it is most probable that the two droplets of nearly equal
size will be produced upon breakup. In all these cases, the resulting droplets may still break

if their Cab is larger than Ca.

3.5.3 Consequences of breakup for drop mobilization
The paths of several drops take as they move through an array is investigated in this
section. Figure 3.19 shows the drop path (dotted line) for a drop in a square array with the

black dot representing the inlet for the syringe and hence the starting point for the drop. The

82

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83
drop follows a straight path parallel with the fluid flow. This straight path will be followed

by the drop regardless of its length, Ca, Vf of the array, breakage, or whether the drop is
Newtonian or viscoelastic. Since the rod gaps are aligned with the flow, there is no force
pushing the drop upward or downward and the drop will move with the fluid flow.

Unlike the square array, there is no readily apparent drop path in the hexagonal array.
When a drop hits a stagnation point, the extensional forces will cause the drop to move over
the drop, under the drop, or it will break the drop in two. Since the drop is density-matched
with the suspending fluid, gravity will not have a significant effect on the drop path. This
is evident with a Newtonian drop that does not break as it moves through the hexagonal
array. The drop took a tortuous path that for the most part followed the centerline of the
array (dotted line). Figure 3.20 shows the path (drawn line) of a 3.50 cm drop at Ca =
0.0064. This particular path is followed by drops of all lengths at Ca < Cab where the drop
does not break. This path is followed because local inhomogeneities in the array makes this
the path of least resistance. The fact that it follows the centerline indicates that gravity has
no effect on this process.

If the drop does break, than the paths of the resulting droplets are less predictable
than if the drop remained unbroken. This would be seen to a greater extent for arrays with
higher Vf. The path depends on the size of the droplets, and the number of times the drop
breaks. If the drops breaks only once and the droplets were moderately close in size, the
droplets would often follow parallel paths one step removed from each other. This can be
seen in Figure 3.21 for a 3.50 cm drop with Ca = 0.012. When the droplets came to

stagnation points, they would both start branching towards one another into the same

84

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Figure 3.20: Path through a hexagonal array of a 3.50 cm Newtonian drop at Ca = 0.0064

85

Drop path

 

 

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Figure 3.21: Path through a hexagonal array ofa 3.50 cm Newtonian drop at Ca

86

adjacent throat. Since one drop was slightly larger, it would move into that throat sooner
forcing the smaller drop into the other adjacent throat. This behavior was also seen when one
drop was significantly larger and was further ahead in the array. The only time this parallel
behavior did not occur was when one of the drops started wrapping around a rod and fell
significantly behind the other drop. When it started to move again, it would not necessarily
follow the parallel path.

If the drop broke multiple times, the paths would become even less predictable.
Figure 3.22 demonstrates this for a 3.50 cm drop at Ca = 0.019. At high Ca, the local array
inhomogeneities which cases slight changes in capillary forces may become insignificant
resulting in more random paths. As can be seen in Figure 3.22, smaller droplets stalled at
various stagnation points as a result of their Ca*’s being less than Ca of the flow.

The trends shown for Newtonian drops moving through a hexagonal array are the
same for a viscoelastic array with two notable differences. The first is that a viscoelastic
drop will tend to wrap around a rod and with one branch retracting more fiequently. The
elastic nature of the drop comes into play here. More significantly, at high Ca, the
viscoelastic droplets often would merge back into a larger drop. This phenomenon was never
seen with Newtonian drops. This may be the result of decreased interfacial tension for the
viscoelastic drops (8.1 vs 17.7 lem for the Newtonian drop). When the leading edges of
two drops come in contact, high viscous forces may shatter the thin film of suspending fluid
between the leading edges allowing the droplets to recombine. Figure 3.23 shows the path

for a 4.66 cm viscoelastic drop at Ca = 0.012. The drop breaks and recombines three times.

87

r
Center [Inc
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Figure 3.22: Path through a hexagonal array ofa 3.50 cm Newtonian drop at Ca = 0.019

88

Center line

Drop paths
Merging point

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Figure 3.23: Path through a hexagonal array of a 4.66 cm viscoelastic drop at Ca = 0.012

89

3.6 Conclusions
From this study of the movement and breakup behavior of drops, the following
conclusions have been determined:

1. In general, drop motion occurs in several stages that may take very different
amounts of time. The drop will form a leading foot through the throat into the next
array. The drop fluid will fill the body gap transversely before another leading foot
penetrates the next throat. When the drop is nearly drained from the first gap, a
trailing foot is formed. Because of the stagnation point found in the hexagonal array,
the drop will tend to wrap around the rod. Viscoelastic drops did not form distinct
feet and had a more continuous motion through the array (this effect was most
noticeable for longer drops).

2. For a square array, breakup occurs as the result of film growth in the throat
pinching off a trailing portion of the drop. The critical capillary number for breakup
(Cab) has a lower and upper limit with the upper limit resulting from low residence
time of the drop. The number of times a drops breaks increases with increasing
length and is at a maximum midway between the lower and upper limits.

3. In a hexagonal array, breakup occurs when the drop wraps around the rod at the
stagnation point. If the leading edges of the resulting branches extends too far, the
bridge connecting them will split resulting in two drops. This bridge snaps at a much
smaller extension for Newtonian drops than for viscoelastic drops. There is no
upper limit for Cab. The number of times the drop breaks increases with increasing

Ca and drop length.

90

4. For all arrays and dr0ps, Cab decreases with increasing drop length. For the square
array, this is the result of increased residence time in the throat. For the hexagonal
array, this results from the greater extensibility of the larger drop around the rod.
5. The path of the drop follows a straight line in the square array. In the hexagonal
array, the drop will follow the path of least resistance if it doesn’t break. If breakage
does occur, the droplets will follow more random paths. This may be the result of
higher Ca reducing the effect of local inhomogeneities in the array.

6. Viscoelastic droplets will often merge in the hexagonal array as the result of

decreased interfacial tension.

4. MOBILIZATION OF CYLINDRICAL DROPS IN

FLOW ACROSS AN ARRAY OF CYLINDERS

4.1 Introduction

This chapter will examine the conditions necessary to mobilize a cylindrical drop in
a fluid flowing across an array of transversely aligned cylinders. The main focus of this
study is the onset of mobilization and the relative velocity or mobility after onset over a
range of Ca. The effect of fluid rheology of the drop, the drop dimensions, and the array
geometry on the relative velocity of the drop will be examined in detail. As in Chapter 3,
Newtonian and viscoelastic drops of different lengths will be compared and contrasted. The
three array geometries from before (square, Vf= 0.40; square, Vf = 0.50; and hexagonal,

Vf=0.40) will be examined. A model for Ca“ was also derived and will be analyzed here.

4.2 Background

During mold filling with a fiber preform, pockets of air (voids) are often tapped
inside the fiber bundle. Binder, which is used to hold the preform together during handling,
is smeared into the fiber tow during consolidation. At higher filling temperatures, the binder

may melt forming small droplets. The presence of voids and binder droplets can be

91

92

detrimental to both the mechanical properties and appearance of the resulting part. During
mold filling, these inhomogeneities can be mobilized and pushed out of the preform by the
viscous forces of the incoming resin.

In the study of drop and bubble movement several parameters are measured. A key
factor is the capillary number (Ca) which is defined as the ratio of viscous forces to capillary

forces and is defined by the following equation:

Ca = (4.1)

H
Y
where n is the drop viscosity, V is the interstitial velocity of the suspending fluid, and y is
the interfacial tension at the drop/fluid interface. In porous media, the pore capillary number

(Cap) is often used since it takes into account the porous media properties:

 

(4.2)

where S" is the surface area to volume ratio of the drop and K is the permeability of the
medium. The drop mobility or relative velocity is the ratio of the drop velocity (U) to V.

In a capillary tube, the dimensionless drop size (A) is defined as the ratio of the drop radius
to the tube radius. For porous media, A can be defined as either 1/(SJ(°-‘) or (1/Svrb ) where
rb is a pore radius. The viscosity ratio (k) is the ratio of the drop viscosity to the suspending
fluid viscosity. Finally, a drop is not in direct contact with the solid surface: instead there
is a thin film of suspending fluid between the drop and the solid surface. The film thickness

is represented by h.

93

There has been extensive work in recent years on the mobilization of bubbles and
drops through various channels such as capillary tubes, tube constrictions, wavy walled
tubes, and packed beds. This study builds on those previous works by studying the
movement of a cylindrical void aligned transversely with the fluid flow in an array of rods.
The Ca values studied here are very low (0.005 < Ca < 0.1) and are often close to the onset

of mobilization.

4.2.1 Flow through capillary tubes and channels

Bretherton [42] did early work on the flow of long bubbles through capillary tubes
at low capillary numbers. He derived an expression for the dimensionless layer thickness
W that depended on the bubble capillary number Cad (defined with bubble velocity U) as

follows:

2
1.29(3Cad)3

V b 2 (4.3)
l-_ : _
u [R]

€

where W

where W is a relation of U to V and R is the tube radius. The above equation agreed well
with experimental results in the range of Cad > 10“ but under predicted to a great extent for
Ca, < 10". This was the result of errors in his assumptions about the fluid film thickness
between the tube and bubble. A more detailed examination of fihn thickness by Schwartz,
et al [43] found that Bretherton had underestimated the film thickness significantly and that

his equation is most accurate for shorter bubbles. They also determined that film thickness

94

depends on Ca, alone in accordance with other work.

The fluid film thickness has a major effect on the mobility of a bubble or drop in a
capillary tube. Olbricht and Kung [29] found that for a Newtonian drop in a capillary tube,
as Ca increased from 0.05 to 0.16, the drop became elongated and the fihn thickness
increased. For very long drops (A > 1.05), changes in length resulted in no significant
change in the film thickness. As Ca increased from 0.035 to 0.110, UN increased by 20%
and leveled off for Ca>>1. Increasing k over the range studied (0.0075 to 2.10) resulted in
a decrease in UN. In all the cases they examined, UN > 1. Higher k causes less extension
of the drop and hence a greater film thickness. In the hexagonal array, the same general
trends were seen. For high k (=3.7), there was no leveling off of UN with respect to Ca.
Except for k=3.7, in other viscosity ratios, increased drop size resulted in increased
mobilization.

