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
ii
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. iii
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 iv
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-Thinning Characteristics...... 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 v
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 vi
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 vii
APPENDIX G: REPEATABILITY STUDY WITH DISTILLED WATER ...... 208
APPENDIX H: EXPERIMENTAL DATA...... 210
VITA...... 221 viii
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 ∆Punsteady and ∆Pc for distilled water ...... 84
5-2. Comparison of ∆Punsteady and ∆Pc 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 value using 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, ∆hy and τ y ...... 175
6-10. Comparison of ∆ht=∞ and ∆hst + ∆hy ...... 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 capillary-tube rheometer with plastic riser tubes...... 215 ix
H-4. A typical experimental data set of human blood obtained by a scanning capillary-tube rheometer with plastic riser tubes...... 218
x
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 and shear-thinning whole blood viscosity ...... 26
2-6. Variation of the relative viscosity of blood and suspension with rigid spheres at a shear rate > 100 s-1 ...... 27
2-7. Rouleaux formation of human red blood cells photographed on a microscope slide 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 column of 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 xi
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 toward the 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 xii
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
xiii
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
xiv
Abstract A 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 xv 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.
1
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]. 2
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].
3
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 4 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.
5
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]:
τ = µγ& (2-1) where µ is the Newtonian viscosity and γ& is the shear rate or the rate of strain. 6
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.
7
(a) 100
50 Shear stress
0 0 50 100 150 Shear rate
(b) 10
Viscosity
0 0 50 100
Shear rate
Fig. 2-1. Flow curves of a Newtonian fluid. (a) Shear stress vs. Shear rate. (b) Viscosity vs. Shear rate.
8
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
9
Table 2-2. Range of shear rates of some familiar materials and processes [Barnes et al., 1989].
Range of Process Application Shear Rates (s-1)
Sedimentation of fine powders 10-6 – 10-4 Medicines, paints in a suspending liquid
Leveling due to surface tension 10-2 – 10-1 Paints, printing inks
Draining under gravity 10-1 – 101 Painting, coating
Screw extruders 100 – 102 Polymer melts, dough
Chewing and swallowing 101 – 102 Foods
Dip coating 101 – 102 Paints, confectionery
Mixing and stirring 101 – 103 Manufacturing liquids
Pipe flow 100 – 103 Pumping, blood flow
Spraying and brushing 103 – 104 Fuel atomization, painting
Application of creams and Rubbing 104 – 105 lotions to the skin
Injection mold gate 104 – 105 Polymer melts
Milling pigments in fluid bases 103 – 105 Paints, printing inks
Blade coating 105 – 106 Paper
Lubrication 103 – 107 Gasoline engines
10
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.
11
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]:
n τ = mγ& (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]:
n−1 η = mγ& (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 12
rate (see Fig. 2-2) rather than to a constant, η0 , 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]:
η −η τ = γ η + 0 ∞ (2-4) & ∞ n 1+ mγ& where
η0 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 (η0 and η∞ ) at both very low and high shear rates.
13
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]:
τ = mB γ& +τ y when τ ≥ τ y , (2-5)
γ& = 0 when τ ≤ τ y , (2-6) where
τ y = a constant that is interpreted as yield stress 14
mB = 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]:
τ = τ y + k γ& when τ ≥ τ y , (2-7)
γ& = 0 when τ ≤ τ y , (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].
15
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]:
n τ = mγ& +τ y when τ ≥ τ y , (2-9)
γ& = 0 when τ ≤ τ y , (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.
16
(a) 100 (b)
(c) 50
stress Shear
0
0 50 100 150
10 (c)
5 (b) Viscosity
(a) 0 0 50 100 150
Shear rate
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).
17
Newtonian regions
η0
Power-law region Viscosity (log)
η∞
Shear rate (log)
Fig. 2-3. Flow curve of a Cross model.
18
100
(a)
(b)
50 mB
Shear stress 1
τ y
0
0 50 100 150
Shear rate
Fig. 2-4. Flow curves of viscoplastic fluids. (a) Casson or Herschel-Bulkley fluid. (b) Bingham plastic fluid.
19
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.
20
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 21 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 22 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 23 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 24 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
2 0.05 dyne/cm (or 5 mPa) [Schmid-Sch&o&nbein 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 25 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.
26
4 Whole blood
Viscosity (cP) Plasma 1
10 100 400 Shear rate (s-1)
Fig. 2-5. Comparison of Newtonian plasma viscosity and shear-thinning whole blood viscosity.
27
Suspension with rigid spheres
100
10 Normal blood Relative viscosity
1
0.2 0.4 0.6 0.8
Particle volume fraction
Fig. 2-6. Variation of the relative viscosity of blood and suspension with rigid spheres at a shear rate > 100 s-1 [Goldsmith, 1972].
28
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]
29
10
Normal blood
Relative viscosity
1
1 10 400 Shear rate (s-1)
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.
30
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 31 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 32 not be discussed in this chapter. As indirect methods, data extrapolation and extrapolation using constitutive models are introduced and discussed in this chapter.
33
Rheometers
Viscosity Yield Stress Measurements Measurements
Drag Pressure- Indirect Direct Flows Driven Flows Methods Methods
Capillary- Data Extrapolation Tube Extrapolation using Viscometer Constitutive Models
Falling/ Rotational Rolling Viscometer Object Viscometer
Fig. 3-1. Rheometers.
34
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
R confined within a narrow gap ( i ≥ 0.99 ) between the inner cylinder rotating at Ω Ro 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]:
M i M o τ (Ri ) = 2 or τ (Ro ) = 2 (3-1) 2πRi H 2πRo H 35
ΩR Ri γ&(Ri ) ≅ γ&(Ro ) = when 1 > ≥ 0.99 (3-2) Ro − Ri Ro where
Ri and Ro = radii of inner and outer cylinders, respectively
R + R R = i o 2
M i and M o = 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 ( ≤ 4o ). 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, M , and given geometric dimensions (see Fig. 3-3) as follows [Macosko, 1994]:
3M Ω τ = and γ& = . (3-3) 2πR 3 β 36
Ω
Ri
H Ro
Fig. 3-2. Schematic diagram of a concentric cylinder viscometer.
37
Torque measurement device
Fluid R Cone
β
Plate
Ω
Fig. 3-3. Schematic diagram of a cone-and-plate viscometer.
38
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. 39
Figure 3-4 shows the schematic diagram of a typical capillary-tube viscometer,
which has the capillary tube with an inner radius of Rc and a length of Lc . 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:
r∆P τ = c (3-4) 2Lc
Rc ∆Pc τ w = (3-5) 2Lc where
τ and τ w = shear stresses at distance r and at tube wall, respectively
r = distance from the capillary axis
∆Pc = 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:
4Q 4V γ&w = 3 = (3-6) πRc Rc where
γ&w = wall shear rate
4 π Rc ∆Pc 2 Q = = πRc ⋅V = volumetric flow rate (Hagen-Poiseuille Equation) 8µ Lc
V = mean velocity. 40
Compressed air
Air
Test fluid
Reservoir tank Capillary tube
Lc 2Rc
Collected test fluid
Balance
Fig. 3-4. Schematic diagram of a capillary-tube viscometer.
41
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.
42
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 43 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
1 1 Casson model can be used from a linear plot of τ 2 versus γ& 2 . 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 44 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].
45
Fig. 3-5. Determination of yield stress by extrapolation [Nguyen and Boger, 1983].
46
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 47 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.
48
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.
49
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 50 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. 51
The length of the capillary tube (i.e., Lc = 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.,
∆ht=∞ > 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 t = 0 , 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 52 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]:
s 1 2 1 2 2 ∂V P1 + ρV1 + ρgh1 = P2 + ρV2 + ρgh2 + ∆Pc + ρg∆ht=∞ + ρ ds , (4-1) 2 2 ∫ s1 ∂t where
P1 and P2 = static pressures at two top points
ρ = density of fluid
g = gravitational acceleration
V1 and V2 = flow velocities at two riser tubes
h1 and h2 = fluid levels at two riser tubes
∆Pc (t) = pressure drop across capillary tube
∆ht=∞ = additional height difference 53
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-
s2 ∂V 1), ρ ds , may be ignored under the assumption of a quasi-steady state. In order ∫ s1 ∂t to make the assumption, one should make sure that the pressure drop due to the unsteady effect is very small compared with that due to the friction estimated from the steady Poiseuille flow in a capillary tube.
The unsteady term can be broken into three integrations that represent the pressure drops due to the unsteady flow along the streamlines at riser tube 1, capillary tube, and riser tube 2 as [Munson et al., 1998]:
s2 ∂V s1′ dV s2′ dV s2 dV ρ ds = ρ r ds + c ds + r ds , (4-2) ∫ s ∫∫ s ∫ s s 1 ∂t 1 dt 1′ dt 2′ dt
where Vr and Vc are mean flow velocities at riser and capillary tubes, respectively.
∂V Since the term of is independent of streamlines, one can simplify the equation as: ∂t 54
s 2 ∂V dVr dVc dVr dVc dVr ρ ds = ρ l1 + Lc + l2 = ρLc + ρ()l1 + l2 , (4-3) ∫ s 1 ∂t dt dt dt dt dt
where l1 and l2 are lengths of the liquid columns whereas Lc is the length of the
2 2 capillary tube as shown in Fig. 4-5. Using the mass conservation, πRc ⋅Vc = πRr ⋅Vr , the pressure drop due to the unsteady effect can be reduced as:
2 s2 ∂V R dV r r ∆Punsteady = ρ ds =ρLc + l1 + l2 , (4-4) ∫ s1 ∂t R dt c where
∆Punsteady = pressure drop due to the unsteady flow
Rr and Rc = radii of riser and capillary tubes, respectively.
In the present experimental set up, l1 , l2 , and Lc are measured to be
approximately 12, 4, and 10 cm, respectively. Since h1 (t) and h2 (t ) are strongly
dependent on each other by the conservation of mass for incompressible fluids, Vr
dh (t) dh (t) dV must be equal to 1 and 2 . In order to calculate the term of r from the dt dt dt experimental values, one could use the following central differential method: dV []h (t + ∆t) − 2h (t) + h (t − ∆t) [h (t + ∆t) − 2h (t) + h (t − ∆t)] r = 1 1 1 = 2 2 2 . (4-5) dt ∆t 2 ∆t 2
For the comparison of ∆Punsteady with ∆Pc , ∆Punsteady was estimated through a curve- fitting process. In order to obtain a smooth curve from raw data, the following exponential equation was used.
2 dVr −bt Error = − a ⋅ e . (4-6) dt 55
Two constants, a and b , were obtained through a curve-fitting process, a least- square method, which minimized the sum of error for all experimental data points obtained in each test.
Typical results showed that the magnitude of the pressure drop due to the
unsteady flow, ∆Punsteady , was always less than 1% of that of pressure drop at capillary
tube, ∆Pc , over the entire shear-rate range. This confirms that the assumption of a quasi-steady state could be used for the present data procedure. The details of experimental results will be discussed in Chapter 5.
Assuming a quasi-steady flow behavior, one may rewrite Eq. (4-1) as follows
[Bird et al., 1987; Munson et al., 1998]:
1 1 P + ρV 2 + ρgh (t) = P + ρV 2 + ρgh (t) + ∆P (t) + ρg∆h . (4-7) 1 2 1 1 2 2 2 2 c t=∞
Since P1 = P2 = Patm and V1 = V2 , Eq. (3-7) can be reduced as:
∆Pc (t) = ρg[]h1 (t) − h2 (t) − ∆ht=∞ . (4-8)
Note that ∆h at t = ∞ contains a height difference due to the surface tension, ∆hst ,
and an additional height difference due to the yield stress, ∆hy , for the case of blood
(i.e., see Fig. 4-3). The next section addresses the mathematical procedure of handling the yield stress.
56
Open to air
3.2 mm
Riser tubes
Capillary tube Stopcock 0.797 mm
100 mm
Fig. 4-1. Schematic diagram of a U-shaped tube set.
57
Riser tube 2 Riser tube 1 (a) at t = 0
(b) at t > 0
∆h t=∞ (c) at t = ∞
Fig. 4-2. Fluid-level variation in a U-shaped tube set during a test. 58
h1 (t)
Height ∆ht=∞
h2 (t)
(a) (b) (c) Time
Fig. 4-3. Typical fluid-level variation measured by a SCTR. (a) at t = 0 , (b) at t > 0 , and (c) at t = ∞ .
59
1 • Wet surface condition
l1
Dry surface
condition • 2
l2 2' L 1' c
Fig. 4-4. Liquid-solid interface conditions for fluid columns of a U-shaped tube set.
60
4.2 Mathematical Procedure for Data Reduction
In Chapter 2, we discussed the non-Newtonian characteristics of whole blood.
This section deals with non-Newtonian constitutive models for blood and their applications to the SCTR. Since blood has both shear-thinning (pseudo-plastic) and yield stress characteristics, three different constitutive models were used for the viscosity and/or yield-stress measurements of blood in this study. Power-law model was chosen to demonstrate the shear-thinning behavior of blood. Casson and
Herschel-Bulkley (H-B) models were selected to measure both shear thinning viscosity and yield stress of blood.
For the purpose of clinical applications, disposable tube sets can be used for the viscosity and yield-stress measurements of blood. Since the disposable tube sets have different surface conditions at riser tube 1 and 2 during the test, one needs to mathematically handle surface tension and yield stress effects in order to measure the viscosity and yield stress of blood using Casson or H-B model. The details of mathematical method of isolating those two effects are shown in this section.
4.2.1 Power-law Model
It is well known that power-law model does not have the capability to handle yield stress. As provided in Chapter 2, the relation among shear stress, shear rate, and viscosity in power-law fluids may be written as follows: 61
n τ = mγ& , (4-9)
n−1 η = mγ& . (4-10)
Since n < 1 for pseudo-plastics, the viscosity function decreases as the shear rate increases. This type of behavior is characteristic of high polymers, polymer solutions, and many suspensions including whole blood.
We consider the fluid element in the capillary tube at time t as is shown in
Fig. 4-5. The Hagen-Poiseuille flow may be used to derive the following relationship for the pressure drop at the capillary tube as a function of capillary tube geometry, fluid viscosity, and flow rate [Fung, 1990; Munson et al., 1998]:
2 2l 2Lc 2µLcγ&w 8µLcQ 8µLc Rr dh ∆Pc = τ = τ w = = 4 = 4 , (4-11) r Rc Rc πRc Rc dt where
r = radial distance
l = length of fluid element
τ and τ w = shear stress and wall shear stress, respectively
4Q γ&w = 3 = wall shear rate πRc
µ = Newtonian apparent viscosity
Lc = length of capillary tube
dh dh dh Q = πR 2 ⋅ 1 = πR 2 ⋅ 2 = πR 2 ⋅ = volumetric flow rate. r dt r dt r dt
The above relationship is valid for Newtonian fluids whose viscosities are independent of shear rate. For non-Newtonian fluids, the viscosities vary with shear 62 rate. However, the Hagen-Poiseuille flow within the capillary tube still holds for a quasi-steady laminar flow. When applying a non-Newtonian power-law model to whole blood, the pressure drop at the capillary tube can be described as follows
[Middleman, 1968; Bird et al., 1987; Fung, 1990]:
n n 2ηw Lcγ&w 2mLcγ&w 2mLc 3n +1 Q ∆Pc = = = 3 Rc Rc Rc n πRc , (4-12) n 2mL 3n +1 R 2 dh = c ⋅ r ⋅ 3 Rc n Rc dt where
ηw = power-law apparent viscosity
3n +1 Q γ&w = 3 . n πRc
It is of note that if n = 1, Eq. (4-12) yields to Eq.(4-11). Applying Eqs. (4-8), (4-11), and (4-12), one can rewrite the energy conservation equation as follows:
2 8µLc Rr dh ρg{}h1 (t) − h2 (t) − ∆ht=∞ = 4 for Newtonian fluids, (4-13) Rc dt
n 2mL 3n +1 R 2 dh ρg h (t) − h (t) − ∆h = c ⋅ r ⋅ {}1 2 t=∞ 3 Rc n Rc dt
for power-law fluids. (4-14)
For convenience, one may define a new function, θ (t) = h1 (t) − h2 (t) − ∆ht=∞ so that
Eqs. (4-13) and (4-14) become as follows: dθ = −α dt for Newtonian fluids, (4-15) θ 63 dθ 1 = −β dt for power-law fluids, (4-16) θ n where
dθ dh dh dh = 1 − 2 = −2 2 dt dt dt dt
4 ρgRc α = 2 4µLc Rr
1 ρgR n c 2mL β = c . 3n +1 R 2 ⋅ r 3 n 2Rc
The above equations are the first-order linear differential equations. Since α and β are constants, these equations can be integrated as follows:
θ (t) = θ (0) e −αt for Newtonian fluids, (4-17)
n n−1 n −1 n−1 θ (t) = θ (0) n − βt for power-law fluids, (4-18) n
where θ (0) = h1 (0) − h2 (0) − ∆ht=∞ : initial condition.
