Blood Rheology

8/19/2019 Blood Rheology

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A Study of Non-Newtonian Viscosity and Yield Stress of Blood

in a Scanning Capillary-Tube Rheometer

A Thesis

Submitted to the Faculty

of

Drexel University

by

Sangho Kim

in partial fulfillment of the

requirements for the degree

of

Doctor of Philosophy

December 2002


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Acknowledgments

I wish to express my sincere gratitude to Dr. Young I. Cho, for his guidance

and inspiration during my entire tenure in graduate school. His experience and idea

have proven to be invaluable. I also wish to thank Dr. David M. Wootton for serving

as my co-advisor, and for his valuable suggestions and guidance on Biofluid

Dynamics.

I wish to express my appreciation to the members of my dissertation

committee, including: Dr. Ken Choi and Dr. Alan Lau from the MEM Department,

and Dr. Peter Lelkes from the School of Biomedical Engineering.

I am deeply indebted to Dr. Kenneth Kensey, Mr. William Hogenauer, and

Dr. Larry Goldstein from Rheologics, Inc. for providing valuable comments on the

test methods and data reduction procedure.

A sincere appreciation is extended to several colleagues whose friendship I

have cherished during my graduate studies, including: Dr.Wontae Kim, Dr. Sunghyuk

Lee, Chagbeom Kim, Giyoung Tak, Dohyung Lim, and Jinyong Wee.

Last but not least, I wish to thank my parents for their unbounded support

throughout my life. Their reliable provision of emotional, spiritual, and financial

support has allowed me to accomplish tasks that would have otherwise been

impossible.


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Table of Contents

LIST OF TABLES.....................................................................................................viii

LIST OF FIGURES ................................................................................................... x

ABSTRACT...............................................................................................................xiv

CHAPTER 1 INTRODUCTION.............................................................................. 1

1.1 Clinical Significance of Blood Viscosity.................................................... 1

1.2 Motivation of the Present Study ................................................................. 3

1.3 Objectives of the Present Study .................................................................. 3

1.4 Outline of the Dissertation.......................................................................... 4

CHAPTER 2 CONSTITUTIVE MODELS.............................................................. 5

2.1

Newtonian Fluid.......................................................................................... 5

2.2 Non-Newtonian Fluid ................................................................................. 10

2.2.1 General Non-Newtonian Fluid........................................................... 10

2.2.1.1 Power-law Model...................................................................... 11

2.2.1.2 Cross Model .............................................................................. 12

2.2.2 Viscoplastic Fluid .............................................................................. 13

2.2.2.1 Bingham Plastic Model............................................................. 13

2.2.2.2 Casson Model............................................................................ 14

2.2.2.3 Herschel-Bulkley Model........................................................... 15

2.3 Rheology of Blood...................................................................................... 19

2.3.1 Determination of Blood Viscosity ..................................................... 19


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2.3.1.1 Plasma Viscosity....................................................................... 20

2.3.1.2 Hematocrit................................................................................. 20

2.3.1.3 RBC Deformability................................................................... 21

2.3.1.4 RBC Aggregation - Major Factor of Shear-ThinningCharacteristics........................................................................... 21

2.3.1.5 Temperature .............................................................................. 22

2.3.2 Yield Stress and Thixopropy ............................................................. 23

2.3.2.1 Yield Stress ............................................................................... 23

2.3.2.2 Thixotropy - Time Dependence ................................................ 24

CHAPTER 3 CONVENTIONAL RHEOMETRY: STATE-OF-THE-ART ........... 30

3.1 Introduction................................................................................................. 30

3.2 Rotational Viscometer ................................................................................ 34

3.2.1 Rotational Coaxial-Cylinder (Couette Type)..................................... 34

3.2.2

Cone-and-Plate................................................................................... 35

3.3 Capillary-Tube Viscometer......................................................................... 38

3.4 Yield Stress Measurement .......................................................................... 41

3.4.1 Indirect Method.................................................................................. 42

3.4.1.1 Direct Data Extrapolation ......................................................... 42

3.4.1.2 Extrapolation Using Constitutive Models................................. 43

3.4.2 Direct Method .................................................................................... 44

3.5 Problems with Conventional Viscometers for Clinical Applications......... 46

3.5.1 Problems with Rotational Viscometers.............................................. 46

3.5.2 Problems with Capillary-Tube Viscometers...................................... 48


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CHAPTER 4 THEORY OF SCANNING CAPILLARY-TUBE RHEOMETER.... 49

4.1 Scanning Capillary-Tube Rheometer (SCTR)............................................ 49

4.1.1 U-Shaped Tube Set ............................................................................ 50

4.1.2 Energy Balance .................................................................................. 51

4.2 Mathematical Procedure for Data Reduction.............................................. 60

4.2.1 Power-law Model............................................................................... 60

4.2.2 Casson Model..................................................................................... 66

4.2.3

Herschel-Bulkley (H-B) Model ......................................................... 72

CHAPTER 5 CONSIDERATIONS FOR EXPERIMENTAL STUDY................... 81

5.1 Unsteady Effect ........................................................................................... 82

5.2

End Effect.................................................................................................... 87

5.3 Wall Effect (Fahraeus-Lindqvist Effect)..................................................... 90

5.4 Other Effects................................................................................................ 95

5.4.1 Pressure Drop at Riser Tube .............................................................. 95

5.4.2 Effect of Density Variation................................................................ 96

5.4.3 Aggregation Rate of RBCs - Thixotropy........................................... 97

5.5 Temperature Considerations for Viscosity Measurement

of Human Blood..........................................................................................101

5.6 Effect of Dye Concentration on Viscosity of Water ...................................104

5.6.1 Introduction........................................................................................104

5.6.2 Experimental Method.........................................................................106

5.6.3 Results and Discussion ......................................................................107

CHAPTER 6 EXPERIMENTAL STUDY WITH SCTR.........................................112

6.1 Experiments with SCTR (with Precision Glass Riser Tubes) ....................112


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6.1.1 Description of Instrument ..................................................................113

6.1.2 Testing Procedure ..............................................................................114

6.1.3 Data Reduction with Power-law Model.............................................116

6.1.4 Results and Discussion ......................................................................117

6.2 Experiments with SCTR (with Plastic Riser Tubes)...................................130

6.2.1 Description of Instrument ..................................................................131

6.2.2 Testing Procedure ..............................................................................132

6.2.3

Data Reduction with Casson Mocel...................................................133

6.2.3.1 Curve Fitting .............................................................................134

6.2.3.2 Results and Discussion .............................................................135

6.2.4

Data Reduction with Herschel-Bulkley (H-B) Model .......................139

6.3 Comparison of Non-Newtonian Constitutive Models ................................158

6.3.1 Comparison of Viscosity Results.......................................................159

6.3.2 Comparison of Yield Stress Results ..................................................162

6.3.3 Effects of Yield Stress on Flow Patterns ...........................................164

CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS .............................180

LIST OF REFERENCES...........................................................................................184

APPENDIX A: NOMENCLATURE.........................................................................194

APPENDIX B: FALLING OBJECT VISCOMETER- LITERATURE REVIEW..............................................................197

APPENDIX C: SPECIFICATION OF CCD AND LED ARRAY............................200

APPENDIX D: BIOCOATING OF CAPILLARY TUBE........................................202

APPENDIX E: MICROSOFT EXCEL SOLVER.....................................................204

APPENDIX F: NEWTON’S METHOD OF ITERATION.......................................206


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APPENDIX G: REPEATABILITY STUDY WITH DISTILLED WATER............208

APPENDIX H: EXPERIMENTAL DATA...............................................................210

VITA..........................................................................................................................221


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List of Tables

2-1. Viscosity of some familiar materials at room temperature............................... 8

2-2. Range of shear rates of some familiar materials and processes ........................ 9

5-1. Comparison of unsteady P ∆ and c P ∆ for distilled water ........................................ 84

5-2. Comparison of unsteady P ∆ and c P ∆ for bovine blood .......................................... 86

5-3. Density estimation............................................................................................. 99

6-1. Comparison of initial guess and resulting value using power-law model.........124

6-2. Comparison of initial guess and resulting value using Casson model ..............144

6-3. Comparison of initial guess and resulting valueusing Herschel-Bulkley model ..........................................................................155

