Extensional flow behavior of turbulent drag reducing polymer solutions

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Title:
Extensional flow behavior of turbulent drag reducing polymer solutions
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xix, 416 leaves : ill. ; 28 cm.
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English
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Shands, Jay Anderson, 1952-
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Subjects / Keywords:
Drag (Aerodynamics)   ( lcsh )
Polymers -- Viscosity   ( lcsh )
Polymers -- Rheology   ( lcsh )
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bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1985.
Bibliography:
Includes bibliographical references (leaves 408-415).
Statement of Responsibility:
by Jay Anderson Shands.
General Note:
Typescript.
General Note:
Vita.

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University of Florida
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 000880464
notis - AEH8283
oclc - 14957881
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AA00003408:00001

Full Text













EXTENSIONAL FLOW BEHAVIOR OF TURBULENT
DRAG REDUCING POLYMER SOLUTIONS










BY

JAY ANDERSON SHANDS


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



UNIVERSITY OF FLORIDA


1985
















ACKNOWLEDGEMENTS


The author would like to express his gratitude to

Professor E.R. "Tex" Lindgren for his supervision,

support, and numerous valuable opinions.

The author would also like to thank Professors U.H.

Kurzweg, E.K. Walsh, and T.T. Bowman for their lectures

and services on the supervisory committee. For joining and

serving on the supervisory committee at a late date, the

author also thanks Professor D.W. Mikolaitis.

For friendship, for many stimulating discussions, and

for assistance in work on various projects, the author

takes the pleasure in thanking Dr. Sergio Zarantonello,

Dr. Henrik Alfredsson, Walter Valerezo, Rikard Gebart, Jan

Hornfeldt, Jean-Marc Holt, Eric Larsson, and Peter

Karsten.

Last, but not least, the author is indebted to his

parents and to his wife, Holly, for their patience and

support.










ii
















TABLE OF CONTENTS


Page


ACKNOWLEDGEMENTS .......................................ii

LIST OF TABLES ....................................... vii

LIST OF FIGURES ..................................... viii

NOMENCLATURE .................. ......................... xii

ABSTRACT ............................................ xvii

CHAPTER

I INTRODUCTION .................................... 1

Summary of Dissertation ....................... 2
Significance of the Problem ................... 4

II LITERATURE REVIEW ............................... 6

Drag Reduction ..... ........................... 6
Main Features of Drag Reduction.............. 7
Relevant polymer-solvent
characteristics..... ...................... 7
Mechanical properties of drag-
reducing solutions....................... 9
Gross aspects of drag reduction .......... 10
Turbulent flow structure ................. 14
Polymer interactions ..................... 18
Theories of Drag Reduction ................. 22
General Characteristics of Viscoelastic
Liquids .................................... 26
Experimental Correlations of Viscoelastic
Characteristics with Drag-Reducing
Ability .................................... 29
Extensional Flow of Viscoelastic Liquids..... 40
Kinematics of Extensional Flows ............ 40
Experimental Methods....................... 41
Comments on the Use of Conical Channels
to Examine Extensional Flow Behavior..... 53


iii










III EXPERIMENTAL ARRANGEMENT ....................... 56

Drag Reduction Apparatus ..................... 56
Conical Channel Rheometers ................... 64
General Arrangement ........................ 64
Design Considerations ...................... 65
Conical Channel Details .................... 68
Sliding Ball Viscometer....................... 70
Polymer Additives ............................. 72
General Structural Characteristics ......... 72
Solution Preparation........................ 74

IV SHEAR VISCOSITY MEASUREMENTS ................... 75

Goal ......................................... 76
Scope ........................................ 76
Viscometer Flow Analysis ..................... 76
Viscometer Calibration ....................... 80
Experimental Procedure....................... 84
Analytical Procedure ......................... 85
Experimental Results.......................... 86
Comments on the Viscometer Flow Analysis..... 91


V DRAG REDUCTION MEASUREMENTS .................... 93

Water Measurements............................. 95
Polymer Solution Measurements for Varying
Reynolds Number ..... .................. .. .. 99
Goal ....................................... 99
Scope......................................... 99
Experimental Procedure ....................100
Analytical Procedure......................101
Experimental Results.......................107
Polymer Solution Measurements for Varying
Concentration at Constant Reynolds Number.125
Goal ...................................... 125
Scope ..................................... 125
Experimental Procedure ....................126
Analytical Procedure....................... 128
Experimental Results ...................... 128
Discussion ................................ 133


VI EXTENSIONAL FLOW MEASUREMENTS .................137

Water Measurements .......................... 138
Preliminary Polymer Solution Measurements...145
Goals ..................................... 145
Scope ................. ..................... 146
Experimental Procedures...................146










Analytical Procedure ...................... 148
Determination of extensional strain
rates ................................. 149
Determination of first normal stress
differences............................ 153
Energy balance........................... 160
Aging analysis .......................... 175
Experimental Results ...................... 177
Discussion ................................ 195
Total pressure drop experiments ......... 195
Aging and shear degradation ............. 196
Comparison with simple fluid theory..... 197
Comparison with theory for a suspension
of elongated large scale particles.... 202
Duration of stretching ..................208
Strain................................... 210
General comments ........................ 212
Comparison with previous experimental
results................................ 213
Summary ................................... 215
General Polymer Solution Measurements.......217
Goal ...................................... 217
Scope ................ ...................... 217
Experimental Procedure .................... 218
Experimental Results......................219

VII COMPARISON OF DRAG REDUCTION, EXTENSIONAL
FLOW, AND SHEAR VISCOSITY MEASUREMENTS...... 235

VIII SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS
FOR FUTURE WORK.............................250

Summary of Results and Conclusions .......... 251
Recommendations for Future Work.............261


APPENDICES

A LIST OF EQUIPMENT USED AND THEIR
MANUFACTURERS/VENDORS ......................... 265

B CALIBRATION OF THE HOPPLER VISCOMETER......... 267

Calibration Measurements ...................268
Determination of Frictional Resistance..268
Determination of Calibration Constants
K and L....... ........................271
Verification of Flow Transition Criterion..275

C VISCOMETER MEASUREMENTS ....................... 277









D PIPE FLOW MEASUREMENTS MADE ON VARIOUS
REYNOLDS NUMBER FLOWS ....................... 284

E PIPE FLOW MEASUREMENTS AT VARIOUS
CONCENTRATIONS FOR CONSTANT REYNOLDS
NUMBER FLOWS ................................ 348

F HEAD-DISCHARGE RELATIONSHIP FOR CONICAL
CHANNEL FLOW AS DETERMINED FROM THE
BOUNDARY LAYER SOLUTION FOR INCOMPRESSIBLE
AXISYMMETRIC FLUID FLOW ..................... 357

G EXTENSIONAL FLOW MEASUREMENTS ................. 362

H DETERMINATION OF LINES OF ACTION OF NORMAL
STRESS DIFFERENCES........................... 403

BIBLIOGRAPHY ......................................... 408

BIOGRAPHICAL SKETCH ....................................416
















LIST OF TABLES


Table Page


3.1 Experiment pipe diameters.................. 60

4.1 Results of shear viscosity measurements.... 87

4.2 Results of shear viscosity measurements.... 89

5.1 Summary of drag reduction data............ 119

6.1 Radial distances associated with the
downstream pressure taps.................. 141

6.2 Ratios of actual to inviscid flow rates,
conical channel flow ...................... 144

6.3 Estimates of effective lengths of
polymer mesh and of the function g(s)
for conical channel flow.................. 207

6.4 Summary of "onset" data for the
extensional flows ......................... 233

C.1 Viscometer calibration measurements ....... 269

E.1 Numerical determination of f[l-f'(n)]dn...361


vii
















LIST OF FIGURES



Figure Page


3.1 Drag reduction apparatus .................... 57

3.2 Schematic of in-line mixer.................. 62

3.3 Dimensions of conical channels .............. 69

3.4 Schematic of Hoppler viscometer............. 71

3.5 Chemical structure of polymer additives..... 73

4.1 Geometry of eccentric annulus ...............78

5.1 Water measurements: pipe flow,
downstream measuring section ...............96

5.2 Water measurements: pipe flow,
upstream measuring section .................97

5.3 Drag reduction trajectories: 5, 15, 30,
and 45 ppm Separan AP-273 solutions....... 109

5.4 Drag reduction trajectories: 5, 10, and
20 ppm Separan AP-30 solutions............ 110

5.5 Drag reduction trajectories: 20 and 30 ppm
Polyox WSR-301 solutions.................. 111

5.6 Drag reduction trajectories: 5 ppm Separan
AP-273 solutions with 10 and 30 rps mixing
and 0.2% and 1% master solutions.......... 113

5.7 Drag reduction trajectories: 1 and 2 ppm
Separan AP-273 solutions .................. 115

5.8 Drag reduction trajectories: 5 ppm Separan
AP-273 solution, upstream and downstream
measuring sections........................ 117


viii









5.9 Wall strain rate at "onset" as a
function of additive concentration........ 121

5.10 Slope difference as a function of
additive concentration..................... 123

5.11 Specific drag reduction versus
concentration: Separan AP-30.............. 129

5.12 Specific drag reduction versus
concentration: Polyox WSR-301............. 131

5.13 Specific drag reduction versus
concentration: Separan AP-273 ............. 132

6.1 Water measurements: conical channels....... 139

6.2 Spherical coordinate system ................ 150

6.3 Control volume for energy analysis......... 162

6.4 First normal stress difference versus
extensional strain rate: 20 ppm Separan
AP-273, Channel A......................... 178

6.5 First normal stress difference versus
extensional strain rate: 20 ppm Separan
AP-273, Channel C ......................... 179

6.6 First normal stress difference versus
extensional strain rate: 20 ppm Separan
AP-273, Channel D......................... 180

6.7 First normal stress difference versus
extensional strain rate: 20 ppm Separan
AP-273, Channel G ......................... 181

6.8 First normal stress difference versus
extensional strain rate: 20 ppm Separan
AP-273, Channels A, C, D, and G........... 184

6.9 First normal stress difference versus
extensional strain rate: 20 ppm Separan
AP-273, Channels G and H.................. 185

6.10 First normal stress difference versus
extensional strain rate: 15 ppm Separan
AP-273, Channels A and G.................. 187

6.11 First normal stress difference versus
extensional strain rate: 45 ppm Separan
AP-273, Channels A and G.................. 188









6.12 Comparison between predicted and
measured overall pressure drops ........... 189

6.13 Pressure drop as a function of elapsed
time ...................................... 190

6.14 Comparison of normal stress differences
for flow being conducted at different
Reynolds numbers in the main experiment
pipe ...................................... 194

6.15 Pressure head difference as a function of
flow rate/extensional strain rate: 10,
15, 20, 30, and 45 ppm Separan AP-273,
channel A ................................. 220

6.16 Pressure head difference as a function of
flow rate/extensional strain rate: 10
and 20 ppm Separan AP-273, channel D......221

6.17 Pressure head difference as a function of
flow rate/extensional strain rate: 5 ppm
Separan AP-273, 10 and 30 rps mixing,
channel D.................................222

6.18 Pressure head difference as a function of
flow rate/extensional strain rate: 1 and
2 ppm Separan AP-273, channel D........... 223

6.19 Pressure head difference as a function of
flow rate/extensional strain rate: 5, 10,
and 20 ppm Separan AP-30, channel D.......224

6.20 Pressure head difference as a function of
flow rate/extensional strain rate: 20 and
30 ppm Polyox WSR-301, channel D.......... 228

6.21 Pressure head difference as a function of
flow rate/extensional strain rate: 30 ppm
Polyox WSR-301, channel D................. 229

6.22 Extensional strain rate at "onset"
as a function of additive concentration...232

7.1 Comparison of the increase in buffer layer
thickness with extensional viscosity:
20 ppm additive solutions ................. 240

7.2 Comparison of the increase in buffer layer
thickness with extensional viscosity:
15 and 45 ppm Separan AP-273 solutions.... 241










7.3 Comparison of the increase in buffer layer
thickness with extensional viscosity: 5,
10, and 20 ppm Separan AP-30 solutions.... 242

7.4 Comparison of the increase in buffer layer
thickness with extensional viscosity:
5 ppm Separan AP-273 solution, 10 and
30 rps mixing............................. 243

7.5 Comparison of the increase in buffer layer
thickness with extensional viscosity:
1 and 2 ppm Separan AP-273 solutions...... 244

C.1 Distilled water shear viscometer
measurements .............................. 272

C.2 Viscometer calibration constant as a
function of effective load ................274

C.3 Viscometer calibration constant as a
^ ^ 2
as a function of (P-Pf)/P ................ 276