Equation (4-18) can be used for curve fitting of the experimental data (i.e.,
h1 (t) and h2 (t) ) to determine ∆ht=∞ , the power-law index, n , and the consistency index, m . A least-square method was used for the curve fitting. The data reduction procedure adopted is as follows:
1. Conduct a test and acquire all data, h1 (t ) and h2 (t ) .
2. Guess values for m , n , and ∆ht=∞ .
3. Calculate the following error values for all data points: 64
2 Error = [{θ (t)}Experimental value − {θ (t)}Theoretical value ] . (4-19)
4. Sum the error values for all data points.
5. Iterate to determine the values of m , n , and ∆ht=∞ that minimize the sum of
error.
6. Let the computer determine whether a test fluid is Newtonian or not.
7. Calculate shear rate and viscosity for all data points as follows:
Rc ρgRc γ&w = ∆Pc = θ (t) for Newtonian fluids, (4-20) 2µLc 2µLc
1 1 R n ρgR n c c γ&w = ∆Pc = θ (t) for power-law fluids. (4-21) 2mLc 2mLc
When n becomes 1 ( ± 0.001), µ is equal to m , whereas when 0< n <1, the viscosity is calculated from Eq. (4-10).
In order to obtain the velocity profile at the capillary tube, which changes with
dV time, using a power-law model, Eq. (4-21) can be used to derive it. Since γ& = − , dr the velocity profile can be expressed as follows:
1 dV (t,r) r n = − ∆Pc (t) , dr 2mLc
1 1 n 1 n n+1 ∆Pc (t) n ∆Pc (t) n n V (t,r) = − ⋅ ∫ r dr = − ⋅ ⋅ r + C , (4-22) 2mLc 2mLc n +1
where C is a constant. Using no-slip condition on the capillary wall, V (t, Rc ) = 0 , the constant can be obtained as: 65
1 ∆P (t) n n n+1 c n C = ⋅ ⋅ Rc . (4-23) 2mLc n +1
Finally, the velocity profile within the capillary tube can be expressed as follows:
1 n ∆P (t) n n+1 n+1 V (t,r) = ⋅ c ⋅ R n − r n c c n +1 2mLc (4-24) 1 n h (t) − h (t) − ∆h n n+1 n+1 = ⋅ 1 2 t=∞ ⋅ R n − r n c n +1 2mLc
where ∆Pc (t) = ρg[]h1 (t) − h2 (t) − ∆ht=∞ . Note that if power-law index becomes zero, n = 1, then the above equation yields to the equation for the Newtonian velocity profile as:
∆P (t) c 2 2 Vc (t,r) = ⋅ ()Rc − r . (4-25) 4µLc
In order to determine the mean flow velocity at the riser tube, one has to find the flow rate at the capillary tube first. The flow rate can be obtained by integrating the velocity profile over the cross-sectional area of the capillary tube as follows:
Rc Q(t) = 2π Vc (t,r)rdr ∫ 0
1 nπ ∆P (t) n 3n+1 c n = ⋅ Rc (4-26) 3n +1 2mLc
1 3n 1 nπ ρg[]h (t) − h (t) − ∆h n + 1 2 t=∞ n = ⋅ Rc 3n +1 2mLc
2 Since Q(t) = πRr Vr (t) , the mean flow velocity at the riser tube can be determined by the following equation: 66
1 3n+1 n ∆P (t) n R n V (t) = ⋅ c c r 2 3n +1 2mLc Rr (4-27)
1 3n+1 n ρg[]h (t) − h (t) − ∆h n R n = ⋅ 1 2 t=∞ c 2 3n +1 2mLc Rr
where Rr is the radius of the riser tube.
4.2.2 Casson Model
The Casson model can handle both yield stress and shear-thinning characteristics of blood, and can be described as follows [Barbee and Cokelet, 1971;
Benis et al., 1971; Reinhart et al., 1990]:
τ = τ y + k γ& when τ ≥ τ y , (4-28)
γ& = 0 when τ ≤ τ y , (4-29) where
τ and γ& = shear stress and shear rate, respectively
τ y = a constant that is interpreted as the yield stress
k = a Casson model constant.
Wall shear stress and yield stress can be defined as follows:
∆Pc (t) ⋅ Rc τ w = , (4-30) 2Lc
∆Pc (t) ⋅ ry (t) τ y = , (4-31) 2Lc 67
where ry is a radial location below which the velocity profile is uniform as shown in
Fig. 4-6, i.e., plug flow, due to the yield stress. Now, for the Casson model, Eq. (4-8)
becomes ∆Pc (t) = ρg[]h1 (t) − h2 (t) − ∆hst , indicating that the effect of the surface tension is isolated from the pressure drop across the capillary tube. Using Eqs. (4-28) and (4-29), one can obtain the expressions of shear rate and velocity profile at the capillary tube as follows:
2 ∆P (t) ⋅ r (t) dVc 1 ∆Pc (t) ⋅ r c y γ& = − = − , (4-32) dr k 2L 2L c c
1 3 3 1 ∆Pc (t) 2 2 8 2 2 2 Vc (t,r) = ⋅ Rc − r − ry (t)(Rc − r ) + 2ry (t)(Rc − r) 4k Lc 3
for ry (t) ≤ r ≤ Rc , (4-33)
1 ∆Pc (t) 3 1 Vc (t) = ⋅ ( Rc − ry (t)) ( Rc + ry (t)) for ry (t) ≥ r . (4-34) 4k Lc 3
For the purpose of simplicity, one may define two new parameters,
r ry (t) C(r) = and C y (t) = , so that Eqs. (4-33) and (4-34) become as follows: Rc Rc
1 3 2 2 2 1 ∆P (t) r 8 ry r ry r V (t,r) = ⋅ c ⋅ R 2 1− − 1− + 2 1− c c 4k Lc Rc 3 Rc Rc Rc Rc
2 1 3 Rc ∆Pc (t) 2 8 2 2 = 1− C (r) − C y (t)1− C (r) + 2C y (t)()1− C(r) 4kLc 3
for ry (t) ≤ r ≤ Rc , (4-35) 68
3 3 r r ∆Pc (t) y 1 y Vc (t) = ()Rc 1− ⋅ Rc 1+ 4kL R 3 R c c c
for ry (t) ≥ r , (4-36) 2 3 Rc ∆Pc (t) 1 = ()1− C y (t) ⋅1+ C y (t) 4kLc 3
ry (t) τ y ∆P(∞) where C y (t) = = = . Rc τ w (t) ∆P(t)
In order to determine the mean flow velocity at the riser tube, one has to find the flow rate at the capillary tube first. The flow rate can be obtained by integrating the velocity profile over the cross-sectional area of the capillary tube as follows:
Rc Q(t) = 2π Vc ⋅ r dr ∫ 0
4 πR ∆P (t) 16 2τ 1 ∆P (t) 1 = c [( c ) − ⋅ ( y ) 2 ( c ) 2 8k Lc 7 Rc Lc 4 2τ 1 2τ ∆P (t) + ⋅ ( y ) − ⋅ ( y ) 4 ( c ) −3 ] 3 R 21 R L c c c (4-37)
4 1 πRc ∆Pc 16 τ y 4 τ y 1 τ y 4 = 1− ⋅ ( ) 2 + ⋅ ( ) − ⋅ ( ) 8kLc 7 τ w 3 τ w 21 τ w
4 πR ∆P 16 1 4 1 4 c c 2 = 1− C y + C y − C y 8kLc 7 3 21
2 Since Q(t) = πRr Vr (t) , the mean flow velocity at the riser tube can be determined by the following equation: 69
4 2τ 1 1 Rc ∆Pc (t) 16 y 2 ∆Pc (t) 2 Vr (t) = 2 [( ) − ⋅ ( ) ( ) 8kRr Lc 7 Rc Lc 4 2τ 1 2τ ∆P (t) + ⋅ ( y ) − ⋅ ( y ) 4 ( c ) −3 ] 3 Rc 21 Rc Lc (4-38)
4 R ∆P 16 1 4 1 c c 2 4 = 2 [1− C y + C y − C y ] 8kRr Lc 7 3 21
where Rr is the radius of the riser tube.
For the purpose of simplicity, Eq. (4-38) can be rewritten to clearly display the unknowns and the observed variables as:
4 R ρg 16 1 V (t) = c [(h (t) − h (t) − ∆h ) − (∆h (h (t) − h (t) − ∆h )) 2 r 8kR 2 L 1 2 st 7 y 1 2 st r c (4-39) 4 1 + ∆h − ∆h 4 (h (t) − h (t) − ∆h ) −3 ] 3 y 21 y 1 2 st
∆P(∞) ∆hy where C y (t) = = . Note that Eq. (4-39) contains two ∆P(t) h1 (t) − h2 (t) − ∆hst
independent variables, i.e., h1 (t ) and h2 (t ) , and one dependent variable, i.e., Vr (t) .
There are three unknown parameters to be determined through the curve fitting in Eq.
(4-39), namely ∆hst , k , and ∆hy . ∆hst is ∆h due to the surface tension, k is the
Casson constant, and ∆hy is ∆h due to the yield stress.
Once the equation for the mean flow velocity, Vr (t) , was derived, one could
determined the unknown parameters using the experimental values of h1 (t ) and h2 (t ) .
A least-square method was used for the curve fitting. For the Casson model, there
were three unknown values, which were k , ∆hst , and τ y . Note that the unknown
values were assumed to be constant for the curve-fitting method. Since h1 (t ) and 70
h2 (t) are strongly dependent on each other by the conservation of mass for
dh (t) dh (t) incompressible fluids, 1 must be equal to − 2 . Therefore, it was more dt dt convenient and accurate to use the difference between the velocities at the two riser
d(h (t) − h (t)) dh (t) dh (t) tubes, i.e., 1 2 , than to use 1 and 2 directly. In order to get dt dt dt the difference between the two velocities from the experimental values, one could use the central differential method as follows: d()h (t) − h (t) []h (t + ∆t) − h (t + ∆t) − [h (t − ∆t) − h (t − ∆t)] 1 2 = 1 2 1 2 . (4-40) dt 2∆t
Using Eq. (4-39), the derivative of the velocity difference can be determined theoretically as follows: d()h (t) − h (t) 1 2 = 2V (t) dt r
4 1 Rc ρg 16 2 = 2 [(h1 (t) − h2 (t) − ∆hst ) − (∆hy (h1 (t) − h2 (t) − ∆hst )) 4kRr Lc 7 4 1 + ∆h − ∆h 4 (h (t) − h (t) − ∆h ) −3 ] 3 y 21 y 1 2 st
(4-41)
where Vr (t) is the mean flow velocity at the riser tube.
In order to execute the curve-fitting procedure, one needs to have a
mathematical equation of Vr for the Casson model. Eq. (4-40) and (4-41) were used for the curve fitting of the experimental data to determine the unknown constants, i.e.,
k , ∆hst , and ∆hy . Note that Eq. (4-41) could be applicable for both Casson-model 71 fluids and Newtonian fluids regardless of the existence of the yield stress. The data reduction procedure adopted is as follows:
1. Conduct a test and acquire all data, h1 (t ) and h2 (t ) .
2. Guess values for the unknowns, k , ∆hst , and ∆hy .
3. Calculate the following error values for all data points.
2 Error = [{2V (t)}Experimental values − {2V (t)}Theoretical values ] (4-42)
4. Sum the error values for all data points.
5. Iterate to determine the unknowns that minimize the sum of the error.
6. Calculate wall shear rate and viscosity for all data points as follows:
2 ρgRc γ&w (t) = ()h1 (t) − h2 (t) − ∆hst − ∆hy , (4-43) 2kLc
ρgRc []h1 (t) − h2 (t) − ∆hst ηw (t) = . (4-44) 2 γ&w (t)Lc
−5 Note that when ∆hy becomes approximately zero (i.e., ≤ resolution of 8.3×10 ), the non-Newtonian viscosity, η , is reduced to k , a Newtonian viscosity. Furthermore, the relation between wall viscosity and shear-rate can be obtained from Eqs. (4-43) and (4-44) as follow:
τ 4kτ y η (t) = k + y + w γ (t) &w γ&w (t) (4-45) ρgR ∆h 2kρgR ∆h c y c y 2L L = k + c + c γ (t) &w γ&w (t)
where k and ∆hy are the fluid properties to be determined using the Casson model. 72
Yield stress could be also determined through the curve-fitting method from
the experimental data of h1 (t ) and h2 (t ) by using the Casson model. Since the
pressure drop across the capillary tube, ∆Pc (t ) , could be determined using Eq. (4-8),
∆Pc (∞) represents the effect of the yield stress on the pressure drop. The relationship
between the yield stress, τ y , and ∆Pc (∞ ) can be written by the following equation:
∆Pc (∞) ⋅ Rc ρg∆hy ⋅ Rc τ y = = . (4-46) 2Lc 2Lc
Therefore, once ∆hy is obtained using a curve-fitting method, the yield stress can be automatically determined.
4.2.3 Herschel-Bulkley (H-B) Model
For a Herschel-Bulkley (H-B) model, the shear stress at the capillary tube can be described as follows [Tanner, 1985; Ferguson and Kemblowski, 1991; Macosko,
1994]:
n τ = mγ& +τ y when τ ≥ τ y , (4-47)
γ& = 0 when τ ≤ τ y , (4-48) where
τ and γ& = shear stress and shear rate, respectively
τ y = a constant that is interpreted as yield stress
m and n = model constants. 73
Since the H-B model reduces to the power-law model when a fluid does not have a yield stress, the H-B model is more general than the power-law model.
For the H-B model, wall shear stress and yield stress can also be defined as follows:
∆Pc (t) ⋅ Rc τ w = , (4-49) 2Lc
∆Pc (t) ⋅ ry (t) ∆Pc (∞) ⋅ Rc ρg∆hy ⋅ Rc τ y = = = , (4-50) 2Lc 2Lc 2Lc
where ry is a radial location below which the velocity profile is uniform due to the yield stress (see Fig. 4-7). Using Eqs. (4-47)-(4-50), one can obtain the expressions of shear-rate outside of the core region as:
1 1 dV ∆P n c c n γ& = − = ()r − ry for ry (t) ≤ r ≤ Rc . (4-51) dr 2mLc
The velocity profile outside of core region can be obtained by integrating Eq. (4-51) as:
1 n ∆P (t) n n+1 n+1 c n n Vc (t,r) = ⋅ ()Rc − ry (t) − ()r − ry (t) n +1 2mLc
for ry (t) ≤ r ≤ Rc . (4-52)
Since the velocity profile inside of the core region is a function of time, t , only, the
profile can be obtained using a boundary condition, Vc (t,r) = Vc (t ) at r = ry .