6-4. Comparison of four unknowns determined with Herschel-Bulkley model

for three consecutive tests..................................................................................157

6-5. Various physiological studies with non-Newtonian constitutive models .........167

6-6. Measurements of water viscosity ......................................................................169

6-7. Measurements of bovine blood viscosity ..........................................................171

6-8. Measurements of human blood viscosity ..........................................................173

6-9. Comparison of model constants, yh∆ and yτ ...................................................175

6-10. Comparison of∞=

∆ t h and y st hh ∆+∆ ..............................................................176

H-1. A typical experimental data set of human blood obtained by a scanning

capillary-tube rheometer with precision glass riser tubes.................................210

H-2. A typical experimental data set of distilled water obtained by a scanning

capillary-tube rheometer with plastic riser tubes..............................................213

H-3. A typical experimental data set of bovine blood obtained by a scanning



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H-4. A typical experimental data set of human blood obtained by a scanning



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List of Figures

2-1. Flow curve of a Newtonian fluid....................................................................... 7

2-2. Flow curve of power-law fluids......................................................................... 16

2-3. Flow curve of a Casson model .......................................................................... 17

2-4. Flow curve of viscoplastic fluids....................................................................... 18

2-5. Comparison of Newtonian plasma viscosity andshear-thinning whole blood viscosity ............................................................... 26

2-6. Variation of the relative viscosity of blood and suspension with rigid spheresat a shear rate > 100 s

-1..................................................................................... 27

2-7. Rouleaux formation of human red blood cells photographed on a microscopeslide showing single linear and branched aggregates and a network................ 28

2-8. Elevated blood viscosity at low shear rates indicates RBC aggregation........... 29

3-1. Rheometers ........................................................................................................ 33

3-2. Schematic diagram of a concentric cylinder viscometer ................................... 36

3-3. Schematic diagram of a con-and-plate viscometer............................................ 37

3-4. Schematic diagram of a capillary-tube viscometer............................................ 40

3-5. Determination of yield stress by extrapolation.................................................. 45

4-1. Schematic diagram of a U-shaped tube set........................................................ 56

4-2. Fluid-level variation in a U-shaped tube set during a test ................................. 57

4-3. Typical fluid-level variation measured by a SCTR........................................... 58

4-4. Liquid-solid interface condition for each fluid columnof a U-shaped tube set........................................................................................ 59

4-5. Fluid element in a capillary tube at time t ........................................................ 79

4-6. Velocity profile of plug flow of blood in a capillary tube................................. 80


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5-1. Pressure drop estimation for distilled water ...................................................... 83

5-2. Pressure drop estimation for bovine blood ........................................................ 85

5-3. Flow-pattern changes due to end effects ........................................................... 89

5-4. Migration of cells toward to the center of lumen (wall effect).......................... 92

5-5. Fahraeus-Lindquist effect due to the reduction in hematocrit in a tube with a

small diameter and the tendency of erythrocytes to migrate towardthe center of the tube......................................................................................... 93

5-6. Viscosity measurements for bovine blood with three different capillary tubes

with ID of 0.797 mm (with length = 100 mm), 1.0 mm (with length = 130 mm),and 1.2 mm (with length = 156 mm) ................................................................ 94

5-7. Viscosity results for human blood with two different capillary tubes with

length of 100 mm (with ID = 0.797 mm) and 125 mm (ID = 0.797 mm) ........100

5-8. Schematic diagram of a U-shaped tube set for temperature measurement........102

5-9. Temperature measurement at a capillary tube during a viscosity test...............103

5-10. Schematic diagram of a scanning capillary-tube

rheometer (SCTR) system................................................................................109

5-11. Variations of both power-law index and consistency index of dye-water

solution due to effects of dye concentrations...................................................110

5-12. Viscosity data for dye-water solution with 6 different dye concentrations

at 25℃..............................................................................................................111

6-1. Schematic diagram of a scanning capillary-tube rheometer

with precision glass riser tubes .........................................................................121

6-2. Curve-fitting procedure with power-law model for mineral oil ........................122

6-3. Curve-fitting procedure with power-law model for human blood ....................123

6-4. Height variation in each riser tube vs. time for mineral oil...............................125

6-5. Viscosity measurement for mineral oil at 25℃ with a scanning

capillary-tube rheometer (SCTR) .....................................................................126

6-6. Height variation in each riser tube vs. time for human blood at 37℃. .............127


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6-7. Viscosity measurement (log-log scale) for human blood at 37℃ with

rotating viscometer (RV) and scanning capillary-tube rheometer (SCTR) ......128

6-8. Viscosity measurement (log-log scale) of unadulterated human blood

at 37℃, measured with scanning capillary-tube rheometer (SCTR) and

cone-and-plate rotating viscometer (RV), for two different donors .................129

6-9. Picture of a SCTR with plastic riser tubes.........................................................141

6-10. Heating pad for a test with unadulterated human blood...................................142

6-11. Curve-fitting procedure with Casson model for distilled water .......................143

6-12. Curve-fitting procedure with Casson model for donor 1..................................145

6-13. Curve-fitting procedure with Casson model for donor 2..................................146

6-14. Height variation in each riser tube vs. time for distilled water at 25℃............147

6-15. Viscosity measurement for distilled water at 25℃ ..........................................148

6-16. Height variation in each riser tube vs. time for bovine blood

with 7.5% EDTA at 25℃.................................................................................149

6-17. Viscosity measurement for bovine blood with 7.5% EDTA at 25℃ using

both rotating viscometer (RV) and scanning capillary-tube

Rheometer (SCTR) ..........................................................................................150

6-18. Height variation in each riser tube vs. time for human blood at 37℃ .............151

6-19. Viscosity measurement for human blood (2 different donors) at 37℃ ............152

6-20. Shear-stress variation vs. shear rate for human blood

(from 2 different donors) at 37℃.....................................................................153

6-21. Curve-fitting procedure with Herschel-Bulkley model for bovine blood ........154

6-22. Viscosity measurements of bovine blood with 7.5% EDTA

at 25℃, analyzed with Herschel-Bulkley model.............................................156

6-23. Test with distilled water at 25℃.......................................................................168

6-24. Test with bovine blood at 25℃ ........................................................................170


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6-25. Test with unadulterated human blood at 37℃..................................................172

6-26. Wall shear stress at a capillary tube vs. shear rate............................................174

6-27. Variations of a plug-flow region at a capillary tube as a function of time

for bovine blood with 7.5% EDTA at 25℃ .....................................................177

6-28. Velocity profiles at a capillary tube for bovine blood

with 7.5% EDTA at 25℃.................................................................................178

6-29. (a) Viscosity, (b) wall shear rate, and (c) wall shear stress

Plotted as a function of mean velocity at a capillary tube using

three non-Newtonian models for bovine blood with 7.5% EDTA ..................179

B-1. Falling cylinder viscometers.............................................................................199

C-1. Cross sectional view of SV352A8-01 module..................................................201

G-1. Repeatability study #1 ......................................................................................208

G-2. Repeatability study #2 ......................................................................................209


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AbstractA Study of Non-Newtonian Viscosity and Yield Stress of Blood

in a Scanning Capillary-Tube Rheometer

Sangho Kim

Professors Young I. Cho and David M. Wootton

The study of hemorheology has been of great interest in the fields of

biomedical engineering and medical researches for many years. Although a number

of researchers have investigated correlations between whole blood viscosity and

arterial diseases, stroke, hypertension, diabetes, smoking, aging, and gender, the

medical community has been slow in realizing the significance of the whole blood

viscosity, which can be partly attributed to the lack of an uncomplicated and clinically

practical rheometer.

The objectives of the present study were to investigate the theoretical

principles of a scanning capillary-tube rheometer used for measuring both the

viscosity and yield stress of blood without any anticoagulant, to experimentally

validate the scanning capillary-tube rheometer using disposable tube sets designed for

daily clinical use in measuring whole blood viscosity, and to investigate the effect of

non-Newtonian constitutive models on the blood rheology and flow patterns in the

scanning capillary-tube rheometer.