Kopf-Sill and Homsy [34] studied air bubble motion as a result of fluid flow in a
horizontal Hele-Shaw cell which consists of two horizontal plates separated by a thin gap.
The bubble would take on seven distinct shapes: near circles, flattened, elongated, long-tail,
short-tail, “Tanveer” (a flattened drop with a concave leading edge), and inverse “Tanveer”.
The resulting shape and UN depended upon Ca and the initial dimensionless bubble radius
(bubble radius/half cell width, a/L). Their results are shown in Table 4.1. Once the flow and
bubble reached a steady state, all of these shapes remained stable. From this table, it can be
seen that larger drops will experience greater distortion at lower Ca than the smaller drops
and these distorted shapes have higher UN. The ranges of Ca and a/L values for producing

different shapes often overlap indicating that the drop can develop into two or more shapes

Table 4.1: Parameters for different bubble shapes in a Hele-Shaw cell

 

 

 

 

 

 

 

 

 

[from Kopf-Sill and Homsy]
Shape Ca a/L UN
Near circle 6x105 - 0.003 0.038 - 0.16 0 - 0.4

Flattened 0.003 - 0.02 0.038 - 0.065 0.4 - 0.7
Elongated 3x104 - 7x10“4 0.10 - 0.16 0.2 - 0.55
Long-tail 2x10‘4 - 0.02 0.038 - 0.16 0.5 - 1.7
Short-tail 0.002 - 0.03 0.065 - 0.16 1.0 - 2.0

Tanveer 5x10‘4 0.16 ~0.2
Inverted Tanveer ---- 0.24-0.27 1.2 - 1.3

 

 

 

 

 

96

under a given set of conditions.

4.2.2 Flow through a constriction

Olbricht and Leal [39] studied the mobility of a Newtonian drop in flow of both
Newtonian and viscoelastic suspending fluids through a wavy walled tube. In all cases
studied, UN was greater than one. At low Ca (<0.083), UN decreased by 13% as A
increased from 0.6 to 1.1. Increasing k from 0.42 to 0.60 resulted in a slight decrease in UN.
At large Ca (>0.9) neither k nor A had a significant effect on UN and doubling Ca fi'om 1.34
to 2.54 increased UN by only 5%. For very small k (0.0014), UN stayed nearly constant
over the A range but increased by 16% when Ca increased from 0.50 to 0.94. Similar trends
were observed with a viscoelastic suspending fluid and the quantitative results for UN were
comparable to those of the Newtonian fluid.

Legait [44, 45] examined the critical capillary number (Ca*) necessary to mobilize
low viscosity Newtonian drops of different A through a capillary tube with a single
constriction. A force balance was derived for the drop resulting in the following relationship

along the length of the drop in the flow direction (x):

v—=—-— (4.4)

where C is the curvature of the drop, pd is the pressure inside the drop and p is the viscous
pressure of the fluid. Ca“ is then determined at the onset of mobilization (U=0) to be the

following:

 

 

where f (k) is a strongly decreasing function of the viscosity ratio (k) and m is an exponent.

Legait performed experiments with non-wetting alkane drops suspended in water
moving through a constricted tube. Ca“ decreased fi'om 20 x 1045 to 5 x 10" as A increased
from 2 to 12. The resulting Ca“ vs A profile showed Ca“ dropping sharply for shorter drops
and gradually level off with increasing A. Increasing k from 0.055 to 26 resulted in Ca“
decreasing by 60% for all drop lengths. These experimental results were in good agreement
with Legait’s model. A tube with a bypass connected downstream and upstream of the

constriction was also used to more effectively model a porous medium.

4.2.3 Flow through a packed bed

Ng [40] investigated the motion of an oil drop displaced by water in a bed of packed
spheres. It was found that the drop will remain immobile until (Ca‘) is reached.
Mobilization will occur only when the hydrodynamic forces of the incoming fluid are greater
than the capillary forces of the drop. Like Legait, NG tried to model Ca" with a force
balance except that Ng’s was based on the overall drop geometry. NG derived the following

equation which relates Ca“ to the geometry of the packed bed:

 

 

k 2
win—vi =["l I»)
y crit Wu 3 cosa (4.7)

where ke is the effective permeability (determined from Kozeny’s equation), 3 is the
separation of the two feet of the drop, or is the angle the drop makes with the direction of
resin flow and r, and r, are the throat and body radii respectively (Figure 2.4). Equation 4.2
predicts that Ca“ will decrease with increasing drop length and decreasing porosity (decrease
in kc, r,, and rb). Experimental results by Ng showed that an increase in drop length leads to
a decrease in Ca". An increase the viscosity ratio (k) of the drop will also cause a decrease
in the critical capillary number.

Liu [41] studied the mobility of initially spherical Newtonian drops flowing through
square (V, = 0.35) and hexagonal (Y = 0.40) arrays of cylindrical rods. In this study,
spherical drops were injected into the cell upstream of the array before coming in contact
with the array. A wide range of Ca (0.1 to 2.3), k (0.07 to 3.7), A (0.47 to 1.75), and S,K°-’
(0.50 to 2.4) values were examined here resulting in UN values ranging from 0.5 to 2.5. For
the square array at low Ca (<1.0), UN increased as Ca increased from 0.15 to 1.0 and
decreased slightly with increasing k. Increased A resulted in a substantial reduction in UN
for all k. [Our results are in contrast to the last trend with drop size and may be the result of
the differences in drop placement (Liu’s gives a higher degree of drop breakup).] For Ca >
1, UN remained constant (at 1.25 for most drops) for increasing Ca. At high k (>1),

viscosity ratio and drop size also had little effect on UN. Only for low viscosity ratio

99

(k=0.07) did drop size have an effect on UN and here increased drop size resulted in an
increase in UN.

In all of the studies that have been reviewed here, the drop fluid did not wet the
individual elements of the packed bed. Rather, there was a thin film of suspending fluid

between them and this film was present over all ranges of Ca studied.

4.2.4 Theory

The majority of work on flow of a bubble or a drop through a constriction dealt with
axisymmetric geometries (constricted tubes). For the study of flow through an array of a
rods, the constriction is two dimensional which will alter the distortion of the drop through
the constriction. Figure 4.1 shows the movement of a drop through a constriction created by
two rods. Two forces act on the drop: the viscous pressure difference pushing the drop

forward and the capillary pressure difference resisting the drop flow. These differences are

defined as follows:

Capillary pressure difference = y(C2 - Cr)

Viscous pressure difference = (I?) V(x2 - xi) (4'8)

where x1 and x2 are the widest sections and of the trailing and leading feet located along the
x direction and Cl and Q are the curvatures of the two feet. At the maximum capillary
pressure (when the foot reaches the throat), C| is related tobr andzC is related ,to r.
Mobilization of the drop will first occur when these pressure differences are equal resulting

in the following general force balance:

100

   

  

///
///////////// ////////////// ///////

////////////// //////////// /////// /////// //////// ////////
///////////////////// ////////////// ///////////////////////
//////////// /////////// ///////////////////
/// // /// /’/ / //

§\\§

   
    

drop

rod

Figure 4.1: Drop moving between two cylinders

101

C - C
(32.) = K(—2——;) (4.9)
Y or (X2 — x1)

This relationship explains the inverse dependence on the extent of the drop along the flow

direction.

4.3 Objectives

This study will examine the mobilization of a drop moving through an array of
cylinders transversely aligned to the fluid flow. The main thrust is to investigate how the
drop mobility is affected by the following factors: capillary number, drop length, array
geometry, and drop rheology. From this mobility study, the critical capillary number (Ca*)
at which the drop just begins to mobilize will be determined.

This study departs from previous studies in several key aspects. The first is that the
rheology of the drop, as opposed to the suspending fluid, will be varied between Newtonian
and viscoelastic behavior. Low capillary numbers (0.005 < Ca < 0.03) will be the focus of
this research. Using an array of cylinders is a departure from most studies which have
focused on capillary tubes with and without constrictions. While Liu’s study was involved
with cylindrical arrays of comparable porosities, she did not examine Ca" or the difference
between Newtonian and viscoelastic drops. Liu also investigated a higher range of Ca (0.1
< Ca < 0.1) than that examined here. Another key difference is with the procedure: in Liu’s
study, a spherical drop was placed in the flow cell upstream from the array; in this study, a

cylindrical drop is impregnated directly into the array. This means that in our study, the drop

102

will be initially at rest in each flow run and will have a cylindrical shape. The drop behavior
will then be more comparable to that of voids and binder droplets trapped in composites

during processing.

4.4 Materials and procedure
The aluminum flow cell and experimental setup used in this study are described in
Chapter 2. The three rod arrays used here are described in detail in Section 2.1.2, with their

geometric specifications being as follows:

Table 4.2: Array specifications

 

 

 

 

Rod Fiber volume 1 r, r,, rb/r, Permeability
alignment fraction (V ,) (cm) (cm) (cm’)

Square 0.40 0.095 0.23 2.42 1.37x10'3

Square 0.50 0.050 0.17 3.40 3.25x104
Hexagonal 0.40 0.12 0.21 1.75 1.25x10'3

 

 

 

 

 

 

 

 

 

The suspending fluid is a chloroparaffrn oil which had a refractive index matched
with that of the cylinders to allow easier viewing of the drops. Air bubbles were first
attempted in the array, but they quickly rose to the top of the array. Instead, drops with
densities matched to the oil were investigated so that gravity effects would be minimal.
Three different drop materials were used here: glycerol, glycerol/water (Newtonian) and
glycerol/water/polyacrylamide (viscoelastic). The key properties of these materials are listed

in the table below:

103

Table 4.3: Properties of the fluids

 

 

 

 

 

 

 

 

 

Material viscosity viscosity surface tension interfacial tension
(mPa-s) at ratio, k (mN/m) with oil (mN/m)
23°C
oil 109 - 33.2 -
glycerol 970 0.55 64.0 13.4
glycerol/water 60 8.9 68.6 17.7
glycerol/water/PAC 1430 13 78.6 8.1

 

 

 

A cylindrical drop was first injected into the fluid-filled array with a syringe. Each
drop occupied the entire space between the rods. Since the initial cross section of the drops
were limited by the array, changes in drop volume resulted in changes in the drops length
transverse to the fluid flow. Because the rods were wetted by the oil, the injected drop was
not be in direct contact with the rods. Instead, a thin oil film was situated between the drop
and the rod.