1 n ∆P (t) n n+1 c n Vc (t) = ⋅ ()Rc − ry (t) for ry (t) ≥ r . (4-53) n +1 2mLc 74
r Again, for the purpose of simplicity, one may define two new parameters, C(r) = Rc
ry (t) and C y (t) = , so that Eqs. (4-52) and (4-53) become as follows: Rc
1 n+1 n+1 n+1 n n n n R ∆P (t) ry (t) r ry (t) V (t,r) c c 1 c = ⋅ − − − n +1 2mLc Rc Rc Rc
1 n R n+1∆P (t) n n+1 n+1 c c 1 C (t) n C(r) C (t) n = ⋅ ()− y − ()− y n +1 2mLc
for ry (t) ≤ r ≤ Rc , (4-54)
1 n+1 n R n+1∆P (t) n r (t) n c c y Vc (t) = ⋅ ⋅1− n +1 2mLc Rc
for ry (t) ≥ r . (4-55) 1 n R n+1∆P (t) n n+1 c c 1 C (t) n = ⋅ ⋅ ()− y n +1 2mLc
In order to determine the mean flow velocity at the riser tube, one has to find the flow rate at the capillary tube first. The flow rate can be obtained by integrating the velocity profile over the cross-sectional area of the capillary tube as follows:
Rc Q(t) = 2π Vc ⋅ r dr ∫ 0
1 ∆P n n n+1 2n+1 c 2 n n = π ⋅[ry ()Rc − ry + ()()Rc + ry ⋅ Rc − ry 2mLc n +1 2n+1 3n+1 n n n n − 2 ry ()Rc − ry − 2 ()Rc − ry ] 2n +1 3n +1
(4-56) 75
1 n+1 2n+1 2 n 3n+1 n 3n+1 n ∆P n ry ry ry ry c n n = π ⋅[Rc 1− + Rc 1+ ⋅1− 2mLc n +1 Rc Rc Rc Rc 2n+1 3n+1 3n+1 n 3n+1 n n ry ry n ry n n − 2Rc 1− − 2Rc 1− ] 2n +1 Rc Rc 3n +1 Rc
1 3n+1 n ∆P n n+1 2n+1 n c 2 n n = πRc ⋅ [C y ()1− C y + ()()1+ C y ⋅ 1− C y n +1 2mLc 2n+1 3n+1 n n n n − 2 C y ()1− C y − 2 ()1− C y ] 2n +1 3n +1
ry (t) τ y ∆P(∞) ∆hy where C y (t) = = = = . Rc τ w (t) ∆P(t) h1 (t) − h2 (t) − ∆hst
2 Since Q(t) = πRr Vr (t) , the mean flow velocity at the riser tube can be determined by the following equation:
3n+1 1 R n n ∆P n n+1 2n+1 V (t) = c ⋅ c [C 2 ()1− C n + ()()1+ C ⋅ 1− C n r 2 n +1 2mL y y y y Rr c (4-57) 2n+1 3n+1 n n n n − 2 C y ()1− C y − 2 ()1− C y ] 2n +1 3n +1
Equation (4-57) can be rewritten to clearly display the unknowns and the observed variables as follows: 76
3n+1 1 R n n ρg[]h (t) − h (t) − ∆h n V (t) = c ⋅ 1 2 st r 2 Rr n +1 2mLc n+1 2 n ∆hy ∆hy ×[ ⋅1− h1 (t) − h2 (t) − ∆hst h1 (t) − h2 (t) − ∆hst 2n+1 n ∆hy ∆hy + 1+ ⋅1− (4-58) h1 (t) − h2 (t) − ∆hst h1 (t) − h2 (t) − ∆hst 2n+1 n n ∆hy ∆hy − 2 ⋅ ⋅1− 2n +1 h1 (t) − h2 (t) − ∆hst h1 (t) − h2 (t) − ∆hst 3n+1 n n ∆hy − 2 ⋅1− ] 3n +1 h1 (t) − h2 (t) − ∆hst
Note that Eq. (4-58) contains two independent variables, i.e., h1 (t ) and h2 (t ) , and
one dependent variable, i.e., Vr (t) . There are four unknown parameters to be
determined through the curve fitting in Eq. (4-58), namely m , n , ∆hst , and ∆hy .
Once the equation for the mean flow velocity, Vr (t) , was derived, one could
determined the unknown parameters using the experimental values of h1 (t) and h2 (t) by using the same curve-fitting method of determining unknowns as in the case of the
Casson model. In the case of the H-B model, there were four unknown values, which
were m , n , ∆hst , and ∆hy . Note that the unknown values were assumed to be constant for the curve-fitting method.
Using Eq. (4-58), the derivative of the velocity difference can be determined theoretically as follows: 77 d()h (t) − h (t) 1 2 = 2V (t) dt r
3n+1 1 2R n n ρg[]h (t) − h (t) − ∆h n = c ⋅ 1 2 st 2 Rr n +1 2mLc n+1 2 n ∆hy ∆hy ×[ ⋅1− h1 (t) − h2 (t) − ∆hst h1 (t) − h2 (t) − ∆hst 2n+1 n ∆hy ∆hy + 1+ ⋅1− h1 (t) − h2 (t) − ∆hst h1 (t) − h2 (t) − ∆hst 2n+1 n n ∆hy ∆hy − 2 ⋅ ⋅1− 2n +1 h1 (t) − h2 (t) − ∆hst h1 (t) − h2 (t) − ∆hst 3n+1 n n ∆hy − 2 ⋅1− ] 3n +1 h1 (t) − h2 (t) − ∆hst
(4-59)
where Vr (t) is the mean flow velocity at the riser tube. In order to execute the curve-
fitting procedure, one needs to have a mathematical equation of Vr for the H-B model.
Eq. (4-58) and (4-59) were used for the curve fitting of the experimental data to
determine the unknown constants, i.e., n , m , ∆hst , and ∆hy . Note that Eq. (4-59) could be applicable for H-B fluids, Shear-thinning fluids, and Newtonian fluids regardless of the existence of the yield stress.
After iterations for the determination of the unknowns that minimize the sum of the error, wall shear rate and viscosity for all data points can be calculated as follows:
1 1 ρgR n c n γ&w (t) = ⋅[]()h1 (t) − h2 (t) − ∆hst − ∆hy , (4-60) 2mLc 78
ρgRc []h1 (t) − h2 (t) − ∆hst ηw (t) = . (4-61) 2 γ&w (t)Lc
−5 Note that when ∆hy becomes approximately zero (i.e., ≤ resolution of 8.3×10 ), the
H-B model is reduced to power-law model. In addition, when n becomes 1, the mathematical form of the H-B model yields to Bingham plastic [Tanner, 1985], which can be described as follows:
τ = mBγ& +τ y when τ ≥ τ y , (4-62)
γ& = 0 when τ ≤ τ y , (4-63) where
τ and γ& = shear stress and shear-rate, respectively
τ y = a constant that is interpreted as the yield stress
mB = a model constant that is interpreted as the plastic viscosity.
Similar to the Casson model, the relationship between wall viscosity and shear-rate using the H-B can be expressed as follows:
n−1 τ y ηw (t) = mγ&w (t) + γ&w (t) (4-64)
ρgRc ∆hy n−1 2Lc = mγ&w (t) + γ&w (t)
where m , n , and ∆hy are the fluid properties to be determined using the H-B model.
79
Capillary tube
R r c
l
(a) Motion of a cylindrical fluid element within a capillary tube.
τ 2πrl
Pπr 2 Flow direction (P − ∆P)πr 2
l
(b) Free-body diagram of a cylinder of fluid.
Fig. 4-5. Fluid element in a capillary tube at time t .
80
Capillary tube
Rc ry
Fig. 4-6. Velocity profile of plug flow of blood in a capillary tube.
81
CHAPTER 5. CONSIDERATIONS FOR EXPERIMENTAL STUDY
Chapter 5 presents the issues and considerations in the experimental study with a scanning capillary-tube rheometer (SCTR). Theoretical and experimental issues involved in the viscosity and yield stress measurements of fluids, such as distilled water, bovine blood, and human blood, are examined.
Sections 5.1, 5.2, and 5.3 address the major assumptions in the study that may affect the rheological measurements in the SCTR: unsteady effect, end effect, and wall effect, respectively. In addition, section 5.4 reports other possible factors that include pressure drop at riser tubes, effect of density variation of blood, and thixotropic effect.
In section 5.5, the temperature consideration for the viscosity measurement of human blood in the SCTR during a test is discussed. The temperature of human blood was checked to see if it could be maintained at a body temperature of 37℃ during a viscosity measurement.
In section 5.6, the study on the effect of dye concentration on the viscosity of distilled water is presented. The objective of the study was to see whether or not the viscosity of distilled water could be altered by the addition of dye.
82
5.1. Unsteady Effect
In order to make the assumption of a quasi-steady flow behavior during a test with the SCTR, one needs to make sure that the pressure drop due to the unsteady state is negligibly small compared to that due to the friction through a capillary tube.
Distilled water and bovine blood were analyzed for the unsteady effects on the viscosity measurements of the fluids.
Figure 5-1 shows the pressure drops due to both unsteady flow and friction at the capillary tube obtained by using Eqs. (4-4), (4-6), and (4-8) in the case of distilled water. Usually, the pressure drop due to the unsteady flow was less than 3 Pa at the beginning of a test while pressure drop due to the friction at the capillary tube was greater than 250 Pa (see Table 5-1).
As shown in Fig. 5-2, in the case of bovine blood, the pressure drop due to the unsteady flow was also much smaller than that at the capillary tube. Typically, the pressure drop due to the unsteady flow was less than 1.2 Pa at the beginning of a test while pressure drop due to the friction at the capillary tube was greater than 700 Pa.
Furthermore, as shown in Table 5-2, the magnitude of the pressure drop due to the
unsteady flow, ∆Punsteady , was always less than 1% of that of the pressure drop at the
capillary tube, ∆Pc , over the entire shear-rate range. This confirms that the assumption of a quasi-steady state could be used for the present data reduction procedure.
83
4 (a) 3 ∆Punsteady
2
1
Pressure drop (Pa) drop Pressure 0
0102030
Time (s)
400
(b) 300 ∆Pc
200 100
Pressure drop (Pa) drop Pressure 0 0102030
Time (s)
Fig. 5-1. Pressure drop estimation for distilled water. (a) Pressure drop due to an unsteady flow in a test with distilled water. (b) Pressure drop at a capillary tube in a test with distilled water.
84
Table. 5-1. Comparison of ∆Punsteady and ∆Pc for distilled water.
∆Punsteady Time (s) ∆Punsteady (Pa) ∆Pc (Pa) ×100 (%) ∆Pc
0.5 2.89 245.46 1.18
1 2.61 222.08 1.18
3 1.56 132.60 1.18
5 0.93 79.40 1.17
10 0.26 22.98 1.13
15 0.07 6.85 1.02
20 0.02 2.02 0.99
85
0.8 (a) ∆P unsteady 0.4
Pressure drop (Pa) drop Pressure 0
0 30 60 90 120 150
Time (s)
800 (b) 600 ∆Pc
400
200
Pressure drop (Pa) drop Pressure 0 0 30 60 90 120 150 Time (s)
Fig. 5-2. Pressure drop estimation for bovine blood. (a) Pressure drop due to an unsteady flow in a test with bovine blood. (b) Pressure drop at a capillary tube in a test with bovine blood.
86
Table. 5-2. Comparison of ∆Punsteady and ∆Pc for bovine blood.
∆Punsteady Time (s) ∆Punsteady (Pa) ∆Pc (Pa) ×100 (%) ∆Pc
0.5 0.54 705.21 0.08
1 0.52 688.11 0.08
5 0.39 535.93 0.07
10 0.28 399.14 0.07
30 0.067 139.24 0.05
60 0.008 42.63 0.02
120 ≈ 0 15.27 ≈ 0
87
5.2. End Effect
Figure 5-3 shows the flow-pattern changes due to end effects at both (a) entrance and (b) exit of a capillary tube. Due to the sudden contraction and expansion, additional pressure drops can occur at the both ends. The most common method used
to estimate these minor pressure drops is to use the loss coefficient, K L , which is defined as [Munson et al., 1998]:
∆P K = End (5-1) L 1 ρV 2 2 c so that
1 ∆P = K ρV 2 (5-2) End L 2 c
where ∆PEnd is the pressure drop due to the end effects and Vc is the mean velocity at the capillary tube.
With the present experimental set-up, the velocity in the capillary tube was approximately 16 times greater than that in the riser tube. Therefore, the energy loss by secondary flow patterns or eddies in the entrance and exit of the capillary tube may appear to be significant in a high shear zone. In the case of a laminar flow, the loss coefficient was reported to be approximately 2.24 [Ferguson and Kemblowski,
1991]. Using the value of the loss coefficient, the pressure loss due to the sudden
changes in geometry, ∆PEnd , became only 1.79 Pa (for distilled water) and 1.88 Pa
(for bovine blood) for the maximum shear rate of 400 s-1 at a corresponding velocity of 0.04 m/s. In contrast, the pressure drops across the capillary tube at the maximum 88 shear rate were 245 Pa (for distilled water) and 705 Pa (for bovine blood), indicating that the loss due to the secondary flow patterns or eddies at both entrance and exit could be neglected.
In these end regions (see Fig. 5-3), the flow is changing from (or to) its previous (or future) distribution outside the capillary tube. The length of an end region is generally a function of tube geometry and some dynamic parameters. The entrance length, the length of tube required to achieve the fully developed simple shear flow, can be estimated by using the following equation [Middleman, 1968]:
L e ≈ 0.035 ⋅ Re (5-3) D
where Le is the entrance region, D (≈ 0.8 mm) is the inner diameter of a capillary tube, and Re is the Reynolds number. The maximum Reynolds number from a typical run in the present study was approximately 28.6. The entrance length in the capillary tube used for the present study was estimated to be 0.0008 m using the above equation. Generally, the ratio of the entrance length to the capillary-tube
L length, e , should be the order of 0.01 in order to assume the effect of entrance Lc length to be negligible [Middleman, 1968]. Since the ratio was 0.008 in the present study, it is reasonable to assume that the entrance length effect is negligibly small.
89
(a) Sudden contraction at entrance of a capillary tube
(b) Sudden expansion at exit of a capillary tube
Fig. 5-3. Flow-pattern changes due to end effects [Munson et al., 1998].
90
5.3. Wall Effect (Fahraeus-Lindqvist Effect)
Apart from end effects, other sources of error in a capillary-tube rheometry should be considered. The wall effect is one of the most important error sources
[Barnes, 1995; Missirlis et al., 2001]. For example, during the flow of a suspension, a thin layer of the solvent whose viscosity is lower than the viscosity of the suspension solution may be formed near the capillary wall, and this wall effect becomes more significant with the decrease in the capillary diameter [Dinnar, 1981; Ferguson and
Kemblowski, 1991; Fung, 1993].
In the case of blood flow, as shown in Fig. 5-4, the wall effect can be described as a tendency for RBCs to move toward the center of the tube or blood vessel [Thomas, 1962; Picart et al., 1998]. The plasma-rich zone next to the solid wall, although very thin, has an important effect on blood rheology. In other words, the plasma-rich layer near the wall must affect the measurement of blood viscosity by any instrument with a solid wall. The reduction in the RBC concentration in this layer near the wall decreases the measured value of blood viscosity, resulting in erroneous viscosity results.
Thus, the apparent viscosity of whole blood decreases with the decrease in tube diameter. However, as shown in Fig. 5-5, the wall effect is reported to be negligibly small when the tube diameter is greater than approximately 0.4 mm
[Fahraeus R and Lindquist, 1931; Dintenfass, 1971; Dinnar, 1981; Stadler et al.,
1990; Pries et al., 1992] or 0.8 mm [Haynes, 1960; Barbee, 1971]. Note that the 91 apparent viscosity of blood decreases to a value close to plasma viscosity if the diameter of the capillary tube decreases below 0.1 mm [Benis et al., 1970].
In order to check whether or not the present capillary tube diameter (with
0.797 mm ID) was large enough to prevent the wall effect, two additional capillary tubes, whose diameters were 1.0 mm (with length = 130 mm) and 1.2 mm (with length = 156 mm), were used for the viscosity measurements of bovine blood with
7.5% EDTA at a room temperature of 25℃. As shown in Fig. 5-6, the experiments performed with three different capillary tubes with ID of 0.797 mm (the standard size of the SCTR), 1.0 mm, and 1.2 mm provided almost identical viscosity results, confirming that the wall effect was negligibly small for the present capillary tube with
ID equal to 0.797 mm.
92
Arterial wall
Flow
RBCs Cell-free region
Fig. 5-4. Migration of cells toward to the center of lumen (wall effect).
93
Blood viscosity (cP)
4
1
10 100 400 800
Tube diameter (microns)
Fig. 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 toward the center of the tube [Fahraeus and Lindquist, 1931; Dintenfass, 1971; Dinnar, 1981; Stadler et al., 1990; Pries et al., 1992]
94
100 0.797 mm 1.0 mm 1.2 mm
10 Viscosity (cP) Viscosity
1 1 10 100 1000 Shear rate (s-1)
Fig. 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).