F.1 Coordinate geometry, conical channel....... 357

H.1 Variation of rN with flow rate:
20 ppm Separan AP-273, channel D.......... 405

H.2 Comparison of methods for determining
representative extensional strain rates
for flow of a Separan AP-273 solution
through channel D......................... 407
















NOMENCLATURE


Page Number
Symbol Meaning at First
Usage in Text


a radius of viscometer sphere ............... 77

a. extensional deformation rates.............40
1
A constant, logarithmic velocity profile.... 14

A constant, friction law...................104

b particle radius .......................... 203

B constant, logarithmic velocity profile....14

B constant, friction law .................... 10

C constant, conical channel head-discharge
relation.................................65

d annular width, viscometer................. 77

d maximum annular width, viscometer.........78
m
D diameter of main experiment pipe.......... 57

D.. components of deformation rate tensor.... 151

e energy per unit mass ..................... 163

ek kinetic energy per unit mass............. 163

e potential energy per unit mass........... 163

e internal energy per unit mass ............ 163

E energy ................................... 161

f friction factor ........................... 95

F (s') relative deformation gradient tensor.....197
-t


xii









Page Number
Symbol Meaning at First
Usage in Text


Functional of the history of the
relative deformation gradient........... 197

g gravitational constant.................... 65

9(s) function of particle aspect ratio........ 203

G function independent of r................ 174

h pressure head .............................66

hI head loss per unit mass.................. 165

I identity tensor.......................... 197

K constant .................................. 77

K constant ..................................79

K kinetic energy flux correction factor.... 164
e
1 length of pipe measuring section.......... 95

1 particle half length..................... 203

it viscometer sphere travel distance.........77

L effective length of viscometer sphere.....78

m constant ..................................79

M constant .................................. 79

n. unit normal vectors...................... 161

N first normal stress difference........... 155

P negative of trace of stress tensor.......153

P viscometer load...........................77

P constant representing frictional
resistance in viscometer................. 81

Q flow rate ................................ 138

Q, heat added to system ......................161


xiii










Page Number
Symbol Meaning at First
Usage in Text


r radial distance from vertex of
conical channel.........................138

r radial distance at the end of the
e conic section of a channel...............69

r radial distance at the center of the
m downstream pressure tap of a conical
channel.................................140

rN radial distance at the "center of
normal stress difference" at the
downstream tap of a conical channel.....182

r radial distance at point of "onset"...... 205

r radial distance at the "center of
P pressure" at the downstream tap of a
of a conical channel.................... 140

r value of r which for which eq. (6.1)
had "best" agreement with conical
channel head-discharge measurements.....141

R Reynolds Number (Ur/v or UD/v)........66, 89

s particle aspect ratio, r/1 .............. 203

s' history parameter, t-t'...................197

S surface area............................. 161

t time ......................................77

t" reference time........................... 197

T. components of stress tensor...............29

T'. components of additional stress.......... 154

u. velocity components.......................40
1
u (U/u,) ....................................14
1/2
u, friction velocity (T /P)...............14


xiv









Page Number
Symbol Meaning at First
Usage in Text


U average flow velocity at a distance y
from the wall, U=U(y), or the main
stream flow velocity in a conical
channel, U=(2gh)1/2..................14, 66

U mean flow velocity.........................89

U constants .................................40
0.
1
V volume ................................... 161

W work done on system......................161

x. coordinates of a material point...........40

y distance from wall........................ 14

y (yu,/' ) ................................... 14

z elevation ................................ 163

a angular coordinate........................77

y shear strain rate in viscometer........... 77

F extensional strain rate...................40

F( ) Gamma function............................ 79

5 boundary layer thickness.................. 14

6 slope difference (A -A ).................106

6.. Kronecker delta.......................... 153
2]
AB increase in buffer layer thickness....... 105

Ap pressure change recorded between taps.....95

e extensional viscosity.....................41

& angular coordinate ....................... 150

0 value of at channel wall................69

V absolute shear viscosity .................. 77

v kinematic shear viscosity................. 14










Page Number
Symbol Meaning at First
Usage in Text


p mass density .............................. 14

T shear stress.............................. 78

T wall shear stress .........................14
w
T. components of deviatoric stress tensor...165

cp angular coordinate ....................... 150

D volume concentration.....................203




Subscripts:


p pertaining to polymer solution flow (with
the exception of r and e )

w pertaining to water flow (with the
exception of Tw )

s pertaining to solvent values

a actual measured values

i inviscid flow values


xvi
















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



EXTENSIONAL FLOW BEHAVIOR OF TURBULENT
DRAG REDUCING POLYMER SOLUTIONS


By


Jay Anderson Shands


May 1985


Chairman: E.Rune Lindgren
Major Department: Engineering Sciences


Turbulent drag reduction by polymer addition may be

defined as the phenomenon by which drag in a turbulent

shear flow is reduced to below that for the solvent alone.

A popular hypothesis for this phenomenon is that polymer

addition leads to the suppression of turbulence production

by increasing fluid resistance to extensional deformations.

Large resistances to extensional deformations have been

observed in drag-reducing solutions; however, it is not yet

clear whether or not such viscoelastic effects are

essential to drag reduction. The purpose of this

investigation has been to further examine the extensional

behavior of such solutions and to determine if their


xvii









extensional characteristics are related to their drag-

reducing abilities.

Conical channel rheometers, designed to conduct nearly

shear-free flow at high extensional deformation rates (of

the order of 1000-10,000 1/s), have been used to examine

extensional flow behavior. Drag reduction has been

measured using a pipe flow apparatus. Results are reported

for aqueous polyacrylamide and polyethylene oxide solutions

ranging in concentration from 1 to 45 ppm.

For the conical channel flows, head-discharge

measurements have been expressed in terms of first normal

stress differences and extensional strain rates.

Comparisons are made with related experimental and

theoretical results. Using conservation of energy

principles, the energy transfer that occurs in a channel

has been examined and an expression has been derived that

relates first normal stress difference in a channel to the

excess pressure drop (relative to solvent flow) occurring

over the entire channel. Good agreement has been found

with experiment. Two distinct types of non-Newtonian

behavior have been observed: for the polyacrylamide

solutions, pressure heads in the conical channels were

lower than for corresponding water flows, while the

polyethylene oxide solutions induced greater pressure

heads. However, without regard to additive type, molecular

weight grade, concentration, and mixing conditions, the

solutions which exhibited the greater non-Newtonian effects


xviii









were found to be the more effective drag-reducers.

Evidence that molecular interaction can play a significant

role in both the drag reduction mechanism and in the

generation of non-Newtonian effects in extensional flows is

also reported.


xix
















CHAPTER I
INTRODUCTION


Turbulent drag reduction by polymer addition may be

defined as the phenomenon by which drag in a turbulent

shear flow is reduced to below that for the solvent

alone. This lowering of turbulent resistance can often

be considerable and may be achieved with very small

amounts of polymer additive. For example, drag in

turbulent shear flow can be reduced by as much as 80%

through the addition of just a few parts per million of

the more effective polymers.

Motivated in part by the enormous potential for

industrial applications, this phenomenon has been

extensively investigated during the past two decades.

Even though many aspects of this phenomenon have been

elucidated, the basic mechanism that is involved is still

not well understood. Presently, the most widely accepted

explanation attributes this phenomenon to the

viscoelastic properties imparted to the fluid by the

additives. In particular, many investigators have

suggested that polymer addition leads to the suppression

of turbulence production by increasing fluid resistance

to extensional deformations. It has been established










that large resistances to extensional deformations can

occur in drag-reducing solutions. However, it is not yet

clear whether or not such viscoelastic effects are

essential to drag reduction. To date, the experimental

evidence has been inconclusive and seemingly

contradictory in nature. The inability of investigators

to establish conclusive results has arisen, in part, from

experimental difficulties in adequately characterizing

the extensional behavior of low concentration polymer

solutions.

The goal of this dissertation is to further examine

the extensional behavior of low concentration polymer

solutions and to determine whether their extensional

characteristics are related to their drag-reducing

abilities.



Summary of Dissertation

In the remaining section of this chapter, the

commercial and theoretical significance of drag reduction

and polymer additive technology is discussed.

Background information pertinent to this

dissertation is presented in Chapter II. First, the

general features of drag reduction are briefly reviewed

and possible explanations for this phenomenon are

mentioned. The general characteristics of viscoelastic

liquids are then discussed and the results of previous









investigations where both the viscoelastic and drag-

reducing characteristics of polymer solutions have been

measured are reviewed. Next, the general aspects of the

extensional flow of viscoelastic liquids are discussed

and the experimental techniques that have been used to

examine such flows are briefly described. Lastly, some

comments on the various techniques for measurement of the

extensional characteristics of low concentration polymer

solutions are made.

The approach used in this work was to conduct shear

viscosity, extensional flow, and drag reduction

measurements and then to examine the results for

correlations. A series of polymer solutions subject to

varied conditions of additive type, molecular weight

grade, concentration, and mixing were used as the test

solutions. The shear viscosity measurements were made

using a sliding ball viscometer; the extensional behavior

of the polymer solutions was characterized using conical

channel rheometers; and drag reduction was determined

using a pipe flow apparatus.

In Chapter III, the experimental apparatus and the

additives used are described. Solution preparation

procedures are also discussed.

The experimental results are presented in Chapters

IV, V, and VI for the shear viscosity, drag reduction,

and extensional flow measurements, respectively. The










general format of these chapters is to first present the

reasons for selection of the experimental apparatus. The

goals and scope of the measurements are then presented

and the experimental and analytical procedures that are

used are discussed. Lastly, the experimental results are

reported and where possible are compared with previous

theoretical and experimental results.

In Chapter VII, the correlations between the results

of the shear viscosity, drag reduction, and extensional

flow measurements are examined.

In Chapter VIII, the results of this investigation

are summarized and recommendations are made for continued

work.



Significance of the Problem

The phenomenon of drag reduction and the use of

polymer additives have been of interest to investigators

from both commercial and theoretical standpoints.

Commercially, the use of drag-reducing polymer

additives has been looked at for

1. More economic transport of liquids over long

distances, e.g. crude oil transport (Burger et

al., 1982);

2. Increasing the capacity of existing pipelines,

e.g. handling peak storm sewer runoff (Sellin et

al., 1982);






5



3. Irrigation and firefighting applications (Union

Carbide Corp., 1966; Fabula, 1971);

4. Reducing the hydrodynamic drag for marine

vessels (Hoyt, 1972).

Polymer additives have also been examined for usage

in non-drag-reducing applications. These include use in

the hydraulic fracturing of oil fields (Melton and

Malone, 1964), in high pressure jet cutting operations

(duPlessis and Hashish, 1978), in hydraulic machines

(Bilgen and Vasseur, 1977), in jet fuels as anti-misting

agents (Peng and Landel, 1983), and in reducing

atherosclerosis (Greene et al., 1980).

Theoretically, understanding drag reduction has

proven to be a difficult task because both the nature of

turbulence and the behavior of macromolecules in a

solution are not well understood. Research on the

interaction between the polymer molecules and the

turbulence should not only produce information concerning

drag reduction, but may also provide further insights

into turbulence and into the rheology of polymer

solutions.
















CHAPTER II
LITERATURE REVIEW


Drag Reduction

Turbulent drag reduction by polymer additives may

be defined as the phenomenon by which drag in turbulent

shear flow of a polymer-additive solution is reduced to

below that for the solvent alone. For the more

effective additives, drag in turbulent shear flow may

be reduced by as much as 80% through the addition of

just a few parts per million of polymer. Drag

reduction can also be observed in solutions containing

fibrous substances and in some soap solutions; however,

the most effective drag-reducing solutions result from

the use of polymer additives.

One of the earliest reports of drag reduction may

be attributed to Forest and Grierson, who in 1931

observed such an effect in wood pulp fiber suspensions

(Zakin and Hunston, 1980a). However, it was not until

1949, through the work of Toms, that drag reduction was

brought to the general attention of scientists. In his

paper, Toms (1949) reported that unusually low friction

coefficients could be obtained for pipe flow of

polymethyl methacrylate in monochlorobenzene. Early










recognition of this phenomenon was also made by K.J.

Mysels, who observed drag reduction in gasoline

thickened with aluminum soaps during World War II

(Agoston et al., 1954). Further recognition of the

drag reduction phenomenon continued in the 1950's, when

both Shaver and Merrill (1959) and Dodge and Metzner

(1959) found during the study of non-Newtonian

turbulent pipe flow, that anomalously low friction

factors occurred for some of the liquids that they were

using. Despite recognition of the phenomenon, research

in the field did not really get started until the early

1960's, when the United States Office of Naval Research

created a program to fund drag reduction work. Since

that time, research in the field has flourished and

much has been learned about the phenomenon. In the

remainder of this section, some of the main features of

drag reduction that have been elucidated will be

briefly reviewed.



