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Dynamic interfacial properties and their influence on bubble motion

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Dynamic interfacial properties and their influence on bubble motion
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Maruvada, Krishna S
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xi, 82 leaves : ill. ; 29 cm.

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Subjects / Keywords:
Bubbles ( jstor )
Film thickness ( jstor )
Fluids ( jstor )
Interfacial tension ( jstor )
Kinetics ( jstor )
Silicones ( jstor )
Surfactants ( jstor )
Thin films ( jstor )
Velocity ( jstor )
Viscosity ( jstor )
Bubbles ( lcsh )
Chemical Engineering thesis, Ph.D ( lcsh )
Dissertations, Academic -- Chemical Engineering -- UF ( lcsh )
Fluid dynamics ( lcsh )
Surface active agents ( lcsh )
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bibliography ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis (Ph.D.)--University of Florida, 1997.
Bibliography:
Includes bibliographical references (leaves 80-81).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Krishna S. Maruvada.

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DYNAMIC INTERFACIAL PROPERTIES AND THEIR INFLUENCE
ON BUBBLE MOTION












By

KRISHNA S. MARUVADA


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1997














ACKNOWLEDGMENTS


During my stay at the University of Florida, I had the privilege to interact with

many talented people who have influenced me in several ways. First among those is my

research advisor Dr. Chang-Won Park whose enthusiasm for research and pursuit of

excellence are admirable. Much of my learning is due to several insightful research

discussions I had with him during my stay here. I wholeheartedly acknowledge the

constant encouragement, guidance and support I have received from him during the

course of this research work. I thank Dr. Ranga Narayanan, Dr. Lewis Johns, Dr. Dinesh

Shah, Dr. Richard Dickinson and Dr. Wei Shyy for serving on my committee and

providing helpful suggestions. I also thank all the members of our research group, Mr.

Charlie Jacobson, Mr. Yongcheng Li, Ms. Keisha Wilson, Mr. John Bradshaw, for their

cooperation and support.

A task of this magnitude can not be completed without the support of my beloved

family members; my parents, my wife, my brother, my sister and brother-in-law who

have helped me in every way they can, to help convert my dream into a reality. With

great happiness, I acknowledge their generous support and encouragement.

Finally I would like to acknowledge financial support by the donors of the

Petroleum Research fund and NASA, Microgravity Science and Applications Division.














TABLE OF CONTENTS


page


ACKN OW LED GM EN TS .............................................................................................. ii

LIST OF FIGU RES ....................................................................................................... v

LIST OF SYM BOLS ........................................................................................................ vii

ABSTRA CT ......................................................................................................................... x

CHAPTERS

1 INTRODU CTION ......................................................................................................... 1

Background .................................................................................................................... 1
M otivation ...................................................................................................................... 4
Dissertation Outline ................................................................................................. 9

2 EX PERIM EN TS ...................................................................................................... 12

The Influence of Surfactants on Bubble M otion ................................................... 12
Experim ents ................................................................................................................. 18
Results and Discussion .......................................................................................... 23
Sum m ary and Conclusions ..................................................................................... 37

3 ESTIMATION OF BUBBLE VELOCITY ............................................................ 38

Introduction .................................................................................................................. 38
Theoretical Analysis .............................................................................................. 38
Com parison with experim ental results .................................................................... 46
Sum m ary ...................................................................................................................... 50

4 DISPLACEMENT OF A POWER-LAW FLUID BY INVISCID FLUID ............ 51
Background .................................................................................................................. 51
Basic Equations ...................................................................................................... 53
Scalings and Regions ............................................................................................... 54
Sm all param eter expansions ................................................................................... 60
Capillary Statics Region : ................................................................................. 60








Transition Region : ............................................................................................ 62
M atching Conditions ........................................................................................ 65
Stability Analysis ................................................................................................... 69
Sum m ary ...................................................................................................................... 76

5 CON CLU SION S ...................................................................................................... 77

REFEREN CES .................................................................................................................. 80

BIOGRA PH ICAL SKETCH ....................................................................................... 82














LIST OF FIGURES
FIGURE PAGE


1.1 Schematic of a moving bubble in a Hele-Shaw Cell (a) View in the xy plane
(b) Bubble in the xz plane ........................................................................................ 2

1.2 Various Bubble shapes observed by Kopf-Sill and Homsy (a) Near-Circle;
(b) Flattened bubble; (c) Tanveer Bubble; (d) Ovoid; (e) Short-tail bubble; and
(f) Long-tail bubble ................................................................................................. 5

1.3 Here U/V vs Ca (Data taken from experiment series 1,2, and 3 of Kopf-Sill and
H om sy) ........................................................................................................................ 7

2.1 A water drop sitting on a Pyrex glass plate immersed in silicone oil .................. 22

2.2 Evolution of water drop shape with increasing capillary number.
(a)Near circular drop; (b) flattened drop; (c) severely stretched drop; (d) ovoid;
(e) transitional shape; and (f) short tail drop ........................................................ 24

2.3 U/V vs. Capillary number for air bubbles ............................................................ 26

2.4 U/V vs. Capillary number for small water drops ................................................. 31

2.5 U/V vs. Capillary number for large water drops ................................................ 33

2.6 U/V vs. Capillary number for Fluorocarbon oil drops ....................................... 35

3.1 Plan view and side view of an elliptic bubble and a control volume moving with
the bubble ................................................................................................................... 42

3.2 Numerical value of the definite integral, Equation 3.11 for various values of the
shape factor k ...................................................................................................... 45

3.3 Comparison of experimental results and the theoretical estimate for pressure
driven flow (SDS concentration : LI 5% ; A 10%; x 20% of CMC) .................... 48

4.1 Schematic of an inviscid fluid displacing a power-law fluid in a Hele-Shaw cell. ..52

4.2 Schematic showing the scales and coordinate systems for different regions.
(A: Transition region; B: Capillary-statics region) ............................................. 57








4.3 Comparison of interface profiles for Newtonian (n=l) and power-law (n = 0.5)
flu ids ....................................................................................................................... 64

4.4 Plot of the Co-efficient Co as a function of the power-law index ...................... 67

4.5 Log (film thickness) vs Log Ca for a power-law fluid.
(to = 100 cP; a = 30 dyns/cm; yo = I sec'; b = 0.06 cm) .................................. 68

4.6 Disturbance growth rate (8) as a function of wave number (k).
(o= 100 cP, y. =1 s", n = 0.6, b = 0.06 cm, a = 30 dyns/cm, U= 0.05 cm/s) ....... 73

4.7 Wave number for the maximum growth rate as a function of the power-law index.
(.o=100 cP, yo= 1 sec', U= 0.05 cm/s, a = 30 dyns/cm, b = 0.06 cm) .............. 74














LIST OF SYMBOLS


a Dimension of the bubble in the flow directions
A Planform area of the bubble
b Dimension of the bubble transverse to the flow direction
b Half-gap thickness as used in Chapter 4
c Constant that appears in Equation 3.5
CO, C1, C2 Constants that appear in Equation 4.27
Ca Capillary number
Ca Ca based on the normal component of the bubble velocity
D Domain encapsulating the bubble
D Rate of strain tensor
f(z) Equation of the interface in z = constant plane (in the plane of the cell)
FD Drag force acting on the bubble
h Gap thickness of the Hele-Shaw cell as used in Chapter 3.
H Interface profile for the transition region in the scaled form
h(x,z) Equation of the interface in y = constant plane (in the gap direction)
I Constant as given in Equation 3.11
k Shape factor for the elliptic bubble
k Exponent that appears in equations 4.9 4.14
K Consistency factor
L Width of the Hele-Shaw cell
1 Location of origin in the transition region
n Unit normal to the surface of the control volume
n Power-law index
n Unit normal to the interface
p Depth averaged pressure filed
p Pressure field in the power-law fluid









q
S
S

s
t
T
til, t2
U
U
U*
U, V, W
V
V
Vavg
X
Greek Letters


a





(D



0

P

U


Scale factor for the elliptic cylindrical coordinate system
Surface of the control volume
= }, 2(Square of the shear rate)

Shift factor
Thickness of the wetting film
Stress tensor
Unit tangent vectors to the interface
Velocity of the drop/bubble
Velocity of the moving interface near the tip
Unretarded bubble velocity in a buoyancy driven flow
Components of velocities in the x,y,z directions
Velocity field in the driving fluid away from the bubble (Chapter 3)
Average velocity of the driving fluid (Chapter 3)
Depth averaged velocity field in the driving fluid
Scaled variable for the x-dimension


Gauge function in the expansions
Retardation factor as given by Equation 3.16
Constant that appears in Equation 4.24
Growth rate of the disturbance
Small parameter (Ratio of natural length scales of Hele-Shaw cell)
Velocity potential
Shear viscosity of the power-law fluid
Viscosity of the driving fluid
Angle of inclination of Hele-Shaw cell
Density of the driving fluid
Equilibrium interfacial tension
Stress tensor
Complex potential
Location of the edge of the control volume
Directions in the elliptic cylindrical Coordinate system









0

Subscript
x,y,z
Superscript

0001etc


Location of the bubble surface
Stream function


Partial differentiations with respect to the variables.


"Over bar" indicates variable in the transition region
Leading order, Order Ca etc..










Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

DYNAMIC INTERFACIAL PROPERTIES AND THEIR INFLUENCE
ON BUBBLE MOTION

By

Krishna S. Maruvada

December 1997

Chairman: Dr. Chang-Won Park
Major Department: Chemical Engineering


The influence of surfactants on the motion of bubbles and drops in a Hele-Shaw

cell is studied by carrying out experiments in a variety of fluid-fluid systems. Saffman-

Taylor theory predicts the translational velocities and shapes of drops and bubbles, not

influenced by surfactants. In order to differentiate the cases with and without surfactants,

the motion of air bubbles and water drops driven by silicone oil was studied. A

controlled amount of surfactant is added to the water drops in order to study its influence

on the bubble motion. In the case of air bubbles in silicone oil, surfactants were likely to

have negligible influence, although they may be present as contaminants. As expected, air

bubbles were moving much faster and their shapes were also in accordance with the

available theories for the surfactant free system. On the other hand, surfactant containing

water drops moved with much smaller velocities and their shapes were also distorted.

These observations are apparently consistent with those of Kopf-Sill and Homsy. When a

surfactant containing drop is set in motion, the resulting spatial variation of surfactants at








the interface induces Marangoni stresses along the interface, and consequently the drop

surface may become rigid. If the surrounding fluid is the wetting phase, the net drag force

experienced by the drop in the wetting film region is significantly high resulting in

smaller drop velocities. Because the surface tension has a spatial variation, to keep the

pressure inside the drop constant, the curvature adjusts itself resulting in a variety of

bubble shapes. Assuming that the bubble surface is rigid due to surfactant influence, the

translational velocity of an elliptic bubble is estimated. The result indicates that the

bubble velocity can decrease by an order of magnitude compared to the prediction of

Taylor and Saffman. The retarded bubble velocity is apparently in reasonable agreement

with the experimental observations suggesting that the puzzling observations of Kopf-Sill

and Homsy are likely to be due to surface active contaminants.

Finally, two phase displacement of a shear thinning power-law fluid by an

inviscid fluid is analyzed. The flow domain is divided into three different regions and a

leading order solution to the flow field and the interface shape is obtained using the

technique of matched asymptotic expansions. In the case of Newtonian fluids, capillary

number uniquely determines the film thickness. On the other hand, in the case of shear

thinning power-law fluids, the relationship is unique to a given fluid and flow conditions.

Finally, the flow field away from the interface in the power-law fluid region is also

solved assuming that the gradients of components of velocities is negligible in the

directions normal to the gap direction. Unlike the Newtonian case, the depth-averaged

velocity field is found be non-Darcian. A stability analysis is also carried out to determine

the wave number with the maximum growth rate. The result indicates that the interface

will be less sinuous for the power-law fluid than the Newtonian fluid.