95
5.4. Other Effects
5.4.1. Pressure Drop at Riser Tube
Since the small diameter of a capillary tube was selected to make sure that the pressure drop at the capillary tube could be dominant, the pressure drops at riser tubes should be negligibly small. It has been suggested that the pressure drop in the reservoir should be estimated by using a power-law model as [Marshall and Riley,
1962; Metzger and Knox, 1965; Macosko, 1994]:
∆Pc Lr ∆Pr = n+3 (5-4) Rr n Lc ( ) Rc where
∆Pr = pressure drop in reservoir
Lr = wetted length in reservoir
Rr = radius of reservoir.
The power-law model is one of the simplest models, which can be used to show non-Newtonian behavior of blood. Furthermore, the power-law model generally provides almost identical viscosity results with both Casson and Herschel-
Bulkley models at the shear rates between approximately 300 and 30 s-1. The viscosity results of blood obtained with those models will be discussed in Chapter 6 in detail. Typically, the power-law index, n , for healthy human blood is 0.75-0.85 at a body temperature of 37℃. 96
Considering the reservoir in Eq. (5-4) as the riser tubes in the present system and n = 0.8 for human blood, one can obtain the following relation between pressure drops at capillary and riser tubes using Eq. (5-4).
1 ∆P ≈ ∆P (5-5) r 500 c
Therefore, in the case of human blood, the sum of the pressure drops at riser tubes is approximately 0.2 Pa at a shear rate of 30 s-1 while the pressure drop at the capillary tube is approximately 93 Pa.
It could be argued that Casson or Herschel-Bulkley model would have a larger pressure drop than the power-law model at a lower shear rate. Therefore, we want to examine whether or not the pressure drop at the riser tube is still negligibly small for
Casson or Herschel-Bulkley model compared to that at the capillary tube at a very low shear rate by looking at the upper bound of the error. It is rather obvious that the pressure drop at the riser tube at a low shear rate (i.e., below 30 s-1) should be smaller than 0.2 Pa. Let’s consider a shear rate of 1 s-1. The pressure drop at the capillary
-1 tube at γ& = 1 s is approximately 15 Pa. Therefore, the pressure drop at the riser tube
0.2 is less than 1.33% (i.e., = 0.0133 ). Hence, it is reasonable to assume that the 15 pressure drops at the riser tubes can be ignored compared to the pressure drop at the capillary tube.
97
5.4.2. Effect of Density Variation
In order to measure the viscosity of blood by using the SCTR, one needs to know the density of blood. However, in the case of human blood, it is not very convenient to measure the density of blood for each viscosity measurement.
Therefore, the following relation [Chien et al., 1987] between hematocrit (Hct as a dimensionless fraction) and blood density ( ρ in kg/m3) was used for the estimation of the density of human blood.
ρ = 1026 + 67Hct (5-6)
Table 5-3 shows the density of blood corresponding to hematocrit. In normal hematocrit concentrations, i.e., 35-45% [Guyton and Hall, 1996], the density variation is less than 1%, which barely affects the viscosity results of blood.
5.4.3. Aggregation Rate of RBCs – Thixotropy
As discussed in Chapter 2, the thixotropic effect on blood viscosity may be more significant at low shear rates than high shear rates. In the SCTR, the shear rate varies from high (approximately 400 s-1) to low (1 s-1) values. At least 50 seconds is required during a test to have the fully aggregated quiescent state at a shear rate near
1 s-1 [Gaspar-Rosas and Thurston, 1998].
In the viscosity measurement of human blood with the SCTR, a typical test duration in which the shear rate decreased from 10 to 1 s-1 was longer than 60 seconds. 98
It is reasonable to assume that the 60-second period is long enough to cause aggregations if the aggregations were going to take place. To further validate the above assumption, a longer capillary tube (125-mm length) was used. Since the test duration increased in the longer capillary tube, an anticoagulant (7.5% EDTA) was added to avoid the blood clotting. As shown in Fig. 5-7, the viscosity results obtained by using the longer capillary tube showed excellent agreements with those obtained by using the capillary tube with 100-mm length (also with 7.5% EDTA). Therefore, it is concluded that, in the present system, the thixotropic effect of blood on the viscosity measurement is negligibly small.
99
Table. 5-3. Density estimation
Hematocrit Density
(%) (kg/m3)
35 1049.5
40 1052.8
45 1056.2
50 1059.5
100
100 100 mm 125 mm
10 Viscosity (cP) Viscosity
1 1 10 100 1000 Shear rate (s-1)
Fig. 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).
101
5.5. Temperature Considerations for Viscosity Measurement of Human Blood
For unadulterated human blood, the temperature of a SCTR was controlled during the test at a body temperature of 37°C by using preheated disposable tube sets and a heating pad installed inside the SCTR. In order to check whether or not the temperature of blood was maintained at a body temperature, a special U-shaped tube set was prepared for the experiment. Figure 5-8 shows the special U-shaped tube set which is basically the same as a standard U-shaped tube set except three additional thermocouples placed at both ends and on the outside surface of the capillary tube.
The temperatures of blood at three predetermined points were measured during a viscosity test. Figure 5-9 provides the temperature measurement results for human blood during the test. The temperature of blood at the exit of the capillary tube was maintained at approximately 38℃, whereas that at the entrance was gradually increased from about 36℃ at the beginning of the test and reached 37.5℃ at the end of the test. Therefore, it is reasonable to say that the temperature of human blood flowing through the capillary tube during the test was maintained at a body temperature of 37℃ with ± 1℃.
102
Capillary Surface Thermocouple
Exit Entrance
Thermocouple Thermocouple
Thermometer
Fig. 5-8. Schematic diagram of a U-shaped tube set for temperature measurement.
103
40
) 38 ℃ 36
34 Entrance Exit
Temperature ( Temperature 32 Capillary Surface
30 0 50 100 150 200 Time (s)
Fig. 5-9. Temperature measurement at a capillary tube during a viscosity test.
104
5.6. Effect of Dye Concentration on the Viscosity of Water
5.6.1. Introduction
Figure 5-10 shows a schematic diagram of a SCTR, which consists of two charge-coupled devices (CCDs) that are positioned vertically, two light-emitting diodes (LEDs), two riser tubes and a capillary tube, a stopcock, and a data-acquisition system. The essential feature in the SCTR is to use two riser tubes, where initial fluid levels are different: one riser tube has a higher fluid level than the other one. Thus, at t = 0, the fluid begins to fall from the riser tube with the high level to the riser tube of low level by gravity. Since the flow rate depends on the pressure head between the two fluid levels, the flow rate gradually decreases with time as the difference between the two fluid level decreases with time. Since the flow rate can be estimated from the time rate of change of the fluid level, one can estimate both flow rate and pressure drop from the measurement of two fluid levels. Then, one can calculate shear rate from the flow rate data and shear stress from the pressure drop data, respectively.
From the shear rate and shear stress, one can determine the viscosity of the liquid.
Thus, the most important experimental variable in the operation of the SCTR is the measurement of two fluid levels in the riser tubes. As shown in Fig. 5-10, the present SCTR uses an optical detector (i.e. CCD sensors and LED array) to measure the fluid-level variations in the riser tubes. The optical detector works as follows: as an opaque fluid level rises in the riser tube, the opaque fluid blocks the passage of the light emitted by the LED. Accordingly, the number of the CCD sensors that receive 105 the light from the LED becomes smaller. Computer software records the changes in the number of CCD sensors that receive the light from the LED. Since the number of the CCD sensors that don’t receive the light from the LED is directly proportional to the fluid level, one can determine the fluid level. In other words, the instantaneous fluid levels are recorded in the form of pixel numbers (i.e., CCD sensors) versus time in a computer data file through an analog-to-digital data–acquisition system. The fluid level data from the two riser tubes were analyzed to determine the viscosity of the fluid.
Therefore, it is essential to have an opaque fluid for the present SCTR operation so that the light from the LED can be blocked by the opaque fluid as the fluid level increases, and vice versa. Of course, one can use a laser light so that a transparent fluid can be used as demonstrated by Kim et. al. (2000b). However, the cost of a SCTR using such a laser-based system became prohibitively expensive, making such a system economically unattractive.
In order to use the SCTR using CCD-LED arrangements for the viscosity measurement of a transparent fluid, one may add dye to the fluid in order to make the fluid opaque. However, the addition of a dye to a transparent fluid may alter the viscosity of the fluid. Furthermore, the addition of the dye may make a transparent
Newtonian fluid such as water a non-Newtonian fluid if the concentration of the dye is sufficiently large [Kim and Cho, 2002].
Therefore, the objective of the study is to investigate the effect of dye concentration on the viscosity of distilled water in the SCTR. More specifically, the 106 present study plans to determine the maximum concentration of dye below which the viscosity of the dye-water solution is not altered.
5.6.2. Experimental Method
Although distilled water is a Newtonian fluid, the aqueous solution of dye- water may exhibit the non-Newtonian characteristics for a sufficiently large dye concentration. Thus, in order to investigate the viscosity characteristics of a dye- water solution, the present study used a non-Newtonian model to reduce experimental data. In the previous chapters, various non-Newtonian models have been introduced for the determination of blood viscosity with the SCTR, which include power-law model, Casson model, and Herschel-Bulkley (H-B) model.
However, it is not very convenient to use the Casson model when a fluid shows only shear-thinning characteristics without yield stress. The H-B model is reduced to a power-law model for the case of fluids with no yield stress. Thus, in the present study, a power-law model was used for the viscosity analysis of the dye-water solution. The procedure of data reduction with power-law model will be discussed in
Chapter 6. Therefore, only the experimental results will be provided and discussed in the next section.
107
5.6.3. Results and Discussion
In this study, six different concentrations (0.5, 1, 2, 3, 4, and 7% by volume) of dye were used for the viscosity measurement of dye-water solution at 25℃. The dye in a liquid form was purchased at a grocery store, which was a vegetable dye produced from brand name, McCormick. For the validation of the method to reduce data for the SCTR, the viscosities of dye-water solution with different dye concentrations were compared with well-accepted reference data for water at 25℃
[Munson et al., 1998].
Figure 5-11 shows the variations of both power-law and consistency indices of the dye-water solution for six different dye concentrations. Both indices were determined through a curve-fitting method. Rectangular symbols indicate the power- law index whereas triangular symbols indicate the consistency index. As shown in the figure, both indices started to vary when the amount of dye used became greater than 2% by volume. When the dye concentration was less than 2%, the power-law index was exactly one. The values of the consistency index for 0.5%, 1%, and 2% of dye concentration cases were 0.890, 0.878, and 0.888, respectively. The distilled water viscosity, which is given in the literature as a function of temperature, was estimated to be 0.892 cP at 25℃ [Munson et al., 1998; Kim et al., 2002]. When the dye concentration was greater than 2%, the power-law index decreased from n = 1 at
2% to n = 0.913 at 7%, whereas the consistency index increased from k = 0.89 at 2% to k = 1.68 at 7%. The present results indicated that the effect of dye concentration 108 on the viscosity of the dye-water solution was negligibly small when the amount of dye used was less than 2% by volume.
Figure 5-12 shows the viscosity data for the dye-water solution with six different dye concentrations. At a high shear-rate of 500 s-1, even with high concentrations (i.e., 3, 4, 7%) of dye, the results showed that the effect of dye concentration on the viscosity of water was very small. However, at low shear-rates such as 1 and 10 s-1, the viscosity of the dye-water solution dramatically increased with increasing dye concentration.
In the present experiment, the maximum concentration of dye, under which the viscosity of the dye-water solution did not change, was approximately 2% by volume. Compared with the reference data for water at 25℃ [Munson et al., 1998], the test results obtained with 0.5%, 1%, and 2% of dye concentrations gave less than
2% error in the entire shear-rate range.
109
LED array
CCD 1
Riser tube 1 Riser tube 2
CCD 2
Computer system for data Test fluid collection
Capillary tube Three-way stopcock
Fig. 5-10. Schematic diagram of a scanning capillary-tube rheometer (SCTR) system.
110
1.04 1.8
1 1.6 1.4 0.96 ) n-1 1.2
0.92 (cPs 1 0.88 Power-law index, n 0.8 Consistency index, k index, Consistency 0.84 0.6 012345678 Dye concentration (%)
Fig. 5-11. Variations of both power-law index and consistency index of dye-water solution due to the effects of dye concentrations.
111
2.5 at 500 1/s 2 at 100 1/s at 10 1/s 1.5 at 1 1/s 1
(cP) Viscosity 0.5
0
012345678
Dye concentration (%)
Fig. 5-12. Viscosity data for dye-water solution with 6 different dye concentrations at 25℃.
112
CHAPTER 6. EXPERIMENTAL STUDY WITH SCTR
Chapter 6 presents the results of viscosity and yield stress measurements with the scanning capillary-tube rheometer (SCTR). Experimental tests were performed with mineral oil, distilled water, bovine blood with 7.5% EDTA, and unadulterated human blood.
Section 6.1 provides the viscosity results of both mineral oil and human blood produced with the SCTR (with precision glass riser tubes) using the power-law model for data reduction.
Section 6.2 gives the test results of distilled water, bovine blood, and human blood obtained with the SCTR (with plastic riser tubes). Casson and Herschel-
Bulkley models were used for data reduction to handle the yield stress of blood.
Section 6.3 reports the effect of the three models on the viscosity and yield- stress measurements of blood with the SCTR as well as on the flow patterns of blood such as velocity profile and wall shear stress in a capillary tube.
6.1. Experiments with SCTR (with Precision Glass Riser Tubes)
The present study measured the viscosity of unadulterated blood at body temperature, 37℃. Blood is a fluid consisting primarily of plasma and cells such as erythrocytes, leukocytes and platelets. Erythrocytes (i.e., red blood cells, RBC) constitute the majority of the cellular content and account for almost one half of the 113 blood volume. The presence of such a high volume of red blood cells makes blood a non-Newtonian fluid whose viscosity varies with shear rate. Whole blood viscosity decreases as shear rate increases, a phenomenon called “shear-thinning characteristics”. In other words, whole blood behavior may be described using a power-law model, a Casson model, or a Herschel-Bulkley model. In the present study, the power-law model was chosen for simplicity.
In order to demonstrate the validity of the scanning capillary-tube rheometer, the viscosity data were compared with data obtained using a cone-and-plate rotating viscometer. Since the rotating viscometer produces only 7 data points at relatively high shear rates due to the torque requirement, the accuracy of the new scanning rheometer at a low shear rate range was demonstrated by comparison with the viscosity of a standard-viscosity oil (a Newtonian fluid) from Cannon Instrument
Company (State Park, PA).
6.1.1. Description of Instrument
Figure 6-1 shows a schematic diagram of the scanning capillary-tube rheometer, which consists of two charge-coupled devices (CCDs) that are positioned vertically, two light emitting diodes (LEDs), two riser tubes and a capillary tube both made of precision glass, two stopcocks, a transfer tube made of tygon, and a computer acquisition system. The inside diameter of both the transfer and riser tubes 114 used in the present tests were 3 mm. The inside diameter and length of the capillary tube were 0.797 mm and 100 mm, respectively.
The essential feature in the scanning capillary-tube rheometer is the use of an optical detector (i.e., CCD sensors and LED array) to measure the fluid level
variations in the riser tubes, h1 (t) and h2 (t ) , every 0.02 s. The instantaneous fluid levels were recorded in a computer data file through an analog-to-digital data acquisition system in the form of pixel numbers vs. time. Since 12 pixels are equal to
1 mm, one could determine the actual height changes in the riser tube with an accuracy of 0.083 mm.
6.1.2. Testing Procedure
Typical tests are conducted as follows: The system was turned on and connected to a computer. The software on the computer was executed, and communication with the viscosity measurement system was properly established. At that point, the computer was ready to acquire data from the CCD sensors in the system. The experimental test run was initiated with a venipuncture on the patient using a 19-gauge stainless steel needle. Fresh blood was first directed from the first stopcock to the second stopcock to collect blood into the syringe. About 5 ml of blood was collected in the syringe for tests with a cone-and-plate viscometer
(Brookfield DV-III) and hematocrit measurements, and the syringe was then removed from the system. Approximately 0.5 ml of this fresh blood from the syringe was 115 immediately transferred to the sample cup of the Brookfield rotating viscometer that was maintained at a constant temperature of 37℃ by a water bath connected to the cup.