The present study introduced detailed mathematical procedures for data

reduction in the scanning capillary-tube rheometer for both viscosity and yield-stress

measurements of whole blood. Power-law, Casson, and Herschel-Bulkley models

were examined as the constitutive models for blood in the study. Both Casson and

Herschel-Bulkley models gave blood viscosity results which were in good agreement


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with each other as well as with the results obtained by a conventional rotating

viscometer, whereas the power-law model seemed to produce inaccurate viscosities at

low shear rates.

The yield stress values obtained from the Casson and Herschel-Bulkley

models for unadulterated human blood were measured to be 13.8 and 17.5 mPa,

respectively. The two models showed some discrepancies in the yield-stress values.

In the study, the wall shear stress was found to be almost independent of the

constitutive model, whereas the size of the plug flow region in the capillary tube

varies substantially with the selected model, altering the values of the wall shear rate

at a given mean velocity. The model constants and the method of the shear stress

calculation given in the study can be useful in the diagnostics and treatment of

cardiovascular diseases.


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CHAPTER 1. INTRODUCTION

1.1. Clinical Significance of Blood Viscosity

The study of hemorheology has been of great interest in the fields of

biomedical engineering and medical research for many years. Hemorheology plays

an important role in atherosclerosis [Craveri et al., 1987; Resch et al., 1991; Lee et al.,

1998; Kensey and Cho, 2001]. Hemorheological properties of blood include whole

blood viscosity, plasma viscosity, hematocrit, RBC deformability and aggregation,

and fibrinogen concentration in plasma. Although a number of parameters such as

pressure, lumen diameter, whole blood viscosity, compliance of vessels, peripheral

vascular resistance are well-known physiological parameters that affect the blood

flow, the whole blood viscosity is also an important key physiological parameter.

However, its significance has not been fully appreciated yet.

A number of researchers measured blood viscosities in patients with coronary

arterial disease such as ischemic heart disease and myocardial infarction [Jan et al.,

1975; Lowe et al., 1980; Most et al., 1986; Ernst et al., 1988; Rosenson, 1993]. They

found that the viscosity of whole blood might be associated with coronary arterial

diseases. In addition, a group of researchers reported that whole blood viscosity was

significantly higher in patients with peripheral arterial disease than that in healthy

controls [Ciuffetti et al., 1989; Lowe et al., 1993; Fowkes et al., 1994].


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Other researchers investigated correlation between the hemorheological

parameters and stroke [Grotta et al., 1985; Coull et al., 1991; Fisher and Meiselman,

1991; Briley et al., 1994]. They reported that stroke patients showed two or more

elevated rheological parameters, which included whole blood viscosity, plasma

viscosity, red blood cell (RBC) and plate aggregation, RBC rigidity, and hematocrit.

It was also reported that both whole blood viscosity and plasma viscosity were

significantly higher in patients with essential hypertension than in healthy people

[Letcher et al., 1981, 1983; Persson et al., 1991; Sharp et al., 1996; Tsuda et al., 1997;

Toth et al., 1999]. In diabetics, whole blood viscosity, plasma viscosity, and

hematocrit were elevated, whereas RBC deformability was decreased [Hoare et al.,

1976; Dintenfass, 1977; Hill et al., 1982; Poon et al., 1982; Leiper et al., 1982].

Others conducted hemorheological studies to determine the relationships

between whole blood viscosity and smoking, age, and gender [Levenson et al., 1987;

Bowdler and Foster, 1987; Fowkes et al., 1994; Ernst, 1995; Ajmani and Rifkind,

1998; Kameneva et al., 1998; Yarnell et al., 2000]. They found that smoking and

aging might cause the elevated blood viscosity. In addition, it was reported that male

blood possessed higher blood viscosity, RBC aggregability, and RBC rigidity than

premenopausal female blood, which may be attributed to monthly blood-loss

[Kameneva et al., 1998].


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1.2. Motivation of the Present Study

The medical community has been slow in realizing the significance of whole

blood viscosity, which can be attributed partly to the lack of an uncomplicated and

clinically practical method of measuring whole blood viscosity. In most clinical

studies, mainly two types of viscometer have been available for general use:

rotational viscometers and capillary tube viscometer, as will be discussed in Chapter

3. These viscometers are used at laboratory only, and are not used in a clinical

environment. Until recently, the most immediate difficulty has been the lack of an

instrument that is specially designed for daily clinical use in measuring whole blood

viscosity.

1.3. Objectives of the Present Study

The objectives of the present study were 1) to investigate the theoretical

principles of a scanning capillary-tube rheometer (SCTR), which is capable of

measuring the viscosity and yield stress of blood without adding any anticoagulant, 2)

to validate the SCTR using disposable tube sets for clinical applications, and 3) to

investigate the effect of non-Newtonian constitutive models on the blood rheology

and flow patterns in the SCTR.

The present study introduced detailed mathematical procedures for data

reduction in the SCTR for both viscosity and yield-stress measurements of blood. In


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experimental studies, distilled water (Newtonian fluid), bovine blood (non-Newtonian

fluid) with 7.5% EDTA, and unadulterated human blood (non-Newtonian fluid) were

used for the measurements of both viscosity and yield stress. Power-law, Casson, and

Herschel-Bulkley models were examined as constitutive models for blood in the study.

1.4. Outline of the Dissertation

Chapter 2 reviews the constitutive models applicable for non-Newtonian

characteristics including shear-thinning and yield stress. Chapter 3 reviews the

conventional rheometers that measure either the viscosity or yield stress of a fluid. In

this chapter, only rheometers that can be applicable to clinical applications are

discussed. Chapter 4 introduces the theory of a scanning capillary-tube rheometer.

Chapter 5 discusses the considerations for the experimental study, which include

unsteady effect, end effect, wall effect, temperature analysis, dye concentration effect,

and other possible factors. Chapter 6 presents the results of experimental studies

performed with a scanning capillary-tube rheometer. Chapter 6 also reports the effect

of non-Newtonian constitutive models on the rheological measurements and flow

patterns of blood in a capillary tube. Chapter 7 gives conclusions of the study and

recommendations for future study.


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CHAPTER 2. CONSTITUTIVE MODELS

This chapter reviews literature on non-Newtonian constitutive models, which

are applicable to the study of blood rheology. Viscous liquids including whole blood

can be divided in terms of rheological properties into Newtonian, general non-

Newtonian, and viscoplastic fluids. The characteristics of blood, which include

shear-thinning, yield stress, and thixotropy, are discussed in this chapter.

2.1. Newtonian Fluid

Fluid such as water, air, ethanol, and benzene are Newtonian. This means that

when shear stress is plotted against shear rate at a given temperature, the plot shows a

straight line with a constant slope that is independent of shear rate (see Fig. 2-1).

This slope is called the viscosity of the fluid. All gases are Newtonian, and common

liquids such as water and glycerin are also Newtonian. Also, low molecular weight

liquids and solutions of low molecular weight substances in liquids are usually

Newtonian. Some examples are aqueous solutions of sugar or salt.

The simplest constitutive equation is Newton’s law of viscosity [Middleman,

1968; Bird et al., 1987; Munson et al., 1998]:

γ τ &= (2-1)

where is the Newtonian viscosity and γ & is the shear rate or the rate of strain.


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The Newtonian fluid is the basis for classical fluid mechanics. Gases and

liquids like water and mineral oils exhibit characteristics of Newtonian viscosity.

However, many important fluids, such as blood, polymers, paint, and foods, show

non-Newtonian viscosity.

Table 2-1 shows the wide viscosity range for common materials. Different

instruments are required to measure the viscosity over this wide range. One

centipoise, 1 cP (= 10-3

Pa·s or 1 mPa·s), is approximately the viscosity of water at

room temperature. Shear rates corresponding to many industrial processes can also

vary over a wide range, as indicated in Table 2-2.


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(a)

(b)

Fig. 2-1. Flow curves of a Newtonian fluid.(a) Shear stress vs. Shear rate.

(b) Viscosity vs. Shear rate.

0

50

100

0 50 100 150

Shear rate

S h e a r s t r e s s

0

10

0 50 100

Shear rate

V i s c o s i t y


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Table 2-1. Viscosity of some familiar materials at room temperature

[Barnes et al., 1989].