Main Features of Drag Reduction

Relevant polymer-solvent characteristics. The

most effective drag-reducing additives are high

molecular weight polymers which have a linear structure

or are only moderately branched. Typically,

macromolecules with molecular weights of greater than

about 105 need to be present in a solution before it










becomes drag reducing. Further, in solutions prepared

with polymer samples having wide molecular weight

ranges (as is usually the case), Berman (1977a) and

Hunston and Reischman (1975) have demonstrated that the

primary contribution to drag reduction will be that

from the small percentage of the molecules having the

highest molecular weights.

Another important factor in drag -reduction is

polymer conformation. Depending on the type of

solvent, polymer molecules will assume different

conformations. Generally, polymer molecules in

solutions at rest tend to form tangled coils. In good

solvents polymer-solvent interactions will be favored

over polymer-polymer interactions, while in poor

solvents, the opposite will be true. As a result,

polymer molecules in good solvents tend to be more

extended than in poorer solvents. Investigators have

found that the use of good solvents produces more

effective drag-reducing solutions than use of poor

solvents. For example, Hershey and Zakin (1967)

observed that polyisobutylene in a good solvent

(cyclohexane) could exhibit twice as much drag

reduction as when in a poorer solvent (benzene).

This effect is also commonly observed in poly-

electrolytes (charged polymers), whose conformations

will change in response to changes in their ionic









environment. An acidic environment causes them to be

highly coiled, while they can be more extended in a

alkaline environment. By varying pH levels, Parker and

Hedley (1972), Kim et al. (1973), and Banijamali et al.

(1974) have found increases in drag reduction in

solvents of increasing alkalinity. Thus, it appears

that extended polymer conformations are more suitable

to drag reduction than more compact structures.

Mechanical properties of drag-reducing solutions.

First, almost all macromolecular solutions exhibit a

strain-rate-dependent shear viscosity. Usually, at low

strain rates, the shear viscosity is constant, then as

strain rates are increased, it decreases until

sufficiently high strain rates are reached, at which

point, it again approaches a constant (but lower)

value. The magnitude of decrease of the shear

viscosity can often be dramatic (of several orders of

magnitude), even for solutions of low additive

concentrations (e.g., see Darby, 1970). However, shear-

thinning effects are generally not considered as being

responsible for drag reduction. Drag reduction has

been observed in many solutions whose shear viscosities

(at the strain rates corresponding to the drag-reducing

flow conditions) were nearly identical to that of their

solvents.










Next, viscoelastic characteristics are often

observed in concentrated macromolecular solutions.

However, such characteristics have been difficult to

observe in the less-concentrated solutions of interest

in drag reduction. In fact, it has not been until

recently (Morgan and Pannell, 1972; Balakrishnan, 1976)

that viscoelastic effects could be measured at all for

solutions at concentrations relevant to drag reduction.

Viscoelasticity now seems to be a concomitant property

of drag-reducing solutions; however, whether or not it

plays a major role in the drag reduction mechanism is

not known. In the following sections, viscoelasticity

will be discussed in more detail.

Drag-reducing solutions also often have the

ability to form thread-like filaments (Gordon et al.,

1973) and they typically exhibit a "slippery to the

touch" consistency. The exact mechanisms involved in

these last two effects are also not well understood.

Gross aspects of drag reduction. Some of the

gross aspects of drag-reducing behavior include the

existence of threshold conditions for onset of the

phenomenon, saturation effects, maximum drag reduction

limits, and degradation effects. In the onset

phenomenon, a certain flow condition must occur before

drag reduction will become apparent. This onset often

occurs after the flow is fully turbulent, although for










certain polymers (e.g., extended polyelectrolytes) at

sufficiently high concentrations, there will not be an

onset in the turbulent regime. Instead, there will be

some sort of direct transition from laminar to

turbulent drag-reducing flow. In the former case,

investigators have recognized that the threshold

condition for onset represents the start of some

significant interaction between the macromolecules and

the turbulent motions of the flow. This interaction

should occur when some certain turbulence scales are of

the same order of magnitude as some corresponding

molecular scales. Length scales, time scales, and

energy scales have been examined, and presently, most

investigators favor the time scaling hypothesis (Tulin,

1980). An example of the experimental support for this

hypothesis may be found in the work by Berman (1977b).

By the addition of various amounts of glycerine to

polyethylene oxide-water and polyacrylamide-water

solutions, he was able to create solutions whose

macomolecules experienced different degrees of coiling,

and hence, different characteristic time scales. It

was found that that increased amounts of glycerine

delayed the onset of drag reduction to larger time

scales and that these results correlated well with

predictions of increased molecular relaxation times

using molecular (Zimm-Rouse) theory. Although the time










scaling hypothesis is widely accepted, it should also

be noted that predictions of characteristic molecular

time scales do not always correlate well with

experimentally measured onset flow time scales. In a

recent study of onset conditions for samples of

polystyrene in different solvents, Zakin and Hunston

(1980b) found that the molecular time scales correlated

differently with the flow time scales, depending on the

solvent type. In poor solvents, the ratios of

molecular to flow time scales at onset were nearly

constant for over a wide range of concentrations for

several different solvents. However, in good solvents,

this same ratio was significantly larger and exhibited

a concentration dependence. Zakin and Hunston

attributed these deviations, in part, to failure of

rheological theories to correctly predict molecular

relaxation times.

In another gross feature of the drag reduction

phenomenon, limits exist as to the extent of drag

reduction that a given polymer solution can attain.

First, drag reduction will increase with concentration,

until at some point a saturation level is reached,

after which further increases in concentration will not

result in increases in drag reduction. The

concentration required to reach this saturation level

will depend on the properties of the polymer solution










and on the flow geometry. Secondly, a maximum drag

reduction limit exists, which is independent of polymer

solution properties. At this second limit, the flow

will consist primarily of large motions and will barely

resemble turbulent flow at all (Rollin and Seyer,

1972). The level of drag reduction at this limit will

correspond to about 80% of that which could be achieved

if the flow were completely laminar (Hoyt, 1972).

Another important feature of the drag reduction

phenomenon is the break-up (degradation) of the long

chain polymer molecules into chains of shorter lengths.

The primary form of such degradation is due to shearing

forces, although both chemical and thermal degradation

can also occur. As a result of scission of the

macromolecules, the molecular weight distribution will

change. The main effect appears to be a shift of this

distribution to lower molecular weight values with the

shape of the distribution undergoing relatively little

change (Zakin and Hunston, 1980b). Typically, with

increasing shear degradation, drag reducing ability

will decrease. This is due to the decrease of the

number of molecules having sufficient molecular weight

to contribute to drag reduction. However, it should be

noted that increased degradation does not always

decrease drag reduction, but sometimes can lead to

increased drag reduction (e.g., see Chang and Darby,









1983). In such cases, it is postulated that even

though the drag reduction contribution from the highest

molecular weight molecules is diminished, increased

drag reduction results from lowered viscous losses

resulting from the scission of the lower molecular

weight molecules (which leads to lower levels of shear

viscosity).

Turbulent Flow Structure. The flow structure of a

turbulent drag-reducing flow will often vary

substantially from that of a non-drag-reducing flow.

Mean velocity profiles, turbulence intensities, and

organized turbulent motions in the flow are all

altered. In fully developed wall-bounded turbulent

flows of non-drag-reducing liquids, a universal

nondimensional velocity is observed. Near the wall,

this profile is composed of three regions: a viscous

sublayer, where u =y for 0
region, where u = A log(y ) + B, for 30
y/6 <0.15; and an interactive or buffer region between
+
them. In the above equations, u is the

dimensionless velocity (u =U/u,), y is the

dimensionless distance from the wall (y =yu*,/),

U is the average flow velocity at a distance y from the

wall, u, is the friction velocity (u,=((w/p) /2),

T is the wall shear stress, 6 is the boundary layer
w
thickness, p is the mass density, and v is the









kinematic viscosity. For Newtonian liquids, A and B

are constants having values of 2.5 and 5.6,

respectively. In a drag-reducing flow, the viscous

sublayer profile remains unchanged. The logarithmic

region is also observed; however, it is usually

displaced to higher velocity levels, causing a

thickening of the buffer region. Generally,

investigators have found that the logarithmic profile

in the drag-reducing flows may be expressed as

u = A log(y ) + B + AB, where AB is an additional

constant representing the thickening of the buffer

region and where the original constants A and B remain

unchanged. Such behavior indicates that the observed

differences between drag-reducing and Newtonian flow

are contained in the buffer region. With increases in

drag reduction, the buffer layer also increases. For

pipe flow, it has been suggested that the condition of

maximum drag reduction corresponds to the point where

the expanding buffer layer reaches the middle of the

pipe (Berman, 1978).

As a measure of the intensity of turbulence,

investigators determine the root-mean-square values of

the fluctuations of the flow velocity components. In a

wall-bounded turbulent flow of a Newtonian liquid,

streamwise turbulent intensities are found to increase

with distance from the wall, reach a peak in the buffer

region at about y =15, and then decrease as the










middle of the pipe is approached. The transverse

intensities are found to gradually increase from the

wall to the middle of the pipe without passing through

a peak and they are of a lower magnitude than the

streamwise intensities. For drag-reducing flows, the

peak of intensities in the streamwise direction still

exists, but occurs further away from the wall. Some

measurements (see Virk, 1975) indicate that these

intensities can reach larger values than in Newtonian

flows; however, Berman (1978) notes that this is

uncertain. Measurements of transverse intensities

indicate similar behavior as for Newtonian flows (with

the magnitudes being slightly lower); but again, it has

been stated that the measurements have not been

sufficiently accurate to determine any trends (Berman,

1978).

In a turbulent boundary layer, the flow has been

observed to be composed of highly organized motions.

These motions, often called coherent structures, can be

significantly altered in drag-reducing flows,

particularly in the near wall region. For a detailed

summary of the general characteristics of these motions

in Newtonian flows, the reader is referred to the

review by Cantwell (1981). Briefly, for a Newtonian

flow, the dominant structures in the viscous sublayer

are fluctuating, counter-rotating pairs of vortices.










These vortices densely cover all parts of the wall and

average about 1000u,/v in streamwise length with

their width being about 100u,/v. Interaction between

these structures and motions in the outer region away

from the wall occurs through a sequence of events

called bursting. In this sequence, a gradual outflow

of liquid occurs as low momentum fluid is swept up into

streaks between the counter-rotating vortices. This is

then followed by lift-up of the streaks, sudden

oscillation, and break-up into intense small scale

motions, which are ejected into the outer regions of

the flow. An inrush of liquid, of velocity greater

than the mean, then occurs and the retarded flow left

by the ejection event is swept away. It has been

suggested that this higher speed liquid has a vortex

ring like structure (Falco and Wiggert, 1980). As this

vortex ring approaches and interacts with the wall,

counter-rotating vortices in the sublayer will again

appear and the entire process will continue. Now, for

a drag-reducing flow, this same sequence of events is

observed, but the frequency of events is diminished and

the spacing of the streaks is increased when compared

to that for a Newtonian liquid at similar mass flow

rates. Further, when the comparison is made after non-

dimensionalization with the wall parameters (u,v ),

the streak spacing is found to increase with increasing










drag reduction; streak streamwise lengths are found to

be longer, but bursting periods appear to remain the

same as for Newtonian flow. In general, the structure

in a drag-reducing flow is one of supressed small scale

motions and enhanced large scale motions. For more

specific results, the reader is referred to the

research papers by Eckelman et al. (1972), Donohue et

al. (1972), Reischman and Tiederman (1975), Achia and

Thompson (1974), Oldaker and Tiederman (1977),

Mizushina and Usui (1977), and Falco and Wiggert

(1980).

Polymer Interactions. Whether the combined

effects of isolated molecules cause drag reduction, or

whether multimolecular structures are responsible for

drag reduction is a controversial topic. As support

for the belief that isolated molecules are responsible

for drag reduction, it has been pointed out that drag

reduction can occur at very low concentrations (of less

than a ppm), where the distances between individual

macromolecules would be so great as to make it unlikely

that molecular interaction would occur. To determine

whether a solution is dilute enough so that

interactions are not likely to occur, investigators

have made calculations of the volume occupied by the

polymer molecules by replacing each molecule with an

equivalent hydrodynamic sphere, having a radius equal










to the rms radius of gyration of the molecule. Now if

the volume fraction occupied by these spheres is larger

than unity, then molecular interactions are expected to

occur, and conversely, for volume fractions

significantly less than one, the occurence of

interactions is less likely. Early investigators found

that drag reduction could occur for solutions having

polymer volume fractions as low as 0.01 (Merrill et

al., 1966), which therefore led to the assumption that

the effects of interactions were negligibly small.