The viscosity measurements with the rotating viscometer were completed within approximately 1 minute from the time when the blood left the human body.
Blood clotting rapidly developed inside the cone-and-plate test section. The rate of blood clotting with time critically depends on the thrombotic tendency of a particular individual’s blood. As soon as blood began to clot, the rotating viscometer flashed an
“EEEE” sign indicating an overloaded torque, and thus tests were stopped. This usually happened within 2 minutes of the test. During the viscosity measurement with the Brookfield rotating viscometer, hematocrit values were determined with a microhematocrit centrifuge (International Clinical Centrifuge).
Immediately following the removal of the syringe, the experiment with the scanning capillary-tube rheometer was continued with the second stopcock turned to a position to allow blood flow to both the capillary tube and riser tube 2. When blood reached a predetermined height of 300 pixels in the riser tube 2, the second stopcock was shut to stop further blood flow into the riser tube 2, and the first stopcock was then turned to direct blood flow into the riser tube 1 up to a height of 1000 pixels. At t = 0, the data acquisition system was enabled, and both stopcocks were adjusted to allow blood to flow from the riser tube 1 to tube 2 as driven by the gravity head. Of note is that the initial pixel difference of 700 was chosen to produce the maximum shear rate of approximately 400 s-1. If a higher shear rate is desired, an initial pixel difference greater than 700 can easily be selected. 116
For the purpose of calibration, the present study used the scanning capillary- tube rheometer to measure the viscosity of mineral oil, which had a standard
Newtonian viscosity of 9.9 cP at 25℃. In the tests with human blood and mineral oil, the capillary tube and major portions of the transfer tube in which test fluids were actually flowing through the capillary tube were placed in a water bath maintained at
37℃ and 25℃, respectively.
6.1.3. Data Reduction with Power-law Model
The mathematical procedure for data reduction using a power-law model was discussed in Chapter 4. The least-square method was used for curve fitting of the experimental data and Eq. (4-18) in order to determine the power-law index, n , and the consistency index, m . A standard software package (Excel-Solver, Microsoft; see Appendix E), which has a formula known as a Newton’s method (see Appendix
F) [Microsoft Corporation, 57926-0694; Harris, 1998; John, 1998; Brown, 2001], was used for iterations to determine the values of n and m that minimize the sum of error
(see Eq. (4-19)).
The analysis of data reduction for a mineral oil is introduced in Fig. 6-2.
Figure 6-2(a) shows both experimental values of θ (t) and theoretical values of θ (t) that were obtained with initial guesses of the two unknowns, whereas Figure 6-2(b) shows the curve-fitting result after iterations to minimize the sum of error. The initial guesses for n and m were 0.8 and 8 (cP·sn-1), respectively, in the case of the mineral 117 oil (see Fig. 6-2(a)). The resulting values of n and m were determined to be 1 and
9.91 (cP·sn-1), respectively, by the iterations using the Excel-Solver.
The analysis of data reduction for human blood is shown in Fig. 6-3. For the human blood case, initial guesses of the two unknowns of n and m were 0.8 and 6
(cP·sn-1), respectively (see Fig. 6-3(a)). After iterations to minimize the sum of error, the unknowns, n and m , were determined to be 0.83 and 9.27 (cP·sn-1), respectively
(see Fig. 6-3(b)). The initial guesses and resulting values of n , m , and ∆ht=∞ for both mineral oil and human blood were reported in Table 6-1.
6.1.4. Results and Discussion
Both mineral oil and unadulterated human blood were used in the present study. The former was specially ordered as a dyed viscosity-standard fluid (i.e., 9.9 cP at 25℃) from Cannon Instrument Company (State Park, PA), and the latter was obtained from donors. For comparison purpose, the viscosity of the human blood was also measured by using a cone-and-plate rotating viscometer (Brookfield model DV-
III) at 37℃. The rotating viscometer used in the present study had an LV-type spring torque with a CP-40 spindle. In order to maintain the preset temperatures, a water bath (PolyScience model 2LS-M) was used, which controlled the temperature with an accuracy of 0.1℃.
Figures 6-4 and 6-5 show test results obtained with the mineral oil at 25℃.
Figure 6-4 shows the fluid level variations in the riser tubes, h1 (t) and h2 (t ) . Both 118 fluid levels converge gradually from the initial fluid level difference and eventually reach an equilibrium fluid level. In the case of mineral oil, 14.5 mm of an initial fluid level difference was used to ensure that viscosity measurements at a low shear rate
range were accurate. It is of note that, for mineral oil, ∆ht=∞ was found to be zero.
Figure 6-5 shows viscosity results for the mineral oil at 25℃ obtained with the SCTR. The power-law index of the mineral oil was determined to be 1.0 by a computer program (Excel-solver), confirming that it was a Newtonian fluid. Based on the present viscosity measurement method, the viscosity of the mineral oil was found to be between 9.86 and 9.91 cP at 25℃, which was a 0.5% difference in the whole range of shear rates from the standard viscosity of 9.9 cP at the same temperature.
Figure 6-6 shows height variations in each riser tube as a function of time for fresh human blood at 37℃. In the case of human blood, about 58 mm of initial fluid level difference was used for the viscosity measurement so that one could obtain the accurate viscosity of human blood over a wide shear rate range as low as 1 s-1. In order to finish a test without using anticoagulants, the test should be completed within
3-4 minutes. Otherwise, blood may begin to clot. In the present study, one test run took less than 2 minutes. For human blood, the trends of fluid level variations were
very similar to those for mineral oil. However, ∆ht=∞ for human blood was not zero but a finite value, which depended on the individual donor. The minimum and
maximum values of ∆ht=∞ were found to be 3.86 mm and 6.22 mm, respectively, 119
among 8 donors. These values of ∆ht=∞ represent the thixotropic characteristics of blood that result in the yield stress.
Figure 6-7 shows the viscosity of unadulterated human blood at 37℃, which was measured with both the SCTR and the cone-and-plate rotating viscometer (RV).
Closed circle symbols indicate viscosity data measured with the SCTR while triangle symbols indicate those measured with the RV. The viscosity of the unadulterated human blood measured with the present SCTR was based on a calculation method that determined the power-law index, n , and consistency index, m . In the case shown in Fig. 5, the values of n and m were 0.828 (dimensionless) and 9.267
(cP ⋅s n-1 ) , respectively.
Compared with the measured data using the RV, the present test results from the SCTR gave excellent agreement with those measured by the RV (i.e., less than
5% difference) in a shear rate range between 30 and 375 s-1. However, as the shear rate decreased below 30 s-1, the RV was not recommended by Brookfield for the measurement of blood viscosity. More specifically, the shear stress should vary from a minimum of 10% to 100% of the full range of the torque sensor used in the rotational type viscometer at a given shear rate for reasonably accurate viscosity measurements [Brookfield, 1999]. Therefore, the minimum shear rate at which the
RV could be used for the viscosity measurement of human blood was 30 s-1.
Blood clotting in the RV was the other reason that one could not obtain more than 7 data points. One could see the effects of blood clotting on viscosity as testing time passed beyond 1 minute with the RV. Since only 0.5 ml was used for the RV test, the blood contact area with the surface of the cone-and-plate was much bigger 120 than that in the case of the SCTR, a condition that might have caused rapid blood clotting.
Figure 6-8 shows the viscosity of unadulterated human blood for two different donors at 37℃, whose hematocrits were Hct = 41 and 46.5. Furthermore, human blood from 8 donors was tested for viscosity measurements in the present study.
Every result using SCTR gave good agreement with that from the RV at high shear rates but had a different slope with respect to shear rate individually. The viscosity for the case with Hct = 46.5 was consistently greater than that for the case with Hct =
41. The difference between the two viscosity data was very small at high shear rates greater than 300 s-1 whereas the difference was significant (i.e., greater than 200%) at a low shear range, indicating the significance of low shear viscosity data.
In fact, it is well known that slip at the wall occurs in the flow of two-phase systems because of the displacement of the disperse phase away from solid surfaces
[Barnes, 1995; Picart et al., 1998a, 1998b]. In the case of blood, a significant amount of slip appears at low shear rates when the size of RBC (red blood cells) is relatively large compared to wall roughness. For a smooth geometry like a glass tube, however, the slip effect begins to be considerable from as low as 0.5 s-1 [Picart et al., 1998a].
Therefore, whole blood that was used in the present study did not show large slip effects since the lowest shear rate data used was 1 s-1. In order to get reliable viscosity data below 0.5 s-1, it may be necessary to use a rough surface capillary.
121
LED Array
CCD CCD
Riser tube 2 Riser tube 1
First stopcock Computer system Blood from vein Transfer tube Blood
Water bath Second stopcock
Capillary tube
Fig. 6-1. Schematic diagram of the scanning capillary-tube rheometer with precision glass riser tubes.
122
16
Experimental data 12 θ Theoretical data
(mm) 8
n = 0.8
m = 8 (cP·sn-1) 4
0
0 20406080100120
Time (s)
(a) With initial guess values
16
Experimental data 12 θ Theoretical data (mm) 8 n = 1 n-1 m = 9.91 (cP·s ) 4
0 0 20 40 60 80 100 120
Time (s)
(b) With final resulting values
Fig. 6-2. Curve-fitting procedure with power-law model for mineral oil.
123
60
Experimental data
40 θ Theoretical data (mm)
n = 0.8 20 m = 6 (cP·sn-1)
0
0 20 40 60 80 100 120 Time (s)
(a) With initial guess values
60 Experimental data
40 Theoretical data θ (mm) n = 0.83 n-1 20 m = 9.27 (cP·s )
0
0 20 40 60 80 100 120 Time (s)
(b) With final resulting values
Fig. 6-3. Curve-fitting procedure with power-law model for human blood.
124
Table. 6-1. Comparison of initial guess and resulting value using power-law model.
Distilled Water Human Blood
n = 0.8 n = 0.8 n-1 n-1 Initial Guess m = 8 (cP·s ) m = 6 (cP·s )
∆h = 0.03 mm ∆h = 3 mm t=∞ t=∞
n = 1 n = 0.83 n-1 n-1 Resulting Value m = 9.91 (cP·s ) m = 9.27 (cP·s )
∆h = 0.0271 mm ∆h = 3.86 mm t=∞ t=∞
125
60
Riser tube 1 40
Riser tube 2
Height (mm) 20
0 050100150 Time (s)
Fig. 6-4. Height variation in each riser tube vs. time for mineral oil (9.9 cP viscosity- standard oil).
126
12
10
SCTR 8 Viscosity (cP) Viscosity
6 0204060 Shear rate (s-1)
Fig. 6-5. Viscosity measurement for mineral oil at 25℃ with a scanning capillary- tube rheometer (SCTR).
127
100 90 80 Riser tube 1 70 60 50 40 Riser tube 2 Height (mm) 30 20 10 0 0 50 100 150 Time (s)
Fig. 6-6. Height variation in each riser tube vs. time for human blood at 37℃.
128
100 SCTV Hematocrit : 40.5 RV
10 Viscosity (cP) Viscosity
1 1 10 100 1000 Shear rate (s-1)
Fig. 6-7. Viscosity measurement (log-log scale) for human blood at 37℃ with rotating viscometer (RV) and scanning capillary-tube rheometer (SCTR).
129
100
SCTV Hematocrit : 46.5 RV
10 Viscosity (cP) Viscosity Hematocrit : 41
1 1 10 100 1000 Shear rate (s-1)
Fig. 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.
130
6.2. Experiments with SCTR (with Plastic Riser Tubes)
A new U-shaped scanning capillary-tube rheometer (SCTR) has been developed from the concept of a conventional capillary-tube viscometer. In general, the capillary-tube viscometer is an attractive technique for several reasons. The basic instrument is relatively inexpensive, easy to construct and simple to use experimentally. Temperature control is relatively easy. The flow in a capillary tube most closely simulates the blood flow in physiological conditions compared with the flows in rotating viscometers and falling-object viscometers. Most of all, since the capillary-tube viscometer uses the gravity as the driving force, it does not require periodic calibration of any components. However, significance of end effect, wall effect, and surface-tension effect should be carefully considered.
The end and wall effects can be made negligibly small by selecting appropriate dimensions for the capillary and riser tubes. However, unlike the case of the SCTR with precision glass riser tubes, the effect of the surface tension is a unique and critical factor in using U-shaped disposable tube sets. Since inexpensive disposable capillary-riser tube sets should be used for clinical applications, it may not be easy to strictly control the surface quality of riser tubes within a certain limit.
Therefore, the effects of the surface tension at the riser tubes as well as the properties of a testing fluid had to be considered in the viscosity and yield stress measurements with the SCTR.
The resistance associated with air-liquid interfaces in plastic riser tubes of a small diameter in the SCTR can be a significant part of the pressure head applied to 131 the SCTR, particularly at low shear rates [Jacobs, 1966; Mardles, 1969; Einfeldt and
Schmelzer, 1982]. The meniscus resistance depends on the surface tension of the fluid-air interface and on the reciprocal of the internal radius of the riser tube.
Throughout the development of a U-shaped scanning capillary-tube rheometer concept, the focus has been on how to isolate the effects of surface tension and yield stress in obtaining low-shear-rate viscosity for non-Newtonian fluids like blood. This study attempted to measure the viscosity of unadulterated blood at a body temperature of 37℃. In the present study, both Casson and Herschel-Bulkley models were selected for the viscosity and yield stress measurements of blood since both models have a yield stress term.
6.2.1. Description of Instrument
Figure 6-9 shows a photograph of the SCTR with plastic riser tubes, which consists of two charge-coupled devices (CCD 1 and CCD 2) that are positioned vertically, two light-emitting diodes (LEDs), two riser tubes made of acrylic plastic and a capillary tube made of glass, a stopcock, and a data-acquisition system. The inside diameter of the riser tubes used in the study was 3.2 mm. The inside diameter and length of the capillary tube were 0.797 mm 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 drop at the capillary tube was significantly greater than those at the riser tubes and connecting fittings. 132
6.2.2. Testing Procedure
Tests with distilled water and bovine blood were performed at the room temperature of 25℃. Riser tube 1 was first filled with the test fluid to the predetermined height of approximately 550 pixels by using a syringe. Once the desired level was reached, the stopcock was turned to a position to allow test fluid to flow to the capillary tube. When the fluid reached the predetermined height of 100 pixels in riser tube 2, the stopcock was shut to stop further fluid movement into riser tube 2. Next, the syringe was removed from the SCTR system, and then the stopcock was turned to allow fluid to move from riser tube 1 to riser tube 2 by gravity. As shown in Fig. 6-9, two CCD sensors were used: one located at the lower (left) side and the other located at the upper (right) side. The difference between the locations of the two CCD sensors was 500 pixels, a number that was taken into account in the height measurements.
For the purpose of comparison, approximately 0.5 ml of bovine blood from the syringe was immediately transferred to a cone-and-plate test cup of a rotating viscometer (Brookfield model DV-III with LV-type spring torque with CP-40 spindle) that was maintained at a constant temperature of 25℃ by a water bath connected to the cup. For unadulterated human blood, the temperature of the SCTR was controlled during the test at a body temperature of 37℃ by using preheated disposable tube sets and a heating pad (shown in Fig. 6-10) installed inside the SCTR.
After the temperature in the SCTR was stabilized, the viscosity measurement was initiated with a venipuncture using a 19-gauge stainless steel needle. About 5 ml of 133 extra blood was collected for hematocrit measurements, which were determined with a microhematocrit centrifuge (International Clinical Centrifuge) for both bovine blood and human blood.
Viscosities of distilled water (a Newtonian fluid), bovine blood containing
7.5% EDTA, and unadulterated human blood were measured over a range of shear rates. Because the CCD sensor requires an opaque fluid, dye was added to the distilled water. The amount of dye used for distilled water was less than 1% concentration by volume, and the dye-effect on the viscosity of distilled water was negligibly small at this concentration. Bovine blood was purchased from Lampire
Biological Laboratories, Inc., and human blood was obtained from two healthy male donors who were 29 and 51 years of age. For comparison purposes, a reference value was used for the distilled water while the viscosity of the bovine blood was independently checked by using the cone-and-plate rotating viscometer.
6.2.3. Data Reduction with Casson Model
The fluid level data from the two riser tubes were analyzed to determine the viscosities of distilled water (a Newtonian fluid) and blood (a non-Newtonian fluid).