Liquid Approximate Viscosity (Pa·s)

Glass 1040

Asphalt 108

Molten polymers 103

Heavy syrup 102

Honey 101

Glycerin 100

Olive oil 10-1

Light oil 10-2

Water 10-3

Air 10-5


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Table 2-2. Range of shear rates of some familiar materials and processes

[Barnes et al., 1989].

ProcessRange of

Shear Rates (s-1)Application

Sedimentation of fine powders

in a suspending liquid10

-6 – 10

-4 Medicines, paints

Leveling due to surface tension

10-2

– 10-1

Paints, printing inks

Draining under gravity 10-1

– 101 Painting, coating

Screw extruders 100 – 10

2 Polymer melts, dough

Chewing and swallowing 101 – 10

2 Foods

Dip coating 101 – 10

2 Paints, confectionery

Mixing and stirring 101

– 103

Manufacturing liquids

Pipe flow 100 – 10

3 Pumping, blood flow

Spraying and brushing 103 – 10

4 Fuel atomization, painting

Rubbing 104 – 10

5

Application of creams and

lotions to the skin

Injection mold gate 104 – 10

5 Polymer melts

Milling pigments in fluid bases

10

3

– 10

5

Paints, printing inks

Blade coating 105 – 10

6 Paper

Lubrication 103 – 10

7 Gasoline engines


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2.2. Non-Newtonian Fluid

Any fluids that do not obey the Newtonian relationship between shear stress

and shear rate are non-Newtonian. The subject of rheology is devoted to the study of

the behavior of such fluids. Aqueous solutions of high molecular weight polymers or

polymer melts, and suspensions of fine particles are usually non-Newtonian.

2.2.1. General Non-Newtonian Fluid

In the case of general non-Newtonian fluids, the slope of shear stress versus

shear rate curve is not constant. When the viscosity of a fluid decreases with

increasing shear rate, the fluid is called shear-thinning. In the opposite case where the

viscosity increases as the fluid is subjected to a high shear rate, the fluid is called

shear-thickening. The shear-thinning behavior is more common than the shear-

thickening.

In general, the Newtonian constitutive equation accurately describes the

rheological behavior of low molecular weight polymer solutions and even high

molecular weight polymer solutions at very slow rates of deformation. However,

viscosity can be a strong function of the shear rate for polymeric liquids, emulsions,

and concentrated suspensions.


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2.2.1.1. Power-law Model

One of the most widely used forms of the general non-Newtonian constitutive

relation is a power-law model, which can be described as [Middleman, 1968; Bird et

al., 1987; Munson et al., 1998]:

nmγ τ &= (2-2)

where m and n are power-law model constants. The constant, m , is a measure of

the consistency of the fluid: the higher the m is, the more viscous the fluid is. n is a

measure of the degree of non-Newtonian behavior: the greater the departure from the

unity, the more pronounced the non-Newtonian properties of the fluid are.

The viscosity for the power-law fluid can be expressed as [Middleman, 1968;

Bird et al., 1987; Munson et al., 1998]:

1−

=

n

mγ η &

(2-3)

where η is non-Newtonian apparent viscosity. It is well known that the power-law

model does not have the capability to handle the yield stress. If n < 1, a shear-

thinning fluid is obtained, which is characterized by a progressively decreasing

apparent viscosity with increasing shear rate. If n > 1, we have a shear-thickening

fluid in which the apparent viscosity increases progressively with increasing shear

rate. When n = 1, a Newtonian fluid is obtained. These three types of power-law

models are illustrated in Fig. 2-2.

One of the obvious disadvantages of the power-law model is that it fails to

describe the viscosity of many non-Newtonian fluids in very low and very high shear

rate regions. Since n is usually less than one, η goes to infinity at a very low shear


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rate (see Fig. 2-2) rather than to a constant, 0η , as is often observed experimentally.

Viscosity for many suspensions and dilute polymer solutions becomes constant at a

very high shear rate, a phenomenon that cannot be described by the power-law model.

2.2.1.2. Cross Model

As discussed in the previous section, the power-law model does not have the

capability of handling Newtonian regions of shear-thinning fluids at very low and

high shear rates. In order to overcome this drawback of the power-law model, Cross

(1965) proposed a model that can be described as [Ferguson and Kemblowski, 1991;

Cho and Kensey, 1991; Macosko, 1994]:

+−+= ∞∞ nmγ η η η γ τ &

& 10 (2-4)

where

0η and ∞η = viscosities at very low and high shear rates, respectively

m and n = model constants.

At an intermediate shear rate, the Cross model behaves like a power-law model as

shown in Fig. 2-3. However, unlike the power-law model, the Cross model produces

Newtonian viscosities ( 0η and ∞η ) at both very low and high shear rates.


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2.2.2. Viscoplastic Fluid

The other important class of non-Newtonian fluids is a viscoplastic fluid.

This is a fluid which will not flow when a very small shear stress is applied. The

shear stress must exceed a critical value known as the yield stress for the fluid to flow.

For example, when opening a tube of toothpaste, we need to apply an adequate force

in order to make the toothpaste start to flow. Therefore, viscoplastic fluids behave

like solids when the applied shear stress is less than the yield stress. Once the applied

shear stress exceeds the yield stress, the viscoplastic fluid flows just like a normal

fluid. Examples of viscoplastic fluids are blood, drilling mud, mayonnaise,

toothpaste, grease, some lubricants, and nuclear fuel slurries.

2.2.2.1. Bingham Plastic Model

Many types of food stuffs exhibit a yield stress and are said to show a plastic

or viscoplastic behavior. One of the simplest viscoplastic models is the Bingham

plastic model, and it can be expressed as follows [Bird et al., 1987; Ferguson and

Kemblowski, 1991; Macosko, 1994]:

y Bm τ γ τ += & when yτ τ ≥ , (2-5)

0=γ & when yτ τ ≤ , (2-6)

where

yτ = a constant that is interpreted as yield stress


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Bm = a model constant that is interpreted as plastic viscosity.

Basically, the Bingham plastic model can describe the viscosity characteristics of a

fluid with yield stress whose viscosity is independent of shear rate as shown in Fig. 2-

4. Therefore, the Bingham plastic model does not have the ability to handle the

shear-thinning characteristics of non-Newtonian fluids.

2.2.2.2. Casson Model

This model was originally introduced by Casson (1959) for the prediction of

the flow behavior of pigment-oil suspensions. The Casson model is based on a

structure model of the interactive behavior of solid and liquid phases of a two-phase

suspension [Casson, 1959]. The model describes the flow of viscoplastic fluids that

can be mathematically described as follows [Bird et al., 1987; Ferguson and

Kemblowski, 1991; Cho and Kensey, 1991; Macosko, 1994]:

γ τ τ &k y += when yτ τ ≥ , (2-7)

0=γ & when yτ τ ≤ , (2-8)

where k is a Casson model constant.

The Casson model shows both yield stress and shear-thinning non-Newtonian

viscosity. For materials such as blood and food products, it provides better fit than

the Bingham plastic model [Fung 1990; Cho and Kensey, 1991; Nguyen and Boger,

1992; Fung, 1993].


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2.2.2.3. Herschel-Bulkley Model

The Herschel-Bulkley model extends the simple power-law model to include a

yield stress as follows [Herschel and Bulkley, 1926; Tanner, 1985; Ferguson and

Kemblowski, 1991; Holdsworth, 1993]:

y

nm τ γ τ += & when yτ τ ≥ , (2-9)

0=γ & when yτ τ ≤ , (2-10)

where m and n are model constants.

Like the Casson model, it shows both yield stress and shear-thinning non-

Newtonian viscosity, and is used to describe the rheological behavior of food

products and biological liquids [Ferguson and Kemblowski, 1991; Holdsworth, 1993].

In addition, the Herschel-Bulkley model also gives better fit for many biological

fluids and food products than both power-law and Bingham plastic models.


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Fig. 2-2. Flow curves of power-law fluids.

(a) shear-thinning fluid ( n < 1).(b) Newtonian fluid ( n = 1).

(c) shear-thickening fluid ( n > 1).

0

5

10

0 50 100 150

Shear rate

(b)

(c)

V i s c o s i t y

(a)

0

50

100

0 50 100 150

(b)

(c)

S h e a r s t r e s s

(a)


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Fig. 2-3. Flow curve of a Cross model.