The flow is then started by opening a pressure pot and is regulated with a needle
valve and rotameter. The drop velocity (U) is found by dividing the distance the drop moved
(usually the span of the window) by the time (taken with a stopwatch). In determining U,
each time the drop completely moves into the next body cavity it is considered to have

moved one step. The point on the drop that is used to determine when it leaves a body cavity

104

is the trailing edge since it is easiest to discern. Timing is started afier the drop moves one
step to eliminate inertia effects from the startup and it allows the shape of the drop to become
firlly developed. The value of U for each condition is the average of the results for 4-10 runs.
These runs are performed for progressively lower fluid flow rates until the drop ceases to

move and this yields the critical capillary nmnber (Ca‘).

4.5 Results and discussion
In this study, the drop shape is a circular cylinder with semi-spherical end caps. The

surface area of the drop is 47tr2+21trL and the volume is (4/3)1tr’+1rr’L. S, for the drops is:

S = surface area = _2_ 2r + L

V
volume r 4
—r + L

3

 

 

(4.10)

Figure 4.2 shows the dependence of S, on drop length for the three arrays and the drop
lengths investigated. For every array, S, decreases with increasing drop length indicating
that the effect of the end caps on S, is minimized for longer drops. ,8 increases with
decreasing rb as would be expected since the drop is thinner. These results were then used
to calculate the dimensionless drop size A (= 1/S,rb). In these studies, the oil will have a two
dimensional flow. The drop, meanwhile will experience three dimensional flow with

transverse and vertical flow as the drop fills the next adjoining cavity.

4.5.1 Newtonian drops

Each portion of this section will examine the mobilization of a Newtonian drop

105

 

 

 

 

 

 

 

 

15
j I rb=0.233cm,square,Vf=O.4O
14.;
. O rb=0.169,square,Vf=0.50
O
l A rb=0.211,hexagonal,Vf=0.40
13-
1 0
. O
- 9 0
E, 12.. O
c?) 11.:
101 I A ‘
i ‘ ‘ ‘
’ I
‘ I
91 I .
8 ““lr I""1""I‘ITT‘l""l'TT’l’TTTlfiTTT
O 0.5 1.5 2 2.5 3 3.5 4 4.5

Drop length (cm)

Figure 4.2: Surface area/volume ratio (8") vs.

cylindrical drop length for drops in various arrays

5

106

through different cylindrical arrays. The results for the three arrays will then be compared
and contrasted. In all cases, a Newtonian liquid (chloroparaffrn oil) is used as the suspending
fluid and the drop consists of the glycerol/water mixture. In this study, the distance between

the two feet (x2 - x,) varies with A [Eq. 4.9].

4.5.1.a V, = 0.40 square array

The first array/drop combination studied involved a Newtonian drop moving through
a square array with V,= 0.40. Drops of six different volumes were injected into the array:
0.10, 0.20, 0.30, 0.40, 0.50 and 0.60 ml which had lengths of 0.60, 1.08, 1.55, 2.03, 2.87, and
3.82 cm respectively. This results in A ranging from 0.427 for the shortest drop to 0.482 for
the longest drop. This is on the high end of the range studied by Liu (0.156 < 1/S,r,, < 0.584)
for a similar square array. In these sets of runs, Ca was varied only by changing the fluid
flow rate. For all runs except those with very low Ca values, each drop moved a total of
eight steps or 5.33 cm (one step is the distance from one body gap to the next, one rod
diameter and throat gap). For very low Ca (approaching Ca"), the drop was timed moving
only one or two steps. Testing was stopped and the drop considered immobile if it failed to
move one step in 20 minutes. Finally, the relative drop velocity is taken as the ratio of the
drop velocity (U) to the interstitial velocity of the fluid (V).

The mobility of drops of different lengths is shown in Figure 4.3. For each drop
length there is a certain Ca value below which the relative dr0p velocity (UN) is zero and
the drop is immobile. This value is the critical capillary number (Ca*). At Ca < 0.010, no

drops of any length will be mobilized. For 0.010 < Ca < 0.022, Ca" is reached for each drop

107

 

 

 

 

 

 

 

0.6
l
1 I 3.82cm
‘ I
0.5. .1. 2.87cm . 4
l A 2.03cm
0-44 x 1.55cm
* i
j C 1.08cm
0
20-3— . 0.60cm I
i at
i l A
0.2-
- I x . .
_ g ‘ O 0
- O O
x O
oi.,..l....,-I'.1z.‘,.0..,,¢,.,....
0 0.005 0.01 0.015 0.02 0.025 0.03
Ca

Figure 4.3: Relative drop velocity vs. Ca for

 

Newtonian drops in a square array (Vf=0.40l

 

108

length with the longer drops having a lower Ca“. When Ca > 0.021, drops of all lengths will
be mobilized. Above the critical flow rate, the drop velocity increases with increasing Ca
which is indicative of the higher hydrodynamic pressure being exerted on the drop. For any
given Ca in the range where mobilization occurs, UN increases with drop length. As an
example, at C a= 0.02, UN is 0.21 for a 3.82 cm drop, 0.10 for a 1.55 cm drop, and UN =
0 for the 0.60 cm drop.

Since the capillary forces which oppose drop movement are exerted on the drop
surface and the viscous forces are related to the drop volume, the surface area/volume ratio
is an indication of drop mobility. Shorter drops will have a larger surface area/volume ratio
which results in greater viscous forces needed to overcome the capillary forces (Eq. 4.9). In
Figure 4.3, it can also be seen that as drop length increases, its effect on UN becomes less
significant with very little difference between drop lengths of 3.82, 2.87 and 2.03 cm. In the
range of flow rates studied, the drop velocity was much smaller than the interstitial flow rate
with UN no greater than 0.6. This was caused by capillary resistance to the drop deforming
and squeezing through the narrow gap (throat) between rods.

Increased U is mostly the result of a reduction in the time for the leading edge of the
drop to bulge and form a leading foot in the next gap (defined as the bulge time). The bulge
time increases with decreasing length. At Ca = 0.0266, on average it took 34.50 seconds for
a 0.60 cm drop to move one step and 98% of this time involved forming a leading foot
(33.73 seconds). For the 3.82 cm drop at the same Ca, the time to move one step was 2.80
seconds with the bulge time accounting for 50% of this period (1.42 seconds). The bulge

time is also reduced with increasing Ca. The bulge time is what causes apparent pauses in

109

the drop motion (particularly for shorter ones) and discontinuous motion.

To determine Ca“, a plot was made of the reciprocal drop velocity (l/U) vs V. Figure
4.4 shows the graph for a Newtonian drop with a length of 2.87 cm. With decreasing V, M}
steadily rises until at a certain value of V, l/U suddenly jumps. This results in an asymptote
being formed and this asymptote is the critical interstitial velocity (V *) since at this point
IN is infinity. Figure 4.5 shows the IN vs V curves for all drop lengths studied. The
asymptotic value (V *) is then used to calculate Ca".

Figure 4.6 shows the relationship of Ca“ to drop length for the V,=0.40 square array.
Ca“ drops from 0.022 for a drop length of 0.60 cm to 0.010 for a drop length of 3.82 cm.
The effect of drop length on Ca“ also decreases with increasing length, with Ca“ leveling off
for longer drops. This is consistent with the experimental results of Legait [45] and Ng [40]
and with the predictions of Eqs. 4.6 and 4.9. The decrease in S, for the drop as shown in

Figure 4.2 results in both the decrease in Ca“ and its leveling off for increasing drop length.

4.5.1.b V, = 0.50 square array

The next array/drop combination studied involved a Newtonian drop moving through
a square array with V,=0.50. Since rb for the 50% array was less than r, for the 40% array
(0.169 vs 0.233 cm respectively), drops of smaller volume were used to get comparable
lengths. Drops of seven different volumes were used: 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, and
0.40 ml which had lengths of 0.57, 1.05, 1.53, 2.00, 2.48, 2.96, 4.03 cm respectively
resulting in A ranging from 0.438 for the shortest drop to 0.487 for the longest drop. The

runs and measurements were carried out in the same manner as before. For most runs

110

900

 

.l

l
800 j 'l'

700i
600i
asymptote

01
O
O
I-

.p.
O
O
1

IIU (slcm)

300;

200.3 +

100. +

: V” +

0"..-.l....l...,,....,..-.'l,'..‘I‘.,.'."..fl!—.-IIL-fl-.—.-.—

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
V(cm/sl

 

 

 

 

Figure 4.4: Determination of V* using 1/U vs. V curve
(drop length of 2.87 cm)

111

 

 

 

 

 

 

 

 

1
3003 -l- I 3.82cm

’ + 2.87cm
700.1

: x A 2.030m
600—: . 8 1.55cm
500 O 1.08am

1 A

1 . 9 0.60cm
400-

I

300-

I
200__ + ‘

‘ 8
1001 :A O.

0 l" 1, 1

0.05 0.1 0315 0.2 0.25 0.3 0.35 0.4 0.45 0.5

V(cm/s)

Figure 4.5: 1/U vs. V for Newtonian drops in a
square array (Vf= 0.40)

08*

 

 

 

 

112
0.025
I
0.02-
0.015; I
‘ I
I
0.01- I I
0.005.:
0‘""l''"l‘lfi'l‘j‘filj"‘l‘file""lfi'_rTl""l“II
0 0511.5 2 2.5 3 3.5 4 4.5 5

Figure 4.6: Ca* vs. drop length for Newtonian

Drop length (cm)

drops in a square array (Vf=0.40l

has...“

 

113

(except when Ca approached Ca“), the time needed for the drop to move 5.33 cm (10 steps)
was used.