CHAPTER 1
INTRODUCTION

1.1 Background

Hele-Shaw cell has been used as a model system to study various physical

phenomena involving multiphase flows in porous media. Although the concept of Hele-

Shaw cell is more than a century old, there has been a lot of interest in recent years in the

motion of two phase flows in a Hele-Shaw geometry."5 The displacement of a more

viscous liquid using a less viscous liquid and the related fingering phenomena have been

extensively studied both experimentally and theoretically, using this model."5 When

the fluid-fluid interface is a closed contour, the inner fluid is referred to as a drop or

bubble depending upon whether the enclosed fluid is a liquid or gas.

A typical Hele-Shaw cell consists of a thin gap between two flat plates. The gap

thickness, which is of the order of a millimeter, is very small compared to the dimensions

of the plates. The gap is filled with a liquid into which a secondary fluid is injected to

form a drop or bubble and its motion can be studied. The injected drop or bubble is

squeezed between the two plates and assumes a circular planform when it is stagnant

because of the interfacial tension. If the surrounding fluid is the wetting phase, a thin film

exists between the bubble surface and the solid boundary of the Hele-Shaw cell. The

bubble can be set in motion either by applying a pressure gradient in a horizontal Hele-

Shaw cell or by imposing a buoyancy force by tilting the cell. If the characteristic length





















\L~ $


t1qu~d


s-


__ _h ;gap

Wetting film Stagnation Points

(b)


Figure 1.1 Schematic of a moving bubble in a Hele-Shaw cell : (a) View in the xy
plane, (b) Bubble in the xz plane









scale of the bubble in the transverse direction is larger than the gap thickness (a/h >>1,

Figure 1.1), the dynamics of the bubble motion can be described by Hele-Shaw

equations6 which are a two dimensional description of the gap-averaged velocity and

pressure fields. This approximation is valid over the entire domain expect for a thin

region (of the order of gap thickness) surrounding the rim of the bubble. The depth

averaged equations resemble the Darcy approximation of the governing equations for the

flow through porous media. Hence the Hele-Shaw cell can also be used to study flow

through porous media.

Taylor and Saffman' have solved the Hele-Shaw equations for the motion of

bubbles in a thin gap with no side walls assuming negligible surface tension. Their

solution predicts the bubble shape as a function of the area and bubble velocity.

Furthermore they have concluded that for a freely suspended bubble, the bubble velocity

(U) is always greater than the surrounding fluid average velocity (V) and the steady

shapes are ellipses. Also if U < 2V these elliptic bubbles travel with their minor axis

parallel to the flow direction and if U > 2V they travel with the major axis parallel to the

flow direction. If U = 2V the bubble shapes are circles. Tanveer3 had incorporated the

effect of finite surface tension to resolve the degeneracy of the Taylor-Saffman solution

and obtained three solution branches. A stability analysis4 of these three solution branches

indicate that only one of them (McLean-Saffinan branch) is stable. The inclusion of

interfacial tension in the theoretical analysis does not alter Taylor-Saffman's conclusion

about the relative velocities of the bubbles and the surrounding fluid.








Bubble motion was experimentally studied be several researchers using a variety

of fluid combinations.5-8 Maxworthy' had experimentally studied the motion of air

bubbles in an oil filled Hele-Shaw cell under the influence of gravity. At smaller values

of the inclination angles, stable circular bubbles or elliptic bubbles elongated in the flow

direction are observed. However at higher angles of inclination, bubbles became unstable

with severe shape distortions. When the bubble size relative to the transverse cell

dimension (a/L) was larger than 0.1, the bubble velocity was slightly larger than the

prediction of Taylor and Saffman.' With decreasing bubble size, however, the bubble

velocity decreased to a value smaller than the predicted one. This discrepancy in the

bubble velocity was apparently resolved by including the viscous dissipation around the

bubble edge in calculating the bubble velocity.


1.2 Motivation

In 1988, Kopf-Sill and Homsy6 conducted similar experiments to investigate the

shapes and velocities of various size bubbles driven by a pressure gradient. They

investigated the motion of air bubbles in a horizontal Hele-Shaw cell filled with a

glycerin-water mixture. Unlike the observations of Maxworthy, a variety of unusual

bubble shapes were observed as shown in Figure 1. 2. Besides the near-circular and

elliptic bubbles, "Tanveer" bubbles, ovoids, long- and short-tail bubbles were observed.

By tracing the locations of the leading and trailing edges of the bubbles for an extended

period of time, they showed that these unusual shapes were not transient but steady.

Furthermore, they found that at low capillary numbers the translational velocities of the

bubbles (near-circle, flattened or elongated ellipses, Tanveer) were much smaller than
























(a) (b) (c)










(d) (e) (f)




Figure 1.2 Various Bubble shapes observed by Kopf-Sill and Homsy: (a) Near-Circle;
(b) Flattened bubble; (c) Tanveer Bubble; (d) Ovoid; (e) Short-tail bubble;
and (f) Long-tail bubble










expected. Especially, the velocity of near-circular bubbles was about an order of

magnitude smaller than the prediction of Taylor and Saffman.

In Figure 1.3, the velocity data taken from Kopf-Sill and Homsy5 are given for

three different bubble sizes. The capillary number in the x-axis is defined as p.V/a where

t and V are the shear viscosity and the average velocity of the glycerin-water mixture,

respectively, and ; is the surface tension. When Ca is small, interfacial tension is

dominant resulting in a circular bubble shape. According to the theoretical prediction,'

the velocity of a circular bubble relative to the average fluid velocity (i.e., UN) should be

2. Contrary to the prediction, the relative bubble velocity was observed to be only about

0.2. The very low bubble velocity at a small capillary number was common for all

bubbles regardless of their size. As Ca was increased, the relative bubble velocity

increased slightly until it jumped suddenly to a much higher value at a certain Ca. The

sudden jump in velocity was apparently accompanied by a drastic change in bubble

shape. While the similar trend was observed with all bubbles of different sizes, the

sudden transition in the bubble shape and velocity occurred at a lower value of Ca for

larger bubbles. They also reported a hysteresis in the transitional behavior.

It should be noted that the Taylor-Saffman solution exists only when the relative

bubble velocity U/V is greater than 1. Although the possibility of U/V < I cannot be

ruled out for non-zero surface tension, Saffman and Tanveer'2 point out that such a low

velocity of small bubbles cannot be explained by a simple inclusion of the surface tension

























A A
0 +
Long tails 0 00 +

Short tails
o+

-A 0 Ovoids +...
A4 0+
Z90 + lattened + +
Near Circles

50 100 150 200 250 3

Ca x 104
A206cis 01.3cms +0.76crs


Figure 1.3 Here U/V vs Ca (data taken from experiment series 1,2, and 3 of Kopf-Sill
and Homsy)


2.50


2.00


1.50
U/V
1.00


0.50


0.00








effect and/or the wetting film effect. They suggested that the low bubble velocity might

be due to a moving contact line dragging along the cell plates. Using an ad hoc boundary

condition, they showed that a moving contact line could account for the very low bubble

velocity. In Kopf-Sill and Homsy's experiment, however, the liquid phase wetted the

solid surface and wetting films were always observed. Therefore, the presence of a

moving contact line was unlikely.

We may note that the surprising variety of the steady bubble shapes and the

striking discrepancy in bubble velocities were observed only in a pressure driven flow.

Unlike Kopf-Sill and Homsy, Maxworthy did not observe any unexpected shapes but

only elliptic bubbles elongated in the flow direction. In his experiments, the bubble

velocities were also in disagreement with the Taylor-Saffman theory. The disagreement,

however, was not as great as in the Kopf-Sill-Homsy experiments and it could be

explained by the inclusion of viscous dissipation at the bubble rim. In the work of Kopf-

Sill and Homsy, the experiment of Eck and Siekmann7 is quoted in which the

translational velocity of small bubbles were also much smaller than the predicted value.

These observations apparently suggest that the pressure-driven and buoyancy-driven

flows may be intrinsically different although Saffman and Taylor2 pointed out the

equivalency between the two flows.

The experiments of Kopf-Sill and Homsy involve several observations that are in

need of explanation. These observations include (1) the velocity of small bubbles which

appear to be an order of magnitude smaller than the theoretical prediction, (2) a surprising

variety of steady bubble shapes, and (3) the sudden transition of bubble velocity and








shape at a certain capillary number. Johann and Siekmann8 have measured the velocities

of air bubbles in a water system containing steric acid. Their reported velocities also

appear to be significantly retarded compared to the theoretical prediction. In all

experimental studies described above, the Hele-Shaw cell was supposed to be clean and

free from any contaminants. Many fluid flow systems, however, are susceptible to

contamination and it seems that the perplexing observations of Kopf-Sill and Homsy may

be explicable if we adopt the notion of the influence of surface-active contaminants on

bubble motion. One of the objectives of this study is to provide a plausible explanation

for the observations of Kopf-Sill and Homsy in the context of surfactant influence.


1.3 Dissertation Outline

In order to investigate the influence of surfactants on the bubble motion, we

conducted experiments using air bubbles, water drops and fluorocarbon oil drops in a

Hele-Shaw cell filled with silicone oil. Chapter 2 presents our experimental findings and

compares them with those of Kopf-Sill and Homsy. Air bubbles were chosen since the

surface properties of an air-oil interface were less likely to be affected by surfactants

which might be present as contaminants. It is well known that unlike the surface tension

of water or the interfacial tension between water and a hydrocarbon, the surface tension

of non-aqueous solutions is not affected significantly by surfactants.17 In case of water

drops, a predetermined amount of surfactant (sodium dodecyl sulfate) was added so that

the surfactant influence could be studied systematically. The motion of the water drops

were found to be drastically retarded by the surfactant, whereas the air bubbles were

moving much faster as we anticipated.








When surface active substances are present in the system, they adsorb near the

fluid-fluid interface. Under dynamic conditions, depending on the surfactant exchange

kinetics, a spatial variation of surface concentration can arise resulting in Marangoni

stresses.'8 As a consequence, the interface can behave as rigid surface.9'20 Hence in the

thin film region, the drag increases significantly causing severe retardation of the bubble.

Chapter 3 presents a theoretical calculation that predicts the extent of retardation and

provides an analytic expression for the drop velocity at low surfactant concentrations. As

the bulk surfactant concentration is increased, Marangoni effects are expected to be

weaker and a complete remobilization is anticipated.2' Contrary to this expectation, the

motion of a drop containing a high concentration of surfactant is also found to be

retarded. Although it is not very clear as to what causes the retardation at high

concentrations, it could possibly be due to the influence of interfacial viscosities. At high

concentrations the interface gets completely saturated and the intrinsic viscosities of the

surface can play a role in retarding the drop motion. An effort to validate this idea

theoretically is only partially successful.

Finally, Chapter 4 presents the analysis of the displacement flow of a shear

thinning power-law fluid by an inviscid fluid in a Hele-Shaw geometry. This problem has

practical applications in several methods of polymer processing such as injection molding

using various gases. The technique of matched asymptotic expansions2 is used to

determine the flow field and the shape of the interface. The flow domain is divided into

three different regions and matching conditions between them provide the film thickness

left behind by the advancing meniscus. Unlike the Newtonian case, the relationship

between capillary number and film thickness is not unique and it varies from fluid to





11

fluid. For power-law fluids, the depth averaged flow field is non-Darcian. A linear

stability analysis is also carried out to determine the dispersion relation for the shear

thinning power-law fluid. Chapter 5 summarizes the findings of this work.














CHAPTER 2
EXPERIMENTS

In this chapter, the influence of surfactant on the bubble motion in a Hele-Shaw cell

is described qualitatively, and the difference between pressure-driven and buoyancy-

driven flows in the presence of surfactant is discussed. Subsequently, the results of the

present experiments are presented to support the arguments described below.