However, Fabula and co-workers (1966) noted that such

low volume concentrations do not necessarily mean that

coil entanglement is unimportant. In a turbulent shear

flow, the polymer molecules should be expected to be

tumbling and elongated, thus, still possibly allowing

frequent contact and interactions to occur. Further,

Fabula et al. suggested that the individual

macromolecular coils are too small to interfere with

the turbulent structure in a particulate manner. For

example, Virk et al. (1966), in a summary of available

data for various polymer-solvent combinations, found

that at onset to drag reduction, the ratios of the rms

end to end lengths of the macromolecular coils to the

turbulent length scales (u /v) ranged from 0.0015 to

0.1145. That is, the turbulent motions were some 200

to 600 times larger than the molecular dimensions at










onset. Such a large size discrepancy between the

molecules and the turbulent length scales led Fabula to

suggest that macromolecular entanglements are an

essential feature of drag reduction. However, more

evidence arose that seemed to justify the consideration

of the hypothesis that isolated molecules are

responsible for drag reduction, when both Virk (1966)

and Paterson and Abernathy (1970) reported measurements

which indicated the possible existence of intrinsic

levels of drag reduction. They observed that for

decreasing concentration (at low concentrations), that

the amount of drag reduction divided by the

concentration (the specific drag reduction) approached

a constant value. Such a result has two implications.

First, it implies that drag reduction exists at

infinite dilutions, and secondly, it implies that

molecular interactions are insignificant. If molecular

interaction were to be significant, then one would

expect, for example, that a doubling of the

concentration (at low concentrations) would more than

double the amount of drag reduction (Lumley, 1973). In

this case, the specific drag reduction would decrease

with concentration, not approach a constant value as

Virk (1966) and Paterson and Abernathy (1970) found.

However, Paterson and Abernathy note that large errors

can occur in the measurements of low values of drag










reduction and they caution that further experiments are

necessary to firmly establish this result.

More recently, experimentalists have found

evidence of interaction, by observing the existence of

multimolecular structures. Stenborg, Lagerstedt, and

Lindgren (1977b) have directly observed polymer

agglomerations in drag-reducing flows through both a

Schlieren set-up and a microscope. Ouibrahim (1977)

has presented photographs, showing network structures

for solutions having additive concentration as low as

10 ppm. Through use of electron microscopes, Dunlop

and Cox (1977) and James and Saringer (1980) have also

presented photographs of multimolecular structures.

However, such direct evidence of molecular interaction

does not necessarily mean that they are essential for

drag reduction.

More recently, experimental evidence indicating

that interactions can affect the mechanism of drag

reduction has been reported (Berman, 1978, 1980). Of

particular interest, is the fact that Berman (1978)

reports that specific drag reduction can decrease with

concentration and not approach a constant value, in

contrast to the earlier work of Virk (1966) and

Paterson and Abernathy (1970). This would be expected

if interactions were to affect drag reduction. In

summary, it is now clear that clusters of










macromolecules exist in drag reducing solutions;

however, whether or not molecular interactions play an

essential role in drag reduction has not yet been

clearly determined.

In this section, some of the main features of the

drag reduction phenomenon have been briefly discussed.

As mentioned earlier, research in this field has been

extensive and the discussion presented here represents

only a few of the highlights of such endeavors. For

more extensive reviews of the work in this field, the

reader is referred to the reviews by Lumley (1969,

1973), Patterson et al. (1969), Hoyt (1972, 1974,

1977), Virk (1975), Little et al. (1975), Berman

(1978), and Sellin et al. (1982). Literature surveys

of work in this field have also been published by

Granville (1976, 1979, 1980, 1981).


Theories of Drag Reduction

The earliest explanation of drag reduction is

attributed to Oldroyd (1949), who suggested that this

phenomenon is the result of wall effects. He envisioned

that the wall would constrain the rotation of the polymer

molecules, which would somehow lead to an abnormally

mobile laminar sublayer, and hence, slip would

effectively occur at the wall. Other early explanations

of the phenomenon are that drag reduction results from










shear-thinning effects (Shaver and Merrill, 1959); delay

of laminar to turbulent transition (Savins, 1964);

anisotropic viscosity effects (Merrill et al., 1966),

where polymer molecules near the wall, which are

elongated and oriented by the flow, would spatially

hinder momentum transport normal to the direction of the

mean flow; and from possible flow interaction effects

from absorbed layers of molecules on the wall (Davies and

Ponter, 1966; Bryson et al., 1971). Up to now, these

explanations have not been substantiated experimentally,

and generally, they are no longer being considered as

possible drag-reducing mechanisms.

That viscoelastic effects are responsible for drag

reduction was probably first suggested by Dodge and

Metzner (1959). Presently, this remains as the most

popular hypothesis for drag reduction, although the

specific mechanisms involved are still not yet clear.

Viscoelastic mechanisms for the explanation of drag

reduction have been suggested for several reasons.

First, nearly all polymer-solvent combinations that are

drag-reducing at low additive concentrations, exhibit

measurable viscoelastic effects at high additive

concentrations. Secondly, large viscoelastic effects

have been measured in low concentration polymer

solutions, while other solution properties (such as shear

viscosity and mass density) remain very close to solvent










values. That drag reduction and viscoelasticity are the

only two large macroscopic effects induced by polymer

addition at low additive concentrations suggests that

they are related. Further, experimental evidence

indicates that the onset of drag reduction occurs when

turbulent time scales match polymer time scales. This

would suggest that some viscoelastic mechanism involving

the stretching of polymer molecules plays a role in drag

reduction.

Possible viscoelastic effects that may be of

importance in drag reduction are effects resulting from

resistance to extensional deformations; stabilization

effects due to stress modifications in transient shear

deformations; and/or energy storage effects of elastic

deformations. As a general mechanism for drag reduction,

most investigators suggest that the polymer molecules

must interfere with the turbulence bursting process; this

would lead to a lower rate of turbulence production, and

hence, lower drag. However, exactly what part of the

bursting process that is involved is not known.

As an example of how viscoelasticity may interfere

with the bursting process, a mechanism proposed by Falco

and Wiggert (1980) will be described. In this proposed

mechanism, it is assumed that streak formation is

initiated by ring-like vortices interacting with the

wall. These vortices approach the wall at an angle and










as they intersect the wall, they are bent parallel to the

wall and stretched. According to these investigators,

the molecules inside these vortices will inhibit the

stretching of the vortices by increasing the resistance

to these motions. As a result, the ring-like vortices

will be stretched less, with the stretching process

taking a longer period of time. Due to this reduced

stretching, the intensities of the vortices will be lower

and they will exhibit lower self-induction effects.

Hence, the vortices will approach the wall at a lower

velocity and the velocity of the fluid picked up from the

wall will also be lower. As a consequence of these

effects, less streak build up will occur, and with the

inhibition of streak formation, the spatially averaged

bursting rate will decrease, turbulence production will

decrease, and drag reduction will result.

Since the ring-like vortices would remain in contact

with the wall for a longer period of time, this mechanism

would explain the observed increase in streak lengths.

Also, with the reduced stretching of the vortices, their

cross-sectional areas should not decrease as much; this

would account for increases in streak spacing. Finally,

because of reduced self-induction effects, these vortices

would not be expected to get as close to the wall and

this would lead to the observed lowering of turbulent

intensities in the region closest to the wall.










General Characteristics of Viscoelastic Liquids

Materials may be rheologically classified according

to their reactions to applied loads. The equations which

describe such material response behavior are called

constitutive equations, or rheological equations of

state. Materials which can be described by a

constitutive equation where the internal stresses are

only a function of the instantaneous magnitudes of the

deformations are defined as being purely elastic. On the

other hand, materials in which the internal stresses are

only a function of the instantaneous deformation rates

are labeled as being purely viscous. Materials which can

exhibit both elastic and viscous characteristics are

classified as being viscoelastic. Of interest here are

viscoelastic liquids, which are primarily viscous, but

are also capable of exhibiting elastic characteristics.

For engineers, probably the most significant

characteristic of a viscoelastic liquid is that its shear

viscosity is often highly dependent on the deformation

rate. For example, polymer solutions (the viscoelastic

liquids of interest in this dissertation) are typically

shear-thinning. They exhibit a constant viscosity region

(i.e., Newtonian behavior) at low strain rates, a region

where the viscosity decreases with strain rate at

intermediate strain rates, and then they exhibit another

Newtonian region at high strain rates. As mentioned










previously, these shear-thinning effects can often be

dramatic; viscosities can decrease to 1/100 1/10,000 of

their zero shear rate values (Bird et al., 1974).

A variety of elastic effects can also be generated

in viscoelastic liquids. Typically, the responses of

these fluids are dependent not only on the instantaneous

deformation conditions, but also on the past history of

deformation. As a result, they often show transient

responses to oscillating and other unsteady flows.

Further, in steady as well as in unsteady flows,

viscoelastic liquids can exhibit elastic effects as a

result of their abilities to sustain anisotropic normal

stresses. Some examples of the effects which can be

caused by anisotropic stresses are the Weissenberg

effect, where liquid climbs a rotating shaft; extrudate

swell, where a jet of a viscoelastic liquid will expand

upon emerging from a tube or die; the development of

unequal pressures on the inner and outer walls in laminar

annular pipe flow; and the tubeless siphon effect, where

a siphon can continue to operate after the upstream end

of the siphon has been withdrawn from the liquid. For

further examples of elastic effects, the reader is

referred to Lodge (1964).

Elastic effects can be generated in both shearing

and extensional flows. For instance, the first three of

the above mentioned examples occur in shearing flows










(flows with transverse velocity gradients), while the

last example describes an extensional flow situation

(flow with a longitudinal velocity gradient). In

general, extensional flows are more capable of generating

stronger elastic effects than shearing flows. To

qualitatively distinguish extensional flows from shearing

flows, consider an idealistic stretching of a dilute

macromolecular solution. In simple shear flow, the

macromolecules will tend to rotate and stretch until they

are aligned with the streamlines. At this point, the

ends of each molecule will be moving at similar

velocities and the stretching will cease. The molecules

will then relax (i.e., return to an unstretched state)

until some force perturbs the molecule out of alignment

with the streamlines, at which point, the stretching

process will start over again. In an extensional flow,

the molecules will also tend to align themselves with the

streamlines; however, the ends of each molecule will now

be in fluid of increasingly different velocity and each

molecule will continue to be stretched (with no periods

of relaxation). Further, if a molecule is disturbed from

such an alignment, the hydrodynamic restoring force will

tend to direct the molecule straight back to the aligned

position. In contrast, if a molecule is perturbed in a

shearing flow so that it rotates in the original

direction of rotation, it will likely continue to keep










rotating until it has shifted position by 180 degrees.

Consequently, the molecules in a shearing flow are likely

to follow a tumbling motion, and at any one time, more

molecules are likely to be out of alignment than for an

extensional flow. Thus, molecules in extensional flows

are more likely to attain greater extensions with a

greater percentage of molecules being stretched than in

shearing flows. For further discussions on the

differences between shearing and extensional motions, the

reader is referred to Pipkin and Tanner (1972), Pipkin

(1977), and Petrie (1979). Lastly, one important point

is that it is not possible to predict behavior in

elongation from behavior in shear (Petrie, 1979).



Experimental Correlations of Viscoelastic
Characteristics with Drag-Reducing Ability

Over the past two decades, investigators have been

attempting to determine whether or not the drag-reducing

abilities of polymer solutions are directly related to

their viscoelastic characteristics. In this section, the

experimental attempts at such correlations will be

briefly reviewed.

In 1964, Metzner and Park measured both first normal

stress differences generated in a shearing flow and




1The first and second normal stress differences in
simple shear flow may be defined as (T11-T22) and










drag reduction characteristics for a 0.3% aqueous

polyacrylamide solution (J-100, Dow Chemical Co.). The

normal stress differences were determined from

measurements of the reductions in thrust (from that for

solvent flow at similar flow rates) of jets issuing from

a smooth cylindrical tube in which laminar shear flow

existed. Drag reduction was determined from measurements

of flow rate and pressure drop in a pipe flow. Their

results indicate that drag reduction may be a function of

the ratio of the elastic to viscous forces developed in

the pipe flow; however, no definite conclusions were

reached due to the lack of sufficiently extensive data.

In 1966, Gadd reported measurements of the pressure

difference between the outer and inner surfaces of an

annular pipe flow for several drag reducing solutions.