Figure 4.7 shows the mobility for the drops moving in a V,=0.50 square array. The
results are qualitatively similar to the results of the 40% square array. Longer drops have a
lower Ca" and a higher mobility than shorter drops. The degree to which drop length affects
mobility is reduced with increasing length. However, there are also a great deal of
differences between the two arrays. The range of values for Ca" is smaller and the values
are lower than for the 40% array. The mobility is also higher in the 50% array. Comparing
two drops of similar lengths (2.03 cm [0.40 ml] for the 40% array and 2.00 cm [0.20 ml] for
the 50% array) at Ca=0.015 gives UN as 0.02 for the 40% array and 0.32 for the 50% anay.
Since the drop volume is half for the 50% case and hence the surface area to volume ratio
is significantly higher (12.4 vs. 9.2), one would expect that the drop in the 50% array would
be more difficult to mobilize. The fact that the rb/r, ratio is larger for the 50% array (3.40 vs.
2.42) resulting in a greater difference in the feet curvatures (Eq. 4.8) would also indicate that
there is greater capillary resistance to the drop squeezing through the throat. These trends
are superseded by the greater pressure gradient associated with the decrease in permeability
when V, increases from 0.40 to 0.50, (1.37x10‘3 cm 2 and 3.25x10“ cm2 respectively). The
viscous pressure will be larger for the 50% array and this will push the drop faster, increasing
its mobility.

Figure 4.8 shows Ca" vs drop length for the V,=0.50 square array. This curve shows
the same basic trend seen with the other square array: Ca“ starts at 0.012 for the 0.57 cm

drop and decreases to 0.0082 for the 4.03 cm drop. In comparison with the results for the

UN

114

 

 

 

 

 

 

 

I 4.03cm :
0.8; .1. 2.96cm 0
A 2.480m
, a 2.00cm . 0
0'6J O 1.53cm
9 1.05cm . g E]
(14; [j 0.57cm +
I 9, .
' :1
0.2- : °
1 I ’ U
1
1:13
00 7% T 0.02 T 003 '0.04

 

 

Figure 4.7: Relative drop velocity vs. Ca for
Newtonian drops in a square array (Vf= 0.50)

Cat

115

 

0.025 .
0.02.:
0.015 _
0.01 _ .

0.005 L

 

 

T I T I Y I I I Y I I I . T T T l T T

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Drop length (cm)

Figure 4.8: Ca* vs. drop length for Newtonian
drops in a square array (Vf=0.501

 

rtTTTlYTITlTTTT

116

V5040 square array, drops of similar lengths had significantly lower Ca" in the \4 =0.50
array. As was stated previously, this was caused by increased viscous forces being placed
on the drop which resulted from increased permeability of the array. It should also be noted
that while Ca" also levels off in this array for longer drops, the overall decrease in Ca“ is not

nearly as large as it was for the V, = 0.40 square array.

4.5.l.c V, = 0.40 hexagonal array

The final array examined with the Newtonian drop is the V,= 0.40 hexagonal array.
The drop volumes used were the same as for the 40% square case but since r, was slightly
smaller (0.211 cm vs 0.233 cm) the drops were longer. Drops with seven different volumes
were examined: 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, and 0.80 ml which had lengths of 0.73,
1.32, 1.89, 2.47, 2.99, 3.50, and 4.66 cm respectively resulting in A ranging from 0.439 for
the shortest drop to 0.486 for the longest drop and is higher than the range studied by Liu
(0.163 < 1/S,rb < 0.606) for a similar hexagonal array. The experimental procedure for these
runs was the same as for the two previous arrays.

The hexagonal array differs from the square arrays in that the drop does not follow
a straight trajectory as it does in the square arrays (see Chap. 3). Figure 4.9.a shows the path
of a drop around a cylinder with the dotted line denoting the center of the drop. The
tortuosity (the ratio of the path length to the distance between the start and end points) of a
hexagonal packing is 1.5. One step (Figure 4.9.b) in this array is defined as the drop moving
just under one cylinder diameter length 30° from vertical and then another cylinder diameter

horizontally with the flow for a total distance of 0.827 cm. For this study, the time for the

 

117

I

l

l

I

I

I
I
s

I
I
I
I
l
I
l
s
s
s

Figure 4.9.a: Drop path (dotted line)

Start

3
O O

B
G-

H

Figure 4.9.b: Definition of a step

Figure 4.9: Path ofa drop in a hexagonal array

118

drop to move 9-10 steps (depending on the starting point defined) was measured (except
when Ca approached Ca“) for a total distance of 7.44-8.27 cm.

Figure 4.10 shows the relative drop velocity vs Ca for the hexagonal array. The drop
behavior in the hexagonal array is similar to its behavior in the square arrays in the following
trends: drop mobility begins at higher Ca for shorter drops; the mobility is greater for longer
drops; and the effect of drop length is less pronounced for longer drops. The drops mobilize
much quicker in the hexagonal array (0.002 < Ca < 0.009) than in the two square arrays. The
mobility is also significantly larger for the hexagonal array, a 1.89 cm long drop has UN =
0.56 for Ca = 0.015 which compares with UN = 0.02 and 0.32 for drops of similar lengths
at the same Ca in the 40% and 50% square arrays respectively.

Since the hexagonal array has a similar permeability (1.25 x 10'3 cm2) and surface
area/volume ratio (Figure 4.2) as the 40% square array, it would be expected that the
mobility should be similar. The increased mobility is the result of the different packing
arrangement for the hexagonal array. Since the rods are arranged so that the fluid has no
straight path through the array (Figure 4.9), it must flow around the rods creating a
stagnation point at the upstream edge of the rod (see Chap. 3). This results in extensional
forces being exerted at that point and these forces may facilitate the movement of the drop
through the array (particularly at lower Ca). The hexagonal array also has the lowest r,/r,,
ratio (1.75) of the three arrays and the drop will experience less lateral distortion as it moves
through the throat. The difference in the curvature of the leading and trailing foot is small
and this will decrease the capillary forces resisting drop movement (Eq. 4.9). The

combination of these two factors serve to increase the drop mobility.

0.6

119

 

0.5.~

 

 

 

 

 

 

 

O
r E- .

. . I

~ +
0.4; I. 8 .

: +0 I 4.66cm

_ A . El El

- I .

3 1
l at El A 2.99cm
O

, O 8 2.47cm
0.2.: i . [j

_ i. . 1.89cm

1
0.1; . El . 1.32cm

[3 0.73am

0” .,. -,.-.,,.,.,,.-,...,

0 0.005 0.01 0.015 0.02 0.025 0.03

Ca

Figure 4.10: Relative drop velocity vs. Ca for
Newtonian drops in a hexagonal array (Vf=0.4l

120

Another distinct feature of the hexagonal array is that at Ca~0.01, mobility levels off
for the longer drops. One reason is that these longer drops start breaking apart forming
smaller droplets with lower mobilities. The breakup process, which involves the drop
wrapping around a rod and splitting in two (see Chap. 3), will also serve to retard the drop
movement. Even if the drop doesn’t break, it may still partially wrap around the rod forming
a finger that will eventually retract back into the main body (Figure 4.11). This wrapping
around occurs more frequently with increasing Ca and with drop length and will counteract
the effects of the increased flow rate on the mobility.

Figure 4.12 shows the effect of drop length on Ca“. Like the square arrays, Ca“
decreases with increasing drop length in the hexagonal array: from 0.0089 for the 0.73 cm
drop to 0.0022 for the 4.66 cm drop. Ca“ levels off for long drops but the ensuing curve is
not as smooth as the 50% square array curve (although it is much smoother than the 40%
square array curve). The values for Ca“ are very low and this is due to extensional flow

around the drop and to a smaller rb/r, ratio.

4.5.1.d Comparison of Newtonian drops in the arrays

Figure 4.13 compares the mobility of long Newtonian drops of similar lengths in the
three arrays. This graph clearly illustrates the trends discussed in the previous sections. The
mobility of drops in the 50% square array is higher than the 40% array as the result of
decreased permeability and hence higher viscous forces. Drops in the hexagonal array are
even more mobile due to extensional forces at the stagnation points in the array. These same

extensional forces serve to level off the drop mobility at higher Ca by forcing the drops to

 

Figure 4.11: Retraction of a drop branch in a
hexagonalarray

Cat

122

0.025

 

0.02;

1

0.015 .
0.01 .1

0.005 -

 

 

 

7 I T T I I I I T I I T I I I T T T l r T

. . 1 . . . 1 . . . . .
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Drop length (cm)

Figure 4.12: Ca* vs. drop length for Newtonian
drops in a hexagonal array (Vf=0.40)

UN

123

 

 

 

 

 

     

 

 

] I Square,Vf=O.40,2.87cm
I Square,Vf=O.50,2.96cm
0.8- A Hexagonal,Vf=0.40,2.990m
0.6-
I

0.4-

1
0.2-

orfi’fil" “"l“"l'rfi'l""
0 0.005 0.01 0.015 0.02 0.025 0.03

Ca

Figure 4.13: Relative drop velocity vs Ca for long
Newtonian drops in the three arrays

124

wrap around the rods. These same trends are seen with the short Newtonian drops in Figure
4.14. The mobility is lower for the shorter drops in all three arrays as the result of higher
surface area/volume ratios.

The relationship of drop length to Ca" for the arrays is compared in Figure 4.15. The
basic relationship remains the same for all three arrays with lower Ca“ for greater drop
length. Ca“ is less in the 50% square array than in the 40% square array. However, this
difference narrows with increasing length: Ca" for the 40% array is 83% greater for the
shortest drop but it is only 22% greater for the longest drop. This could be the result of the
higher capillary resistance in the 50% array (as the result of increased curvature difference,
Eq. 4.9) counteracting the higher viscous forces. This would result in a flatter Ca“ curve for
the longer drops. Ca" for the 40% hexagonal array is a fraction of that for the 40% square
array (0.0089 vs 0.022 respectively for the short drops and 0.0022 vs 0.010 for the long
drops). Figure 4.16 shows Ca“ vs A (1/S,r,,) for the three arrays. The same general trends
are seen as before except with that the three curves are more linear. Also, it should be noted
that a small increase in A (from 0.42 to 0.46) will result in large drops in Ca" for all three

array geometries examined.