2.1 The Influence of Surfactants on Bubble Motion


As mentioned in Chapter 1, the bubble can be set in motion either by applying a

pressure gradient in a horizontal Hele-Shaw cell or by imposing a buoyancy force by

tilting the cell. When the bubble is driven by buoyancy through a stagnant fluid in a Hele-

Shaw cell, the sweeping motion of the surrounding liquid in a bubble fixed coordinate

system is always from the front to the trailing end of the bubble. If the bubble is driven

by the surrounding liquid, on the other hand, two different flow situations may arise

depending on the relative magnitude of the bubble velocity U to the average velocity of

the surrounding fluid V. If U > V, the sweeping motion of the surrounding liquid is

toward the back of the bubble as in the buoyancy driven flow. Whereas if U < V, the

sweeping motion along the edge of the bubble is from the rear to the leading end of the

bubble. In the absence of surface-active substances, the surface tension is constant and

the two flow situations may be indifferent. They are, however, distinctly different if there

exists a spatial variation of surface tension due to the presence of surfactant.








When U > V, surfactants will be accumulated at the trailing end of the bubble thus

lowering the surface tension at the back of the bubble. When U < V, the flow situation is

more complicated. At the edge (or rim) of the bubble, the surrounding fluid is convected

from the back to the front of the bubble. Thus, surfactants are convected toward the

bubble front along the edge. In the thin film region between the wall and the bubble, on

the other hand, the dominant convection is always toward the back of the bubble since the

solid wall is moving backward relative to the bubble and the liquid film is very thin.

Therefore, the surfactant distribution is complicated and may depend on the balance

between the two competing effects. The studies of Park and Homsy9 and Burgess and

Foster" indicate that stagnation points (or rings) are present around the bubble at the

transition region between the thin film region and the cap region (i.e., points s in

Figurel.1). These stagnation points are similar to those in the front and the back of the

Brethertons bubble in a capillary tube." Although the location of the stagnation points

may move toward the cap region when surfactants are present, the existence of the

stagnation points (or rings) may result in three regions of surfactant accumulation; one

near the front cap of the bubble and the other two in the thin film regions (top and

bottom) near the back of the bubble. It should be noted that this complicated flow

situation cannot occur in a buoyancy driven flow. Therefore, the pressure-driven and

buoyancy-driven bubble motions can be different from one another if surfactants are

present in the system.


The most interesting bubble shapes that were observed by Kopf-Sill and Homsy may

be the ovoids with the sharper end pointed in the flow direction and the bubbles with a








long- or a short-tail. When the average liquid velocity V was low (i.e., at a low Ca where

the surface tension is dominant), the bubbles were observed to be near circular. With

increasing V, they became slightly deformed to take on the shape of an ovoid. As V was

further increased, the bubble shape went through a sudden transition to the long-tail (or

short-tail) shape. Interestingly, a hysteresis was observed in that the backward transition

from long-tail to ovoid occurred at a lower value of V than the forward transition from

ovoid to long-tail for the same bubble. One important aspect which should be noted is

that the ovoids apparently moved with a velocity smaller than the average liquid velocity

(i.e., U < V) whereas the velocities of the short- or long-tail bubbles were larger than V.

That is, the sudden increase (or decrease in case of a reverse experiment) in the bubble

velocity cuts across the UN=I line in Figure 1.3. Thus, the flow situations for the ovoids

and the bubbles with a tail are different in that the direction of the sweeping motion is

opposite. We may point out, however, that a group of four data points in Figure 1.3

(a=0.76 cm) does not follow this argument as their relative velocities were reported to be

about 0.8 although the shapes were short-tail. It may be possible that this discrepancy is

due to measurement error.


When the surrounding liquid sweeps the bubble from back to front along the edge

(i.e., when U < V), the local interfacial tension may be lower in the front region of the

bubble due to the surfactant accumulation there. Since the pressure inside the bubble is

constant, the radius of curvature at the bubble front should be smaller than that at the

back to compensate for the lower interfacial tension. Therefore, assuming a constant

curvature in the gap direction (i.e., the z-direction in Figure 1.1), an ovoid with the








sharper end pointing in the flow direction is a plausible shape. The same argument may

be applicable even in the absence of surfactants. In that case, however, the difference in

curvature between the front and rear ends of the bubble will not be as large as the

surfactant-influenced case. By applying the similar argument to the case of U > V, we

may also speculate that a drastic curvature change in the rear part of the bubble (hence

the long-tail or short-tail bubbles) is possible if surface-active substances are present.

This qualitative explanation does not rule out the possibility that the long-tail bubbles can

also exist in a buoyancy driven flow since U is always greater than V in that case. A

probable reason why it was not observed in Maxworthy's experiment is that silicone oil

was used instead of glycerine-water mixture in his experiments. As it was pointed out

previously, the air-oil interface is less likely to be affected by surface-active

contaminants.


Under the assumption that surface-active contaminants were present, the overall

behavior of a bubble observed by Kopf-Sill and Homsy may be explained as follows.

When an air bubble is stagnant in a contaminated Hele-Shaw cell, its entire surface will

be covered with surface-active contaminants. At a very low Ca, the low shear stress may

induce only a small change in surfactant distribution. Thus the Marangoni effect may be

global. Although the Marangoni stress may be small due to the small surface tension

gradient, the shear stress is also small and the entire bubble surface may act as a rigid

24
surface. In the presence of the stagnation rings near the top and bottom wall (Figure

1.1), three separate regions of high surface concentration can coexist as pointed out

previously. Along the edge of the bubble, surfactants may be swept toward the bubble








front and consequently, the Marangoni effect may assist the bubble motion. In the thin

film region at the top and bottom of the walls, on the other hand, surfactants are swept

backward retarding the bubble motion. When Ca is small, the viscous stress along the

bubble edge may not be as significant as that in the thin film region which is inversely

proportional to the film thickness. Therefore the drag in the wetting film is dominant

retarding the bubble motion significantly. Consequently, the relative bubble velocity

(U/V) may remain much smaller than 1. Although the bubble motion can be severely

retarded, the shape change may be insignificant due to the small spatial variation of the

surface tension. Consequently, the bubble shape may be near-circular.


As Ca is increased, the larger viscous stress along the bubble edge increases the

surface tension gradient resulting in larger Marangoni stress. In the thin film region, on

the other hand, the relative increase of the drag may not be as significant since the film

thickness increases with the bubble velocity. Therefore, the relative bubble velocity may

increase with increasing Ca although its rate of increase may not be very large.

Furthermore, the surfactant concentration at the leading end of the bubble will also

increase with Ca. Consequently, the surface tension will become lower at the leading end

creating a larger curvature. Therefore, the ovoid with the sharper end pointing in the

direction of the flow may be a favorable shape.


With increasing Ca, the film thickness increases and the location of the stagnation

rings may move toward the tip of the bubble edge. Eventually the two stagnation rings

may collapse into one along the tip of the bubble edge." Then the three spearate regions

of high surfactant concentration cannot coexist. Consequently, the large drag in the thin








film region may decrease significantly inducing higher bubble velocity which may be

greater than the average velocity of the surrouding fluid (i.e., UN > 1). Once the relative

bubble velocity becomes greater than 1, all surfactants will be collected at the trailing end

of the bubble lowering the surface tension there. Consequently, a larger curvature is

required forming a high curvature tail. The size (or length) of the tail may vary

depending upon the surfactant distribution at the rear end. In an extreme case where the

bubble is large enough and the surfactant transport is limited by bulk diffusion, the

leading end of the bubble may be virtually free from surfactant due to convection whereas

the trailing end is full of surfactants.19'2'26 In such a case, a long cylindrical tail may be

conceivable as a form of minimizing viscous dissipation while maintaining a large

curvature for the pressure balance across the interface. The Marangoni effect is greater if

the surface curvature is larger. Consequently, its strong influence may persist to a higher

value of Ca if the bubble is smaller. Therefore, the critical value of Ca where the

transition occurs may increase with decreasing bubble size.


If the capillary number is decreased gradually from a large value to a small value, the

reverse trend may be observed in which the bubble goes through a transition from a long-

or short-tail bubble to an ovoid and eventually to a near-circle. The forward and

backward transitions, however, are not necessarily reversible as the adsorption and

desorption kinetics of surfactants may not be reversible. Thus, the hysteresis in the

transitional behavior can occur if such a change is driven by the presence of surfactants.

The argument presented above is rather simple and only qualitative. Nevertheless, it

apparently accounts for most of the observations of Kopf-Sill and Homsy. In the








following section, the results of the present experiments are given as a supporting

evidence.

2.2 Experiments

The Hele-Shaw cell consisted of two 1/2-in thick Pyrex glass plates separated by a

rubber gasket of 0.9 mm or 1.8 mm in thickness. The effective cell dimension was 17.8

cm by 86.4 cm. A silicone oil with the measured viscosity and surface tension of 97 cp

and 21 dyn/cm was used as the driving fluid. The cell had a deep groove channel at both

ends so that a uniform distribution of the oil along the transverse direction could be

ensured when it was injected. The cell also had an injection port at one end of the top

plate so that an air bubble or a water drop could be introduced into the cell using a

syringe without creating extra bubbles or drops. The planform diameter of the stagnant

bubbles and drops was controlled to be at about 1.3 cm and 2.1 cm throughout the

experimental study. These sizes were chosen to be about the same as those of Kopf-Sill

and Homsy for direct comparison of the results. While the size variation among the small

bubbles was small as they ranged between 1.3 cm and 1.4 cm, the size control was more

difficult with larger bubbles and their size ranged between 2.2 cm to 2.5 cm.


Water drops containing an organic surfactant at various levels of concentration were

prepared by dissolving sodium dodecyl sulfate (SDS) in distilled water. Since the

surfactant influence is more significant at low concentrations, the SDS concentration was

set at 5%, 10%, 20%, and 100% of the critical micelle concentration (CMC), respectively.

The CMC of SDS in water at 25'C is 8.2 mmol/I which is equivalent to 0.236% by

weight. The interfacial tension measured at room temperature by the Wilhelmy plate








method is listed in Table 2.1 at various SDS concentrations. As the present data for air-

21
water interface is found to be consistent with those reported in literature, the data for the

silicone oil-water interface may also be reliable. It can be deduced from the data that the

rate of change of interfacial tension is greater at a lower concentration indicating a

stronger Marangoni effect at a lower concentration.


Once an air bubble (or a water drop) of predetermined size was positioned at one end

of the cell, the silicone oil was driven by a peristaltic pump at various flow rates to induce

the bubble motion. The translational velocity and the shape of the bubble (or drop) were

then recorded for each value of oil flow rate. For the prescribed dimensions, the values of

both a/L and h/a were smaller than 0.14. Thus the influence of the side wall (i.e., the

rubber gasket) may be small and the depth-averaged Hele-Shaw equations may be

applicable to describe the bubble motion if the system is free from surfactants. In order

to avoid an excessive pressure drop at high flow rates, most of the experiments were

conducted with the thick gasket (1.8 mm) and only a part of the air bubble experiments

used the thin gasket (0.9 mm) for comparison.


The Bond number (Bo) which is defined as pgh2/C is a measure of the relative

importance of gravity to surface tension, and a large Bo may indicate asymmetry of the

bubble in the gap. In the experiments of Kopf-Sill and Homsy, Bo was as small as 0.1

whereas it was about 1.5 in Maxworthy's. In the present experiment, it was either 0.4 for

the 0.9 mm thick gasket or 1.5 for the 1.8 mm thick gasket. The large Bo was caused not

only by the thick gasket but also by the low interfacial tension. While the gravity effect








might not be significant when Bo = 0.4, uncertainty existed when Bo was as large as 1.5.

The experimental results for air bubbles, however, were indifferent in terms of both

bubble velocity and shape to whether Bo was 0.4 or 1.5. Thus, the gravity effect might

not have significant influence even when Bo = 1.5. In reporting the results in the

following section, no distinction has been made between the data for Bo = 1.5 and 0.4 in

order to avoid unnecessary complication.