Shear rate (log)

Viscosity

(log)

0

η

∞η

Power-law region

Newtonian regions


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Fig. 2-4. Flow curves of viscoplastic fluids.

(a) Casson or Herschel-Bulkley fluid.

(b) Bingham plastic fluid.

0

50

100

0 50 100 150

(b)

S h e a r s t r e s

s

(a)

yτ

Bm

Shear rate

1


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2.3. Rheology of Blood

Blood behaves like a non-Newtonian fluid whose viscosity varies with shear

rate. The non-Newtonian characteristics of blood come from the presence of various

cells in the blood (typically making up 45% of the blood’s volume), which make

blood a suspension of particles [Fung, 1993; Guyton and Hall, 1996]. When the

blood begins to move, these particles (or cells) interact with plasma and among

themselves. Hemorheologic parameters of blood include whole blood viscosity,

plasma viscosity, red cell aggregation, and red cell deformability (or rigidity).

2.3.1. Determinants of Blood Viscosity

Much research has been performed to formulate a theory that accounts

completely for the viscous properties of blood, and some of the key determinants

have been identified [Dinnar, 1981; Chien et al., 1987; Guyton and Hall, 1996]. The

four main determinants of whole blood viscosity are (1) plasma viscosity, (2)

hematocrit, (3) RBC deformability and aggregation, and (4) temperature. The first

three factors are parameters of physiologic concern because they pertain to changes in

whole blood viscosity in the body. Especially, the second and third factors,

hematocrit and RBC aggregations, mainly contribute to the non-Newtonian

characteristics of shear-thinning viscosity and yield stress.


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2.3.1.1. Plasma Viscosity

Plasma is blood from which all cellular elements have been removed. It has

been well established that plasma behaves like a Newtonian fluid. Careful tests

conducted using both rotating and capillary tube viscometers over a range of shear

rates (i.e., from 0.1 to 1200 s-1

) found no significant departures from linearity.

Therefore, its viscosity is independent of shear rate. Figure 2-5 illustrates this clearly

in the horizontal viscosity line for plasma [Dintenfass, 1971; Dinnar, 1981]. Since

blood is a suspension of cells in plasma, the plasma viscosity affects whole blood

viscosity, particularly at high shear rates.

2.3.1.2. Hematocrit

Hematocrit is the volume percentage of red blood cells in whole blood. Since

studies have shown normal plasma to be a Newtonian fluid [Fung, 1993], the non-

Newtonian features of human blood undoubtedly come from suspended cells in blood.

The rheological properties of suspensions correlate highly with the concentrations of

suspended particles. In blood, the most important suspended particles are the red

blood cells (RBC). Hematocrit is the most important determinant of whole blood

viscosity [Benis et al., 1970; Thurston, 1978; Fung, 1993; Picart et al., 1998; Cinar et

al., 1999]. The effect of hematocrit on blood viscosity has been well documented.

All studies have shown that the viscosity of whole blood varies directly with


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hematocrit at all cell concentrations above 10%. In general, the higher the hematocrit,

the greater the value of whole blood viscosity [Dintenfass, 1971; Dinnar, 1981; Chien

et al., 1987; Guyton and Hall, 1996].

2.3.1.3. RBC Deformability

Deformability is a term used to describe the structural response of a body or

cell to applied forces. The effect of RBC deformability in influencing general fluidity

of whole blood is clearly revealed in Fig. 2-6. This figure shows the relative viscosity

of blood at a shear rate >100 s-1

(at which particle aggregation is negligible, isolating

RBC deformability) compared with that of suspensions with rigid spheres. At 50%

concentration, the viscosity of a suspension of rigid spheres reaches almost infinity so

that the suspension is not able to flow. On the contrary, normal blood remains fluid

even at a hematocrit of 98%, on account of the deformability of its RBCs [Fung,

1993].

2.3.1.4. RBC Aggregation - Major Factor of Shear-Thinning Characteristic

Since red cells do not have a nucleus, they behave like a fluid drop [Dinnar,

1981]. Hence, when a number of red cells cluster together as in the flow of a low

shear rate, they aggregate together. Accordingly, human RBCs have the ability to


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form aggregates known as rouleaux. Rouleaux formation is highly dependent on the

concentration of fibrinogen and globulin in plasma. Note that bovine blood does not

form rouleaux because of absence of fibrinogen and globulin in plasma [Fung, 1993].

Various degrees and numbers of rouleaux in linear array and branched network are

pictured in Fig. 2-7.

Figure 2-8 shows the relationship between blood viscosity and rouleaux

formation. Rouleaux formation of healthy red cells increases at decreasing shear

rates. As red cells form rouleaux, they will tumble while flowing in large vessels.

The tumbling disturbs the flow and requires the consumption of energy, thus

increasing blood viscosity at low shear [Fung, 1993]. As shear rate increases, blood

aggregates tend to be broken up, resulting in drop in blood viscosity (see Fig. 2-8). In

short, rouleaux formation increases blood viscosity, whereas breaking up rouleaux

decreases blood viscosity.

2.3.1.5. Temperature

Temperature has a dramatic effect on the viscosity of any liquid, including

whole blood and plasma. As in most fluids, blood viscosity increases as temperature

decreases [Fung, 1993; Guyton and Hall, 1996]. In blood, reduced RBC

deformability and increased plasma viscosity particularly elevate whole blood

viscosity at low temperatures [Barbee, 1973]. Consequently, precise control of the

sample temperature is necessary to measure viscosity accurately in vitro. It is


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preferable and is a standard in hemorheologic studies to carry out blood viscosity

measurements at body temperature of 37℃. Typically, blood viscosity increases less

than 2% for each℃ decrease in temperature [Barbee, 1973].

2.3.2. Yield Stress and Thixotropy

2.3.2.1. Yield Stress

In addition to non-Newtonian viscosity, blood also exhibits a yield stress. The

source of the yield stress is the presence of cells in blood, particularly red cells.

When such a huge amount (40-45% by volume) of red cells of 8-10 microns in

diameter is suspended in plasma, cohesive forces among the cells are not negligible.

The forces existing between particles are van der Waals-London forces and

Coulombic forces [Cheng and Evans, 1965; Mewis and Spaull, 1976]. Hence, in

order to initiate a flow from rest, one needs to have a force which is large enough to

break up the particle-particle links among the cells.

However, blood contains 40-45% red cells and still moves relatively easily.

The healthy red cells behave like liquid drops because the membranes of red cells are

so elastic and flexible. Note that in a fluid with no suspended particles, the fluid

starts to move as soon as an infinitesimally small amount of force is applied. Such a

fluid is called a fluid without yield stress. Examples of fluid with no yield stress

include water, air, mineral oils, and vegetable oils. Examples of fluids having the


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yield stress include blood, ketchup, salad dressings, grease, paint, and cosmetic

liquids.

The magnitude of the yield stress of human blood appears to be at the order of

0.05 dyne/cm2 (or 5 mPa) [Schmid- nbeinoSch && and Wells, 1971; Walawender et al.,

1975; Nakamura and Sawada, 1988; Fung, 1993; Stoltz et al., 1999] and is almost

independent of temperature in the range of 10-37℃ [Barbee, 1973].

2.3.2.2. Thixotropy - Time Dependence

The phenomenon of thixotropy in a liquid results from the microstructure of

the liquid system. Thixotropy may be explained as a consequence of aggregation of

suspended particles. If the suspension is at rest, the particle aggregation can form,

whereas if the suspension is sheared, the weak physical bonds among particles are

ruptured, and the network among them breaks down into separate aggregates, which

can disintegrate further into smaller fragments [Barnes, 1997].

After some time at a given shear rate, a dynamic equilibrium is established

between aggregate destruction and growth, and at higher shear rates, the equilibrium

is shifted in the direction of greater dispersion. The relatively long time required for

the microstructure to stabilize following a rapid change in the rate of flow makes

blood thixotropy readily observable [How, 1996].

This effect on viscosity has been studied using a steady flow [Huang et al.,

1975]. At high shear rates, structural change occurs more rapidly than that at low


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shear rates. In their study, the first step was from the no-flow condition to a shear

rate of 10 s-1

. They found that blood viscosity decreased over a period of

approximately 20 seconds at the shear rate of 10 s-1

before the final state was attained.