In order to measure blood viscosity using a U-shaped SCTR, one needs to isolate the effects of both surface tension and yield stress on the viscosity of blood. The details of mathematical procedure for curve-fitting using the Casson model have already 134 been introduced in Chapter 4. Thus, in this section, the procedure to determine the unknown values for the Casson model is discussed.
6.2.3.1. Curve Fitting
As discussed in Chapter 4, there are three unknown values, i.e., k , ∆hst , and
∆hy , to be determined through the iterations using the same software package (Excel
Solver; see Appendix E) used in the power-law model case. The least-square method was used for curve fitting of the experimental data and Eq. (4-41) to obtain the three unknowns involved in the Casson model.
The procedure of data reduction for distilled water is shown in Fig. 6-11.
Figure 6-11(a) shows mean-velocity variations at a riser tube which were obtained experimentally and theoretically. In the case of theoretical values, the initial guesses for the three unknowns were used to estimate the values. In Fig. 6-11(b), the curve- fitting results after iterations to minimize the sum of error that were calculated by Eq.
(4-42) are shown. The initial guesses and final values of k , ∆hst , and ∆hy for distilled water are shown in Table 6-2.
Figures 6-12 and 6-13 show the curve-fitting procedures for human blood obtained from two donors. As shown in Table 6-2, the same initial guesses of the three unknowns were used for the two different bloods. However, the resulting values of the three unknowns in the Casson model for the two donors were very different, validating the present curve-fitting method. 135
6.2.3.2. Results and Discussion
Figures 6-14 and 6-15 show test results obtained with distilled water at 25℃.
Figure 6-14 shows the fluid-level variations in the two riser tubes, h1 (t ) and h2 (t ) .
Both fluid levels converged gradually from the initial difference to an equilibrium
state. Even for the distilled water, ∆ht=∞ was not zero due to the difference in the surface tension between the two riser tubes. Unless the wetting conditions of the liquid-solid interface at the two riser tubes are exactly same, the surface-tension difference always exists.
Figure 6-15 shows viscosity results from two tests for distilled water at 25℃ obtained with the SCTR, together with the reference data for comparison. The values
of ∆hst were determined to be approximately 4 mm for both tests whereas the values
of ∆hy were determined to be zero by a computer program (Microsoft Excel-solver), validating the data reduction procedure involving the yield stress. Based on this viscosity measurement method, the viscosity of the distilled water was found to be between 0.876 and 0.878 cP at 25℃. The solid line indicates the viscosity of the distilled water calculated by using the so-called Andrade’s equation. Since the water viscosity data are given in the literature as a function of temperature, the exact viscosity of water at 25℃ was calculated using the Andrade’s equation [Munson et al.,
1998]:
B µ = D ⋅ e T (6-1) 136 where D and B are given as 9.93×10−4 mPa·s and 2026.57 K, respectively, for water in a temperature range between 20 and 30℃.
Based on Eq. (6-1), the water viscosity was estimated to be 0.892 cP at 25℃.
The test results obtained with the SCTR gave less than 2% error in the entire shear rate range, validating the test methods and data reduction procedure. Thus, it is concluded that it is extremely important to consider the effect of the surface tension on the viscosity measurement using a gravity driven, U-shaped capillary-tube system.
Figure 6-16 shows height variations at the two riser tubes as a function of time for bovine blood with 7.5 % EDTA at a room temperature of 25℃. The trends of fluid-level variations for the bovine blood were very similar to those for the distilled
water. As expected, ∆ht=∞ for the bovine blood was not zero but a finite value that is slightly greater than that for the distilled water. The height difference due to surface
tension, ∆hst , and the height difference due to yield stress, ∆hy , were determined to be in the range of 5.7-6.1 mm and 0.52-0.59 mm, respectively. The hematocrit of the bovine blood was measured to be 35 percent.
Figure 6-17 shows the viscosity of the bovine blood with 7.5% EDTA at 25℃, which was measured with both the SCTR (indicated by circles and triangles) and the rotating viscometer (RV; indicated by diamonds). Compared with the measured data using the RV, the test results from the SCTR gave excellent agreement within 3% in a shear rate range between 15 and 300 s-1. However, as the shear rate decreased below
15 s-1, the viscosity data measured from the RV seemed to be incorrect. More specifically, the torque for viscosity measurements with the RV should be greater than 10% of the full scale (as suggested by Brookfield) at a given shear rate for 137 reasonably accurate viscosity measurements. In the case of the bovine blood, the minimum shear rate that Brookfield recommended, based on the 10% criterion, was approximately 30 s-1. In contrast, the SCTR gave a consistent viscosity measurement over a range of shear rates as low as a shear rate of 1 s-1. Related to the uncertainty in measuring blood viscosity at low shear rates, wall slip is an issue. Since the minimum shear rate in this study was 1 s-1, as discussed earlier, one could assume that the slip effect was negligibly small [Picart et al., 1999a].
Figures 6-18 and 6-19 show test results of fresh, unadulterated human blood at a body temperature of 37℃. The test was completed within 2-3 min to avoid blood clotting, which might have altered the viscosity of the blood. Figure 6-18 shows height variations in the two riser tubes as a function of time for the fresh human blood
at 37℃. The value of ∆ht=∞ for the fresh human blood had a finite value, which
depended on individual donors. The values of ∆hst for donors 1 and 2 were found to be approximately 8.5 and 9.6 mm, respectively.
In the use of the SCTR, two phenomena of particular importance should be pointed out in regards to clinical hemorheology: one is the carry-over effect and the other is the surface tension effect. First, at the completion of a measurement, a thin layer of the test blood sample was always retained on the tube wall, unless the tube was very carefully washed and dried. This residual layer can be called carry-over.
Hence, for unadulterated blood-viscosity measurements, it was necessary to use disposable capillary-riser tube sets to avoid the carry-over phenomenon. Second, the difference between surface tensions at the two riser tubes may vary from one 138 disposable set to another. Although the riser tubes were made of the same material
(acrylic plastic), the surface tensions were slightly different from set to set.
The values of ∆hy representing the effect of the yield stress for donors 1 and 2 were found to be 0.7 and 0.2 mm, respectively. These results were consistent with hematocrit data for donors 1 and 2, which were 42 and 35 percent, respectively.
Donor 2 (a physician) practices therapeutic bloodletting periodically, which explains
why the hematocrit of donor 2 was unusually low. It is of note that the value of ∆hy represents the thixotropic characteristics of fresh human blood that is closely related to the yield stress.
A thixotropic blood exhibits a high viscosity when first sheared from rest.
The viscosity continues to decrease as shearing continues. Thixotropy is usually a result of the partial destruction, by shearing, of the internal liquid structure. While at rest, the internal structure made of suspended cells and plasma may form to create aggregations of RBCs, for example. This phenomenon is generally referred to as
‘structure viscosity’.
In fact, the phenomenon of RBCs aggregations at low shear rates is well known but not well understood so far. The forces leading to aggregations are weak, so if a sample of normal blood is subjected to increasing shear rate, the aggregates progressively break up and are generally monodispersed at a shear rate greater than
10 s-1. However, it is important to note that there could be two kinds of yield stresses: a start-up yield stress and a stopping yield stress [Cho and Choi, 1993]. The yield stress determined in this study was the stopping yield stress, an important phenomenon in clinical hemorheology and studies of cardiovascular disease. In the 139 viscosity measurement with whole blood, a typical test duration in which the shear rate decreased from 10 to 1 s-1 was longer than 60 seconds. It is reasonable to assume that the 60-second period is long enough to cause aggregations if the aggregations were going to take place.
Figure 6-19 shows the viscosities of the two donors at 37℃ measured with the
SCTR. The viscosities for donor 2 were significantly lower than those for donor 1 due to the difference in hematocrit. In addition, the viscosity curve for donor 2 was
flatter than that for donor 1 since donor 2 had a smaller value of ∆hy and a lower hematocrit. Figure 6-20 shows shear stress variations against shear rate for both donors. Like the case of viscosity, shear stress for donor 2 was significantly lower than that for donor 1 due to differences in hematocrit and yield stress. The values of the yield stress determined from Casson model were approximately 14 mPa and 5 mPa for donor 1 and 2, respectively. The difference between the viscosity data increased as the shear rate decreased indicating that the viscosity was more influenced by hemorheological parameters such as hematocrit and yield stress at low shear rates than at high shear rates.
6.2.4. Data Reduction with Herschel-Bulkley (H-B) Model
The detailed mathematical procedure for curve-fitting using a Herschel-
Bulkley (H-B) model was provided in Chapter 4. As in the case of the Casson model, the H-B model can also handle the yield stress of blood. However, in the data 140 reduction with the H-B model, there are four unknown values to be determined through the curve-fitting technique, whereas the Casson model has only three unknowns.
The four unknown values, i.e., m , n , ∆hst , and ∆hy , are to be determined through a least-square method using the same software package (Excel-Solver,
Microsoft; see Appendix E) that uses the formula of Newton’s method. In the process of curve fitting, Eq. (4-59) was used for theoretical values. In the case of the
H-B model, the experimental data of bovine blood with 7.5% EDTA were used to validate the method of data reduction.
Figure 6-21(a) shows mean-velocity values at a riser tube which were obtained both experimentally and theoretically. In the case of theoretical values, the initial guesses for the four unknowns, which are shown in Table 6-3, were used. In
Fig. 6-21(b), the curve-fitting results after the iterations to determine the four
unknowns are shown. The initial guesses and final values of m , n , ∆hst , and ∆hy for bovine blood with 7.5% EDTA using the H-B model are shown in Table 6-3.
Figure 6-22 shows the viscosity of the bovine blood with 7.5% EDTA at 25℃, which was measured with the SCTR. Three consecutive tests were performed with bovine blood. As expected, the H-B model also produced very accurate and repeatable results. The final values of four unknowns for each test are reported in
Table 6-4. The effects of constitutive models on the viscosity and yield stress measurement of blood will be further discussed in the next section.
141
CCD-LED arrays
Riser tube 1 Riser tube 2
Test Fluid Stopcock Capillary tube
Fig. 6-9. Picture of a SCTR with plastic riser tubes.
142
Rheometer System
Heating Pad
Fig. 6-10. Heating pad for a test with unadulterated human blood.
143
0.025 Experimental data 0.02 Theoretical data
2Vr 0.015 (m/s) 0.01
0.005
0 010203040 Time (s)
(a) With initial guess values
0.025 Experimental data 0.02 Theoretical data 2V r 0.015 (m/s) 0.01
0.005
0 0 10203040 Time (s)
(b) With final resulting values
Fig. 6-11. Curve-fitting procedure with Casson model for distilled water.
144
Table. 6-2. Comparison of initial guess and resulting value using Casson model.
Distilled Water Human Blood
k = 1 (cP·s) k = 1 (cP·s) Initial Guess ∆hy = 0.5 mm ∆hy = 0.5 mm
∆hst = 3 mm ∆hst = 5 mm
Donor 1 Donor 2 k = 0.878 (cP·s) k = 2.743 (cP·s) k = 2.121 (cP·s) Resulting Value ∆hy = 0.00 h = 0.74 mm h = 0.16 mm h = 4.02 mm ∆ y ∆ y ∆ st ∆h = 8.47 mm ∆h = 9.58 mm st st
145
0.008 Experimental data 0.006 Theoretical data
2Vr (m/s) 0.004
0.002
0 0 20 40 60 80 100 120 Time (s)
(a) With initial guess values
0.008 Experimental data 0.006 Theoretical data
2Vr (m/s) 0.004
0.002
0 0 20406080100120 Time (s)
(b) With final resulting values
Fig. 6-12. Curve-fitting procedure with Casson model for donor 1.
146
0.008 Experimental data 0.006 Theoretical data
2Vr (m/s) 0.004
0.002
0 0 20 40 60 80 100 120 Time (s)
(a) With initial guess values
0.008 Experimental data 0.006 Theoretical data 2Vr (m/s) 0.004
0.002
0 0 20406080100120 Time (s)
(b) With final resulting values
Fig. 6-13. Curve-fitting procedure with Casson model for donor 2.
147
0.1
0.08 Riser tube 1 0.06
0.04
Height (m) Riser tube 2 0.02
0 0 50 100 150 Time (s)
Fig. 6-14. Height variation in each riser tube vs. time for distilled water at 25℃.
148
1.5 Test 1 1.3 Test 2 Reference 1.1
0.9
Viscosity (cP) Viscosity 0.7
0.5 0 100 200 300 400 Shear rate (s-1)
Fig. 6-15. Viscosity measurement for distilled water at 25℃. 1 cP = 1 mPa·s.
149
0.1
0.08 Riser tube 1
0.06
0.04 Height (m) Riser tube 2 0.02
0 0 30 60 90 120 150 Time (s)
Fig. 6-16. Height variation in each riser tube vs. time for bovine blood with 7.5% EDTA at 25℃.
150
100 Test 1 Test 2 RV 10 Viscosity (cP) Viscosity
1 1 10 100 1000 Shear rate (s-1)
Fig. 6-17. Viscosity measurement for bovine blood with 7.5% EDTA at 25℃ using both rotating viscometer (RV) and scanning capillary-tube rheometer (SCTR). Hematocrit was 35. 1 cP = 1 mPa·s.
151
0.1 0.09 A : Donor 1 0.08 B : Donor 2 0.07 0.06 0.05 B 0.04 A Height (m) 0.03 0.02 0.01 0 0 306090120150 Time (s)
Fig. 6-18. Height variation in each riser tube vs. time for human blood at 37℃.
152
100 Donor 1 Donor 2
10 Viscosity (cP) Viscosity
1 1 10 100 1000 Shear rate (s-1)
Fig. 6-19. Viscosity measurement for human blood (2 different donors) at 37℃. Hematocrits for donors 1 and 2 were 42 and 35, respectively. 1 cP = 1 mPa·s.
153
10000
1000
100
10 Donor 1
Shear stress (mPa) stress Shear Donor 2 1 1 10 100 1000 Shear rate (s-1)
Fig. 6-20. Shear-stress variation vs. shear rate for human blood (from 2 different donors) at 37℃.
154
0.005 Experimental data 0.004 2Vr Theoretical data (m/s) 0.003
0.002
0.001
0 04080120 Time (s)
(a) With initial guess values
0.005 Experimental data 0.004 Theoretical data 2Vr (m/s) 0.003
0.002
0.001
0 04080120 Time (s)
(b) With final resulting values
Fig. 6-21. Curve-fitting procedure with Herschel-Bulkley model for bovine blood.
155
Table. 6-3. Comparison of initial guess and resulting value using Herschel-Bulkley model.
Bovine Blood
n = 0.8 n-1 m = 5 (cP·s ) Initial Guess ∆hy = 1 mm
∆hst = 6 mm
n = 0.875 n-1 m = 8.6 (cP·s ) Resulting Value ∆hy = 1.2 mm
∆hst = 5.8 mm
156
100 Test #1 Test #2 Test #3
10 Viscosity (cP) Viscosity
1 1 10 100 1000 Shear rate (s-1)
Fig. 6-22. Viscosity measurements of the bovine blood with 7.5% EDTA at 25℃, which were analyzed with Herschel-Bulkley model.
157
Table. 6-4. Comparison of four unknowns determined with Herschel-Bulkley model for three consecutive tests.
Test #1 Test #2 Test #3
n = 0.875 n = 0.876 n = 0.872 n-1 n-1 n-1 m = 8.6 (cP·s ) m = 8.7 (cP·s ) m = 8.76 (cP·s ) ∆hy = 1.2 mm ∆hy = 1.2 mm ∆hy = 1.24 mm
∆hst = 5.8 mm ∆hst = 5.8 mm ∆hst = 5.8 mm
158
6.3. Comparison of Non-Newtonian Constitutive Models
It is well known that blood has both shear-thinning (pseudoplastic) characteristics and yield stress. The present study examined the capability of three constitutive models in handling data from the scanning capillary-tube rheometer
(SCTR) for the viscosity and yield-stress measurements of blood. Power-law model was chosen for the shear-thinning behavior of the blood. Casson and Herschel-
Bulkley (H-B) models were selected to measure both the shear-thinning viscosity and yield stress of the blood.