The size of air bubbles changed slightly during the experiment due to the pressure

drop in the cell. Kopf-Sill and Homsy observed similar behavior and reported the volume

change of air bubbles up to 20%. In the present experiment, not only was the gap

thickness of the cell slightly larger but also the liquid viscosity was lower. Furthermore,

the liquid flow rate was 0.75 cm3/sec when the capillary number was as high as 110.

Consequently, the pressure drop was smaller and the change in the bubble volume was

less than about 5%. Despite the change in the bubble volume, variation in the bubble

velocity was not detectable. In case of water drops, neither the volume nor the velocity

changed.


The description in Section 2.1 regarding the surfactant influence on the bubble

motion assumes the existence of a thin oil film between the wall and the moving bubble

or drop. When oil is the wetting phase, the film thickness is uniform in the direction of

bubble or drop motion (i.e., x direction in Figure 1.1) whereas it varies in the cross-stream

direction (i.e., y direction in Figure 1.1). Therefore, it may be possible in principle to

prove the existence of the thin film by an optical method which creates interference

fringes. Our attempt to observe the interference fringes, however, was unsuccessful due








to various difficulties involved with the large size of the cell. Thus we provide only a

qualitative evidence to support the assumption.


A photograph converted from a video image is shown in Figure 2.1 for a water drop

of about 2 mm in diameter which is sitting on a Pyrex glass plate immersed in the

silicone oil. In order to improve contrast, a blue food coloring was added to the water

drop. Due to the difficulty in obtaining perfectly horizontal alignment for the contact

angle measurement, the water drop was viewed with a small positive angle from the

horizontal and a shadow of the drop was created in front by indirect back lighting. The

lower part of the picture is the shadow of the drop and the bright line between the drop

and its shadow represents the three-phase contact line. At the edge of the contact line, the

apparent static contact angle is shown to be greater than 900 when measured from the

water-drop side. This may suggest that silicone oil is the wetting phase under dynamic

conditions.


When a water drop is stationary in the Hele-Shaw cell at the beginning of each

experiment, the water drop may have direct contact with the Pyrex glass wall creating a

three-phase contact line. As we induced the drop motion by injecting the silicone oil into

the cell, the trailing edge of the drop exhibited briefly a ragged contact line pinned on the

wall. The contact line soon detached from the wall and the water drop took on a steady

shape with a perfectly smooth edge around the entire drop (Figure 2.2). If a moving

contact line existed, its motion might have been jerky at times unless the solid surface

was perfectly homogeneous. In the present experiments, the drop motion was always

smooth excluding the brief moment at the start of motion. In case of air bubbles, even the










































Figure 2.1 A water drop sitting on a Pyrex glass plate immersed in silicone oil.








start-up motion was smooth without any evidence of contact line. These observations

along with the observation on the apparent static contact angle seem to support the

assumption regarding the existence of thin oil films.

2.3 Results and Discussion

In Figure 2.2, shapes of a water drop containing surfactant are given sequentially

with increasing capillary number. These pictures were converted from the video images

of a water drop which was 2.5 cm in diameter and contained sodium dodecyl sulfate

(SDS) at 20% of the critical micelle concentration (CMC). The direction of the flow is

downward from top to bottom of the figure. In order to improve contrast a blue food

coloring was added to the drop. It was confirmed by measurements that the food coloring

did not affect the surface tension. All shapes were observed to be steady excluding (2.2e)

which was a transitional shape from (2.2d) to (2.2f). Figure 2.2a is the image of a

stationary drop at Ca = 0. Due to the typical distortion of a video image in the vertical

direction, this circular drop appears to be slightly elliptic. As Ca was increased, the drop

was elongated sideways (2.2b). Further increase in Ca resulted in further stretching of

the drop in the transverse direction (2.2c). This shape is apparently asymmetric about the

vertical axis (or the x-axis in Figure 1.1). Nevertheless, its dimensions remained

unchanged and moved with a steady velocity. Hence a steady shape.

It should be pointed out that the severely stretched drop shape (2.2c) occurred

only when the drop was larger than about 2.1 cm in diameter and when it contained SDS

at a concentration between 5% and 20% of CMC. It did not occur with the pure water

drop nor with the one containing SDS at 100% CMC although their sizes were larger than




















(a) (b)


(c) (d)


(e)


(f)


Figure 2.2 Evolution of water drop shape with increasing capillary number. (a) Near
circular drop; (b) flattened drop; (c) severely stretched drop; (d) ovoid; (e)
transitional shape; and (f) short tail drop








2.1 centimeters. Furthermore, when Ca was increased further from the value for Figure

2.2c, the drop broke into two instead of showing a transition to an ovoid or short-tail. In

order to obtain the steady shapes at a higher Ca (i.e., Figure 2.2d or 2.2f) with the large

drops containing SDS at 5% to 20% CMC, the Ca had to be set to a large value from the

beginning. Then the drop evolved directly from (2.2a) to (2.2d) or (2.2f) bypassing the

severe sideways stretching (2.2c). In case of small drops (1.3 or 1.35 cm in diameter) or

the large drops with 0% or 100% CMC, on the other hand, the severely stretched shape

(2.2c) did not occur and the evolution of steady shapes was smooth from (2.2a) to (2.2f)

with the gradual increase of Ca. It may be interesting to note that the drop shape (2.2c)

has a negative curvature at the leading edge as in the case of "Tanveer" bubbles although

the trailing edge also shows a negative curvature.


The steady shape (2.2d) which occurred at a higher Ca than (2.2b) or (2.2c) is

different from others as the curvature of its leading edge (i.e., the lower part of the drop in

Figure 2.2d) is apparently larger than that of the trailing edge. We may call this shape

"ovoid" with its sharper end pointing in the direction of flow which is equivalent to the

sketch given in Figure 1.2d. The relative velocity (i.e., UN) of this drop was still smaller

than 1. When Ca was further increased, the ovoid went through a rapid transition to a

short-tail (2.2f) which moved faster than the average velocity of the surrounding fluid

(i.e., UN > 1). This transition always occurred through a transitional shape given in

Figure 2.2e.


In Figure 2.3, 2.4, and 2.5, the bubble or drop velocity relative to the average

velocity of the surrounding fluid is given as a function of capillary number Ca. Also



















2.5

2.0 t A A A
X A X O

1.5 A 0
UN/v
1.0
i mm m om i IN
0.5 .x

0.0
0 50 100 150
Cax 104
1 I.27 cms M 1.3 cms (Kopf-Sill) A2.54 cms x2.06 cms (Kopf-Sill)


U/V vs Capillary number for air bubbles.


Figure 2.3









specified in the figures are the steady shapes of the bubbles or drops classified according

to Figure 2.2. The capillary number in these plots is defined as iV/a where p and V are

the viscosity and the average velocity of the silicone oil respectively and a the

equilibrium interracial tension. In case of air bubbles (Figure 2.3), a is 21 dyn/cm

whereas a for the water drops (Figures 2.4 and 2.5) varies depending on the SDS

concentration. The equilibrium values at each SDS concentrations are given in Table 2.1

and these respective values have been used in calculating the capillary number.


The relative velocities of air bubbles described in Figure 2.3 are for two different

sizes (a=l.27 cm and 2.54 cm). Overlaid in the figure are the data taken from Kopf-Sill

and Homsy for similar size bubbles. In their experiments, both small and large bubbles

experienced a sudden increase in the relative velocity accompanied by the shape change.

The transition for the small bubble occurred when Ca was about 10-2 whereas it occurred

at Ca z 5 X 104 for the large bubble. Such transitions, however, were not observed in the

present experiment which covered the range of Ca between 2 X 10.3 and 1.1 X 102. The

relative velocity was always larger than one for both bubbles, although the larger bubble

was moving slightly faster than the smaller one. Furthermore, the bubble shapes were

always near circular unlike the experiments of Kopf-Sill and Homsy.


The lower limit of Ca (2 X 10") was set by the sensitivity of the pump whereas the

upper limit (1.1 X 10"2) was determined by the maximum flow rate which was imposed

by the pump capacity or the upper limit of pressure drop in the cell. The lower limit is

well below the critical Ca of Kopf-Sill and Homsy at which the small bubble experienced








the velocity/shape transition. The critical value for the large bubble (Ca -_ 5 X 10 4), on

the other hand, was smaller than the lower limit. Thus, the present experiment could not

have detected the transition if it had occurred at a very low Ca. It should be noted,

however, that in the experiments of Kopf-Sill and Homsy, the velocity transition was

always accompanied by the shape change and the bubbles with the relative velocity (U/V)

greater than one had either a long- or a short-tail. In the present experiments, on the

other hand, the bubble shapes were near-circular although their relative velocities (U/V)

were greater than about 1.5. Thus, it is unlikely that these near-circular bubbles had

experienced the sudden transition at a smaller Ca than the lower limit or they would

experience it at a higher Ca than the upper limit.


Considering the slight deviation from a circular shape and the viscous dissipation at

the bubble edges which was pointed out by Maxworthy,5 the bubble velocities in the

present experiments appear to be in reasonable agreement with the prediction of Taylor-

Saffman. The distinct differences between the results of the present experiments and of

Kopf-Sill and Homsy are apparently in accordance with the argument described in section

2.1 that they were probably caused by the influence of surface-active contaminants.

In Figures 2.4 and 2.5, the results for water drops are given for two different sizes.

In these experiments the 1.8 mm gasket was used to minimize the pressure in the cell at

high flow rate, and the maximum attainable capillary number was determined by the

capacity of the pump (Cole-Parmer, Masterflex Model 7520-25). Due to the variation of

the equilibrium interfacial tension with the SDS concentration, the maximum capillary


















Table 2.1 Variation of the interfacial tension with sodium dodecyl sulphate
concentration at 25C


SDS concentration in water

(% of CMC)


air-w


Interfacial Tension (dyn/cm)

rater Silicone oil-water


72.8

58.7

53.8

41.7

33.0


38.8

34.3

32.1

25.9

13.7









number also varied accordingly for the given maximum flow rate. In case of the 100%

CMC water drop, the equilibrium interfacial tension was as low as 13.7 dyn/cm whereas

that of pure water and oil was 38.8 dyn/cm (Table 2.1). Consequently, the maximum

capillary number for the former (3.7 X 10-2) was higher than that for the latter (1.3 X 10-2)

by a factor of about 2.8.


The relative velocities of the small water drops (a = 1.30 cm or 1.35 cm) are shown

in Figure 2.4. The labels (a)-(f) indicate the drop shapes as indicated in Figure 2.2.

Unlike the air bubbles, they were moving very slowly at low capillary numbers as in the

observations of Kopf-Sill and Homsy. It is interesting to note that the water drops which

were supposed to be pure were also moving very slowly at low Ca although they were

expected to be as fast as the air bubbles in Figure 2.3. A plausible explanation for this

behavior may be that the water drops still contained some surfactant despite the

precaution we had taken to prevent contamination. When the SDS concentration was

between 5% and 20% of CMC, the relative velocities were smaller than one even at the

upper limit of Ca. Furthermore, the drop velocities were indistinguishable and the shapes

were near-circular or elliptic (Figures 2.2a and 2.2b). Although it was expected that the

relative velocities of these drops would increase with Ca and become eventually larger

than one, the transition could not be observed as it might occur at a higher Ca than the

attainable upper limit.

In case of the "pure" drop or the drop with SDS at 100% CMC, on the other hand,

the transition could be observed in that the drops took on the short-tail shape (Figure 2.2f)

and moved with a relative velocity greater than one. Unlike the observations of Kopf-Sill




















2.0


1.5


U/V1.0


0.5


0.0


.- (0

0 m (d)
xx x


[M ... *

L (a)


200


300


400


Ca x 104
QO/ (13.5mm) n5% (135mm) A9.1*(13mm) x2-) (13mm) 00 /, (13.5m) Kopf-Sii (13 mm)


U/V vs Capillary number for small water drops.


Figure 2. 4









and Homsy, however, the change (or the transition) was gradual rather than abrupt.