These pressure differences were used as a measure of

second normal stress differences. Aqueous solutions,

each being supposedly of the same drag reducing ability,

of 40 ppm Polyox WSR-301 (Union Carbide), of 72 ppm

Separan AP-30 (Dow Chemical Co.), and of 120 ppm Guar gum

were studied. Substantial pressure differences were

observed for the Polyox solutions, but not for the other




(T 22-T 33), where T,1' T22' T33 are the components
of stress in the direction of flow, in the direction of
the displacement gradient, and in a "neutral" direction,
respectively.









two solutions. Gadd (1966) concluded that there was no

obvious correlation between turbulent drag reduction and

normal stress differences in laminar shear flow. In a

follow-up experiment with similar solutions, Brennen and

Gadd (1967) demonstrated that both the level of second

normal stress differences and the differences in pitot

tube pressure between corresponding polymer solution and

solvent flows (also thought to be a viscoelastic effect

(Metzner and Astarita, 1967)) could be significantly

diminished without apparently impairing their drag-

reducing abilities. This would further seem to indicate

that drag reduction does not correlate with solution

viscoelasticity.

Correlations of Deborah number with drag reduction

have also been examined. The Deborah number may be

defined as the ratio of a characteristic time parameter

of the fluid to that of the flow and a correlation of it

with the level of drag reduction would indicate that

fluid elasticity would be likely to play a role in drag

reduction. Seyer and Metzner (1967) have examined such a

possible correlation for pipe flow of several aqueous

polyacrylamide solutions ranging in concentration from

0.2% to 0.6% for Reynolds numbers ranging from 5000 to

60,000. In their determination of the Deborah numbers,

the characteristic time of the flow was taken as the

reciprocal of an estimate of the dissipative frequency of









the turbulence and the characteristic solution time was

taken as a molecular relaxation time of a Maxwell fluid.

These investigators found that for a given Reynolds

number, the level of drag reduction when plotted as a

function of the Deborah number was independent of

concentration. However, this relationship was found to

vary with Reynolds number. These results would seem to

indicate that fluid elasticity may be involved in the

drag reduction mechanism; however, the lack of a complete

correlation does not allow any definite conclusions.

Other investigators (e.g., Elata et al., 1966; Rodriguez

et al., 1967) have also examined the correlation of drag

reduction with fluid elasticity through examination of

the Deborah number. Generally, these correlations, which

have utilized different formulations of the Deborah

number, have shown fair agreement between drag reduction

and Deborah number; however, no complete correlations

have been found. It should also be noted here that

theoretical attempts to explain drag reduction using

viscoelastic models have also been tried (e.g., Patterson

and Zakin, 1968; Ruckstein, 1971, 1973; Ting, 1972; Hinch

and Elata, 1979); however, no one model has proven

entirely satisfactory.

In the early 1970's, investigators found the first

indications that very large viscoelastic effects could

occur in relatively low concentration polymer solutions










(of the order of 100 ppm). In studying extensional flow

through orifices, Metzner and Metzner (1970) and Oliver

and Bragg (1973) estimated that extensional viscosities

(defined as the first normal stress difference divided by

the deformation rate in the direction of the flow) as

large as 1000 to 10,000 times their corresponding shear

viscosities could occur. In both sets of experiments,

aqueous polyacrylamide solutions of greater than 100 ppm

were studied at deformation rates of the order of 10,000

1/s. The normal stresses were determined using momentum

balance analyses and measurements of the reduction in

thrust of the jets issuing from the orifices. The

extensional deformation rates were determined from

estimates of the geometry of the flow entering the

orifices. No correlations of extensional viscosity with

drag-reducing ability were made; however, just the

existence of such large values of extensional viscosity

has led many investigators to suggest that drag reduction

somehow arises from such large increases in resistance to

extensional deformations.

The first viscoelastic measurements for very low

concentration (as low as 1 ppm) polymer solutions were

reported by Morgan and Pannell (1972). The reductions in

jet thrust were measured for laminar shear flow exiting

from capillary tubes. Drag reduction measurements were

also made in capillary tubes at similar wall shear rates










as experienced in the jet thrust measurements. The

measured values of the reduction in jet thrust were

compared at three different wall shear rates for

solutions containing just enough polymer to give maximum

drag reduction. Several aqueous polyethylene oxide and

polyacrylamide solutions were examined and the

measurements were conducted at wall strain rates of

64,000, 100,000, and 178,000 1/s. The concentrations

that yielded maximum drag reduction ranged from 1 to 220

ppm. These investigators found that, regardless of the

polymer type or concentration, that at each particular

wall shear rate, the reductions in jet thrust in each of

these solutions were roughly equal. As a result, they

concluded that a general relationship exists between the

level of first normal stress difference and drag reducing

ability.

Gordon, Balakrishnan, and Pahwa (1975) investigated

the importance of filament formation in drag reduction.

The ability of many drag reducing solutions to form

filamentary structures, called pituity, is thought to be

a viscoelastic effect arising from the high extensional

viscosity of such solutions (Chang and Lodge, 1971).

Gordon et al. devised a pituity test in which a rod was

suddenly removed from a beaker of polymer solution. This

caused a thread of solution to be formed between the rod

and beaker. The time it took for the thread to break was










then taken as a measure of pituity. They examined

aqueous solutions made with several different molecular

weight grades of a polyethylene oxide and of a

polyacrylamide. They found that filament forming ability

loosely correlated with drag-reducing ability for each

series of these two polymers, however, but not between

them. They tentatively concluded that drag reduction and

filament formation are not manifestations of the same

mechanism. However, one drawback to their correlations

was that their pituity tests were conducted at additive

concentrations of 0.825% (the high concentrations were

necessary to observe any pituity), while the drag-

reducing tests were run at much lower concentrations

(0.01%).

In his doctoral dissertation, Balakrishnan (1976)

examined the freely converging flow of several aqueous

polyacrylamide solutions. Extensional viscosities were

calculated using a momentum balance with measurements of

orifice pressure drop, flow rate, and the angle of

convergence of the upstream flow into the orifice.

Various conditions of polymer molecular weight,

concentration, and conformation were studied. Generally,

the more effective drag reducing solutions were found to

have the higher extensional viscosities. Balakrishnan

concluded that drag reduction was a manifestation of the

large extensional viscosities that dilute polymer










solutions can possess. In this investigation,

extensional viscosities as large as 3000 times their

corresponding shear viscosities were calculated for

additive concentrations as low as 10 ppm.

Further measurements of normal stress differences

generated in laminar shearing flows were reported by

Hasegawa in 1978. Several polyethylene oxide and

polyacrylamide solutions ranging in concentrations from 2

to 80 ppm were examined. First normal stress differences

were determined using a jet thrust technique for wall

shear strain rates ranging from 1000 to 100,000 1/s. In

general, the polymer solution flow behaved in nearly the

same manner as for water flow through the apparatus.

Hasegawa concluded that normal stress effects were not

responsible for drag reduction.

Scrivener et al. (1979) report an investigation in

which both the extensional and drag reducing behavior

were measured for several aqueous polyethylene oxide and

polyacrylamide solutions. The extensional behavior of

these solutions was examined using birefringence

measurements for flow through a cross cell, with the

presence of flow birefringence being taken as evidence of

macromolecular stretching. Drag reduction was measured

using a separate pipe flow apparatus. In particular, the

influence of polymer molecular weight was examined. Both

drag reduction and the intensity of birefringence were










found to increase with molecular weight. Further, it was

determined that the birefringence effects were caused by

the highest molecular fraction of the polymer, as has

previously been found in drag reduction. The drag

reduction experiments were run for concentrations ranging

from 10 to 80 ppm, while the phenomenon of flow

birefringence could only be observed above a certain

minimum additive concentration (500 ppm) and molecular

weight (450,000). Despite these concentration

differences, Scrivener and co-workers concluded that drag

reduction is strongly related to the elastic properties

of the macromolecules having the longest chain lengths.

Chang and Darby (1983) report an investigation where

the effects of shear degradation on both the drag-

reducing and rheological properties of several aqueous

polyacrylamide solutions were studied. Drag reduction

was measured using a pipe flow apparatus and first normal

stress differences were determined using a cone and plate

shear flow apparatus. Additive concentrations ranging

from 100 to 500 ppm were examined in both set-ups; and in

the cone and plate apparatus, shear strain rates were

varied from 1 to 100 I/s. Shear viscosity measurements

were also reported for strain rates ranging from 0.01 to

10,000 1/s. As a result of the applied shear

degradation, the magnitudes of both the elastic and

viscous properties of the test solutions were found to be










reduced. However, the degree of drag reduction was found

to increase with degradation. Since the fluid elasticity

was found to decrease in solutions of increasing drag-

reducing ability, it would appear that these results

indicate that fluid elasticity does not play an important

role in drag reduction. However, Chang and Darby offer

an explanation for these results which still supports the

viscoelastic hypothesis for drag reduction. They note

that a decrease in shear viscosity will tend to increase

the apparent level of drag reduction and they argue that

the effect of lowered shear viscosity was relatively more

significant in their experiments than decreased drag

reduction effects due to lowered fluid elasticity. Thus,

fluid elasticity may still be responsible for drag

reduction, even though drag reduction was found to

increase with degradation in these experiments.

In summary, no direct experimental evidence exists

which conclusively shows whether or not the drag-reducing

abilities of polymer solutions are related to their

viscoelastic characteristics. The results of the

investigations of Metzner and Park (1964), Seyer and

Metzner (1967), Morgan and Pannell (1972), Balakrishnan

(1976), and Scrivener et al. (1979) suggest that such a

correlation exists, while Gadd (1966), Brennen and Gadd

(1967), Gordon et al. (1975) and Hasegawa (1978) present

contrary evidence.










These investigations have established that

viscoelastic effects can occur in solutions of the low

additive concentrations (of the order of just a few ppm)

of interest in drag reduction. However, whether the

viscoelasticity measured in these instances is merely a

concomitant property of the solution, or an essential

element of drag reduction is not yet known.

Further, it may be noted from these experiments that

the extra normal stresses generated in the extensional

flows were generally larger and occurred at lower

deformation rates than in shearing flows. For example,

Balakrishnan reported a normal stress difference of

250,000 dynes/cm2 at an extensional deformation rate of

5000 1/s for a 20 ppm Separan AP-30 solution. While,

Morgan and Pannell reported normal stress differences of

only 5400 dynes/cm2 at a shear deformation rate of

100,000 1/s for a 17 ppm Separan AP-30 solution. (In

general, results from different investigations should not

be directly compared due to the many variables that can

affect additive behavior; however, this comparison is

used only as an example of the general trend that

extensional flow effects can be much larger than shearing

effects.) Thus, it would seem likely that extensional

effects, instead of shearing effects, would play a more

significant role in drag reduction, if viscoelasticity

were to indeed be be responsible for the phenomenon.










Extensional Flow of Viscoelastic Liquids

In this section, the experimental methods that have

been used to examine extensional flow behavior are

examined. First, however, the kinematics of such flows

are described.



Kinematics of Extensional Flows

A flow is extensional if a fixed Cartesian

coordinate system exists where the velocity field may be

expressed as



u. = a.x. + U
1 1 1 0.
1


where u. are the components of velocity at a material
1

point xi, a. are extensional deformation rates, and

U are constants. (The summation convention is not
O.
applied in this equation.) For steady flow, ai are

constants and if the flow is incompressible, then



al+a2+a3 = 0



Now, if there is positive extension in only a single

principal direction (e.g., (al,a2,a3) = ( F,- F/2,

- F/2)), then the flow is classified as uniaxial

extension. Another possible type of extension is equal

biaxial extension, where there is equal positive










extension in two principal directions (e.g.,

(ala2,a3)=( F F ,-2r )). If a flow is two

dimensional and extensional, then the flow is classified

as strip biaxial (or also called planar, pure shear, or

hyperbolic) extension. An example of this type of flow

is (al,a2,a3)=( F ,-r ,0). For further details

concerning the possible types of extensional flows, the

reader is referred to Dealy (1971), Tanner and Huilgol

(1975), Tanner (1976), and Powell (1983).



Experimental Methods

To describe the extensional behavior of viscoelastic

liquids, investigators typically determine the stress-

strain rate response for a liquid in an extensional flow

and then they express their results as an extensional (or

elongational) viscosity, which is then recorded as a

function of either the stress or extensional strain rate.

These extensional viscosities are defined as a ratio of a

normal stress difference to the extensional strain rate.

For example, the extensional viscosity, Te, may be

defined for uniaxial sink flow as



ne = (T11 T22)/F



where T11 and T22 are components of the stress tensor

in the directions of flow and orthogonal to the flow and










where r is the magnitude of the strain rate in the

direction of the flow.