As shown in Table 6-5, many researchers used non-Newtonian constitutive models for the investigations on blood rheology and flows which include power-law,
Casson, and H-B models. In addition, it has been pointed out that the selection of a constitutive model could be very significant in analyzing blood flows [Tu and Deville,
1996; Siauw et al., 2000]. Siauw et al. (2000) performed a comparative study of non-
Newtonian models for the prediction of unsteady stenosis flows by using power-law and Casson models, whereas Tu and Deville (1996) used H-B, Bingham, and power- law models for their study on stenosis flows.
The objective of the present study was to investigate whether or not the results of blood rheology and flow in the SCTR could be significantly altered by constitutive models. Hence, the study investigated the effect of the three models on the viscosity and yield-stress measurements of blood using the SCTR as well as on the flow patterns of blood such as velocity profile and wall shear stress in a blood vessel.
159
6.3.1. Comparison of Viscosity Results
Figures 6-23(a) and 6-23(b) show the calibration results obtained in the SCTR with distilled water at 25℃. Figure 6-23(a) shows the fluid-level variations in the
two riser tubes, h1 (t ) and h2 (t ) . Both fluid levels converged gradually from the initial difference to an equilibrium state. Even for the distilled water, the difference
in the two fluid-levels at t = ∞ , ∆ht=∞ , was not zero due to the difference in the surface tension between the two riser tubes.
Figure 6-23(b) shows viscosity results for distilled water at 25℃ obtained with the SCTR using the three different non-Newtonian constitutive models, together
with the reference data for comparison. The values of ∆hst were determined to be
approximately 3.3-3.4 mm for the three models, whereas the values of ∆hy were determined to be zero for all models, validating the data reduction procedure involving the yield stress. The viscosity of distilled water was found to be between
0.884 and 0.905 cP at 25℃ for all three models. As shown in Table 6-6, the test results obtained with the SCTR gave less than 2% error over the entire shear rate range, validating the experimental procedure and data reduction method using the three non-Newtonian constitutive models. Furthermore, the viscosity measurement of water confirmed that one needs to consider the effect of the surface tension on the viscosity measurement in the SCTR.
Figures 6-24(a) and 6-24(b) show test results obtained with bovine blood at
25℃. Figure 6-24(a) shows height variations at two riser tubes as a function of time 160 for the bovine blood with 7.5% EDTA as an anticoagulant. The trends of fluid-level variations for the bovine blood were very similar to those for the distilled water. As
expected, ∆ht=∞ for the bovine blood was not zero, but a finite value that is slightly greater than that for the distilled water. The hematocrit of the bovine blood was measured to be 35%.
Figure 6-24(b) shows the viscosity of the bovine blood at 25℃, which was measured with both the SCTR and a rotating viscometer (RV). The SCTR results obtained by using the three constitutive models showed very good agreement among themselves at shear rates higher than approximately 30 s-1. However, the power-law model maintained the rate of the viscosity change (the slope of a viscosity curve) constant whereas both the Casson and H-B models increased the rate of viscosity change as shear rate decreased, indicating the existence of yield stress in the bovine blood. Moreover, the viscosity results using the Casson model showed very good agreement with those using the H-B model.
The test results from the SCTR using the three non-Newtonian models gave excellent agreement with the measured data using the RV within 5% in a shear-rate range between 30 and 300 s-1. However, as the shear rate decreased below 30 s-1, the viscosity data measured from the RV seemed to be incorrect. More specifically, the torque for viscosity measurements with the RV should be greater than 10% of the full scale (as suggested by Brookfield) for reasonably accurate viscosity measurements.
In the case of the bovine blood, the minimum shear rate that Brookfield recommended, based on the 10% criterion, was approximately 30 s-1. In contrast, the
SCTR gave a consistent viscosity measurement over a range of shear rate as low as a 161 shear-rate of 1 s-1. In addition, the viscosities measured using both the Casson and H-
B models in the SCTR seemed to be more accurate than the power-law viscosity at low shear rates. Table 6-7 provides the viscosity results at the several shear rates, which were produced by using the three non-Newtonian constitutive models.
Figures 6-25(a) and 6-25(b) show test results of fresh, unadulterated human blood at a body temperature of 37℃. The test was completed within 2-3 minutes to avoid blood clotting, which might have altered the viscosity of the blood. Figure 6-
25(a) shows height variations in the two riser tubes as a function of time for the fresh
human blood at 37℃. The value of ∆ht=∞ for the fresh human blood was measured to be approximately 8.5 mm. The hematocrit of this donor’s blood was 42%.
Figure 6-25(b) shows the viscosity of the human blood at a body temperature of 37℃, which was measured with the SCTR. The SCTR results obtained by using the three constitutive models showed very good agreement among themselves at shear rates higher than approximately 20 s-1. However, like the bovine blood case, the power-law model maintained the rate of the viscosity change constant whereas both the Casson and H-B models increased the rate of viscosity change as shear rate decreased, indicating the existence of yield stress in the human blood. Table 6-8 shows the viscosity results of the unadulterated human blood for the three constitutive models.
For cases of the bovine blood and fresh human blood, the power-law model maintained constant slopes in the viscosity curve, while both the Casson and H-B models showed rapid increases in the viscosity as the shear rate decreased. However, 162 the viscosities obtained from the three different constitutive models gave good agreement at a high shear rate zone.
6.3.2. Comparison of Yield Stress Results
Over the years, many researchers have reported on the yield stress of blood.
The measurements of the rheological properties of fluids having yield stress are summarized by Nguyen et al. (1983). According to their classifications, the yield- stress measurement with the SCTR can be classified as indirect methods rather than direct methods since the yield stress of a fluid can be obtained in the SCTR by using constitutive models. Bingham plastic, Casson, and H-B models were used in their study to describe the rheological behavior of yield-stress fluids.
Figures 6-26(a), 6-26(b), and 6-26(c) show the results of shear stress versus shear rate for distilled water, bovine blood, and human blood, respectively. In the case of the distilled water, the shear stresses obtained using the three different non-
Newtonian models were almost identical. The yield stress can be graphically described as the intersecting point where the shear stress-shear rate curve meets with the y-axis (i.e., at zero shear rate). Table 6-9 shows the yield-stress results together with the model constants of the three constitutive models. As expected, the distilled water shows no yield stress, while both the bovine and human bloods show finite values of the yield stress for the cases of Casson and H-B models. Note that the yield 163 stress values for the Casson and H-B models were calculated using the following equation:
ρg∆hy ⋅ Rc τ y = (6-2) 2Lc
The yield stress of the human blood was consistently greater than that of the bovine blood although the human blood was tested at a high temperature of 37℃. It might be due to both the difference in hematocrit and the RBC aggregations of the human blood at low shear rates. The yield stress values of the human blood with hematocrit of 42% were measured to be 13.8 and 17.5 mPa for the Casson and H-B models, respectively. Note that the yield stress measured in the present study was the stopping yield stress, an important phenomenon in clinical hemorheology and treatments of cardiovascular disease. The yield stress values vary from 1 mPa to 30 mPa for normal human blood with hematocrit of 40% [Chen et al., 1991; Picart et al.,
1999a], supporting the validity of the method of the present yield-stress measurement using the SCTR.
The yield stress obtained with the H-B model was consistently greater than that obtained with the Casson model in the cases of both bovine and human bloods as shown in Table 6-9. In order to evaluate which model produces more accurate yield
stress results, the experimentally measured values of ∆ht=∞ were compared with those
of ∆hst + ∆hy determined analytically through the curve-fitting procedure for the bovine and human bloods (see Table 6-10). The value of the fluid-level difference in
riser tubes 1 and 2 at a time of 180 seconds was taken as a measure of ∆ht=∞ . Hence, 164
the experimental values (i.e., ∆ht=∞ ) should be bigger than those (i.e., ∆hst + ∆hy ) to be obtained analytically.
As shown in Table 6-10, the values of ∆hst + ∆hy obtained with the Casson model were consistently smaller than the experimentally measured values while the values obtained with H-B model were bigger. Based on the comparison, one may conclude that the Casson model does a better job in determining the yield stress of blood than the H-B model. However, it is of note that both models produced almost
identical values of ∆hst , thus almost identical surface tensions, for the bovine and human blood.
6.3.3. Effect of Yield Stress on Flow Patterns
Figure 6-27 shows the variations of C y (t) for the bovine blood with 7.5%
EDTA, indicating that the plug-flow region grows at the capillary tube with increasing time. Due to the difference in yield stress values for the Casson and H-B models, the size of the plug-flow region estimated from the two models start to differ after approximately 30 seconds. Note that the H-B model predicts a much larger plug-flow region than the Casson model.
Figures 6-28(a), 6-28(b), and 6-28(c) show velocity profiles at the capillary tube for the bovine blood at room temperature of 25℃, which were plotted at three mean velocities of 3, 0.3, and 0.03 cm/s, respectively. At a relatively high velocity of
3 cm/s, the three constitutive models predicted identical velocity profiles. However, 165 as the mean velocity (i.e., shear rate) decreased, the clear deviation among the three models started to appear near the center of the tube at 0.3 cm/s and became bigger at
0.03 cm/s, a phenomenon which can be attributed to the difference in yield stress values. Therefore, it can be concluded that the yield stress plays an important role in the determination of both the blood viscosity and velocity profiles in a blood flow.
Figure 6-28(c) also shows that the size of the plug-flow region at the center of the tube for the H-B model is much larger than that for the Casson model at approximately 0.03 cm/s.
The shear rate, viscosity, and shear stress obtained with both the Casson and
H-B models for the bovine blood were plotted as a function of the mean velocity at the capillary tube in Figs. 6-29(a), 6-29(b), and 6-29(c), respectively. Wall shear stress represents the friction exerted on the vessel wall by moving blood. It has been shown in a number of studies that wall shear stress may play an important role in endothelial cell morphology and functions influencing the production of substances such as nitric oxide, prostacyclin, and endothelin [Baldwin and Thurston, 1995;
Usami et al., 1995; Fung, 1996; Mitsumata et al., 1996; Samijo et al., 1998; Frame et al., 1998; Cotran et al., 1999; Kensey and Cho, 2001].
As shown in Fig. 6-29(c), the value of the wall shear stress is almost independent of the selection of a constitutive model. Due to the difference in the size of the plug-flow regions, the wall shear rates for the Casson model at low mean velocities were much lower than those for the H-B model, resulting in consistently higher wall viscosity for the Casson model as shown in Fig. 6-29(b). Since the wall shear stress can be calculated from the product of viscosity (shown in Fig. 6-29(b)) 166 and wall shear rate (shown in Fig. 6-29(a)) at a given mean velocity, the difference in the wall shear stress between the two models is very small.
167
Table. 6-5. Various physiological studies with non-Newtonian constitutive models.
Researchers Power-law Casson Herschel-Bulkley
Chakravarty and Datta (1992) x x
Siauw et al. (2000) x x
Tu and Deville (1996) x x
Liepsch and Moravec (1984) x
Walburn and Schneck (1976) x
Rohlf and Tenti (2001) x
Misra et al. (2000, 2002) x
Das and Batra (1995) x
Dash et al. (1996) x
Walawender et al. (1975) x
Rodkiewicz et al. (1990) x
Misra and Kar (1991) x
Chakravarty and Datta x (1989, 1992)
168
100 (a)
80 Riser tube1 60
40 Riser tube 2
Height (mm) 20
0
0 30 60 90 120 150
Time (s)
1.5 (b) power-law 1.3 H-B
Casson 1.1 Reference
0.9
(cP) Viscosity 0.7
0.5 0 100 200 300 400
Shear rate (s-1)
Fig. 6-23. Test with distilled water at 25℃. (a) Fluid-level variations in two riser tubes. (b) Viscosity results.
169
Table. 6-6. Measurements of water viscosity.
Model Viscosity (cP) *Error
Power-law 0.905 1.46%
Herschel-Bulkley (H-B) 0.884 0.90%
Casson 0.886 0.68%
*Error is based on the comparison with the reference value (0.892 cP at 25℃)
170
100 (a)
80 Riser tube 1
60
40
Height (mm) Riser tube 2 20
0
0 30 60 90 120 150
Time (s)
100 (b) power-law H-B
Casson 10 RV
(cP) Viscosity
1 1 10 100 1000 -1 Shear rate (s )
Fig. 6-24. Test with bovine blood at 25℃. (a) Height variations in riser tube vs. time for bovine blood with 7.5% EDTA. (b) Viscosity measurement for bovine blood with 7.5% EDTA using a rotating viscometer (RV) and a scanning capillary-tube rheometer (SCTR).
171
Table. 6-7. Measurements of bovine blood viscosity.
Viscosity Viscosity (cP) from SCTR Shear rate (cP) (s-1) from RV Power-law Casson H-B
300 4.43 4.39 4.49 4.28
150 4.78 4.75 4.84 4.71
90 5.11 5.03 5.18 5.09
45 5.75 5.44 5.85 5.71
30 6.25 5.7 6.38 6.2
15 8.81 6.16 7.7 7.21
7.5 17 6.67 9.7 8.9
3 7.4 14.5 12.8
8.38 22.5 18.55 Lower than 3 (at 1 s-1) (at 1.35 s-1) (at 1.55 s-1)
172
(a)
100
80 Riser tube 1
60
40
Height (mm) Riser tube 2 20
0 0 30 60 90 120 150 Time (s)
(b)
100 power-law H-B Casson 10 Viscosity (cP) Viscosity
1 1 10 100 1000 Shear rate (s-1)
Fig. 6-25. Test with unadulterated human blood at 37℃. (a) Height variations in riser tubes vs. time. (b) Viscosity results.
173
Table. 6-8. Measurements of human blood viscosity.
Viscosity (cP) from SCTR
Shear rate (s-1)
Power-law Casson H-B
300 3.89 4.11 4.09
150 4.45 4.47 4.57
90 4.93 4.85 4.97
45 5.67 5.59 5.63
30 6.15 6.16 6.15
15 7.06 7.65 7.3
7.5 8.12 9.95 9.1
9.76 14.73 12.9 Lower than 5 (at 3 s-1) (at 3.33 s-1) (at 3.3 s-1)
12.17 27.26 21.9 Lower than 3 (at 1 s-1) (at 1.18 s-1) (at 1.32 s-1)
174
(a) 1000 power-law Water
H-B 100 Casson
10
(mPa) stress Shear 1 110100
1000 (b) Bovine Blood
100
power-law 10 H-B Casson (mPa) stress Shear 1 110100
1000 (c) Human Blood
100
power-law 10 H-B
Casson (mPa) stress Shear 1
1 10 100 -1 Shear rate (s )
Fig. 6-26. Wall shear stress at a capillary tube vs. shear rate. (a) for distilled water at 25℃. (b) for bovine blood at 25℃. (c) for human blood at 37℃.
175
n-1 Table. 6-9. Comparison of model constants, ∆hy , and τ y . (Note that [m] = cP·s )
Power-law H-B Casson
n = 1 n = 1 k = 0.886 cP Distilled water m = 0.905 m = 0.884
(25℃) ∆hy = 0 ∆hy = 0 ∆hy = 0
τ y = 0 τ y = 0 τ y = 0 n = 0.8866 n = 0.8753 k = 3.7302 cP Bovine blood m = 8.3771 m = 8.599
(25℃) ∆hy = 0 ∆hy = 0.8 mm ∆hy = 0.52 mm
τ y = 0 τ y = 16.4 mPa τ y = 10.7 mPa n = 0.7991 n = 0.8601 k = 3.2896 cP Human blood m = 12.171 m = 8.9721
(37℃) ∆hy = 0 ∆hy = 0.85 mm ∆hy = 0.67 mm
τ y = 0 τ y = 17.5 mPa τ y = 13.8 mPa
176
Table. 6-10. Comparison of ∆ht=∞ and ∆hst + ∆hy .
H-B Casson
∆h t=∞ 6.5 mm 6.5 mm (experimental) Bovine blood
(25℃) 6.6 mm 6.26 mm ∆hst + ∆hy ( ∆hst = 5.8 mm ( ∆hst = 5.74 mm (analytical) ∆hy = 0.8 mm) ∆hy = 0.52 mm)
∆h t=∞ 9.13 mm 9.13 mm (experimental) Human blood
(37℃) 9.25 mm 9.07 mm ∆hst + ∆hy ( ∆hst = 8.4 mm ( ∆hst = 8.4 mm (analytical) ∆hy = 0.85 mm) ∆hy = 0.67 mm)
177
1 Casson 0.8 H-B
0.6 y C 0.4
0.2
0 0 50 100 150 200 Time (s)
Fig. 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℃.