Although it is not clear what caused this difference in the transitional behavior, it may be

partly due to the differences between the two experiments. In the current experiment of

water drop and silicone oil, the surfactant was present inside the water drop whereas in

case of Kopf-Sill and Homsy's experiment of air bubble and aqueous solution, the

surfactants (or surface-active contaminants) might be present in the aqueous solution

outside the bubble. Therefore, the mechanism for surfactant transport to the bubble or

drop surface could be different. Furthermore, considering the differences in the fluid

properties, quantitative match between the two experiments may not be expected. These

differences could be also the reason why long-tail drops were not observed in the present

study.


The large water drops showed the similar trend as the small ones in that the relative

velocities were smaller than one at low Ca (Figure 2.5). Unlike the case of small drops,

however, the transitional behavior is evident for all water drops in Figure 2.5. This

observation is consistent with the trend in Figure 1.3 in that the critical Ca for the

velocity/shape transition decreased with increasing bubble size. As we mentioned in

section 2.1, the surfactant influence (or the Marangoni effect) should be greater if the

surface curvature is larger. Consequently, the strong influence of surfactant may persist

to a higher value of Ca if the drop is smaller. Apparently, the critical Ca for the large

drops was smaller than the upper limit of Ca of the present experiments whereas that for

the small ones was larger than the upper limit. The variation in the drop shape was also




















2.0
13
0 0
1.5 XX
.1u X X
&X
U/N 1.0

... ... .. X d
DXA...
(b))
0.5 _6() c
)(a)
0 .0 .
0 100 200 300 400
Ca x 104
0/.cmc (25mm) g 5% c (24mm) & 91% cc (21.5mm)X 20% cmc (23.5mm)'
0 100/ cmc (24m) Ann (20.6mm)


U/V vs Capillary number for large water drops.


Figure 2. 5









consistent in that it was near-circular, elliptic or ovoid when U/V < 1, whereas all drops

became short-tailed when UV > 1. Figure 2.5 also indicates that the critical Ca for the

velocity and shape transition increases with increasing SDS concentration with the

exception of the drop at 100% CMC. Although only a quantitative analysis will provide

proper answer to such a trend, a qualitative explanation may be given as follows. For

most organic surfactants the adsorption kinetics is fast. Therefore, if the surfactant

concentration is low, its transport may be limited by bulk diffusion. At a reasonably high

Ca, the surfactant distribution may be localized due to convection and consequently, the

Marangoni effect is also restricted to the local region. As long as the surface

concentration gradient is smaller than the maximum allowable value before the

monolayer collapse for the given surfactant, the area covered with the surfactant may

grow with increasing bulk concentration. Consequently, the Marangoni effect may also

increase with the bulk concentration. If the bulk concentration is higher than a certain

limit, however, surfactant transport may not be limited by bulk diffusion any more and

the surface concentration gradient will start to decrease. Thus the surfactant influence

will also diminish with further increase in bulk concentration. It seems that the critical

bulk concentration for the present experiment at which the surfactant influence starts to

decrease may be between 20% and 100% of CMC. In an extreme case, if the bulk

concentration is very high, the surface may be remobilized due to the lack of surface

concentration gradient. We have also conducted an experiment with a water drop which

contained SDS at 10 times the CMC to investigate whether the remobilization would























2.0
A
A
A
1.6 3A
0 0

1.2 o
0
U/V
0.8


0.4


0.0
0 200 400 600 800

Ca x 104

06ram 09mm A 13m


Figure 2.6 U/V vs Capillary number for Fluorocarbon oil drops.









occur at that concentration. The result, however, almost coincided with that for 100%

CMC drop in Figure 2.5 in that the relative drop velocity was still very low at small Ca.

Finally, an experiment is conducted using an oil-oil system. Although, surface active

substances are present as contaminants in this system, they are not expected to adsorb at

the interface hence their influence on the drop motion will be insignificant. Fluorocarbon

oil is chosen because it is immiscible with silicone oil. The interfacial tension was

measured by Wilhelmy plate method and is found to be close to 5 dyns/cm. Pressure

gradient is imposed to study the motion of a Fluorocarbon oil drop placed in Hele-Shaw

cell filled with silicone oil. Figure 2.6 presents the normalised velocity of fluorocarbon

oil drop as a function of capillary number. A nominal value of 5 dyns/cm is used in

calculating the capillary number. These results indicate that the normalised velocity is

always greater than one and the drop velocity is close to the predictions of Taylor-

Saffman theory. Unlike the water drops, the shape of the fluorocarbon oil drop is always

found to be circular. These results reinforce the idea that the retardation experienced by

drops or bubbles in an aqueous environment is due to the influence of surfactants.


2.4 Summary and Conclusions

Under the assumption that surface-active contaminants were the primary reason for

the interesting behaviors of the bubbles in a Hele-Shaw cell observed by Kopf-Sill and

Homsy, similar experiments were conducted using the fluid combinations of air-silicone

oil and water-silicone oil. These systems were chosen to delineate the surfactant

influence systematically. In case of air bubbles in silicone oil for which the surfactant

influence was likely to be insignificant, the relative velocities were observed to be as high








as the Taylor-Saffman prediction. Furthermore, the unusual bubble shapes such as long-

or short-tail bubbles were not observed. In case of water drops which contained

predetermined amount of an organic surfactant (sodium dodecyl sulfate), on the other

hand, very low translational velocities as well as the unusual shapes of Kopf-Sill and

Homsy were observed.


Although it is impossible to confirm the presence of surfactants in the past

experiments, the present results are consistent with those of Kopf-sill and Homsy and it

appears that most of the observations by them may be due to the influence of surface-

active contaminants.














CHAPTER 3
ESTIMATION OF BUBBLE VELOCITY

3.1 Introduction

This chapter presents a theoretical calculation for the translational velocity of a

bubble which is retarded by the surfactant influence. In order to simplify the analysis, the

bubble shape is assumed a priori to be elliptic. It is also assumed that the surrounding

fluid wets the solid surface thereby forming a thin film between the bubble and the plates

of the Hele-Shaw cell. For a relatively small bubble (i.e., b/L << 1) yet large enough to

neglect the edge effect (h/b << 1), the entire bubble surface tends to be rigid due to the

surface active substance. Consequently, a large drag arises in the thin film region thus

retarding the bubble motion significantly. The calculated velocity of the bubble is

apparently in reasonable agreement with our own experimental observations and also

with those of Kopf-Sill and Homsy.6 In addition, the present calculation provides an

explanation for the evolution of the bubble shape with increasing capillary number.


3.2 Theoretical Analysis

A bubble or drop in a Hele-Shaw cell assumes a circular plan form when it is stagnant

because of the interfacial tension. Once the bubble is set in motion, its shape deviates

from the circular plan form and takes on a steady shape depending on the flow conditions

(Figure 1.1). In the absence of surface tension effect, the analysis by Taylor and

Saffman' predicts an elliptic shape if the bubble size is much smaller than the width of








the cell (i.e., b/L << 1). If surface active substances are present, the bubbles are not

necessarily elliptical and may take on various interesting shapes depending on the flow

conditions6. Nevertheless, we assume an elliptic plan form since the shape distortion is

small at low capillary numbers. While the bubble shape should be determined as a part of

solution in a rigorous analysis, the present analysis is an approximation in which the

bubble shape is assumed a priori. Unlike a full three-dimensional analysis that may be

possible only numerically, the present analysis provides an analytic description for the

translational velocity of the bubble.

As indicated in Figure 1.1, the two principal axes of the elliptic bubble are assumed to

be aligned with the flow and the transverse directions. The surrounding fluid wets the

solid surface thus forming a thin liquid film between the plates and the bubble. In the

presence of surface active substances, the Marangoni effect resulting from the surface

tension gradient may complicate the flow field near the bubble. When the bubble is

small, however, it may be simplified since the entire bubble surface may become

rigid9-2". Here we consider the case of a small bubble in which the entire bubble surface

is assumed to be rigid.


The flow field slightly away from the bubble is known to be parabolic in the xz-plane

(i.e., across the gap). Thus, the velocity field can be given as1


v 4z ) 3.1
v l5avg 1- h 2


Here vavg is the gap (or depth)-averaged velocity field that satisfies the following Hele-

Shaw equations:





40


V"Vavg = 0 3.2


Vp = v 3.3

Here p is the gap-averaged pressure field, t is the fluid viscosity and h is the gap of the

Hele-Shaw cell. It is apparent from Eqns. (3.2) and (3.3), that the average flow field in the

xy-plane can be described as a potential flow. In an elliptic cylindrical coordinate system

depicted in Figure 3.1, the complex potential Q for the flow past an elliptic cylinder in a

bubble fixed frame of reference is given by6

Cl=ID+i = -b(V-(k+l)Vk2 ,7-1 k+I
2 [k+ k- ].

where y = + i i1 and k the shape factor defined as k = a/b. V and U are the average

velocity of the surrounding fluid and the bubble, respectively. From the gradient of the

velocity potential cD or the stream function ', the velocity components in and ti

directions are given as


2q Lk+1lj- q


Vg=_C(VU)k+lF e' e- l UC 3.6
I1 2q Lk+l+ k-1 sn -U sirq



Here c = a2- b2j and q is the scale factor for the elliptic cylindrical coordinate system

defined as q2 = c2 (sinh2 +sin2Tl). As it was pointed out previously, Eqns (3.5) and (3.6)

are valid only in the region which is O(h) distance away from the bubble rim.








In the thin film region between the plates and the bubble, the flow field is

represented by a Couette flow if the bubble surface is rigid. In this case the thickness of

the wetting film can be given as10,20,23,27

= h pun_ 3
t = 1.337 (2Ca,,)3 where Ca,, 3.7
2 "



Here Can is the capillary number based on the component of the bubble velocity normal

to the bubble surface in the xy-plane, and a the equilibrium interfacial tension. In the

absence of surfactant, the bubble surface is stress-free and the film thickness can be

expressed by the same equation but with the constant 2 in the parenthesis replaced by 1.


Equations (3.1), (3.5), (3.6) and the Couette flow along with the film thickness

described by Equation (3.7) represent the three-dimensional velocity field around the

translating bubble excluding the small region of O(h) in the immediate vicinity of the

bubble rim. Using this velocity field, an analytic description of the translational velocity

of the elliptic bubble can be obtained from the x-directional momentum balance on a

control volume surrounding the bubble.


An integral form for the x-directional momentum balance for a domain D which

encapsulates the bubble and moves with the same velocity U as the bubble may be

written as (Figure 3.1)

d fpvdV = -f {p[n. (v U)]v, + (np n. c)dS 3.8
dtD S


This integral is zero at steady state and the surface integral can be evaluated using the















V


V


U


STop
SBubble


film region

I x Sedge
2a
S Bottom


Plan view and side view of an elliptic bubble and a control volume moving
with the bubble


Figure 3.1









prescribed velocity field over the four separate surfaces as shown in Figure 3.1; top,

bottom, edge of the control volume D, and the bubble surface.


On the bubble surface, v = U and the integral over the bubble surface represents the

drag force (FD) acting on the bubble:

f (np- n. -)xds = -FD 3.9
Sbbbk



On the top and bottom surfaces of D, the first term of the integrand in Equation (3.8)

is zero, since (v U) is perpendicular to n. The second term is evaluated separately for

the film region directly above (or below) the bubble and the region which is O(h) distance

away from the bubble rim to give


(np- n "t)xdS 12 c2 {(e2 0" V 3.10
SUt + S h 4 4
_(e-2'_ e-2 #V _U)e2o+ U] 161pU T(Al
2.122hCa (Al




As noted in Figure 3.1, 4= and = represent the edge (or rim) of the control volume

and the bubble, respectively. The last term in Equation (3.10), which is the contribution

from the Couette flow in the thin film region, is zero if the bubble surface is clean and

stress-free. Here A is the plan form area of the bubble (i.e., A = cab) and Ca the capillary

number based on the bubble velocity U. I is a constant given as a definite integral which

accounts for the film thickness variation in the transverse direction (i.e., y-direction):








1_r ,+ (k2 -1)y2]-[1 3.11




Once k(= a/b) is specified, I can be determined numerically and the results are given in

Figure 3.2 for various values of k. When the bubble is circular (i.e., k = 1), I = 0.91.