Next, when the shear rate stepped from 10 to 100 s-1

, almost no time was required to

reach the microstructual equilibrium after the change of shear rate.

Gaspar-Rosas and Thurston (1988) also investigated on erythrocyte aggregate

rheology by varying shear rate from 500 s-1

to zero. Based on their results, it can be

concluded that the recovery of quiescent structure requires approximately 50 seconds

while the high shear rate structure is attained in a few seconds. In other words, in

order to minimize the effect of the thixotropic characteristic of blood on the viscosity

measurement between the shear rates of 500 and 1 s-1

, at least 50 seconds should be

allowed during the test to have the fully aggregated quiescent state at a shear rate near

1 s-1

.


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Fig. 2-5. Comparison of Newtonian plasma viscosityand shear-thinning whole blood viscosity.

Shear rate (s

-1

)

40010

4

1 V i s c o s i t y ( c P )

100

Whole blood

Plasma


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Fig. 2-6. Variation of the relative viscosity of blood and suspension with rigid spheres

at a shear rate > 100 s-1

[Goldsmith, 1972].

Particle volume fraction

0.80.2

10

1

R e l a t i v e v i s c o s i t y

0.4

100

0.6

Normal blood

Suspension with rigid spheres


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Fig. 2-7. Rouleaux formation of human red blood cells photographed on a microscope

slide showing single linear and branched aggregates (left part) and a network (right

part). The number of cells in linear array are 2, 4, 9, 15 and 36 in a, b, c, d, and f,respectively. [Fung, 1993; Goldsmith, 1972]


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Fig. 2-8. Elevated blood viscosity at low shear rates indicates RBC aggregation

(rouleaux formation). Blood viscosity decreases with increasing shear rates as RBC

aggregations breaks up to individual red cells.

1

R e l a t i v e v i s c o s i t y

10

Normal blood

Shear rate (s-1

)

4001 10


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CHAPTER 3. CONVENTIONAL RHEOMETRY: STATE-OF-THE-ART

This chapter reviews literature on conventional rheometries. Section 3.1

briefly introduces conventional rheometers. In sections 3.2 and 3.3, viscometers

commonly used for the viscosity measurements of fluids, which have been used for

hemorheology studies, are demonstrated. Section 3.4 provides conventional methods

of measuring yield stresses of fluids. Section 3.5 presents the drawbacks of

conventional viscometers for clinical applications.

3.1. Introduction

Numerous types of rheometers have been used to measure the viscosity and

yield stress of materials [Tanner, 1985; Ferguson and Kemblowski, 1991; Macosko,

1994]. In the present study, rheometer refers to a device that can measure both

viscosity and yield stress of a material, whereas viscometer can measure only the

viscosity of the material. In addition, only shear viscometers will be discussed in the

study since the other type, extensional viscometers, are not very applicable to

relatively low viscous fluids, such as water and whole blood.

Typically, shear viscometers can be divided into two groups [Macosko, 1994]:

drag flows, in which shear is generated between a moving and a stationary solid

surface, and pressure-driven flows, in which shear is generated by a pressure

difference over a capillary tube. The commonly utilized members of these groups are


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shown in Fig. 3-1. Numerous techniques have been developed for determining the

yield stress of fluids both directly and indirectly.

Most of these viscometers can produce viscosity measurements at a specified,

constant shear rate. Therefore, in order to measure the viscosity over a range of shear

rates, one needs to repeat the measurement by varying either the pressure in the

reservoir tank of capillary tube viscometers, the rotating speed of the cone or cup in

rotating viscometers, or the density of the falling objects. Such operations make

viscosity measurements difficult and labor intensive. In addition, these viscometers

require anticoagulants in blood to prevent blood clotting. Hence, the viscosity results

include the effects of anticoagulants, which may increase or decrease blood viscosity

depending on the type of anticoagulant [Rosenblum, 1968; Crouch et al., 1986;

Reinhart et al., 1990; Kamaneva et al., 1994].

Drag-flow type of viscometers includes a falling object (ball or cylinder)

viscometer and a rotational viscometer. However, the falling object viscometer is not

very convenient to use for clinical applications. In the case of the falling object

viscometer, the relatively large amount of a test fluid is required for the viscosity

measurement. In addition, since the testing fluid is at a stationary state initially, the

type of viscometer is not very applicable to a thixotropic fluid like whole blood. The

principle of the falling object viscometer is provided in Appendix B.

For the yield measurement of blood, most researchers have used indirect

methods rather than direct methods for practical reasons [Nguyen and Boger, 1983;

de Kee et al., 1986; Magnin and Piau, 1990]. Thus, the details of direct methods will


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not be discussed in this chapter. As indirect methods, data extrapolation and

extrapolation using constitutive models are introduced and discussed in this chapter.


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Fig. 3-1. Rheometers.

Rheometers

ViscosityMeasurements

Yield Stress

Measurements

Drag

FlowsPressure-

Driven Flows

Indirect

MethodsDirect

Methods

Falling/

RollingObject

Viscometer

Capillary-Tube

Viscometer

Data

Extrapolation

Extrapolation

usingConstitutive

Models

Rotational

Viscometer


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3.2. Rotational Viscometer

In a rotational viscometer, the fluid sample is sheared as a result of the

rotation of a cylinder or cone. The shearing occurs in a narrow gap between two

surfaces, usually one rotating and the other stationary. Two frequently used

geometries are Couette (Fig. 3-2) and cone-and-plate (Fig. 3-3).

3.2.1. Rotational Coaxial-Cylinder (Couette Type)

In a coaxial-cylinder system, the inner cylinder is often referred to as bob, and

the external one as cup. The shear rate is determined by geometrical dimensions and

the speed of rotation. The shear stress is calculated from the torque and the

geometrical dimensions. By changing the speed of the rotating element, one is able to

collect different torques, which are used for the determination of the shear stress-

shear rate curve. Figure 3-2 shows a typical coaxial-cylinder system that has a fluid

confined within a narrow gap ( 99.0≥o

i

R

R) between the inner cylinder rotating at Ω

and the stationary outer cylinder.

Once the torque exerting on either inner or outer cylinder is measured, the

shear stress and shear rate can be calculated as follow [Macosko, 1994]:

H R

M R

i

i

i 22)(

π τ = or

H R

M R

o

o

o 22)(

π τ = (3-1)


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io

oi R R

R R R

−

Ω=≅ )()( γ γ && when 99.01 ≥>

o

i

R

R (3-2)

where

i R and o R = radii of inner and outer cylinders, respectively

2

oi R R R +

=

i M and o M = torques exerting on inner and outer cylinders, respectively

H = height of inner cylinder

Ω = angular velocity.

3.2.2. Cone-and-Plate

The common feature of a cone-and-plate viscometer is that the fluid is sheared

between a flat plate and a cone with a low angle; see Fig. 3-3. The cone-and-plate

system produces a flow in which the shear rate is very nearly uniform. Let’s consider

a fluid, which is contained in the gap between a plate and a cone with an angle of β .

Typically, the gap angle, β , is very small ( o4≤ ). The shear rate of the fluid depends

on the gap angle, β , and the linear speed of the plate. Assuming that the cone is

stationary and the plate rotates with a constant angular velocity of Ω , the shear stress

and shear rate can be calculated from experimentally measured torque, , and given

geometric dimensions (see Fig. 3-3) as follows [Macosko, 1994]:

32

3

R

M

π τ = and

β γ Ω=& . (3-3)


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Fig. 3-2. Schematic diagram of a concentric cylinder viscometer.

H

o R

i R

Ω


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Fig. 3-3. Schematic diagram of a cone-and-plate viscometer.

Ω

R

β

Cone

Plate

Torque measurement device

Fluid


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3.3. Capillary-Tube Viscometer

The principle of a capillary tube viscometer is based on the Hagen-Poiseuille

Equation which is valid for Newtonian fluids. Basically, one needs to measure both

pressure drop and flow rate independently in order to measure the viscosity with the

capillary tube viscometer. Since the viscosity of a Newtonian fluid does not vary

with flow or shear, one needs to have one measurement at any flow velocity.

However, for non-Newtonian fluids, it is more complicated because the viscosity

varies with flow velocity (or shear rate).