The first measurement of extensional viscosity was

made by Trouton (1906). He placed rods of pitch, pitch-

tar mixtures, and wax under constant loads and measured

their velocities of extension. He computed coefficients

of viscous traction extensionall viscosities) for these

substances and found them to be three times greater than

their corresponding shear viscosities. This behavior is

now known as Trouton's rule and is followed in most cases

by Newtonian fluids (Reiner, pp.172,175-178, 1949).

For viscoelastic liquids, the types of experimental

methods which can be used to determine extensional

viscosities are dependent on the level of mobility of the

test liquid. For highly viscous liquids (having shear

viscosities of order 104-109 Ns/m2), investigators

have conducted extension tests by stretching cylindrical

rods of test liquid in uniaxial extensions and by using

bubble inflation techniques to obtain equal biaxial and

strip biaxial extensions. Typically, a uniaxial

extension test consists of clamping a test sample at one

end and then extending it in some programmed manner,

usually to yield a constant strain rate. The bubble

inflation techniques involve clamping a strip of test

material between two flat plates, in which a hole has

been cut (circular for equal biaxial extension or









rectangular for strip biaxial extension) and then

applying a differential pressure across the sample to

produce an expansion. The rates of strain that can be

obtained in these and the uniaxial tests range from about

0.001-1 1/s and the uses of these techniques are

restricted to materials having zero strain rate shear

viscosities of greater than about 10 Ns/m2. Due to

commercial interest in synthetic polymers, an extensive

amount of research has been conducted with polymer melts

using such techniques. For reviews of the experimental

work conducted in this area, the reader is referred to

Dealy (1971, 1978) and Petrie (1979).

For liquids of more moderate shear viscosities,

investigators have studied extensional behavior using

controlled fiber spinning and related techniques, film

blowing processes, stagnation flows, converging flows,

and some various unsteady flows. First, in controlled

fiber spinning tests, the test liquid is usually extruded

downward through a die and is taken up on a rotating

spool. In order to stretch the liquid, the take-up speed

is set higher than that which the liquid would naturally

attain as a result of gravity alone. Deformation rates

are calculated from measurements of the change in

diameter along the thread, usually being determined from

photographs. Stresses are calculated using measurements

of the take-up force and applying force balances along










the thread. Fiber spinning can be used for both polymer

melts and solutions, with the least viscous liquids that

have been spun having zero strain rate shear viscosities
2
of about 0.1 Ns/m The extensional strain rates that

have been obtained using such techniques range from about

1 to 100 1/s. For a detailed review of spinning

experiments, the reader is referred to the monograph by

Petrie (1979).

In another technique closely related to fiber

spinning, investigators have examined tubeless siphon

flow (or sometimes called Fano flow, named for the

investigator for whom its discovery is attributed). In

this flow, a siphon is set up with its inlet being

positioned above the surface of the material being

siphoned. The resulting flow is a rising column of

liquid, which has a decreasing cross-sectional diameter

as column height increases. The kinematics and dynamics

of the flow are determined, in a manner similar to

spinning flows, through the use of photographs and

measurements of the force required to support the siphon.

Tubeless siphon flow has been examined for extensional

strain rates ranging from 0.1-100 1/s for liquids having

zero strain rate shear viscosities as low as 0.02
2
Ns/m2. For more information concerning such flow, the

reader is referred to Astarita and Nicodemo (1970),

Acierno et al. (1971), and Nicodemo et al. (1975).










Still another technique related to fiber spinning is

the triple-jet method developed by Oliver and co-workers

(Oliver and Bragg, 1974; Oliver and Ashton, 1976a).

Here, a jet of liquid issues from a capillary tube with

another set of higher speed jets of the same liquid set

to impinge on the central jet at some downstream

location. The result of the impingement of the higher

speed jets is to stretch the central jet. Stresses are

determined by measuring the reduction in thrust on the

capillary tube and extensional strain rates are

determined using photographs. Oliver and co-workers used

this method to examine liquids having shear viscosities

of the order of 0.03 Ns/m2 for extensional strain rates

ranging from 100-800 1/s. Strain rates of greater than

1000 1/s could not be attained due to slip developing at

the jet impingement point.

The extensional behavior of viscoelastic liquids in

film blowing processes has also been experimentally

examined (Farber and Dealy, 1974; Han and Park, 1975;

Gupta et al., 1982). In such processes, molten polymer

is extruded from an annular die and is drawn upward by a

take-up device. At the bottom of the die, air can be

introduced to inflate the tube of polymer film into a

bubble. If the air pressure is set so that there is no

pressure difference across the film, then the tube will

not inflate and the result will be a uniaxial extensional










flow. By increasing the air pressure in the tube, the

extruded film will also be stretched in the transverse

direction to the fluid motion and biaxial extensions can

be obtained. The kinematics of such flows may be

determined from the geometry of the film, which is

usually determined using photographs, and from

measurements of mass flow rate and film thickness.

Stresses may be calculated using force balances with

measurements of the take-up force and pressure

differential across the film. In such flows, extensional

deformation rates ranging from about 0.005 to 5 1/s have

been attained.

At this point, it should be noted that the fiber

spinning and related techniques and the film blowing

processes are materially unsteady flows. In these flow

geometries, a fluid element will experience changing

deformation rates as it moves and, in general, the stress

that it experiences will not have sufficient time to

reach a constant level. As a result, the extensional

viscosities determined for such flows are not only

dependent on the rate of deformation, but also on the

entire strain history of that element. Therefore, while

often being a convenient comparative parameter,

extensional viscosities measured in materially unsteady

flows should not be interpreted as representing material

properties.









To try to attain constant deformation rates for over

the entire flow field, investigators have examined a

class of flows called stagnation flows. In such flows,

extensional flow fields may be set up by forcing streams

of liquid to collide. The stagnation flows that have

been examined have been generated through use of opposed

coaxial jets (Frank et al., 1971; Mackley and Keller,

1973), through the use of counter rotating cylindrical

rollers (Taylor, 1934; Peterlin, 1966, Frank and Mackley,

1976; Crowley et al., 1976; Mackley, 1978; Leal et al.,

1980) and through the use of cross flow cells (Scrivener

et al., 1979; Lyazid et al., 1980; Cressely and Hocquart,

1980). To obtain stagnation flow in the roller

experiments, symmetrically placed rollers are rotated in

a reservoir of liquid at equal rates, with each roller

being rotated in an opposite sense to the adjacent

rollers. The resulting flow, for example, if using four

rollers, will have liquid being pumped into the region

between the rollers from two opposing directions, with

the flow exiting in the other two directions. The flow

pattern in a cross flow cell is similar. Here, a chamber

is formed by the meeting of four channels, each at right

angles to the adjacent channels. The flow enters the

chamber from two opposing channels and exits through the

other two channels. The kinematics of such flows have

been determined through photography of trace particles










and through use of Laser Doppler anemometry. To date, a

primary goal of experiments using such flow geometries

has been to examine the degree of stretching that

macromolecules may attain; and this stretching has been

determined through birefringence measurements. Stress

measurements, however, have not been made in such set-ups

and hence extensional viscosities have not be determined

using such flow geometries. For a further discussion of

the kinematics and dynamics of stagnation flows and the

experiments that may be run to realize such flows, the

reader is referred to the report by Winter et al. (1979).

Another class of flows that investigators have used

for the examination of the extensional behavior of

viscoelastic liquids is that of converging flows. The

investigations in this area have utilized two types of

flows: one, where the flow is conducted from a reservoir

through an abrupt contraction (free convergence), and

secondly, where the flow is conducted through converging

channels (constrained convergence). In the first case,

investigators have utilized the fact that some polymer

melts and solutions will exhibit dramatically different

flow through a contraction than Newtonian liquids. For a

Newtonian liquid, flow will enter a contraction from

almost all angles; while for some viscoelastic liquids,

flow entering a contraction is restricted to a conical

region resembling a wine-glass-stem (Tordella, p.72,










1969; Metzner et al., 1969). This wine-glass-stem shaped

region is surrounded by a large torroidal region of

slowly recirculating flow. Within the conical region,

the flow is almost purely extensional and may be

approximated by the irrotational sink flow equations

(Metzner et al., 1969). In the experiments conducted by

Metzner and co-workers (1969), it was determined that the

sink flow equations applied to about 70% of all fluid

entering their contraction. After assuming sink flow

conditions, the deformation rates can easily be

determined from measurements of the angle of flow

convergence and of the mass flow rate. To determine

normal stress differences, investigators have applied

momentum balance analyses which have utilized three types

of measurements: measurements of the thrust exerted by

the viscoelastic jets emerging from the contractions

(Metzner and Metzner, 1970; Oliver and Bragg, 1973),

measurements of total pressure drop across the

contractions (Balakrishnan, 1976), and measurements of

the difference in pressure drop between the flow of the

polymer solution and its solvent at a similar flow rate

(Fruman and Barigah, 1982). These flows are also

materially unsteady, and hence, the extensional

viscosities determined using such techniques will again

not represent true material properties. These techniques

are still valuable, however, as they provide information










about the elasticity of the liquids at higher strain

rates ( of the order of 10,000 1/s) than can be attained

by using any of the previously discussed methods.

Generally, liquids of any level of shear viscosity can be

used in such techniques; however, this method is

restricted to liquids which exhibit wine-glass-stem

shaped flow and not all polymer solutions are capable of

exhibiting such flow (especially solutions of low

additive concentrations).

In constrained convergence, investigators have

examined flows through channels of conical and wedge

shaped geometries and also for channels profiled to give

constant average extensional deformation rates. Liquids

ranging from viscous polymer melts to low viscosity

polymer solutions have been studied using such channels.

The major difference between free and constrained

convergence is that shearing effects can be more

prominent for flow constrained by channel walls. To

avoid such effects, lubricated dies have been used with

viscous polymer melts and channels with dimensions

designed to minimize viscous losses have been used for

lower viscosity polymer solutions. If viscous losses are

negligible, then the irrotational sink flow equations may

again be used to evaluate the deformation rates. As for

the free convergence flows, normal stress differences

have been evaluated by using jet thrust techniques. In










addition, normal stresses (and hence, normal stress

differences) can now also be determined from pressure

measurements obtained from taps positioned along the

channel walls (James and Saringer, 1980; Winter et al.,

1979). For a review on such flows, with an emphasis on

polymer melts, the reader is referred to the paper by

Cogswell (1978). For work using less viscous polymer

solutions, the reader is referred to reports by James and

Saringer (1980, 1982), where both conical and planar

wedge flow have been examined, and to a paper by Oliver

and Ashton (1976b), where flow through channels, which

were designed to yield constant average deformation rates

throughout the flow, was examined.

Investigators have also developed techniques for

examining the extensional behavior of viscoelastic

liquids which make use of unsteady flows. One such

technique developed by Pearson and Middleman (1977)

utilizes the collapse of a gas bubble in a reservoir of a

viscoelastic liquid. In this method, a bubble is brought

to rest on top of a capillary tube situated in the

reservoir; a rapid change in pressure is then applied to

the bubble through the tube which causes the bubble to

collapse. As a result, uniaxial extensional flow is set

up in the surrounding liquid. Kinematics of the flow are

determined from high speed photography. Pearson and

Middleman were able to obtain nearly constant strain










rates ranging from 0.1-10 1/s for liquids having shear

viscosities of about 100 Ns/m2. Another technique

utilizing an unsteady flow is the method of controlled

jet instabilities developed by Schummer and Tebel (1983).

This method is based on the behavior of viscoelastic jets

emerging from a harmonically vibrating cylindrical

nozzle. The resulting jet consists of alternating

filament and drop regions. Due to higher surface

tensions in the filaments, a mass flow is set up from the

filaments to the adjacent drop regions. The result is an

extensional deformation of the filaments, with the

diameter of each filament decreasing until the filament

breaks away from the adjacent drops. The kinematics of

the flow are again determined using photographs.

Schummer and Tebel examined solutions having zero strain

rate shear viscosities as low as 0.074 Ns/m2 for

extensional strain rates ranging from about 10-150 1/s.

In both of these set-ups, the investigators have found

that nearly constant deformation rates could be attained;

however, the flows were maintained for only a short

period of time and it is doubtful that constant levels of

stress were attained. Recent measurements indicate that

this was probably the case, as extensional viscosities

determined using the bubble collapse method were less

than those determined by the stretching of a cylindrical

rod of the same material (Middleman and Munstedt, 1982).









Comments on the Use of Conical Channels to Examine
Extensional Flow Behavior

In the present investigation, conical channels have

been used to examine the extensional characteristics of

the polymer test solutions. This type of flow geometry

was selected for several reasons. First, conical

channels can be designed to conduct flow at the high

extensional deformation rates (of the order of 1000-

10,000 1/s) of interest in this investigation.