4.5.2 Viscoelastic drops

In this section, the mobilization of a viscoelastic dr0p (composed of the
glycerol/water/polyacrylamide solution) through different cylindrical arrays was examined.
The V,= 0.40 square and hexagonal arrays were studied here (the V, = 0.50 array cracked

during the Newtonian test and broke when it was removed from the flow cell making it

UN

125

 

 

 

 

 

I Square,Vf = 0.40.0.600m
I Square,Vf = 0.50.0.57cm
0,3_ A Hexagonal,Vf=0.40,0.73cm
0.6 ..
0.4 _
0.2 _
O 1 1 ,

 

 

0 0.005 0.61 026150621 0625’ '06:; ‘ 0035
Ca

Figure 4.14: Relative drop velocity vs Ca for short
Newtonian drops in the three arrays

Cat

126

 

 

 

 

 

 

0.025
1
A I Square,Vf=O.4O
I
0 02; . Square,Vf =0.50
‘ A Hexagonal, Vf=0.40
0.015 .1 I
A I
‘ I
0.01 _ . I .
“ A . . . . .
0.005 Q
. A
« A
. ‘ A A ‘
.1
O I II I I ...... m I r. . .

 

“‘ "' ‘TIHH ""l“"l""l
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Drop length (cm)

Figure 4.15: Ca* vs drop length for Newtonian
drops in the three arrays

 

5

C8*

127

 

 

 

 

 

 

 

0.025
,vr=o.4
I I square
I s uare,Vf=0.50
o.oz_ ' q

~ A hexagonal,Vf=O.4O
0.015; '

‘ I

0 I
0.01.: g I I

I

A A o . .. .
0.005;

_ A

A
A ‘AA A
I
O I I I ' ' I ' ' I I ' I I I
0.4 0.42 0.44 0.46 0.48 0.5
= 1/(Svrb)

 

Figure 4.16: Ca* vs dimensionless drop length (A) for

Newtonian drops in the three arrays

128

unavailable for these nms). After a drop was injected into the array and the syringe needle
was removed, a thread was drawn out of the drop. The run was delayed by a few minutes
to allow the thread to retract back into the drop. The viscoelastic drop volumes studied here
were the same as those in the Newtonian drop study. All other procedural aspects of this

study were identical to those for the Newtonian drop study.

4.5.2.a Vf = 0.40 square array

Figure 4.17 shows the motion of a viscoelastic drops of two different lengths through
the array. Flow of a viscoelastic drop differs from a Newtonian in that the motion is much
smoother and continuous. Leading and trailing feet are not as clearly developed. This
difference is especially obvious for shorter drops.

Figure 4.18 displays the relative drop velocity (UN) vs Ca for the viscoelastic drops.
The mobility behavior of the viscoelastic drops contrasts sharply with that of the Newtonian
drops in several respects. The most significant difference involves the effect of drop length
on UN. Longer drops are mobilized at lower Ca" and close to the onset, mobility increases
with increasing drop length, which would be expected based on the Newtonian results.
However, in the range of 0.025 < Ca < 0.032, the dependence of drop mobility on length
becomes inverted and by Ca > 0.04, mobility increases with decreasing length (the crossover
is more clearly illustrated in Figure 4.19). The effect of drop length on mobility is still most
pronounced for shorter drops with UN of the shortest drop (0.60 cm) being over 50% higher
than UN for the second shortest drop (1.08 cm). The mobility of the viscoelastic drops is

far lower than it is for Newtonian drops. The highest mobility reached by the Newtonian

129

 

Figure 4.17.b: Drop length = 2.87 cm

Figure 4.17 : Movement of viscoelastic drops through a Vf = 0.40
square array at Ca = 0.0907

UN

130

 

 

 

 

 

 

 

 

0.35
3 I 3.82cm
0.3- CI
~ + 2.87cm
0251 A 2.03cm D
I
. 1. 0
. x 55cm D CI
0.2.: . 1.08cm D . a
. [j 0.6cm Cl *
0.15; o 3“ I
.l
O :3 a: :
1 El A
0.1 - z r A I
: i + i
. I
it
- . I
J
O r T g’fu I T I I I fl
0 O 02 0.04 0 06 0 08 0.1
Ca

Figure 4.18: Relative drop velocity vs. Ca for a
viscoelastic drop in a square array (Vf=0.40l

131

0.35

 

 

 

 

 

I
i 1
o.3i
: 2
0.25 _
j 3
0.2 _‘ 4
~ 5
E — e
0.15 L
0.1 L
.l
0.05;
/- /
0 // I I I I I
0 O 02 0 04 0.06 0 08 0 1
Ca
1-0.60 cm 4-2.03 cm
2-1.08 cm 5-2.87 cm
3-1.55 cm 6-3.82 cm

Figure 4.19: Relative drop velocity vs. Ca for a
viscoelastic drop in a square array (Vf=0.40) [Redrawn]

132

drops is UN = 0.54 at Ca = 0.028 for a drop length of 3.82. For the viscoelastic drops, the
maximum UN is 0.32 at Ca = 0.090 for a drop length of 0.60. This would be expected since
the viscosity ratio (R) for the viscoelastic drop is 13 and this is much higher than it is for the
Newtonian drop (k = 0.55). For viscoelastic drops, the bulge time is much lower (the drop
moves in a more continuous fashion) than for the Newtonian drops and does not have as
great an effect on U.

The cross over occurred at Ca values between 0.025 and 0.032 and UN values
between 0.025 and 0.040. At this low range, U is very small for the drops, between 0.0055
and 0.010 cm/s. The thin film of oil between the rod and the drop will act as a lubricant and
the drop will move through the throat as a cylindrical plug. As a result, the shear forces on
the drop fluid are minimal across the width of the throat. However, there is a velocity
gradient along the flow direction with the flow being a maximum at the throat and at a
minimum in the center of the body gap. This will result in extensional stress being placed
on the drop parallel to the fluid flow. The following equations are used to calculate De

where x is 0.333 cm (the distance between the maximum and minimum velocities):

Dc : A.
6d
(4.11)
where 0d = 3—
U

The relaxation time (A) is taken at the zero shear limit and is 0.46 seconds. At low flow, the
residence time of the drop in the throat is approximately 40% of the time for the drop to

move one step. For the cross over range, the actual velocity of the drop goes from 0.0138 to

133

0.025 cm/s. The corresponding range of Deborah numbers (De) is calculated as 0.019 to
0.035.

At this low De range, the drop would be expected to behave in a Newtonian fashion
(liquids at De <0.50 should behave in a viscous manner) however the experimental data
indicates otherwise. The lower mobility for the viscoelastic drops indicates that viscoelastic
stresses in the drop (as well as capillary resistance) oppose drop flow through the throat. The
transience of the drop shape as it moves through the throat is accentuated by extension.
Since smaller drops have shorter residence times their flow will be increased by the
extensional stresses to a greater extent than longer drops. This results in shorter drops having
higher mobilities in the array and an inversion point occurs.

The Ca“ dependence on drop length for the viscoelastic drops is shown in Figure
4.20. The onset for the viscoelastic drops is similar to the onset for the Newtonian drops:
Ca" decreases with increasing drop length and for longer drops, Ca‘ levels off. Ca“ falls
from 0.024 for the shortest drop to 0.014 for the longest drop. However, as will be seen in
Section 4.5.3, Ca* is larger for the viscoelastic drop than it is for the Newtonian drop. This
results from the combination of viscoelastic stresses and capillary forces resisting drop flow

through the throat.

4.5.2.b V, = 0.40 hexagonal array
The UN vs. Ca graph for these runs is shown in Figure 4.21. At high flow rates (Ca
> 0.45), the same trend is seen as in the square array: mobility increases for shorter drops.

Unlike the square array, the region where inversion occurs is wider and not as well defined

08*

134

 

0.025 -
o.02- I

0.015 _ I
0.01 _

0.005 Q

 

 

 

rrTIIITTIT‘TVTIIITITI rTlIIIrlrrrrerTrlert

. . . l . .
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Drop length (cm)

Figure 4.20: Ca* vs. drop length for viscoelastic
drops in a square array (Vf= 0.40)

135

 

0.2
o_13_ I 4.660m
3.5cm
0.16- +
A 2.47cm
0-14i x 1.89cm g
‘ D
0.12: . 1.32cm * A
Z 01— El 0.73cm a A +
D ' I I
008— ‘ i
0.06- ‘ E]
.l .= .
004— 9 E5
. ha a
0.02; J x8
I x D
o‘ , ,°, , w fifl

 

 

 

 

 

 

 

Figure 4.21: Relative drop velocity vs. Ca for a
viscoelastic drop in a hexagonal array (Vf=0.4)

136

(Figure 4.22 displays this more clearly) with the inversion ranging from Ca=0.021 to Ca =
0.041. The De value range was slightly higher for the hexagonal array than the square array
(0.020 to 0.052 and 0.019 to 0.035 respectively) but it was still very low. At Ca below the
inversion (approaching Ca*) the trend for drop length was not as clear although the shortest
drop does have the lowest mobility. Above the inversion, mobility is not as dependant on
drop length with the UN values being closer together. Since r/rb is smaller for this array,
the velocity gradient along x is smaller and there is less extensional stresses occurring. This
will have a smaller effect on the transient shape of the drop and while the inversion will
occur, it will not be as acute as it was for the square array. The mobility also levels off and
this is similar to the leveling off seen with the Newtonian drop (although not as acute).
Factors such as the drop wrapping around the rod and breaking will serve to impede drop
movement as in the Newtonian case. An additional factor involves the thread formed when
a viscoelastic drop wraps around a rod and forms droplets (Chap. 3). This thread may exert
an extensional force on the two resulting droplets impeding their movement through the
array until the thread breaks apart.

Figure 4.23 is the graph of Ca“ vs drop length for the viscoelastic drop in the
hexagonal array. As with all of the previous studies, Ca" is highest for the shortest drop
(Ca* = 0.012) and then falls off to a minimum for the longest drop (Ca* = 0.0060). Ca“
drops quickly from the shortest drop and there was little difference between Ca“ for the
second shortest drop (Ca* = 0.0067) and the longest dr0p. Since De approaches 0 at Ca“,
viscous behavior will dominate and factors such as surface area/volume ratio will have the

greatest influence on the mobilization of the drop.