Finally at the edge of the control volume, the only non-zero contribution to the

surface integral is due to the pressure which is given as




Jnp) ds = l2 2 (v-U)+ Ve2_ -e2 U+ (VU)k+1 3.12
h 4 1k-I k -

From Equation (3.9), (3.10) and (3.12), the final form of the x-directional momentum

balance for a large control volume (i.e., -+oo) is obtained as



h 4 k16Uk+ Ib 3k"1 2.122hCaI

For a freely suspended bubble with zero inclination angle (i.e., 0 = 0 in Fig. 1), FD= 0 and

Equation (3.13) reduces to

2
U =(k +I)V Ca3 3.14
Ca 3 + 0.2kI

When the bubble is driven by gravity only, V = 0 and FD = -Apg(7rabh)sin0, since the

drag force is balanced by the buoyancy force. Thus,
2

U=kU 2Ca where U* Aph gsin0 3.15
CaI + 0.2kI



















1.5

1.3




0.9

0.7


0.5
0.0


1.5 2.0


Shape factor (k)



Figure 3.2 Numerical value of the definite integral, Equation 3.11 for various values of
the shape factor k.










When the bubble is driven by both pressure gradient and buoyancy, Equation (3.14) and

(3.15) are combined to give a general expression for an elliptic bubble as

2
U=8 [(k + 1)V + kU*] where = Ca3 3.16
Cai + 0.2kI
As Figure 3.2 indicates, I is 0(1) unless k is extremely large. Thus 0 is O(Ca3).

This order of magnitude decrease in the bubble velocity is due to the large drag in the thin

film region where the bubble surface is rigid. Since this drag is proportional to the film

thickness, the bubble velocity is also proportional to Cda. In the absence of Marangoni

effect, on the other hand, the bubble surface in the thin film region is stress free. In this

case, the expression for the bubble velocity is equivalent to setting 03 = 1 in Equation

(3.16) since the last term in Equation (3.13) is absent. Thus the result of Taylor and

Saffman is recovered in which U = 2 V for a circular bubble, U > 2 V and U < 2 V for an

elliptical bubble elongated in the flow direction (i.e., k > 1) and in the transverse

direction (k < 1), respectively.


3.3 Comparison with Experimental Results

Retarded motion of aqueous drops containing sodium dodecyl sulfate (SDS) in an oil-

filled Hele-Shaw cell was reported in the previous chapter for a pressure driven flow. In

Figure 3.3 the present predictions for ellipses with various aspect ratios are compared

with experimental results for a drop which is 13 mm in diameter (b/L = 0.04). The drop

contained SDS at 5, 10 and 20% of the critical micelle concentration. At these

concentrations of SDS, the data sets are indistinguishable from one another and show








reasonable agreement with current predictions for ellipses that are elongated in the

transverse direction (i.e., k < 1). While the drops in the experiment were not exactly

elliptical (Figure 2.2), they were nearly elliptical and elongated in the transverse

direction.


The present analysis assumes the bubble shape a priori and consequently, the

evolution of bubble shape with increasing capillary number cannot be predicted.

However, an interesting observation may be made in Figure 3.3 When the drop was

stagnant at Ca = 0, its shape was circular (i.e., k = 1.0). As it was pushed by the

surrounding fluid, the k value decreased initially from 1.0 prior to exhibiting an

increasing trend with increasing Ca. The initial decrease in the k value occurred at a very

low Ca and in some cases, the drops simply elongated in the transverse direction without

moving until the average velocity (V) of the surrounding fluid reached a certain value.

Unfortunately, not all k values are available for the experimental data in Figure 3.3, and

those that are available have been specified in the figure. Although a quantitative

agreement is not expected between the observed and calculated k values, it is interesting

to note that they are not very far apart.


It should be pointed out that the retardation factor 3 in Equation (3.13) is independent

of bubble size suggesting that the same level of retardation may occur regardless of the

bubble size as long as their shapes are identical. One of the simplifying assumptions in

the present study, however, is that the entire bubble surface is rigid. While this

assumption is plausible for small bubbles, it may fail for large bubbles since a stress-free

mobile surface region may emerge in the front part of the thin film region".




























U/V


150


0.2
0.3
0.5
1 .0


200


Ca x I 04


Comparison of experimental results and the theoretical estimate for pressure
driven flow (SDS concentration: 5% ; A 10%; x 20% ofcMc)


Saffman-Taylor Theory k=1.0



k=-0.4



A a ,A I X x ....
K ...k-- .3


Figure 3.3









Consequently, big bubbles may move with a higher velocity than the present estimate.

Furthermore, as pointed out in the previous chapter, bubbles which are large enough to

have a mobile surface region for a given surfactant concentration, may experience a

transition at a certain value of Ca in that the bubble velocity becomes greater than the

average velocity of the surrounding fluid (i.e., U > P). Such a transition, then, will make

it possible for the bubble to take on an unusual shape such as long- or short-tails as

observed previously.6 This argument, of course, is only qualitative and remains to be

proven rigorously.


Experiments with air bubbles in an oil-filled Hele-Shaw cell have shown that the

bubbles move with a velocity comparable to the predictions of Taylor and Saffman. In

addition, they are always elongated in the flow direction unlike the bubbles influenced by

surfactants. This difference in the steady shapes between clean bubbles and surfactant-

laden bubbles may also be explained by the current study at least qualitatively. When

two identical elliptic bubbles move with the same velocity but with different orientation

(i.e., one moving with its longer axis in the flow direction and the other with its shorter

axis in the flow direction), our calculation indicates that the prolate bubble experiences

smaller drag force (or smaller energy dissipation) than the oblate one in the absence of

surfactant effect. In case of surfactant-laden bubbles, on the other hand, the prolate

bubble results in a larger energy dissipation than the oblate one due to the larger drag in

the thin film region. Thus, the elongation in the flow direction (i.e., prolate bubble) may

be a preferable shape for clean bubbles whereas the oblate shape is preferable for

surfactant-laden ones.








3.4 Summary

A theoretical calculation has been presented to estimate the translational velocity of

bubbles or drops in a Hele-Shaw cell under the influence of surface active substances.

Assuming that the solid wall is wet by the surrounding fluid and that the bubble shape is

elliptical, an analytic expression has been derived for the bubble velocity. The calculated

velocity is apparently in reasonable agreement with our own experimental findings and

also with those of Kopf-Sill and Homsy. This result along with our experimental

findings suggest that most of the observations by Kopf-Sill and Homsy are probably due

to the influence of surface active substance which may be present in the system as

contaminants.














CHAPTER 4
DISPLACEMENT OF A POWER-LAW FLUID BY INVISCID FLUID


4.1 Background

Two phase displacement flow is a problem of industrial importance in areas such

as enhanced oil recovery, separations etc., and has been studied by various researchers in

the past4. When the displacing fluid is of low viscosity, then instabilities can result

causing viscous fingering. Park and Homsy have theoretically analyzed the two-phase

displacement of Newtonian fluids in a Hele-Shaw geometry using the technique of

matched asymptotic expansions.'5 Analytic expressions for both film thickness and

pressure drop and their dependence on capillary number were derived. When the

displaced fluid is a shear thinning fluid, these results can be very different. In this chapter,

the displacement of a power-law fluid by an inviscid fluid (i.e. air/gas) in a Hele-Shaw

cell is analyzed using the same technique of matched asymptotic expansions. The present

analysis is aimed at deriving the film thickness and understanding the stability issue

involved in this process. This problem has practical applications in the general area of

polymer processing and specifically in applications such as gas assisted injection

molding.

Figure 4. 1 is a schematic of an inviscid fluid displacing a power-law fluid in a

Hele-Shaw geometry. It is assumed that the displaced fluid wets the wall and there will be

a thin film left behind after the displacement. The displacement is slow enough to ignore






















Re ion x = f(z)
Region II
Region III


x Fluid 2 (air/gas) ) 2b Fluid 1 (Power-law fluid)

z


Figure 4. 1 Schematic of an inviscid fluid displacing a power-law fluid in a Hele-Shaw
cell








the effects of inertia and the gap thickness is small enough to ignore the effects of gravity.

Under these assumptions, the interface remains symmetrical about the center plane. The

tip of the interface is assumed to be moving with a constant velocity U and the analysis is

carried out in a frame of reference moving with the tip velocity.


4.2 Basic Equations

The governing equations relevant to the problem are as given below.

VOv=0 4.1

Vp = T](S)V2v + 2D e Vil 4.2


Where D = (Vv+VvT) 4.3a


S= 2(D :D) 4.3b

n-I
ri=KS 2 4.3c
K is called the consistency factor and n is called the power law index.

Boundary conditions are

u=-U, v,w=0 at y=b 4.4a



nev=0
tTn=0 at y =h(x,z) 4.4b
t2oTon=0
noTn=c(VOn



Here rl is the viscosity of the power-law fluid which depends on the shear rate and

(u,v,w) are the components of velocities in the (x,y,z) directions respectively. n, t,, and t2








are the unit normal and unit tangential vectors to the interface, cr is the interfacial tension

and T is the stress tensor. The origin for the reference frame is placed at the tip which is

moving with a velocity U. b and h(xz) are the half gap thickness and the location of the

interface respectively. f(z) is the projection of the tip of the interface onto the xz plane.

Since the interface is symmetrical about the center plane, only the bottom half (i.e.

y = -h(x, z))is considered for the analysis. The following two dimensionless parameters

appear during the scaling of the above problem.

b K___b_-_
6 = and Ca =
L a


c is a parameter which represents the ratio of the two characteristic length scales of Hele-

Shaw geometry and capillary number, Ca, represents the ratio of viscous to interfacial

tension forces. A complete solution to this problem is very complicated to obtain, and

only an asymptotic solution using the above mentioned two small parameters is attempted

in the following sections.


4.3 Scalings and Regions

As shown in Figure 4. 1, the fluid domain is divided into three different regions;

Constant film thickness region (Region I), front meniscus region (Region II), and power-

law fluid region (Region III). The flow profile in Region III can be obtained by straight-

forward integration of the governing equations and is given in the next section. In the

constant film region the flow field resembles plug flow in the moving reference frame.

The flow profile in Region II can be understood only after a rigorous analysis as

described below. The following scalings are used to non-dimensionalize the equations.









(u,v,w)-(U,U,cU)

(x,y,z) (b,b,L )

-a )
p t-e S (Ub)e
Using the above scalings, the Equations 4.1- 4.4 are written as given below.


n-I
px = Ca S (Ux3 + Uyy


U, +V y +9 2Wz =0


n-3 f


2uxSx + (Uy +* V)Sy

+6 2 (U" + WX)Sz I


py :


+ Vyy +6 2Vz ) +


n-3
S 2
2+ +


p =Ca S2(wx + Wy +62 n Sn {2 3-(1 + W ,)Sx + 2}
Whe2 v + wy)SY + 2wzS


(U w 2) + (uy + Vx)' + +w ) 62(U. )
Where S = 2(u2 + c v +Wx 6(Z+W


at y = -1,


at y = -h(x,z),


I +(hkv -hv.)