Secondly, measurements could be conducted at very

low additive concentrations for almost all types of

solutions. This is an advantage compared to the

techniques which utilize freely converging flow, which

are restricted to fluids exhibiting wine-glass-stem

shaped flow. At the low additive concentrations (of a

few ppm) of interest in this investigation, most drag-

reducing solutions do not exhibit this type of flow.

A further advantage of the use of conical flow

geometry is that only the gross flow measurements of flow

rate and pressure drop are needed to estimate

viscoelastic parameters. Since such measurements can be

quickly and easily made, extensional flow experiments can

be easily conducted in conjunction with the drag

reduction experiments.

In addition to these advantages, some drawbacks also

exist in the use of conical channels as rheometers. The









principal disadvantage is that the flow is materially

unsteady. Thus, any viscoelastic parameters that may be

determined from the flow measurements in general will not

represent material properties. Now, channels can be

constructed to yield constant average extensional strain

rates (i.e., materially steady flow) and still retain

some of the advantages of conical channel flow. However,

the use of such channels would also introduce some

further complications. In such channels the walls are no

longer aligned with a principal plane of stress (as in

conical channel flow). As a result, any pressure

measurements made using taps situated along the channel

wall would include effects from more than one principal

stress and measurements from more than one location would

be needed to extract the magnitudes of the principle

stresses (and hence, normal stress differences). More

difficulties and uncertainties would arise in the

interpretation of pressure measurements in terms of

first normal stress differences than for conical channel

flow.

For the purposes of the present investigation, the

use of conical channels was selected as the best possible

method for characterization of the extensional behavior

of the polymer test solutions. This technique will

produce viscoelastic parameters that will be useful for

comparative purposes; however, due to the materially






55


unsteady flow conditions, these parameters will not be

able to be interpreted as representing material

properties.















CHAPTER III
EXPERIMENTAL ARRANGEMENT


The experimental arrangement consists of set-ups to

measure the drag-reducing abilities, the extensional flow

behavior, and the shear flow characteristics of highly

dilute polymer solutions. These set-ups include a pipe

flow apparatus to record drag reduction, conical channel

rheometers for examination of extensional flow behavior,

and a sliding ball viscometer for characterization of

shear behavior. In this chapter, descriptions of each of

these set-ups, of the polymer additives used, and of the

polymer solution preparation procedures are presented.



Drag Reduction Apparatus

In this investigation, drag reduction was determined

from measurements of gross-flow rate and pressure drop in

turbulent pipe flow. The polymer test solutions, which

were used on a single-pass basis, were prepared "on-the-

fly" by the injection of a concentrated master solution

into the pipe flow.

In the experimental arrangement, shown in Fig. 3.1,

filtered tap water (the solvent in all cases) was

delivered to a constant head overflow tank. The excess

water was passed directly to the drain, while the
































oz









0




MO



I^m


MU
D m
m


z
0



z
P1-1


zK
z
<

u .

< ^
u ^-
S-)



0
u


'-4


-.
s








r-
-*
5

I csl






1-


1-1


z
1-1
o<
E-ir










regulated flow was conducted to the experiment pipe via a

strain gage flow rate meter. The desired polymer

additive, in the form of a concentrated master solution,

was injected into the flow at a mixer located just

upstream of the experiment pipe. Flow pressure drop was

measured at two locations along the experiment pipe using

separate strain gage differential pressure transducers.

Both pressure transducers and the flow rate meter were

connected to potentiometer recorders. The flow rate was

adjusted by means of two valves downstream of the

experiment pipe. These valves were connected in parallel

and both conducted the flow into calibrated compartments

of known volume in a storage tank. From the storage

tank, the fluid was passed directly to the drain.

The experiment pipe was 5.5 m long with a nominal

diameter of 16 mm. It consisted of three equal lengths

of acrlyic tubing joined together by flanges. A 6 mm

diameter orifice was located at the pipe inlet to provide

high entry disturbance levels. The upstream pressure

taps for the two pressure transducers were located 147

and 261 diameters downstream of the pipe inlet. The

distance between taps for each transducer was 44

diameters. Prior to the drilling of the taps, the mean

diameters of the pipe sections were determined from

volumetric measurements of their interiors. Each pipe

section was mounted vertically, filled with water, and










then a portion of it of measured length was drained, with

the water being collected and weighed. Several

measurements were made for each pipe section by draining

the entire section at a time, and then, additional

measurements were conducted by draining the pipes in 10

cm increments. The results of these measurements are

presented in Table 3.1. The average diameters, based on

measurements where the entire pipe sections were drained

at once, were found to be 1.5920+0.0009, 1.5964+0.0006,

and 1.5981+0.0011 cm for the upstream, middle, and

downstream sections, respectively. These averages were

each based on four separate measurements and the + values

indicate the maximum observed deviation of the measured

values from the corresponding average value. Average

pipe diameters for each section were also determined by

taking a weighted average of the diameters determined

from the incremental measurements. These values were

found to agree within 0.2% of the previous results. It

may also be noticed that from either set of diameter

values, the average diameters between pipe sections could

vary by 0.4%. Further, based on the incremental

measurements, it was found that the diameter could vary

along a pipe section by up to about +1% of the mean

diameter of that section. The average diameters

associated with the upstream and downstream test

sections, as determined from the incremental


















41) 1-
- i -i

0 C


3 C
r- x -r
C) (c 0

O E

O4 O
E) m


S0) 0)

4 0
0) 0




O )
44- 4

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> -4)
o0 0c




SZO
0 n3
+1>

C) () ^3


(40 -)


4-J 0
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C) 4-) ()
04' (0 4-)
*-H U -H (C)
- .Q > ,




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4J -H r3

E 0) U 10
1-4l > U)(

040 0 (n C)
X -H X! >
W 04 0 (0


>1
-4


r-

F-t
O

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u -H





0
,-4

0 ()
04








,C D1


4-1

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measurements, were found to be 1.592 and 1.597 cm

respectively. These values are used in all following

calculations requiring diameter measures.

A peristaltic pump was used to inject the

concentrated polymer master solution into the main flow.

In this pump, flexible tubing was encased in a housing

where rollers, driven by a variable speed motor, rotated

to squeeze the fluid through the tubing. In the present

investigation, two pump heads which used tubing of 0.8

and 1.7 mm in internal diameter were used.

Approximately, 20 cm of 6 mm diameter flexible tubing was

used to connect the master solution container to the pump

and another equal length of similar tubing was used to

connect the pump to the main flow set-up.

The polymer solution was injected into the main flow

at the upstream side of an in-line rotational mixer.

This mixer, shown in Fig. 3.3, consisted of a Plexiglass

housing, 25 cm in length and 4 cm in diameter, in which

three sets of impeller blades were mounted on a drive

shaft. This shaft was aligned so that the blades rotated

in a plane perpendicular to the flow. A variable speed

motor was used to rotate the shaft and impellers, with

rotation rates of up to 50 revolutions per second (rps)

being attainable. To prevent a general swirl of the flow

and to assist in the break-up of the polymer additive

solution, honeycomb sections were affixed to the


























IMPELLER
BLADES


FILTERED
TAP WATER




>s ADDITIVE
SOLUTION






\ HONEYCOMB
/ SECTIONS


I
TO
EXPERIMENT PIPE


Fig. 3.2. Schematic of in-line mixer.









mixer housing following each set of impeller blades.

These sections, made of 0.1 mm sheet aluminum, were 5 cm

long with individual cross dimensions of about 5 mm.

The differential pressure transducers used in this

set-up had adjustable measuring spans, ranging from 10 to

125 cm of water. For the drag reduction measurements,

the spans were set at 25 cm of water. From calibration

by using a manometer set-up, the accuracy of the

transducers (defined as being the maximum amount by which

the measured values differed from those of the

calibration curve) was determined as being better than

+0.3% of full span values. The flow rate meter used in

this investigation had a flow range of 20 to 600 cm /s.

Flow rates (during calibration) were determined by

measuring the time for the flow to fill compartments of

known volume in the storage tank. The accuracy of the

flow rate readings was also determined as being better

than +0.3% of full span. For further detail on these

measuring instruments and on the other commercially

available equipment used in the drag reduction apparatus,

the reader is referred to the manufacturers' literature.

A list of the commercially available equipment used in

this set-up and their manufacturers is presented in

Appendix A.

For further discussion on the use of the drag

reduction apparatus, the reader is referred to Chapter V.










Conical Channel Rheometers

General Arrangement

The conical channel rheometers were connected to the

drag reduction apparatus (cf. Fig. 3.1) in such a manner

so that the fluid that passed through the drag reduction

experiment pipe could also be conducted through the

conical channels. The conical channel set-up was

attached to the main flow set-up at a point immediately

following the main experiment pipe. The set-up consisted

of a needle valve for flow rate control, followed by

about 10 cm of 2.5 cm diameter tubing onto which the

various channels could be attached (at one at a time).

After passing through a channel, the flow exited to the

open atmosphere.

Flow rates and pressure heads at points just

upstream of the channel exits were the measured variables

in this set-up. The flow rates were determined by using

a catch and weigh technique, while pressure heads were

measured using the same transducer that was used to

measure the downstream pressure drops in the main

experiment pipe. Three-way valves, located in the

pressure lines, allowed connection to either set-up as

desired.

For some preliminary polymer solution measurements,

an additional separate conical channel set-up was also

employed. In this separate set-up, the conical channels










were connected to a small cylindrical reservoir (10 cm in

diameter) by a vertical section of flexible tubing (2.5

cm in diameter). Various lengths of tubing were used,

ranging from 0 to 2 m in length. For the experiments

conducted with this set-up, the polymer test solutions

were supplied to it by manual transfer of solutions from

the main flow apparatus. Pressure heads and flow rates

were measured in the same manner as in the other channel

set-up.



Design Considerations

Present usage of conical channels to measure normal

stress effects in dilute polymer solutions is limited;

hence, some mention should be given to the basic

considerations involved in their design. First, it is

desirable to achieve a velocity distribution as near to a

frictionless flow as possible. Secondly, the deformation

rate in the flow needs to be high enough to create

measurable non-Newtonian effects. Lastly, the pressure

decrease in a channel due to inertial forces should not

be so large as to mask the pressure change due to the non-

Newtonian effects.

First, to determine the level of frictional effects

in the flow, consider the head-discharge relationship



Qa = Qi{1- }
R1/2










which has been developed from the boundary layer solution

for flow of a Newtonian fluid through a conical channel

(cf. Appendix F). Here, Qa is the actual flow rate

through the channel, Qi is the corresponding flow rate

if the flow were frictionless, C is a constant depending

on channel geometry, and R is a Reynolds number which may

be defined as R=Ur/v where U=(2gh)1/2, h is the

pressure head at a distance r from the imaginary apex of

the channel, g is the gravitational constant, and v is

the solution kinematic viscosity. Thus, to diminish the

effects of friction, the channels should be designed to

conduct flows at high Reynolds numbers. Since the

constant C is of the of the order of unity, a Reynolds

number of Ur/v-1000 was selected as being sufficiently

large to minimize the influence of frictional effects

upon the present investigation.

In conical sink flow, the deformation rate in the

direction of flow is 2U/r. For laminar flow of polymer

solutions, the onset of non-Newtonian behavior often

occurs at a deformation rate of -1000 1/s (e.g., see

James and Saringer, 1980). Thus, to satisfy the second

design requirement, 2U/r should be of the order of 1000

1/s.

The appropriate channel dimensions in terms of r,

can now be determined by combining the first two design

requirements. For the present design, r is found to be










r -(2/\ )1/2 and with v ~ 0.01 cm 2/s, the desired

dimensions attain a value of some 0.1 cm.

The flow of dilute polymer solutions through a

channel of such designed dimensions should exhibit non-

Newtonian effects. However, they may be masked by

pressure changes due to inertial forces. The pressure

decrease along a channel will be -pU 2/2, and if the

pressure -change due to non-Newtonian stresses is small

compared to this inertial pressure change, difficulties

will be encountered in observing and determining the non-

Newtonian effects. Therefore, the designed channel

dimensions should be checked to see if such masking can

be avoided. For the presently designed channel

dimensions of -0.1 cm and Reynolds number of -1000, the

inertial pressure change should be of the magnitude of

1000 N/m2. Since dilute polymer solutions can exhibit

widely varying non-Newtonian behavior, it is difficult a

priori to determine whether or not the inertial pressure

effects will dominate the non-Newtonian effects. For

example, Balakrishnan (1976) found that a 20 ppm

polyacrylamide solution exhibited extra stresses of about

35,000 N/m2 at strain rates of 5000 1/s, while James

and Saringer (1980) measured non-Newtonian stresses of

some 100 N/m2 for a 20 ppm polyethylene oxide solution

at similar strain rates. Based on these reported

observations, it seems that the non-Newtonian effects










should easily be observable and should not be masked by

inertial pressure drops for the presently designed

channels. However, if it is found that only very weak

non-Newtonian effects are generated, then the channels

will have to be re-designed to smaller dimensions (at the

sacrifice of introducing more frictional effects).