137

 

O)

 

 

 

0 08 _ /
I /
o 06 _ /
. /
0.04 a
0.02 _
1
0 I I f r l 1 r I w r r r
0 0 01 0.02 0 03 0.04 0.05 0 06 0.07
Ca
1 - 0.73 cm 4 - 2.47 cm
2-1.32cm 5-3.50 cm
3 -1.89 cm 6 - 4.66 cm

Figure 4.22: Relative drop velocity vs. Ca for a viscoelastic
drop in a hexagonal array (Vf=0.4) [Redrawn]

Cat

138

0.025

 

0.02.:
0.015 .I
0.01 L

0.005 j

 

 

[4,III- WT.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Drop length (cm)

Figure 4.23: Ca* vs drop length for viscoelastic
drops in a hexagonal array (Vf=0.40)

 

5

139

4.5.2.c Comparison of viscoelastic drops in the arrays

The relative drop velocity for viscoelastic drops of a similar length is compared for
the square and hexagonal arrays in Figure 4.24. This graph shows that the drops have higher
mobilities in the hexagonal array: at Ca=0.42, UN =0] 1 for the hexagonal array and
UN =0.068 for the square array. The difference between the mobilities was not as significant
as it was for the Newtonian case. This smaller difference in mobility between the arrays can
be attributed to the drops wrapping around the rods to a great extent and to the formation of
threads that link the resulting droplets in the hexagonal array.

Figure 4.25 compares the Ca“ values for viscoelastic drops in the square and
hexagonal arrays. With increasing drop length, Ca“ declines from 0.024 to 0.014 for the
square array and from 0.012 to 0.0060 in the hexagonal array. In both cases, Ca" values
level off at higher drop lengths. Although Ca“ is higher, these trends are similar to those
seen for the Newtonian drops. Since De=0 at Ca“, the viscoelastic drops are expected to act
in a Newtonian fashion. The difference in Ca“ between the two arrays can be accounted for

by the extensional flow and lower rb/r, ratio present in the hexagonal array.

4.5.3 Comparison of Newtonian and viscoelastic drops

4.5.3.a Square array

The motion of a viscoelastic drop is significantly different from that of a Newtonian

drop with comparable Ca. At low Ca (< 0.05), drops would form leading and trailing feet

140

 

 

 

 

 

 

 

0.35
- I Square,Vf = 0.40,2.03cm
0.3;
. . Hexagonal,Vf = 0.40,1.89cm
0.25 .i
1
0.2;
0.15 .1
i I
0.1 _
0.05 d. .
o i , I - I .
0 0 02 O 04 0.06 0.08 0.1

Figure 4.24: Relative drop velocity vs Ca for
viscoelastic drops in the Vf= 0.40 arrays

 

141

 

 

 

 

 

 

 

0.025
4 I
I square,Vf=0.4
- h |,V=.
0.02_ I O exagona f 04
I
" I
0.015_
I I .
« O
1
0.013
O O
- ‘ e o
0.005_
0‘. ..,,.,-,,,,.,....,..,,,..,.,,..-,-,”we.
l
O 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Drop length (cm)

Figure 4.25: Ca* vs drop length for viscoelastic

drops in the V = 0.40 arrays

f

5

142

in both cases but the feet formed by viscoelastic drops were much wider transversely and not
as distinct. For Ca > 0.05, the feet tended to disappear altogether for the viscoelastic drop
as it formed an elliptic like shape that displayed continuous motion through the array. This
behavior became more apparent with increasing drop length. For shorter drops, the motion
along the array tended to be more uneven with pauses occurring as the leading edge
penetrated the throat. The short viscoelastic drops however displayed a much more
continuous motion than the short Newtonian drops with briefer pauses and a more constant
shape.

Figure 4.26 compares the mobility of the Newtonian and viscoelastic drops in the Vf
= 0.40 square array for drops with lengths of 0.60 cm and 3.82 cm. The most significant
difference between them is the effect of drop length on the mobility. For the Newtonian
drops, UN increases with increasing drop length and this trend remains constant for all Ca.

For the viscoelastic drops, this trend becomes inverted at 0.025 < Ca < 0.32 and above this
range, UN increases with decreasing drop length (see Sec. 4.4.2.a). For Newtonian drops,
the increased mobility results from smaller Sv which results in larger viscous forces pushing
the drop in comparison to smaller capillary forces resisting flow. For viscoelastic drops, the
inversion results from extensional forces which have a greater effect on the transience of
shorter drops and results in greater mobility. For this range of Ca, 0.019 < De < 0.035
indicating that the viscoelastic nature of the drop will have a significant effect on its mobility
even as Ca approaches the onset of mobility.

For both drop lengths, the mobility is of an order of magnitude less for the

viscoelastic drops. At Ca = 0.0279, for the 3.82 cm drops, UN was 0.0236 and 0.513 and

143

 

 

 

0.6
4 I viscoelastic, 3.82 cm
: [3 Newtonian, 3.82 cm
0'5", A viscoelastic, 0.60 cm
A Newtonian, 0.60 cm
0.4;
2 0.3;
D .
0.2; A ‘
J
0.1 ._
A
O -

 

 

 

 

 

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Ca

Figure 4.26: Relative drop velocity vs. Ca for Newtonian
and viscoelastic drops in a square array (Vf= 0.40)

144

for the 0.60 cm drops, UN was 0.0132 and 0.175 for the viscoelastic and Newtonian drops
respectively. This is due in part to the greater viscosity ratio (k) of the viscoelastic drop
(13.0 vs 0.55 for the Newtonian). These results are in agreement with previous studies ([29],
[39], [41]) that have shown that an increase in k will result in a reduction in film thickness
and thus will reducing the drop mobility. In those studies, however, increased k resulted in
only a small decrease in mobility, much smaller than what was seen here. However, those
studies only examined Newtonian drops.

For a better comparison of the effect of k on mobility, an abbreviated test with
Newtonian drops of 1.55 cm length and composed of pure glycerol (k = 8.9) was conducted.
The mobility for drops composed of the three fluids is shown in Figure 4.27. When Ca is
low the mobility of the glycerol drop was higher than the mixture (Newtonian) drop
indicating that the glycerol drop has a lower Ca". Above Ca = 0.016, the mobility of the
glycerol drop was less than the mixture drop but much greater than the viscoelastic drop. At
Ca=0.023, the mobilities were 0.140, 0.112, and 0.0227 for the mixture, glycerol, and
viscoelastic drops respectively. For the Newtonian drops, a 16 fold increase in k results in
a mobility reduction of ~20% which is comparable with the previous studies cited. The large
reduction in mobility between the Newtonian and the viscoelastic drops (there is a 24 fold
increase in k for the viscoelastic drop) is the result not only of increased k but also of
viscoelastic stresses resisting drop flow through the throat. The various phenomena
examined in the previous paragraphs will cause the viscoelastic drop to take on a more
continuous motion through the array as was described earlier.

Figure 4.28 shows Ca“ for the three drops of various lengths. For drops with a 1.55

145

 

    

 

I Newtonian, k = 0.55
O Newtonian, k = 8.9

A viscoelastic,k = 13

 

 

 

 

 

 

 

....... ,
o 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Ca

Figure 4.27: Relative drop velocity vs. Ca for three drops
of length 1.55 cm and different k in a square array

(Vf= 0.40)

Cat

146

 

 

 

 

 

 

 

0.025
. O
I
0.02: e
9 .
1 I .
0.015
T . e
l I
I
0.01.: 8 I I
I
: I Newtonian,k=0.55
0.005 _
_ O viscoelastic,k=13
8 Newtonian,k=8.9
O l 'TTT 7' Tl""l“*rl""l"“

....,..ee
0 0.5 1 1.5

2 3,5... 3 3.5 4 4.5

Drop length (cm)

Figure 4.28: Ca* vs drop length for Newtonian and
viscoelastic drops of different k in a square array
(Vf = 0.40)

 

147

cm length, Ca“ is 0.0136, 0.0106, and 0.0176 for the mixture, glycerol and viscoelastic drops
respectively. The reduction in Ca" with increased k for the Newtonian drops is in good
agreement with the result seen by Legait [45] and N6 [40]. The viscoelastic drop, despite
having the higher k in comparison to the Newtonian (mixture) drop (13 vs 0.55), has higher
Ca“ values. For the shortest dr0p Ca“ was 0.022 and 0.024 for the Newtonian and visco-
elastic drops respectively. F or the longest drop, Ca" was 0.010 and 0.014 respectively.
This is in contrast to Legait’s and Ng’s results and once again is the result of viscoelastic

stresses acting on the drop.

4.5.3.11 Hexagonal array

Because of the reduced rb/r, ratio (and hence smaller curvature difference), motion is
more continuous for both types of drops in the hexagonal array. If drop breakup did not
occur, then the main difference in flow behavior was that the viscoelastic drops tended to
wrap around the rod at its stagnation point more readily than the Newtonian drops. The
major discrepancy occurred when the drop broke in two. The drop breaks by wrapping
around the rod to form two branches, the branches continue to extend along the flow
direction until the connection between them breaks forming two droplets. For the Newtonian
drop, this break occurs cleanly forming two separate droplets. In the case of a viscoelastic
drop, as the branches extend forward, the connection gets thinner and thinner until it forms
a thin thread that is wrapped around the rod and connects the resulting droplets on their
trailing edges. This thread will continue to be extended (often across three rod diameters)

until it becomes unstable and shatters. The length to which it is extended increases with

148

increasing Ca.

A comparison of the Newtonian and viscoelastic drop mobilities in the hexagonal
array is displayed in Figure 4.29 for drop lengths of 0.73 cm and 4.66 cm. This graph is very
similar to Figure 4.26 with the mobility of the Newtonian drops being much greater than that
for the viscoelastic drops. At Ca = 0.00717 for the 4.66 cm drops, UN was 0.0203 for the
viscoelastic drop and 0.396 for the Newtonian drop. At Ca = 0.0185, for the 0.73 cm drops,
UN was 0.0331 for the viscoelastic drop and 0.432 for the Newtonian drop. As with the
square array, this reduction is only partially explained by an increase in k. Viscoelastic
stresses once again plays a major role in retarding drop movement. The extensional flow in
the hexagonal array at the stagnation point causes the viscoelastic drops to wrap around the
rods more frequently (because of its elastic nature) than Newtonian drops and this will stall
the drop reducing its mobility. The connecting thread formed when two droplets are formed
can also impede movement of the droplets by exerting extensional forces on the drops. The
inversion for the viscoelastic drops occur because of the reduced transience of the shorter
drops caused by extensional forces in the fluid flow direction.