In a capillary-tube viscometer, the fluid is forced through a cylindrical

capillary tube with a smooth inner surface. The flow parameters have to be chosen in

such a way that the flow may be regarded as steady-state, isothermal, and laminar.

Knowing the dimensions of the capillary tube (i.e., its inner diameter and length), one

can determine the functional dependence between the volumetric flow rate and the

pressure drop due to friction. If the measurements are carried out so that it is possible

to establish this dependence for various values of pressure drop or flow rate, then one

is able to determine the flow curve of the fluid.

For non-Newtonian fluids, since the viscosity varies with shear rate, one needs

to vary the pressure in the reservoir in order to change the shear rate, a procedure that

is highly time-consuming. After each run, the reservoir pressure should be reset to a

new value to obtain the relation between flow rate and pressure drop. In order to

determine the flow curve of a non-Newtonian fluid, one needs to establish the

functional dependence of shear stress on shear rate in a wide range of these variables.


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Figure 3-4 shows the schematic diagram of a typical capillary-tube viscometer,

which has the capillary tube with an inner radius of c R and a length of c L . It is

assumed that the ratio of the capillary length to its inner radius is so large that one

may neglect the so-called end effects occurring in the entrance and exit regions of the

capillary tube. Then, the shear stress at the tube wall can be obtained as follows:

c

c

L

P r

2

∆=τ (3-4)

c

ccw

L

P R

2

∆=τ (3-5)

where

τ and wτ = shear stresses at distance r and at tube wall, respectively

r = distance from the capillary axis

c P ∆ = pressure drop across a capillary tube.

It is of note that the shear stress distribution is valid for fluids of any rheological

properties.

In the case of a Newtonian fluid, the shear rate at tube wall can be expressed

by taking advantage of the well-known Hagen-Poiseuille Equation as:

cc

w R

V

R

Q 443 ==

π γ & (3-6)

where

wγ & = wall shear rate

V R L

P RQ c

c

cc⋅=

∆=

2

4

8π

µ

π = volumetric flow rate ( Hagen-Poiseuille Equation)

V = mean velocity.


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Fig. 3-4. Schematic diagram of a capillary-tube viscometer.

Test fluid

Capillary tube

Air

Balance

Collected

test fluid

c R2c L

Reservoir

tank

Compressed

air


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3.4. Yield Stress Measurement

Whether yield stress is a true material property or not is still a controversial

issue [Barnes and Walters, 1985]. However, there is generally an acceptance of its

practical usefulness in engineering design and operation of processes where handling

and transport of industrial suspensions are involved. The minimum pump pressure

required to start a slurry pipeline, the leveling and holding ability of paint, and the

entrapment of air in thick pastes are typical problems where the knowledge of the

yield stress is essential.

Numerous techniques have been developed for determining the yield stress

both directly and indirectly based on the general definition of the yield stress as the

stress limit between flow and non-flow conditions. Indirect methods simply involve

the extrapolation of shear stress-shear rate data to zero shear rate with or without the

help of a rheological model. Direct measurements generally rely on some

independent assessment of yield stress as the critical shear stress at which the fluid

yields or starts to flow.

The value obtained by the extrapolation of a flow curve is known as

“extrapolated” or “apparent” yield stress, whereas yield stress measured directly,

usually under a near static condition, is termed “static” or “true” yield value.


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3.4.1. Indirect Method

Indirect determination of the yield stress simply involves the extrapolation of

experimental shear stress-shear rate data at zero shear rate (see Fig. 3-5). The

extrapolation may be performed graphically or numerically, or can be fitted to a

suitable rheological model representing the fluid and the yield stress parameter in the

model is determined.

3.4.1.1. Direct Data Extrapolation

One of most common procedures is to extend the flow curve at low shear rates

to zero shear rate, and take the shear stress intercept as the yield stress value. The

technique is relatively straightforward only if the shear stress-shear rate data are

linear. With nonlinear flow curves, as shown in Fig. 3-5, the data may have to be

fitted to a polynomial equation followed by the extrapolation of the resulting curve fit

to zero shear rate. The yield stress value obtained obviously depends on the lowest

shear rate data available and used in the extrapolation. This shear rate dependence of

the extrapolated yield stress has been demonstrated by Barnes and Walters (1985)

with a well-known yield stress fluid, Carbopol (carboxylpolymethylene). They

concluded that this fluid would have no detectable yield stress even if measurement

was made at very low shear rates of 10-5

s-1

or less. This finding should be viewed

with caution, however, since virtually all viscometric instruments suffer wall slip and


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other defects which tend to be more pronounced at low shear rates especially with

yield stress fluids and particulate systems [Wildermuth and Williams, 1985; Magnin

and Piau, 1990]. Thus, it is imperative that some checking procedure should be

carried out to ascertain the reliability of the low shear rate data before extrapolation is

made.

3.4.1.2. Extrapolation Using Constitutive Models

A more convenient extrapolation technique is to approximate the experimental

data with one of the viscoplastic flow models. Many workers appear to prefer the

Bingham model which postulates a linear relationship between shear stress and shear

rate. However, since a large number of yield stress fluids including suspensions are

not Bingham plastic except at very high shear rates, the use of the Bingham plastic

model can lead to unnecessary overprediction of the yield stress as shown in Fig. 3-5

[Nguyen and Boger, 1983; de Kee et al., 1986]. Extrapolation by means of nonlinear

Casson model can be used from a linear plot of 21

τ versus 21

γ & . The application of

Herschel-Bulkley model is less certain although systematic procedures for

determining the yield stress value and the other model parameters are available

[Heywood and Cheng, 1984].

Even with the most suitable model and appropriate technique, the yield stress

value obtained cannot be regarded as an absolute material property because its

accuracy depends on the model used and the range and reliability of the experimental


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data available. Several studies have shown that a given fluid can be described equally

well by more than one model and hence can have different yield stress values

[Keentok, 1982; Nguyen and Boger, 1983; Uhlherr, 1986].

3.4.2. Direct Method

Various techniques have been introduced for measuring the yield stress

directly and independently of shear stress-shear rate data. Although the general

principle of the yield stress as the stress limit between flow and non-flow conditions

is often used, the specific criterion employed for defining the yield stress seems to

vary among these techniques. Furthermore, each technique appears to have its own

limitations and sensitivity so that no single technique can be considered versatile or

accurate enough to cover the whole range of yield stress and fluid characteristics.

Usually, the direct methods are used for fluids having yield stresses of greater than

approximately 10 Pa [Nguyen and Boger, 1983]. Therefore, as mentioned earlier, the

direct method is not very convenient to use for the yield stress measurement of blood

since the yield stress of human blood is approximately 1 to 30 mPa [Picart et al.,

1998].


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Fig. 3-5. Determination of yield stress by extrapolation [Nguyen and Boger, 1983].


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3.5. Problems with Conventional Viscometers for Clinical Applications

3.5.1. Problems with Rotational Viscometers

Over the years, rotational viscometers have been the standard in clinical

studies investigating rheological properties of blood and other body fluids. Despite

their popularity, rotational viscometers have some drawbacks that limit their clinical

applicability in measuring whole blood viscosity. They include the need to calibrate a

torque-measuring sensor, handling of blood, surface tensions effects, and the range of

reliability.

The torque-measuring sensor can be a conventional spring or a more

sophisticated electronic transducer. In either case, the sensor requires a periodic

calibration because repeated use of the sensor can alter its spring constant. The

calibration procedure is often carried out at manufacturer’s laboratory because it

requires an extremely careful and elaborate protocol, requiring the viscometer unit to

be returned for service.

Another concern is the need to work with contaminated blood specimens.

After each measurement, the blood sample must be removed from the test section,

and the test section must be cleaned manually. Not only is this procedure time-

consuming, but also it poses a potential risk for contact with contaminated blood.

Surface tension effects arise in the use of the coaxial-cylinder viscometer

because surface tension is relatively high for blood and macromolecular solutions.

The contact area between the blood and an inner cylinder is not uniform along the


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periphery. The bob (inner cylinder) is pulled in different directions and revealed in

fluctuating torque readings, introducing serious errors in viscosity measurement.