Conversely, if very strong non-Newtonian effects are

generated, then larger channels may be needed to avoid

possible secondary flows.



Conical Channel Details

A series of channels, 4 mm long with exit dimensions

of the order of 0.1 cm, were constructed for use in this

investigation. Two sets of channels, one with conical

half-angles of 29 and the other of 440, were cut

with a lathe into 6 mm thick Plexiglass sheets.

Downstream pressure taps were located just upstream of

the channel exits and the upstream taps were placed on

the face of the Plexiglas sheet (facing the oncoming

flow). The exact dimensions and pressure tap locations

for the channels used in this investigation are presented

in Fig. 3.3. After drilling the pressure taps, the

interior channel walls were manually polished to remove

burrs and then were inspected for smoothness using a

microscope. Lastly, 10 cm long, 2.5 cm diameter rigid

tubing was fastened to the upstream face of the










FLOW








o PRESSURE
TAPS








e\/ r r r
r e r 1r 2











CHANNEL r r r 2
e 1 2
(cm) (cm) (cm) (deg.)



A 0.120 0.130 0.175 29.5
C 0.120 0.145 0.190 29.0
D 0.090 0.100 0.145 29.0
F 0.110 0.110 0.155 29.0
G 0.070 0.095 0.135 44.0
H 0.075 0.115 0.155 44.0


Dimensions of conical channels.


Fig. 3.3.










Plexiglass sheets, with the channels being centered with

respect to these tubes. At this point, the channels were

ready for attachment to the main flow set-up.



Sliding Ball Viscometer

The shear viscosity measurements were conducted

using a Hoppler Rheo-Viscometer. A schematic of this

viscometer is shown in Fig. 3.4. This instrument

consists of a vertical cylindrical glass container,

within which a closely fitted glass sphere is driven

downward (without rotation) through the liquid by a known

force. This force is transmitted to the sphere via a

rigid shaft, which is attached to the sphere at one end

and fastened to the balance arm assembly at the other.

Various weights were loaded on to the balance arm to

provide the driving force. The viscometer measurements

consisted of measuring the time required for the sphere

to travel a set distance (usually 3 cm) for a given

loading. The distance the sphere traveled was monitored

using a gage connected to the balance arm. Temperature

control during the measurements was maintained by a Haake

constant temperature circulator (model F, number 423).

For further discussion on the use of this viscometer,

including flow analysis, procedures for calculating shear

viscosities, and instrument calibration, the reader is

referred to Chapter IV and Appendix B.


















7 12

9 3



12




6 .



10







(1) Balance Arm
(2) Sphere Rod
(3) Precision Thermometer
(4) Measuring Vessel
(5) Saucer for Weights
(6) Set of Weights
(7) Gage Indicating Sphere
Travel Distance
(8) Movable Weight for
Buoyancy Compensation
(9) Eccentric Cam for Adjustment of
Sphere Location in Measuring Vessel
(10) Box Level
(11) Level Adjustment Screws
(12) Balance Arm Pivot Point


Fig. 3.4. Schematic of Hoppler viscometer.










Polymer Additives

General Structural Characteristics

Two molecular weight grades of a polyacrylamide,

Separan AP-30 and Separan AP-273 (Dow Chemical Co.), and

a polyethylene oxide, Polyox WSR-301 (Union Carbide

Corporation), were selected as the polymer additives to

be used in the present investigation. These additives,

which are widely used by investigators studying drag

reduction, are water soluble, linear, high molecular

weight polymers which can be used to produce very

effective drag-reducing solutions. Diagrams illustrating

their chemical compositions are shown in Fig. 3.5. It

should be noted that unmodified polyacrylamide molecules

are essentially non-ionic; however, typically some of the

amide groups (-NH 2) are hydrolyzed to anionic carboxyl

groups (-COO-Na ) (Dow Chemical Co., 1975). Thus,

such polymers in neutral solutions may be classified as

anionic polyelectrolytes.

In a state of rest, polyethylene oxide molecules are

thought to form randomly coiled spherical structures.

The polyacrylamide molecules are also thought to be

coiled, but instead of being spherical, they are thought

to be more elongated due to their ionic character

(Berman, 1978).

Due to the polymerization process and to

manufacturer's blending procedures, each additive


















[ CH2-CH2-O ] -



Polyethylene Oxide












-[ ( CH2-CH ) x- CH2-CH ] -

C=O C=O
I I1
+
NH2 0 Na



Partially Hydrolyzed Polyacrylamide


Fig. 3.5. Chemical structure of polymer additives.










contains a broad molecular weight spectrum. Typically

listed values for average molecular weights of these

additives are 4x106 for Polyox WSR-301 (Berman, 1980),

4x106 for Separan AP-30 (Berman, 1980), and 5x106

for Separan AP-273 (Lagerstedt, 1979).



Solution Preparation

Polymer solutions used in this investigation were

prepared "on-the-fly" by injection of a concentrated

master polymer solution into the main flow. The master

solutions, containing 1% additive by weight, were

prepared in about 10 kg quantities by sifting the

additives into tap water and manually stirring.

Thereafter, solutions were stirred a few minutes at a

time on a daily basis. This continued until the

additives were dissolved and the solutions visually

appeared homogeneous, which required about 2 weeks. The

solutions were then stored in covered containers until

use.















CHAPTER IV
SHEAR VISCOSITY MEASUREMENTS



Shear viscosities were measured using the Hoppler

Rheo-Viscometer (cf. Chapter III). The reasons for the

selection of this instrument include speed and simplicity

of operation, use of small sample sizes (less than 20

ml), and excellent reproducibilty of results. Thus, the

shear viscosity measurements could easily be conducted as

supplementary measurements to the primary experiments

involving drag reduction and extensional flow. A further

advantage of this instrument is that it is capable of

measuring fluid responses at high shear strain rates (of

the order of 10,000 1/s). For the present investigation,

this was desirable from a comparative standpoint, since

the results from the extensional flow experiments were

also conducted at similar high deformation rates.

The principal disadvantage of this instrument is

that neither the shear stresses nor the shear rates are

constant within the viscometer. As a result, while the

sliding ball viscometer has been used to determine shear

viscosities of Newtonian liquids, it has generally not

been deemed an appropriate instrument for the

determination of the rheological properties of non-










Newtonian liquids. However, recent analysis of such

viscometer flow by Schonblom (1974) now allows the

determination of shear stress-strain rate relationships

for non-Newtonian, as well as for Newtonian liquids.



Goal

The goal of the shear viscosity measurements was to

provide a quantitative measure of the shear behavior of

the polymer test solutions for comparison with their drag-

reducing and extensional flow behavior.



Scope

Shear viscosity measurements were conducted for the

same polymer test solutions as used in the drag reduction

and extensional flow experiments. Specifically, results

are reported for Separan AP-273 solutions of additive

concentrations of 5, 15, 20, 30, and 45 ppm; for 5, 10,

and 20 ppm Separan AP-30 solutions; and for 20 and 30 ppm

Polyox WSR-301 solutions.


Viscometer Flow Analysis

The Hoppler Rheo-Viscometer was developed in

at the onset of World War II for the measurement

viscosities of Newtonian liquids. According

manufacturer (VEB Prufgerate-Werk Medingen, Sitz

D.D.R.), the viscosity of a Newtonian liquid,


Germany

of shear

to the

Freital,

U, may










be obtained using the expression



= KPt (4.1)



where K is a calibration constant, P is the force

applied to the viscometer sphere divided by the sphere's

cross-sectional area, and t is the time required for the

ball to travel a set distance (usually 3.00 cm).

In 1974, Schonblom analyzed the flow in such a

viscometer and developed procedures to determine shear

viscosities for non-Newtonian liquids. In the remainder

of this section, Schonblom's analysis will be briefly

reviewed. First, the main assumptions in his analysis

were that (1) the speed of the falling sphere was

negligible when compared to the flow of the liquid and

that (2) the flow was steady and parallel to the outer

cylinder walls. In other words, the flow was effectively

modeled as steady fluid flow through an eccentric

annulus. With these assumptions and the application of

conservation of mass and momentum equations across an

annular fluid element, Schonblom derived the viscometer

equation



1/t = 2 ^2 (a+ ) wTy(T)dTda (4.2)
T alt 0 0









where T is the shear stress with T being its value at

the wall, y is the shear strain rate, a is radius of the

viscometer sphere, d is the annular width at angular

location a, 1t is the distance through which the sphere

travels in time t, and P is again the force applied

to the viscometer sphere divided by its cross-sectional

area. A definition sketch for the geometrical parameters

a, d, and a is presented in Fig. 4.1. In arriving at eq.

(4.2), Schonblom set the pressure gradient in the

direction of the flow equal to the load, P, on the

sphere divided by an effective sphere length, L. This

effective length, which has to be determined

experimentally, was assumed to be a constant.

For a Newtonian fluid, the viscometer response may

now be determined by substituting y(T)=T/p into the

viscometer eq. (4.2), and integrating. After assuming

the half annular width, d/2, was negligible when compared

to the sphere radius, a, and after the wall shear stress

was written in terms of the applied load, P,

Schonblom found the Newtonian viscometer response to be


Fig. 4.1 Geometry of eccentric annulus.









= (5d3Pt)/(96altL) (4.3)



where d is the maximum annular width and the other
m
variables are as previously defined. In comparison with

eq. (4.1), we see that the flow response predicted by

Schonblom is identical to the manufacturer's supplied

expression for the shear viscosity when the ratio

(5d /96altL) is replaced by the manufacturer's

constant K.

In his dissertation, Schonblom also presents

evaluations of viscometer equation, eq. (4.2), for

several cases of non-Newtonian behavior, including that

for power law liquids. In one dimensional form, the

constitutive relation for a power law liquid may be

written as



T = K ym (4.4)



where both K and m are constants. The viscometer

response for such liquid a may be determined by

substituting eq. (4.4) into eq. (4.2) and integrating.

Schonblom found the resulting response to be



dM (M+) 1
m 2 1/m (4.5)
1/t = (P)1/m (4.5)
M-2 al LM-2MK-2 /m (M+l)










where F(M+1/2) and r(M+l) are Gamma functions with

M=(l+2m)/m and where the other variables are as

previously defined. Thus, the viscometer response in

terms of the measurable variables of applied load P

and fall time t should also follow a power law type

behavior. It may also be noted that for the case where

m=l, the Newtonian solution, eq. (4.3), is recovered

(with K becoming the shear viscosity coefficient).

For viscometer measurements which do not follow a

power law response, Schonblom devised another more

general procedure to determine the shear stress-strain

rate relationship. It consisted of first determining the

functional relationship between P and t, and then

using it to solve eq. (4.2). For the details of this

procedure, the reader is referred to Schonblom's

dissertation. For the low additive concentration

solutions used in this dissertation, it is expected that

the laminar shear flow behavior exhibited in the

viscometer will be nearly Newtonian (for the deformation

rates to be examined) and that either eq. (4.1) or (4.4)

should be adequate to describe shear behavior.



Viscometer Calibration

In order to use either eqs. (4.1), (4.3), or (4.5),

the viscometer needs to be calibrated with a fluid of

known viscosity to determine either the constant K for










use with eq. (4.1) or to determine the effective length L

for use with eqs. (4.3) and (4.5).

Before determination of these calibration constants,

it is necessary to discuss two limitations of the Hoppler

Rheo-Viscometer. First, previous investigators (Lindgren,

1957; Schonblom, 1974) have noticed the presence of

frictional resistance in the main ball bearing of the

balance arm. This resistance reduces the effective load

on the sphere and, if not corrected for, manifests itself

as an apparent decrease in the calibration constant K for

decreasing loads. Lindgren (1957) suggested that the

true value of the calibration constant should be that

which was associated with an infinite load. Schonblom

(1974) followed this idea and determined that the bearing

friction may be adequately accounted for by replacing the

applied loading on the viscometer sphere with an

effective load, (P-Pf), where Pf is a

constant representing the amount of frictional

resistance. Thus, for more accurate representation of

the true loads on the viscometer sphere, the effective

load (P-Pf) should be used in place of P in

eqs. (4.1)-(4.5).

The resistance load, Pf, may easily be

determined through measurements made with a Newtonian

liquid. After replacing the applied load, P, with

the effective loading (P-Pf) in eq. (4.1) and