4.6a


4.6b




4.6c


u=-l, v=w=O


uh. + v + ewh = 0


4.8a




4.8b










n-1 (h + 62h { h (U + h.u) 4.8c
+ V)+g-:( v: +w)U =0




n-1
-h,(l++2h)-2h(l+h2)+2e2hh 2CaS 2-A
Ap = +-
(1+h + 2h2)i (1+h+e2h ) 4.8d

where

A=[-h(u + V 2 h v(v + W)_ 2hx h(u. + wx) vy h ehw,]


The subscripts x,y,z in the above equation represent partial differentiation of the

respective variables. To solve the above set of equations, we can begin by setting both ,

and Ca equal to zero. The resulting equations indicate that pressure is hydrostatic and is

independent of (x,y,z). The shape of the interface is a portion of a circle which can be

obtained by solving the normal stress condition. Hence this sub-region (Outer region of

Region II) may be referred to as capillary statics region where pressure and interfacial

tension are both important. In order to match the solution smoothly to the constant film

region (Region I), there should be a second sub-region (Inner region of Region II) where

the interface profile is smoothly matched to a constant value. This may be called as

transition region. In the transition region both pressure and viscous effects are important

and the variables need to be rescaled accordingly. Under the lubrication approximation,

the scalings relevant for this region can be derived by balancing the pressure and viscous

terms in the momentum equation and by balancing the pressure and interfacial tension

forces in the normal stress condition. These scalings are different from the ones used for






















ly



III III




Figure 4. 2 Schematic showing the scales and coordinate systems for different regions.
(A: Transition region; B: Capillary-statics region)








the Newtonian case and they depend upon the power-law index and capillary number as

mentioned below.


_)(v wheek
u, v, w ,w where k = -
=W),Cak 2n+1

Ix z=x+1 +
y Cak Ca2k )

-h ; p = Ca4kS
ca2k P P; S=aS


The variables in the transition region are denoted by an overbar. As indicated in Figure

4.2, the origin for the transition region is x = -1 and y = -1. The unknown I will be

determined later using the matching conditions. Using the newly scaled variable, the

governing equations valid for the transition region are given below.


uX + Vy + C2Cak w = 0



[ ca2ku (
- -n- nI-n-3
p-= S 2 +U- +
\+E2Ca4k u 2


4.9


4.10a


_ Y=n-I
py; = Ca2k-S 2


n-I
cakp =S 2


n-3
+ nICa22
2


3 Ca2k (Cak uz + wx )SX +1
n 2 S2~ a -+- z y)-

+22Ca4k wZSz


4.10b


4.1 Oc










Where S =


at y = O,


at y -z),


u=-1, v=w=0


uh v + 2Cak whz = 0


'(akhzuy- hxw)+ 2Ca'*kh-x hz-(s2Ca

+Ca2k(Cak -- + -)(i 2Ca2kjKJ


+ ca3k (h _x =z 0


Ca 2k (h2+,2Ca2kh2J


hz(Ca3k vz


2

Ca Ca -


2


- hx(u + Ca 2k VX)c 2Ca k h (Ca 3k VZ +w)


4.12c


+ 2Ca4k-hJ + e2Ca2kh(1 + Ca2kh) 2e2Ca4kh-hzh


(1 + Ca2kh2 + 62Ca4k-h2)2


n-I
S 2


+ Ca2kV ) + 2Cak (Ca3k + Wy

c 2Ca 3kh;-(Ca k + )-v;

_ Ca2k-h2- _4 Ca5k-i2w-J
x z


n-I
S2


4.11


4.12a


4.12b


n-i
S 2


+2 Ca2k


Ap=


2Ca-2kn+k+l


I + Ca2k-h2 + ,2Ca4k-h 2


.1 2d








4.4 Small Parameter Expansions
To obtain the solution, all the variables (u, w, p, h etc..) are expanded as given

below.

p=(p00 + CaapOl+ Ca2aP02 +...)+82(p1O+Caapll+ Ca2apI2 +...) +. 4.13


The choice of gauge function, a is obtained by comparing all the exponents that appear

in the governing equations and boundary conditions and choosing the least common

multiple of them so that matching is possible at every order. For example, the choice of a

few gauge functions are as given below.


a ..2nk ifn= 1; etc. 4.14

nk if n = I ;I;-etc.

Since the gauge function depends on the power-law index, only the leading order solution

is attempted in the following sub-sections. Once the choice of gauge function is known,

higher order corrections can be obtained following the procedure of Park and Homsy.

Consequently, in the present analysis the dynamic pressure jump across the interface can

not be determined since it appears at a higher order.

4.4.1 Capillary Statics Region :

Substituting 4.13 into 4.5-4.8, the leading order governing equations and

boundary conditions for the capillary statics region are as given below.

T 00 =0
uO + =y 4.13

00 00 004
PX PY PZ 4.14









Boundary conditions, u00 = -1, v oo = 0 at y = -1 4.15a

at y = -h(x,z) uh + v = 0 4.15b


ApOO -h 4.15c

(1+ hxo2 )2

[(h uyoo +oo + 2h~hu + h u.o2 (u? + w ) + (h2v hv)] 0 4.15d
h UO+ vOO) + 2h n ( V, h,(uoo V0]

4.15e


As evident from the above equations, the viscous force term in the momentum equations

is negligible and the pressure is independent of (x, y, z). The shape of the interface can be

obtained by integrating the normal stress balance and is given as below.



=oo 00 [1 {Apoo(x -f(z))+ 1}12]2 4.16


The two boundary conditions that were used in the integration and which apply at all

orders are


hx -oo as x f(z) 4.17
h = 0 at x = f(z)

The solution 4.16 indicates that the tip of the interface is a portion of a circle modulated

in the z-direction by f(z). This solution contains one unknown Apo, which will be

determined using the matching conditions. Equation 4.16 is not uniformly valid in the

entire region and it needs to matched with the constant film region (Region I).








4.4.2 Transition Region

Variables in Equations 4.9-4.12 are expanded using 4.13 and the leading order

terms are gathered to obtain the governing equations for the transition region.

-00 -00 4.18
ux +Vy =0

0(-oo,, 4.19a
0 Y 4.19b
pyOO = 0

-00 4.19c
-0000
0=w + (n-1) -- -wy
Uy
--00 -00 00 4.20
at y=O u =-1, v =w 0
at y h oo(x ) -ooo =0O 4.21a
aty= hx, z) u hX -00_0 = 4.1
-00 -00 42l
Ap = hoo 4.21 b
00
u7 =0 4.21c


By solving these equations and matching with the constant film thickness region (Region

I) the velocity profile can be determined. Once the velocity field is known, then normal

stress along with kinematic condition results in an evolution equation for the shape of the

interface as given below.

n
-oo = 2n +I n0oo -t
xxX nh i 4.22


When n is set equal to one, the above equation reduces to the Newtonian case (Park and

-00
Homsy 1984). t represents the leading order approximation for the constant film

thickness which will be determined later using the matching conditions. For numerical








integration, the above equation is transformed into a canonical form by using the

following transformations.

-00
H00 = 4.23
-00
t
x+s n+2
X where n+2 4.24

(t 00) 3

s is an arbitrary shift factor which is determined by higher order matching conditions.

With these transformations, the evolution equation is written as follows.


H00 =(2n+l (H00 1) 4.25


The condition for matching with region I now becomes

H -- 1 as X -- -oo 4.26

Equation 4.25 can be numerically integrated, using small slope and small curvature as

initial conditions. A 4thorder Runge-Kutta routine is used to do the numerical integration.

The shape of the interface depends on the power-law index and it is plotted in Figure 4.3

for the case where n = 0.5 along with the profile for a Newtonian case (i.e. n = 1.0). Since

as X -+ +o the interface profile H also goes to infinity, it is possible to approximate the

profile using a quadratic form as given below.


H0(x,z) = 1COX2 + Cj(z) X + C2(z) 4.27
2

Equation 4.27 can be written in terms of the original scalings as given below.



















5000


4000 N
Power-law
n=0.5
3000
nOO

2000
nl
1000


0
0 20 40 60 80
x


Figure 4. 3 Comparison of interface profiles for Newtonian (n=l) and power-law
(n=0.5) fluids.











2t(z-C 002,6-1 0 2,8-1 -, +1 4.28



The unknown t will be determined by the following matching conditions.


4.4.3 Matching Conditions

The solution in the transition region has to be matched with the solution in the

capillary statics region using the following equation.


Jim { Ca2 z =im h(x,z) 4.29

This limits are to be interpreted in terms of the matching principle by Van Dyke (1964).

By expanding h(x,z) about x = -1 using Taylor series expansion, rewriting the expansion

in inner variables and comparing it term by term with the left-hand side, matching

conditions for each order can be determined as given below. Higher order matching

conditions can be derived once the choice of ac is known.

OOCao) hoo(_o,Z) = 1 4.30

0cak) hxO(-10,z)=0 4.31


OCa2k h. 000 3
h0(-_0' Z) = _C0(j) 4.32


Physically these conditions mean that the outer static solution meets the wall with zero

slope and the curvature matches with that of transition region solution. Solving the above

equations, the following unknowns in the solution can be determined.








A00 = I 4.33a
0 = 1-(z) 4.33b

-00 3
t = (Co)2,,+1 4.33c


The above solution determines the location of the origin and also the film thickness to a

leading order. Since the film thickness is stretched by Ca2k, the dimensionless film

thickness is given by

3[U~bn n 4.34
too= (Co)21 K -bI 2n+1


Figure 4.4 displays the dependence of the constant Co on the power-law index n. As

n -> 1 the constant C. approaches 1.337 which is the value for the Newtonian case. In

Equation (4.34), if K is replaced by go /7 o where go is apparent viscosity and 70 is a

reference shear rate, then the film thickness is given as follows.


32(1-n) 2n 4.35
to = (C0) 2nLl Ca,2n+' where Ca' atg



Figure 4.5 is a plot of Equation (4.35) which is a straight line on a log-log scale. All the

variables influence the intercept while only the power law-index influences the slope of

the straight line plot. In the Newtonian case, all the fluids have a master curve (i.e. the

plot for n = 1) whereas each power-law fluid has a separate curve uniquely defined for the

given operating conditions.






















2.5
0


2.0

C0 o.
'01
1.5 0
1.337 for Newtonian Fluid 0



1.0
0 0.2 0.4 0.6 0.8
Power Law Index (n)


Figure 4. 4 Plot of the Coefficient Co as a function of the power-law index


























0~


7 -6 -5 -4 -3 -2 -1


*A
A X
X


X x


X


A
A

x x


n = 1.0

n = 0.7

A n = 0.5


X


n = 0.3


Log Ca




Figure 4. 5 Log (film thickness) vs Log Ca for a power-law fluid. ( Io = 100 cP; a = 30
dyns/cm; y, = 1 sec'; b = 0.06 cm)









4.5 Stability Analysis

In order to carry out the stability analysis, it is important to derive the flow field in

the power-law fluid region. Although far away from the interface, the flow is only along

the x-direction, near the interface a two dimensional flow field is considered. The solution

to this region is fairly straight forward. It is assumed that the gradient of components of

velocities in xz direction are negligible and only the gradients in the gap direction (i.e. y

direction) are important. The simplified governing equations and boundary conditions are

given below.


Px = K S2 where S= Uy 2y
y 4.36a

n-1
p = K S-2-Wy
y 4.36b

py = 0 4.36c

at y=b u=w=0 4.37a

at y=O Uy,Wy = 0 4.37b


The above set of equations can be solved to obtain the velocity field as given below.

n 21-- (n n+l +
U. =, n I + P 2 2 y n b n4 .3 8 a
n [ 2
n [PX2]K r ~ .8



W n Pz 1-] n n-bn
Wn +IPx l 2, b 4.38b








The components of velocity are averaged in the gap direction (i.e. y direction) and the

resulting velocity profile is written in the vector form as given below.


-v nb (b P11-I
v 2n+ 1K) p 4.40



When n = 1, the above equation reduces to Darcy's law. However, in the case of power-

law fluids, it may be noted that the average velocity field is non-Darcian. The interface is

assumed to be flat across the gap and moving with a constant velocity U. Then, the

average velocity in the power-law fluid, ug, can be related to the velocity of the

interface, U, as given below. The average velocity in the lateral direction is zero.