Another inherent difficulty in measuring whole blood viscosity using

rotational viscometers is the limited shear rate range. In the extremes of the reputed

range (whether high shear or low shear, depending on the instrument), the detected

torque values do not have sufficient accuracy. Usually, manufacturers recommend

discarding viscosity data if the torque is less than 10% of the maximum value of the

sensor. This restriction is a major concern. For example, in the case of Brookfield

rotational viscometer, the minimum shear rate is often limited at approximately 30-50

s-1

due to the 10% restriction.

There are other clinical, practical considerations in using the rotational

viscometer. For example, it is usually necessary to treat the blood sample with a

measurable amount of anticoagulant, such as ethylenediaminetetraacetic acid (EDTA)

or heparin, to prevent coagulation during viscosity measurements. The reason for this

is that the contact area among blood, rotational viscometer component, and air is

relatively large for the size of the blood sample, and it usually takes a relatively long

time to complete viscosity measurements over a range of shear rates. Treating blood

with such anticoagulants results in an altered sample, and subsequent viscosity

measurements do not reflect the intrinsic values of unadulterated blood.


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3.5.2. Problems with Capillary-Tube Viscometers

There are some drawbacks in the use of conventional capillary-tube

viscometers for clinical applications. The range of shear rate is limited to high shears

over 100 s-1

. Although one can produce viscosity data at lower shear rates below 100

s-1

with a sophisticated vacuum system, the capillary tube system is basically designed

and operated to obtain viscosity at the high shear range. Since it is essential to obtain

blood viscosity at low shear rates below 10 s-1, the traditional capillary tube

viscometer is not suitable for measuring the viscosity at low shear rates. However,

capillary-tube viscometer is simple in its design and uses gravity field to drive test

fluid such that there is no need for calibration.

It takes a relatively long time to complete viscosity measurements over a

range of shear rates because at each shear rate, a sufficient quantity of a fluid sample

must be collected for an accurate measurement of flow velocity. After the

measurement at one shear rate, the pressure at the reservoir tank must be readjusted to

either increase or decrease shear rate. Then, the next shear rate case resumes. Thus,

anticoagulants must be added to whole blood for the viscosity measurement over a

range of shear rates.


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CHAPTER 4. THEORY OF SCANNING CAPILLARY-TUBE RHEOMETER

Chapter 4 presents the theory of scanning capillary-tube rheometer (SCTR).

Mathematical procedures for both viscosity and yield-stress measurements were

demonstrated in detail using power-law, Casson, and Herschel-Bulkley (H-B) models.

Section 4.1 provides a brief introduction to the SCTR. In section 4.1.1, the

description of a U-shaped tube set is reported. In addition, this section shows how the

dimensions of the disposable tube set were determined. Section 4.1.2 demonstrates

the equations for the energy balance in the disposable tube set.

Section 4.2 provides the mathematical details of data reduction for both

viscosity and yield-stress measurements. Sections 4.2.1, 4.2.2, and 4.2.3 deal with

the mathematical modeling in the data reduction by using the power-law, Casson, and

H-B models, respectively. Especially, in sections 4.2.2 and 4.2.3, the yield stress as

well as the viscosity of blood was considered in the data reduction.

4.1 Scanning Capillary-Tube Rheometer (SCTR)

One of the drawbacks of using conventional capillary viscometers is that one

needs to change the pressure in the reservoir tank in order to measure the viscosity at

a different shear rate. Viscosity can only be measured at one shear rate at a time in

the conventional system. Similarly, in other types of viscometers such as rotating

viscometers and falling object viscometers, the rotating speed has to be changed or


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the density of the falling object has to be changed in order to vary shear rate as

mentioned in Chapter 3. Such operations can make viscosity measurements time

consuming and labor intensive. Because of the time required to measure viscosity

over a range of shear rates, it is necessary to add anticoagulants to blood to prevent

clotting during viscosity measurements with these conventional viscometers. The

present study introduces an innovative concept of a new capillary tube rheometer that

is capable of measuring yield stress and viscosity of whole blood continuously over a

wide range of shear rates without adding any anticoagulants.

4.1.1 U-Shaped Tube Set

Figure 4-1 shows a schematic diagram of a U-shaped tube set, which consists

of two riser tubes, a capillary tube, and a stopcock. The inside diameter of the riser

tubes in the present study is 3.2 mm. The inside diameter and length of the capillary-

tube are 0.797 and 100 mm, respectively. The small diameter of the capillary tube,

compared with that of the riser tubes, was chosen to ensure that the pressure drops at

the riser tubes and connecting fittings were negligibly small compared to the pressure

drop at the capillary tube [Kim et al., 2000a, 2000b, and 2002].

Furthermore, the inside diameter of the capillary tube was chosen to minimize

the wall effect which is often known as Fahraeus-Lindqvist effect [Fahraeus and

Lingqvist, 1931]. The details of the wall effect will be discussed in Chapter 5. In the

present study, the wall effect was found to be negligibly small.


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The length of the capillary tube (i.e., c L = 100 mm) in the U-shaped tube set

was selected to ensure that the end effects would be negligible [Kim et al., 2000a,

2000b, and 2002]. The end effects at the capillary tube will be also reported in

Chapter 5. In addition, the capillary-tube dimensions in the SCTR were selected to

complete one measurement within 2-3 min, a condition that is desirable when

measuring the viscosity of unadulterated whole blood in a clinical environment.

Figure 4-2 shows sketches of the fluid levels in the U-shaped tube set as time

goes on. The fluid level in the right-side riser tube decreases whereas that in the left-

side riser tube increases. As time goes to infinity, the two fluid levels never become

equal due to the surface tension and yield stress effects as shown in Fig. 4-2(c) (i.e.,

∞=∆ t h > 0). While a test fluid travels through the capillary tube between riser tubes 1

and 2, the pressure drop caused by the friction at the capillary tube can be obtained by

measuring the fluid levels at riser tubes 1 and 2. In Fig. 4-3, a typical fluid-level

variation measured by the SCTR is shown. Points (a), (b), and (c) represent the three

moments indicated in Fig. 4-2 (i.e., at 0=t , t > 0, and ∞=t , respectively).

4.1.2 Energy Balance

Figure 4-4 shows the liquid-solid interface condition for each fluid column of

a U-shaped tube. A falling column (right side) always has a fully wet surface

condition, while a rising column (left side) has an almost perfectly dry surface

condition at the liquid-solid interface during the entire test. Therefore, the surface


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tension at the right side was consistently greater than that at the left side since the

surface tension of a liquid is strongly dependent on the wetting condition of the tube

at the liquid-solid interface [Jacobs, 1966; Mardles, 1969; Kim et al., 2002]. The

height difference caused by the surface tension at the two riser tubes was one order of

magnitude greater than the experimental resolution desired for accurate viscosity

measurements. Thus, it is extremely important to take into account the effect of the

surface tension on the viscosity measurement using the disposable tube set.

The mathematical model of the flow analysis began with the equation of the

conservation of energy in the form of pressure unit, where the surface-tension effect

was considered between the two top points of the fluid columns at the riser tubes (see

Fig. 4-4). Assuming that the surface tension for the liquid-solid interface at each riser

tube remains constant during the test, one may write the governing equations as [Bird

et al., 1987; Munson et al., 1998]:

dst

V h g P ghV P ghV P

s

st c ∫ ∂

∂+∆+∆+++=++ ∞=

2

1

2

2

221

2

112

1

2

1 ρ ρ ρ ρ ρ ρ , (4-1)

where

1 P and 2 P = static pressures at two top points

ρ = density of fluid

g = gravitational acceleration

1V and 2V = flow velocities at two riser tubes

1h and 2h = fluid levels at two riser tubes

)(t P c∆ = pressure drop across capillary tube

∞=∆ t h = additional height difference


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V = flow velocity

t = time

s = distance measured along streamline from some arbitrary initial point.

In Eq. (4-1), the energy emitted from LEDs was ignored since the energy transferred

from the LEDs, which can affect the temperature of a test fluid, was negligible small.

In order to ensure that the amount of the heat emitted from the LEDs is very small,

the temperature of bovine blood was measured during a room-temperature test. The

results showed no changes in temperature during the test, indicating that the energy

emitted from LEDs might be negligibly small.

For the convenience of data-reduction procedure, the unsteady term in Eq. (4-

1), dst

V s

s∫ ∂∂2

1

ρ , may be ignored under the assumption of a quasi

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Blood Rheology

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