Figure 4.30 shows the Ca“ vs drop length data for both types of drops in the
hexagonal array. As with the square array, the viscoelastic drop has higher Ca“ values
despite having a higher k. Ca" falls from 0.012 to 0.0060 with increasing drop lengths for
the viscoelastic drop compared to it falling from 0.0089 to 0.0022 for the Newtonian drop.

This is the result of viscoelastic resistance in the throat.

149

 

 

 

 

 

 

 

 

0.6
l
I viscoelastic, 4.66 cm
0.5 _ a Newtonian, 4.66 cm
j A viscoelastic, 0.73cm
0,4; A Newtonian, 0.73cm
A
2 0.3 ..
D ..
0.2;
0.1 L A
I
O ‘ ' T 1 fl ’ ‘ l ' ‘ ' l ' ‘ ‘
0 0.02 0.04 0.06 0.08
Ca

Figure 4.29: Relative drop velocity vs. Ca for
Newtonian and viscoelastic drops in a hexagonal
array (Vf = 0.40)

Cat

150

 

 

 

 

 

0.025
I
_ I Newtonian,k=0.55
0'02? I viscoelastic,k=13
I
0.015.
I
0.01..
‘ I
_ I I . . .
0.005-
- II
I
I I I I
O I 'l jTIH"Irrjfilfl"lTHTI"le""I'fi‘

 

 

 

0 0.511 1.5 2 2.5 3 3.5 4 4.5
Drop length (cm)

Figure 4.30: Ca* vs drop length for Newtonian and
viscoelastic drops in a hexagonal array (Vf= 0.40)

151

4.6 Drop mobilization model

A model was developed to predict Ca“ for an elliptical Newtonian drop in a
cylindrical array based on the geometric properties of the array and the drop. The general
approach was to extend Ng’s derivation method [40] for an elliptical drop (he examined
spherical drops). The complete derivation is outlined in Appendix B and the resulting final

equation (model) was:
4r +1 4r +1 I:
(13* = [1!) = _'__‘_ — _‘.’_L [J] (4.12)
O crit 2r,2 +r,lt 21‘: +rblb lb

where lb is the transverse length of the drop in the body cavity and It is the transverse length
of the leading foot in the throat.

Figure 4.31 compares the experimental results (black symbols) to the model results
(white symbols). For both square arrays, the model works well for the short drops and over
predicts Ca“ by under 10%. However, with increasing length, the model quickly deviates
from the experimental results and the model value is just 40% of the experimental value for
the longest drop. This model works best for the hexagonal case, over predicting Ca“ by 15%
for shorter drops and matching almost exactly for the longer drops. Qualitatively, this model
does a good job of predicting trends such as the decrease in Ca" with increasing permeability
or the leveling off of Ca“ values with increased drop length. The model is very good for the
hexagonal array. This model is inadequate for square arrays, particularly for long drops.
One possible reason is that this model does not take into account transverse flow of the drop

fluid as it moves from one body gap to the next.

08*

152

 

 

 

 

 

 

0.025
. I Square, Vf = 0.40
CI
I I Square, Vf = 0.50
0.02 _ A Hexagonal, Vf = 0.40
. U E] Square, Vf = 0.40
« S uare,Vf=O.50
0.015_ I 0 q
~ 1% I A Hexagonal, Vf = 0.40
8 u
0.01- e . D l I
f ‘ 6 9 [1:11 o o I
- 6 C]
0.005- 6 En
- A 2% Q U D U D
- m 6 6 6 A
o.,,.,.,.,,.,,.,..,,,.
0 1 2 3 4
Drop length (cm)
Black-experimental

White-model results

Figure 4.31: Model results vs experimental
results for Ca*

 

153

4.7 Conclusions
From this study of drop mobilization and mobility, the following conclusions can be
rendered:
1. For both Newtonian and viscoelastic drops, mobility (UN) increases with
increasing flow (Ca) as the result of increased viscous forces.
2. UN increases going from a square to hexagonal array as the result of decreased
capillary forces. UN levels off for long drops in the hexagonal array due to breakup.
3. For both Newtonian and viscoelastic drops, Ca“ decreases with increasing drop
length as the result of decreased 8,.
4. Increased Vf causes decreased permeability, increased viscous forces and finally
results in increased mobility.
5. For Newtonian drops, UN increases with increasing length for all Ca. An
inversion in this trend occurs for viscoelastic drops just above Ca" resulting in UN
increasing for decreasing length. These trends occur in both array geometries.
6. The viscoelastic drops have a smoother shape and a more continuous motion than
the Newtonian drops. This the result of the plug flow through the throat and the
extensional forces which accentuate the transient shape of the viscoelastic drops.
7. Drop breakup is detrimental for the mobilization of Newtonian drops but
beneficial for the mobilization of viscoelastic drops provided that Ca > Ca“ for the
resulting droplets.

8. UN decreases only slightly with increasing k for Newtonian drops. Viscoelastic

154

drops have much lower mobilities than Newtonian drops.

9. In both arrays, Ca“ is higher for a viscoelastic drop even though it has a higher k.

10. Although the model derived here was not completely satisfactory, it did predict

the trends seen in the experimental results. This combined with the agreement

between our results and those of Legait indicate that the balance of forces on the drop
and not the film thickness is most important for the mobilization of cylindrical drops
at low Ca.

The reduction in mobility of the viscoelastic drop is the result of two factors:
increased k which leads to a decrease in film thickness and viscoelastic stresses which work
with capillary forces to impede drop movement. The inversion seen for viscoelastic drops
results from the transience of the shorter drops being accentuated to a greater extent by
extensional forces along the flow direction. These extra phenomena will give the

viscoelastic drop a more continuous movement through the array.

5. CONCLUSIONS

This study examined the mobilization, motion, and breakup of a drop moving through
an array of transversely aligned cylinders. Three arrays were used: V, = 0.40 square and
hexagonal arrays, and a V, = 0.50 square array. The suspending fluid was a chloroparaffin
oil that was refractive index matched with the PMMA cylinders. Two types of drops were
studied: glycerol/water Newtonian drops and glycerol/water/PAC viscoelastic drops. Both
types were neutrally buoyant and immiscible in the suspending fluid. Most runs were done
at low capillary numbers (0.005 < Ca < 0.1).

In both arrays, the drop moved in an amoeba like fashion, forming a leading foot that
penetrates the next body gap in the array and a trailing foot as it leaves the previous body
gap. The drop fills the body gap transversely before it penetrates through the next throat.
Drops in the hexagonal array will tend to wrap around a rod as the result of a stagnation point
formed at its upstream end. Increased Ca will lead to greater elongation of the drop in the
flow direction. Viscoelastic drops move in a similar fashion except that the leading and
trailing feet are not as well defined.

The critical capillary number (Ca‘) is defined in this study as capillary number at
which the drop first becomes mobilized. For Newtonian drops, Ca“ decreases with

increasing drop length and levels off for the longest drops studied. This results from a

155

156

decrease in the surface to volume ration (S) of the drop which leads to increased viscous

forces relative to capillary forces being applied on the drop. Increased V, increases the
pressure drop along the length of the array and hence the viscous forces on the drop which
reduces Ca". Ca“ is lower for the hexagonal array as a result of extensional forces at the
stagnation points forcing the drop forward and a smaller rb/r, ratio which reduces the capillary
forces resisting drop flow.

These same trends were seen for viscoelastic drops. For both square and hexagonal
arrays, Ca“ was higher for the viscoelastic drops. This may have been the result of increased
k but a test with a Newtonian drop of pure glycerol, which also had a higher k than the
Newtonian mixture, found that Ca‘ decreased if just the viscosity ratio was increased. The
higher Ca* for viscoelastic drops was probably the result of increased stiffness of the drop
which results in viscoelastic stresses opposing drop flow. This occurs even though De is
nearly zero at Ca“ where Newtonian behavior is expected.

The mobility of a drop as it moves through the array depends on several
characteristics of the drop, array, and flow. The mobility of Newtonian drops increases with
increasing interstitial velocity, drop length, V, (for a given packing), and is higher for square
as opposed to hexagonal packings. Mobility in the hexagonal array tapers off at high Ca as
a result of drop breakage. In both arrays, the relationship of mobility to drop length was
different for viscoelastic drops. At low Ca near Ca“, mobility increases with increasing
length as before. However, with increasing Ca, there is a point at which this trend becomes
inverted and mobility increases with decreasing length. This inversion occurs at low

Deborah numbers (0.02 < De < 0.05) where one would expect the drop to behave in an

157

essentially Newtonian manner. The velocity gradient of the dr0p fluid in the flow direction
resulted in extension forces pulling the drop forward. The transience of the drop in the throat
is then accentuated. Since smaller drops have shorter residence times they are subjected to
higher stretching rates than longer drops. The mobility of viscoelastic drops are much lower
than for Newtonian drops and this results from both a higher drop/suspending fluid viscosity
ratio (k) and viscoelastic stresses impeding the drop flow.

Drop breakup is the result of different mechanisms for different array geometries.
In the square array, breakup results from film growth in the throat that pinches off the trailing
foot of the drop. In hexagonal arrays, breakage occurs as the result of extensional forces at
the stagnation point that extends branches of the drop around the rod and eventually splits
the drop in two. Viscoelastic drops differ from Newtonian drops in that a thread is formed
that wraps around the rod and connects the resulting droplets. Breakup often results in the
formation of very small droplets that become demobilized in the array.

The capillary number at which drop breakup first occurs is defined as the critical
capillary number for breakage (Cab). For the square array, there is a lower and upper Cab
between which breakage occurs. Breakage occurs when the snap off time of the fluid film
is less than the residence time of the drop in the throat. Below the lower limit, the film is
relatively stable and the snap off time is too long. Above the upper limit, the residence time
becomes very short, and the drop moves through the throat before the film can snap off the
drop’s trailing foot. Within the limits, snap off time is less than the residence time and
breakage occurs. N0 breakage of viscoelastic drops in a square array was observed in this

study. In the hexagonal array, there is only one Cab above which breakage occurs. Ca, in the

158

hexagonal array is only slightly lower for the viscoelastic drops. In both cases, with
increasing length Cab decreases and the number of times a drop breaks increases indicating
that longer drops are more unstable.