~ where b film thickness 4.41

b

The location of the interface is denoted by x = rio which is a function of time and without

loss of generality the pressure inside the inviscid fluid may be set equal to zero. Now,

equation (4.41) is substituted in equation (4.40) and the steady state solution to the depth

averaged pressure field can be obtained as given by the following equation.

p S = _((2n + 1)_ U K( 1. )
nb ) b -)o 4.42


Equations (4.41) and (4.42) constitute the base state solution or the steady state solution

for the stability analysis. The stability of this solution is analyzed following the standard

procedure for the linear stability analysis. The steady state velocity and pressure field is

perturbed by introducing disturbances in all the physical variables as given below.









u = U + U' w= w' p=pS +p, x = 776 + i'(z, t) 4.43

Here the superscript prime indicates that it is a perturbation from the steady state solution.

By substituting in equation (4.40), the relation between the perturbed velocity and

pressure can be written as follows.


I = bU[(2n+l)TU]-P'x 4.44a


w' bU [(2n+l)TU -nP' 4.44b
K [ nb 4.4

According to the linear stability theory, the perturbation in the velocity field can be

written as follows.

u'= y(x) exp[ikz+ 8t] 4.45

Here k is the wave number and 8 is the growth rate of the disturbance. Using equation

(4.45) and the continuty equation (i.e. u + w' = 0), an expression for w' can be

obtained. By substituting u', w' in equation (4.44) and eliminating pressure the following

ordinary differential equation is obtained.

y" nk2Y =0 4.46

Far away from the interface, into the power-law fluid region, the disturbance dies out.

Hence the above differential equations can be solved assuming that as x -+ 0, T + 0.

The resulting solution is as given below.


y(x)= a exp(-nk2 x) 4.47


Now, the components of the perturbed velocity, u', w', are determined to a constant. By

substituting them in equation (4.44) and integrating, the pressure field in the power-law








fluid region can be determined. As indicated in equation (4.43), the instantaneous

location of the interface is given by x, which can be differentiated to give instantaneous

velocity in the x-direction as mentioned below.

(x r76)t = 17t = U'x= o +rf 4.48

Equation (4.48) can be integrated with respect to time to obtain an expression for the

perturbed location of the interface (i.e. rq'). The pressure jump across the interface can be

written as


Pi P' = r zz
4

The constant ;'/4 appearing in equation (4.48) was first derived by Park and Homsy

(1984) as an 0(82) correction that accounts for the curvature change in the z-direction.

Substituting the deviation variables in the above boundary condition, the dispersion

relation for the growth rate of the disturbance can be written as follows.


15 kI_ nb b____k 4.50
T=n (2n+1) U 4K .50


When n is set equal to one, the dispersion relation for the Newtonian case is recovered

(Saffman & Taylor 1958). Figure 4.6 is a graph of 8 as a function of the wave number. If

8 is positive then the disturbance grows at an exponential rate and the interface is

unstable to small disturbances. If 6 is negative, then the disturbance is damped at an

exponential rate and the motion is stable to small disturbances. As is evident from the

graph, there is an optimum value for the wave number for which the growth rate is

maximum and beyond which value, the growth rate monotonically decreases. So 5 = 0





















0.040
0.035
0.030
0.025 Decreasing n
0.020
0.015
SO~lO /./ \\ kcut-.9
o0.010
0.005 k

0.000

-0.005
-0.010
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Wave number k







Figure 4.6 Disturbance growth rate (8) as a function of wave number (k). (It = 100 cP,
y'=1 s-', n = 0.6, b = 0.06 cm c = 30 dyns/cm, U = 0.05 cm/s)





















0.8



0.7



0.6



0.5



0.4
0.0


0.2 0.4 0.6 0.8


Power-law index (n)





Figure 4.7 Wave number for the maximum growth rate as a function of the power-law
index. (0.=100 cP, y-= 1 sec1, U = 0.05 cm/s, a = 30 dyns/cm, b = 0.06 cm)








represents the cut-off wave number whose expression is as given below.


kc 4K J(2n+)t 451
ut- nb


Hence the wave numbers greater than the cut-off wave numbers die out while the smaller

ones grow at different growth rates. The wave number with the maximum growth rate can

be obtained by differentiating 8 with respect to k and setting it equal to zero and it is

related to the cut-off wave number as given below.


k. ktff 4.52


In equation (4.51), the consistency index K can be replaced by apparent viscosity (g.) and

a reference shear rate(y) as K = ,Uo/yn-I, then equation (4.52) can be plotted for

various power-law fluids as given in Figure 4.7. While making this comparison, it should

be kept in mind that the reference shear rate should be high enough so that power-law

model is applicable to all the fluids being compared. Because, one of the limitations of

power-law model is that it can not describe the viscosity behavior at low shear rates.

The result in Figure 4.7 indicates that, the wave number for maximum growth rate

(kn,) decreases as the power-law index reduces. In other words, the wave length for the

maximum growth (km) is greater for power-law fluids. To experimentally observe this

wave length, the width of the Hele-Shaw cell (L) should be larger than Xm. Hence, for

fluids with similar viscosity at a constant reference shear rate, it may be concluded that

the interface will be less sinuous for a power-law fluid than for a Newtonian fluid.








4.6 Summary

The displacement of a power-law fluid by an inviscid fluid is analyzed and a

leading order solution is obtained using the technique of matched asymptotic expansions.

For slow displacement, the shape of the interface in the capillary statics region is a

portion of a circle which is smoothly matched to a parabolic profile in the transition

region. Under the lubrication approximation, the shape of the interface in the transition

region is obtained by numerically integrating the evolution equation. Matching conditions

with the capillary statics region determine the film thickness of Region I. The flow field

in Region III is determined and found to be non-Darcian. Assuming that the interface is

flat, the steady state solution for the velocity and pressure field is determined. Stability

analysis for this steady state solution is carried out to determine the wave number with

the maximum growth rate. The result indicates that the interface will be less sinuous for

power-law fluids than Newtonian fluids.














CHAPTER 5
CONCLUSIONS


The experimental findings of Kopf-Sill and Homsy on the motion of air bubbles

in a Hele-Shaw cell filled with Glycerin-water mixture have raised the following issues.

" The velocities of small air bubbles were found to be an order of magnitude smaller

than the theoretical predictions of Saffman and Taylor.

" A variety of unusual steady bubble shapes and a sudden transition of bubble velocity

and shape at a critical capillary number were observed.

The present study explains the above observations in the context of surface active organic

contamination.

To study the influence of surfactants, experiments were conducted both using air

bubbles and water drops containing surfactant in an oil-filled Hele-Shaw cell. While the

motion of air bubbles is in agreement with the predictions of Saffman-Taylor theory, the

velocity of water drops is an order of magnitude smaller than the predicted value. Unlike

the air bubbles, the water drops exhibited a variety of shape transitions similar to the ones

observed by Kopf-Sill and Homsy. In the presence of surfactants, the bubble motion can

induce a surface concentration variation resulting in Marangoni stresses along the drop

interface. These stresses can cause the drop surface to be rigid. In the presence of a

wetting film, the dissipation in the thin film region can be significantly high and

consequently the velocity of the water drops is decreased by an order of magnitude. The








observed shapes can also be explained using the same argument. The motion of the

bubble creates a spatial variation of surface tension. In order to keep the pressure

constant, the curvature normal to the gap direction adjusts itself resulting in a variety of

shapes. Theoretical modeling was done for an elliptic bubble assuming that the entire

surface is rigid. The theoretical predictions agree with the experimentally observed

velocities. This calculation also suggests that a sideways elongated shape is the preferred

one in the presence of surfactants.

In Chapter 4, the displacement of a shear thinning power-law fluid by an inviscid

fluid is analyzed. The flow domain is divided into three different regions and a leading

order solution is obtained using the technique of matched asymptotic expansions. Since

the displacement is very slow, in the capillary statics region the shape of the interface is

nearly circular. Under the lubrication approximation, the interface shape in the transition

region is obtained by numerically integrating an evolution equation. Matching condition

with the capillary statics region provides an expression for the film thickness in the

constant film thickness region. In the case of Newtonian fluids, irrespective of individual

physical properties, capillary number uniquely determines the film thickness. On the

other hand, in the case of shear thinning power-law fluids, the relationship is unique to a

given fluid and flow conditions. Just as in the Newtonian case, as the Capillary number is

increased, the film thickness increases linearly on logarithmic scale. However, at a given

capillary number, fluids with a lower power-law index have a smaller film thickness. The

flow field away from the interface in the power-law fluid region is also solved assuming

that the gradients of components of velocities are negligible in the directions normal to

the gap direction. Unlike the Newtonian case, the depth averaged velocity field is found





79


be non-Darcian. A stability analysis is carried out for a flat interface and a dispersion

relation is derived for the growth rate as a function of wave number. The result indicates

that the wave number for the maximum growth rate decreases as the power-law index

decreases. In other words, the dominant wave length increases as the fluid becomes more

and more non-Newtonian. Hence, the interface will be less sinuous for the power-law

fluid













REFERENCES


1 G. I. Taylor and P. G. Saffman, "A note on the motion of bubbles in a Hele-Shaw
cell and porous medium," Q.J.Mech.Appl.Math. 12, 265 (1959)

2 P.G. Saffman, and G.I.Taylor, "The penetration of a fluid into a porous medium or
Hele-Shaw cell containing a more viscous liquid," Proc. R. Soc. London Ser. A
245, 312 (1958).

3 S. Tanveer, "The effect of surface tension on the shape of a Hele-Shaw cell
bubble," Phys.Fluids 29, 3537 (1986)

4 S. Tanveer, "New Solutions for steady bubbles in a Hele-Shaw cell," Phys.Fluids
30, 651 (1987)

5 T. Maxworthy, "Bubble formation, motion, and interaction in a Hele-Shaw cell,"
J.Fluid.Mech. 173, 95 (1986)

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Phys.Fluids 31, 18 (1988)

7 W. Eck and J. Siekmann, "On bubble motion in a Hele-Shaw cell, a possibility to
study two-phase flows under reduced gravity," Ing. Arch 47, 153 (1978)

8 W. Johann and J. Siekmann, "Migration of a bubble with adsorbed film in a Hele-
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10 D. A. Reinelt, "Interface conditions for two-phase displacement in Hele-Shaw
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Fluids 30, 2624 (1987).

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Thin film and contact angle effects," Phys. Fluids A 1, 219 (1989).

13 D. Burgess, and M.R.Foster, "Analysis of the boundary conditions for a Hele-
Shaw bubble," Phys. Fluids A, 2, 1105 (1990).









14 G.M.Homsy, "Viscous Fingering in porous media," Ann.Rev.Fluid Mech.,19:271-
311 (1987)

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New York, 1988).

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capillary tube," Phys.Fluids A 4, 2335 (1992)

21 K.J.Stebe, S.Y.Lin, and C.Maldarelli, "Remobilizing surfactant retarded fluid
particle interface. I. Stress-free conditions at the interfaces of micellar solutions of
surfactants with fast sorption kinetics," Phys. Fluids A 3, 3 (1991).

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BIOGRAPHICAL SKETCH


Krishna Maruvada was born in Andhra Pradesh, India. He got his bachelors

degree in chemical engineering from Birla Institute of Technology & Science (BITS),

Pilani, India, in 1990. Subsequently he worked for 2 years for a chemical company called

Sriram Fibers, at New Delhi as a projects engineer. Later he joined BITS, Pilani and

served as a lecturer for one year. In 1993, he joined the graduate program at the

University of Florida, Gainesville, and started working toward his Ph.D. degree.









I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.



Chang-*j Park, Chairman
Associate'lrofessor of Chemical Engineering

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.



Dinesh 0. Shah
Professor of Chemical Engineering

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.


Ranga Narayanan
Professor of Chemical Engineering

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.



Lewis Johns 4
Professor of Chemical Engineering

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.



Wei Shyy
Professor of Aerospace Engineering,
Mechanics and Engineering Science








I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.



Richard B. Dickinson
Assistant Professor of Chemical Engineering


This dissertation was submitted to the Graduate Faculty of the College of
Engineering and to the Graduate School and was accepted as partial fulfillment of the
requirements for the degree of Doctor of Philosophy.

December, 1997


Winfred M. Phillips
Dean, College of Engineering




Karen A. Holbrook
Dean, Graduate School




Full Text

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