Ng’s model was further developed for the Newtonian cylindrical drops studied here.
The model was adequate in qualitatively predicting the effect of drop length and array
geometry on Ca“. Quantitatively, the model over predicted the effect of drop length on Ca“
for the square arrays. The model was in good agreement with the experimental results for
short drops but with increasing length it quickly deviated and significantly under predicted
Ca“ for the longer drops. The model was in good agreement with the experimental data for
the hexagonal array for all drop lengths. Overall, this model is not sufficiently accurate for
predicting Ca".

Many of the trends for Ca" listed above match those observed by Legait and Ng and
would lend credence to the force balance approach. However, the effect of k (which is not
factored into Ng’s model) and the mechanism for drop breakup in a square array indicate that
even at low Ca, the film thickness does play a role in the mobilization and mobility of a drop.

The shape of the drop as it moves through an array is determined by a combination
of initial drop and array geometries, the drop fluid, the force balances on the drop, and the
film thickness. This drop shape will determine the mobility, Ca“, and the breakup behavior

of the drop as it moves through the cylindrical array.

6. RECOMMENDATIONS

To better understand the flow of a drop through an array of cylinders, several further
experiments should be conducted. Flow studies of a drop moving through a wavy walled
channel (analogous to Olbricht and Leal’s wavy walled tube) would give a more detailed
account of the mechanism of drop motion. Since the refractive index and buoyancy are not
as important with this geometry, greater flexibility can be used in choosing the suspending
and drop fluids. Different fluids (both viscoelastic and Newtonian) with different properties
can be examined. This would allow for a more extensive examination of factors such as
shear viscosity ratio, and drop viscoelasticity or extensional viscosity. With a transparent
cross section, detailed observations of the fluid film can be recorded to determine its effect
on drop movement. The channel can also be used to study actual air bubbles since they
won’t rise out of the array. A bypass would be need to mimic the flow through an actual
array more effectively.

For viscoelastic drops, a more detailed examination of their properties is needed to
determine the precise mechanism that causes the anomalies in drop mobility described
earlier. A new analytical model could then be developed to relate the many diverse factors

in drop mobility.

159

APPENDICES

APPENDIX A:

Determination of interfacial tension

The interfacial tension between two liquids is related to the surface tensions of each

liquid as follows:

712 .—. Y1 + Y2 - WaLL 06"”

where 7,, is the interfacial tension, 7, and y, are the surface tensions of the liquids and W,LL
is the work of adhesion at the liquid-liquid interface. W,ILL is found using one of the

following equations [46,47]:

war. - 2[(Y§Yi)m + (Y§Y§)"2l (A.2)
4y??? + 4V???
Y‘I'w‘g' 7M;

WILL .—.

 

(A.3)

where y,“ and 72" are the dispersion components and y,” and 72" are the polar components of
the surface tensions. Equation A2 is the geometric-mean equation and is used for non-
polymer interfaces. Equation A3 is the harmonic-mean equation is used for interfaces where
at least one of the liquids is a polymer solution.

W,LL cannot be found directly so, 7 and y and their components must first be

160

161

determined by performing the Sessilc drop method (Figure A.1). In this procedure, a drop
of fluid of known volume is placed on a horizontal solid surface (Fig. A.1.a). The drop is
then observed using a magnifying eyepiece with two cross-hairs (Fig. A. l .b). The horizontal
cross-hair is aligned with the solid surface and the vertical cross-hair is aligned with the drop
surface at the solid surface and the angle between these two cross-hairs is the contact angle
(0) (Fig. A. l .c). For each fluid/surface system looked at, this process was repeated 10 times

and the contact angles were averaged.
In the Sessile drop method, the work of adhesion for the solid-liquid interface (WAL)

is first found by using modified versions of equations A2 and A3 and the contact angle

measurements I

210271)” + (Yfo)"’] (A.4)
47373 + 47M
Yin/1' Yin/f

W.SL - 71(1 +cose)

€

.31. — Y,(1 +cose)

 

 

(A.5)

where y, and y, are the surface tensions of the liquid and the solid respectively. In order to
find WasL, one must first find the polar and dispersion components of y for the liquid and the
solid. This is done by measuring 0 for the liquid on two different solid surfaces and using
either equations A.4 or A5 or a combination of both to determine 7, (which is the sum of the
polar and dispersion components). Eq. A.4 is the geometric-mean equation and is used for
inorganic or polymeric liquids on high energy materials (glass, metals). Eq. A5 is the
harmonic mean equation and is used for a polymer in contact with another polymer or a low

energy liquid (water, organics).

162

Drop

Solid surface

\/\/

Figure A.1.a: Drop on horizontal surface

Cross-hairs
1
I
I
I

Figure A.1.b: Cross-hairs (seen through optical piece)

I
I
I

I
m
I

Figure A.1.c: Contact angle measurement with the cross-hairs

 

Figure A.l: Sessilc drop method for determination 0
contact angle

163

In this study, PMMA (used for the array rods) and glass were used as the solids.
Since their dispersion and polar components were not known, the components had to be
determined by contact angle measurements with known liquids which in this case were
glycerol and water. For glass, two geometric-mean equations (one for each liquid) were

combined and rearranged so that the components were found analytically:

 

 

Yd: YZ/Yax'Q (A6)
. WHEN") - I?
w w 3
2
_ d d
v? = Q Y'Y‘ (A?)
Y:
X = c0892+ LYw Q : cos92+ 17: (A.8)

where 78 and W were the surface tension of glycerol and glass respectively. For PMMA, two

harmonic-mean equations were needed:

 

 

 

 

d d
f = by“ + YzY: - Y M (A 9)
1 d + d p p 3 4 .
Y, Y. Y, + Y.
6 d p 9
f2 = deYs + Y'Y' - M (A.10)

n+7? film? W 4

but these functions couldn’t be combined and solved analytically like they could for the

164

glass. Instead these equations were solved numerically using the IMSL program. The results

for both solid surfaces were as follows:

Table A.1: The surface tension for the solid surfaces

 

 

 

 

 

 

material 6 (water) 0 (glycerol) 7“ (mN/m) y“ (mN/m) y (mN/m)
water - - 21.8 51.0 72.8

glycerol - - 34.0 30.0 64.0
glass 34.6° 39.l° 6.92 57.8 64.7

PMMA 73 .4° 642° 17.1 18.9 36.0

 

 

 

 

 

 

 

These results were then used to determine the components for the three unknown liquids
examined here: the chloroparaffin oil, the glycerol/water Newtonian drop liquid, and the
glycerol/water/polyacrylamide viscoelastic drop liquid. For each liquid, one harmonic
equation (PMMA) and one geometric equation (glass) would be solved numerically using

IMSL:

y, - yf (A.11)

 

(cosfi +1) Y; + Y: 721* Y1)

r 2 \
f2 = —— Ma? + W37?) - vi' - yr (A.10)

(c036 +1)

I 4 ‘ Yin/1' + YZ.YI]_ ..

 

 

where 78, and yp. were the surface tensions for glass and PMMA respectively and 7. was for
the liquid. For the two drop solutions, this gave the components but for the oil they did not
converge despite trying several different initial guesses. Instead two geometric equations

were used to get an analytical solution. The results were:

165

Table A.2: The surface tension for the liquid materials

 

 

 

 

fluid 0 (glass) 0 (PMMA) y" (mN/m) y" (mN/m) y (mN/m)
oil 32.8° 10.8° 29.8 3.4 33.2
Newtonian 39.4° 67.9° 32.1 36.5 68.6
viscoelastic 45.8° 59.8° 39.0 14.5 53.5

 

 

 

 

 

 

 

 

The Wll values for the oil/Newtonian drop and oil/glycerol interfaces were found using
geometric mean equation (eq. A2). The harmonic mean equation (eq. A3) was used for the
oil/viscoelastic drop interface. These values were then plugged into equation A.1 to get the

interfacial tension. The interfacial tension results were:

Table A.3: The interfacial tensions for the liquid-liquid systems

 

 

 

 

 

 

 

 

 

Suspending fluid/drop y. (mN/m) 72 (mN/m) W, (mN/m) 7,2 (mN/m)
system
oil/glycerol 33.2 64 83.8 13.4
oil/Newtonian 33.2 68.6 84.1 17.7
oil/viscoelastic 33.2 53.5 78.6 8.1

 

These results were then used to for determining Ca in the flow experiments.

 

APPENDIX B:

Derivation of cylindrical drop model

The derivation of the relationship of Ca“ to the geometric properties of an array of
cylinders is analogous to that used by Ng [40]. The critical condition for mobilization of a
drop is that the hydrodynamic pressure difference (Darcy’s law) between two feet of a drop
is just greater than the net capillary pressure difference between the two feet. The

hydrodynamic pressure difference is hence written as:

Ap = |Vp|l cosa - nvlkc-lcosa (3.1)

vise

where l is the separation of the two feet, a is the angle they make with the flow, and ke is
the effective permeability.
A drop in this system would have a cylindrical shape with hemispherical end caps

(Figure 3.1). The surface area of this drop would then be:

s = 2an + 4an (13.2)
513— = 2m + 8m (3.3)
dr

The force balance is then defined as:

p[41tr2 + 21trl]dr = [8m- + 21:1]‘ydr (3.4)

which can be rearranged to give:

166

 

 

hemispherical end caps

Figure B.l: Cylindrical drop with round ends

E Rod

 

 

 

 

 

 

I

Figure B.2: Dimensions ofa drop moving through a rod constriction
(throat)

168

 

Y (3.5)

_ 4r+l
p— 2
2r +rl

 

The net capillary pressure between the two feet is defined as:

4r,I +14 4": + 1n

Ap = - (B.6)

21': + rdld 2r“2 + ’qu

 

 

where rd and [d is the radius and length downstream of the flow and ru and In is the radius and
length upstream of the flow. The maximum net capillary difference occurs at the maximum

drop constriction between the two cylinders:

 

D 4r + 1 4r + l
_ r r b b
2rt + r,lt 2rb + rblb

 

where rt and l, are the radius and length of the leading foot of the drop in the constriction
between two cylinders (or throat) and r, and 1., are the radius and length of the portion of the
drop main space between rods (or body) (Figure 3.2) . Equating eqs. 3.] with B.7 and

rearranging gives:

 

 

t t _ b b ] kc (3.8)

For a square array, cos a=1. This equation can then be used to model the mobilization of

a drop through an array of cylinders.

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