178
6 3 cm/s (a)
4
power-law 2 Casson Velocity (cm/s) H-B 0 0 0.01 0.02 0.03 0.04
0.6 0.3 cm/s (b) 0.4
power-law 0.2 Casson
(cm/s) Velocity H-B 0
0 0.01 0.02 0.03 0.04 0.06 0.03 cm/s (c) 0.04
power-law 0.02 Casson Velocity (cm/s) H-B 0
0 0.01 0.02 0.03 0.04 Radius (cm)
Fig. 6-28. Velocity profiles at a capillary tube for the bovine blood with 7.5% EDTA at 25℃: (a) at a mean velocity of 3 cm/s. (b) approximately 0.3 cm/s. (c) approximately 0.03 cm/s. 179
) 1000
-1
(a) 100
10 Casson 1 H-B
Wall shear rate (s 0.1 0.01 0.1 1 10
1000 Casson (b) 100 H-B
10
Viscosity (cP) Viscosity 1
0.01 0.1 1 10
100
(c) ) 10 2
1
Casson (dyne/cm 0.1 H-B Wall stress shear 0.01 0.01 0.1 1 10 Mean velocity (cm/s)
Fig. 6-29. (a) Wall shear rate, (b) Viscosity, 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. Note that 1 dyne/cm2 = 102 mPa. 180
CHAPTER 7. CONCLUSIONS AND RECOMMENDATIONS
The present study introduces a scanning capillary-tube rheometer to measure liquid viscosity over a range of shear rates continuously from high to low shear rates
(as low as 1 s-1). The feasibility and accuracy of the new viscosity measurement technique has been demonstrated for a standard-viscosity oil and unadulterated human bloods by comparing the results obtained with a power-law model against an established viscosity measurement technique. One of the advantages of this new rheometer is that one can measure the viscosity of whole blood without using anticoagulants. In addition, the viscosity measurement of whole blood can be completed within 2 minutes in a clinical setting, rendering viscosity results over a wide range of shear rates. The viscosity data from the new rheometer gave excellent agreement with those measured within 1 minute by the rotating viscometer. The rotating viscometer could not be used after 1 minute of use with an unadulterated blood sample due to blood clotting.
The present study introduces a mathematical method to isolate the surface- tension and yield-stress effects on the viscosity measurement in using a SCTR. The feasibility and validity of the method to reduce data for the SCTR have been demonstrated for distilled water and bovine blood by comparing with reference data and the results from a Brookfield cone-and-plate rotating viscometer. The viscosity of unadulterated human blood has also been measured using the SCTR. 181
The effect of the surface tension was taken into account by using an additional
term, ∆hst , while the effect of the yield stress was considered as a model constant in either Casson or Herschel-Bulkley model. For the SCTR using gravity as a driving force, it was necessary to consider the effect of the surface tension even for
Newtonian fluids. Furthermore, in the case of a thixotropic liquid like whole blood, the surface-tension effect should be isolated from the yield-stress effect to obtain accurate viscosity data over a range of shear rates using the SCTR. In order to avoid the influence of the carry-over phenomenon on viscosity measurements, disposable tube sets were used for tests with fresh human blood.
The present study also investigated the effect of dye concentration on the viscosity of a dye-water solution using a SCTR. In the experiment, six different concentrations (0.5, 1, 2, 3, 4, and 7% by volume) of dye were used at 25ºC. When the dye concentration was greater than 2%, the viscosity of the dye-water solution could be significantly altered particularly at low shear rates. Based on the experiment with the SCTR, one can conclude that the maximum 2% concentration of dye by volume can be used to make a transparent aqueous solution opaque for the operation of the SCTR.
The present study investigated the effects of three non-Newtonian constitutive models on the viscosity and yield stress measurements in a scanning capillary-tube rheometer: power-law, Casson, and Herschel-Bulkley models.
For a Newtonian fluid (i.e., distilled water), all three models produced excellent viscosity results. For non-Newtonian fluids (i.e., bovine and human bloods), both Casson and H-B models gave viscosity results which are in good 182 agreement 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 due to its inability to handle the yield stress of blood.
The yield stress values obtained from the Casson and H-B models for the human blood were measured to be 13.8 and 17.5 mPa, respectively. The two models showed some discrepancies in the yield-stress values. The results from the Casson model seemed to be more accurate than those from the H-B model.
The ability to estimate the wall shear stress in various arterial vessels could be a significant step in clinical hemorheology. In the present study, the wall shear stress was found to be almost independent of a constitutive model, whereas the size of the plug flow region 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.
Recommendations for Future Research
- The present study developed a new rheometer that was specially designed for
measuring unadulterated human blood. However, the measurement was not
strictly in vivo. It would be very useful to develop a method to measure the
viscosity of human blood in vivo. 183
- The present study focused on the method of measuring the viscosity of
unadulterated human blood. As discussed in Chapter 2, whole blood could be
affected by several factors such as RBC deformability and aggregation. The
effects of RBC deformability and aggregation on the blood viscosity should
be studied.
- The present study measured the viscosity and yield stress of human blood
without adding any anticoagulants. The study on the effect of thrombotic
tendency of each individual person on both viscosity and yield stress of blood
should be conducted.
- The two yield stress models, Casson and Herschel-Bulkley models, gave
different yield stresses for blood in the present study. It is not very clear
which model is more accurate. An experimental method of measuring
velocity profiles should be developed to determine the more accurate model
for characterizing blood sample.
184
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194
APPENDIX A. Nomenclature
English Letters
C dimensionless radius at τ = τ , r y R
r C dimensionless radius at τ = τ , y y y R g gravitational acceleration [m·s-2]
H height [m] h fluid height [m]
∆h fluid height difference [m] k constant for Casson model [cP]
K L loss coefficient
L length of tube [m] l length of fluid element [m] m consistency index in Power-law and Herschel-Bulkley models [cP·sn-1]
M torque [N·m] n power-law index
P pressure [Pa]
∆P pressure drop [Pa]
R radius [m] 195
R mean radius [m] r radial distance [m]
ry radial distance at τ = τ y [m]
Re Reynolds number
Q volumetric flow rate [m3·s-1] s distance measured along stream line [m] t time [s]
V flow velocity [m·s-1]
V mean flow velocity [m·s-1]
-1 v∞ terminal velocity [m·s ]
Greek Letters
τ shear stress [Pa] or [cP·s-1]
-1 γ& shear rate [s ]
µ Newtonian viscosity [cP]
η non-Newtonian viscosity [cP]
η0 constant viscosity near zero shear rate [cP]
η∞ constant viscosity near infinite shear rate [cP]
ρ density [kg·m-3]
Ω angular velocity [rad·s-1] 196
β angle [rad]
Subscripts
atm atmosphere
B Bingham plastic model c capillary tube e entrance
End end effects
f fluid i inner cylinder
L loss n needle o outer cylinder r reservoir or riser tube st surface tension t time unsteady unsteady state w wall y yield stress
∞ infinity
197
APPENDIX B. Falling Object Viscometer – Literature Review
Falling Cylinder/Needle
Figure B-1 shows a schematic diagram of a falling cylinder moving in a fluid filled in another cylinder. In case that the gap between two cylinders is very small, then simple shear can be obtained as indicated in Fig. B-1(a). As the cylinder is dropped in a closed cylinder, the displaced fluid must flow back, which results in the velocity profile shown in Fig. B-1(b). Typically, a small diameter needle is dropped in a large cylinder of the test fluid. After the needle falls for a distance great enough
to allow the fluid to reach a steady state, the terminal velocity, v∞ , is determined by timing between two marks. This generally limits the technique to transparent fluids.
Assuming a wide gap, i.e., κ << 1, the relations for shear stress, τ , and the
Newtonian viscosity, µ , are as follows [Macosko, 1994]:
(ρ − ρ )gR τ = n f (B-1) 2κ
(ρ − ρ )gR 2 µ = n f (1+ lnκ ) (B-2) 2v∞ where
ρ n = density of needle
ρ f = density of fluid
g = gravitational acceleration 198
R = inner radius of outer cylinder
v∞ = terminal velocity of needle
Park and Irvine (1988) gave relations for a power-law fluid. They also demonstrate that one can easily change the density of the needle and thus the shear stress, τ , by using a hollow tube filled with various amounts of dead weight. In that way, one can obtain non-Newtonian viscosity, η , as a function of shear rate, γ& .
199
n
Fig. B-1. Falling cylinder viscometers. (a) open ends for high viscous samples (b) closed end, free falling
200
APPENDIX C. Specification of CCD and LED Array
Description
Syscan’s SV352A8-01 Contact Image Sensor (CIS) is a black/white linear image sensor module, which is originally designed for scanning a document. Figure
C-1 illustrates the cross sectional view of the SV352A8-01 module. The module consists of a LED light source to illuminate the document, a one-to-one erect graded index micro lens array to focus the document image on the photo detector array, an array of linear MOS image sensors to convert the image into an electronic signal, a glass cover to protect the sensor array, micro lens array, and LED light source from dust, 8-pin connector for input/output connections and a protective case to house all of the components.
Key Features
• Compact size: 12 mm height × 15 mm width × 70 mm length
• Resolution: 12 dots/mm (304dpi)
• Scanning length: 2.2 inch or 3.2 inch
• Scanning speed: 2.5 ms/line
•Single power supply (+5V) 201
Cover Glass Plastic Housing
Rod Lens LED Light Source
Connector
Sensor Array PCB Substrate
Fig. C-1. Cross sectional view of SV352A8-01 module.
202
APPENDIX D. Biocoating of Capillary Tube
For the experiment with unadulterated human blood, capillary tubes used in a
SCTR have been coated with biocompatible materials. The coating work was carried out by a company called Biocompatibles in Farnham, Surrey (U.K.). The following procedure was employed to coat the inner surface of the capillary tube:
1. Prepare 100 ml of a 10mg/ml PC1036 solution in 99% Hexane and 1%
Ethanol.
2. Clean the capillary tube lumen by using a 20ml syringe to pull and push the
Hexane through the lumen vertically.
3. Pass compressed air vertically through both ends of the capillary tube lumen
for 2 seconds at a flow rate of 30-35 liters per minute to remove any
remaining traces of Hexane.
4. Coat the capillary tube by using a 20 ml syringe to pull and push the PC1036
polymer solution through the lumen vertically.
5. Immediately after coating, pass compressed air vertically through both ends of
the capillary tube lumen for 2 seconds at a flow rate of 30-35 liters per minute
to remove any remaining traces of PC1036 polymer solution.
6. Place the capillary tube horizontally in an incubator preheated to 70℃ for 4
hours to allow the coating to cure. 203
The capillary tube lumen was reported to be free from blockage after the coating procedure. In addition, the thickness of the polymer coating cured on the inner surface of the capillary tube was reported approximately 1 µm.
204
APPENDIX E. Microsoft Excel Solver
A powerful tool that is widely available in a spreadsheet format provides a simple means of fitting experimental data to both linear and non-linear functions
[Microsoft Corporation, 57926-0694]. The curve-fitting method by using Excel
Solver is well-known and widely used in scientific researches [Harris, 1998; John,
1998; Brown, 2001]. The procedure and its mode of operation are very easy and obvious. Frequently in engineering, science and business, data is collected and plotted as a graphical representation of the variables involved. The next step is to create an association between the variables by connecting the points with a line.
Once drawn, the line is examined and a model which best fits the data points assumed when the theoretical solution for the data points is not available. Then, this is fitted and used to replace the existing set of data points as the appropriate model. However, in case that the theoretical solution is available, this procedure can also be used to determine the unknowns in the solution.
In order to fit a curve to a data series, using the excel solver is simplicity itself.
If a data series contains the x and y values, and an appropriate model has been available. Fitting the chosen model is then as follows [Harris, 1998; John, 1998;
Brown, 2001]:
1. Enter the known x and y values as a data series onto the spreadsheet. 205
2. Add a further column containing a chosen model. The parameters
(unknowns) of the chosen model are estimated and located in any free cells.
These are the ‘Change cells’.
3. Add a further column which expresses the squared error between the known
y values and the assumed model values.
4. Sum the squared error column in an appropriate free cell.
5. Evoke Solver by selecting the Tolls menu and Solver to present the Solver
dialogue box.
6. In the dialogue box, enter the sum of the squared error cell as the target cell.
7. Set the Equal to option to Min.
8. Enter the selected ‘Change cells’ to the ‘By changing cells’ box.
9. Include any constraints and modify the options as necessary.
10. Select the Solver button to initiate the curve fitting.
The values of the assumed model parameters (unknowns) will then be adjusted in each of the ‘Change cells’ until the Target cell value is a minimum. Excel Solver uses Newton’s method of iteration to determine the best combination of unknowns that fit into the model [Microsoft Corporation, 57926-0694].
206
APPENDIX F. Newton’s Method of Iteration
The Newton method is one of the most widely used methods for root finding.
It can be used for the problem to find solutions of a system of non-linear equations
[Young and Gregory, 1988; Isaacson and Keller, 1994; Hildebrand, 1987]. Consider the general problem of fitting a function of the following type: y = f (X ; A) (F-1)
where X (Variables) = (x1 , x2 ,..., xn ) and A (Parameters) = (a1 ,a2 ,...,am ) .
Choosing the parameters, A = (a1 ,a2 ,..., am ) , which minimize the sum of error, the least-square error function, E(A) , can be expressed by using the following equation:
l 2 E(A) = ∑[]f (X ( j) ; A) − y ( j) (F-2) j=1
A necessary condition that the parameters corresponding to a minimum is that they are a stationary point. Therefore, the following system of equations should be satisfied:
∂E ∂E ∂E = 0, = 0 , …, and = 0 . (F-3) ∂a1 ∂a2 ∂am
Note that some or all of the equations in Eq. (F-3) may be non-linear. Applying the chain rule to the definition of the error function E , one may rewrite Eq. (F-3) in the following forms: 207
l ∂f (X ( j) ; A) ∑[]f (X ( j) ; A) − y ( j) = 0 j=1 ∂a1 l ∂f (X ( j) ; A) ∑[]f (X ( j) ; A) − y ( j) = 0 j=1 ∂a2 (F-4) M l ∂f (X ( j) ; A) ∑[]f (X ( j) ; A) − y ( j) = 0 j=1 ∂am
Considering the non-linear cases, the standard form for these problems is Eq. (F-4).
F1 (A) = 0 F (A) = 0 2 i.e., M
Fm (A) = 0
F(A) = 0 (F-5) where F and A are vectors.
(0) (0) (0) Supposed that an initial approximation, A0 = (a1 ,a2 ,...,am ) , to a solution of the system is provided, the Newton’s method can be used. The Newton’s method is based on the Taylor Expansion, which can be expressed in matrix form as follows:
F(A) = F(A0 ) + J (A0 ) ⋅ (A − A0 ) + higher order terms (F-6)
where J is the Jacobian matrix whose elements are evaluated at A0 . Since F(A ) should vanish, and the higher order terms can be assumed to be negligible, Eq. (F-6) can be reduced to:
J (A0 ) ⋅ (A − A0 ) = −F(A0 ) (F-7)
The above equation is a linear system of equations, so one can solve it for A1 − A0 .
208
APPENDIX G. Repeatability Study with Distilled Water
1 Test #1 0.98 Test #2 0.96 Test #3 0.94 Test #4 Test #5 0.92 Reference (0.892 cP) 0.9 0.88
Viscosity (cP) Viscosity 0.86 0.84 0.82 0.8 0 100 200 300 400 500 Shear rate (s-1)
Fig. G-1. Repeatability study #1.
209
1 Test #1 0.98 Test #2 0.96 Test #3 Test #4 0.94 Test #5 0.92 Reference (0.892 cP) 0.9 0.88
Viscosity (cP) Viscosity 0.86 0.84 0.82 0.8 0 100 200 300 400 500 Shear rate (s-1)
Fig. G-2. Repeatability study #2.
210
APPENDIX H. Experimental Data
Table H-1. A typical experimental data set of human blood obtained by a scanning capillary-tube rheometer with precision glass riser tubes. One out of 100 data points is selected from an original data set.
Time (s) Pixel Number at Riser Tube 1 Pixel Number at Riser Tube 2 0.00 1183 399 2.00 1040 460 4.00 1036 509 6.00 1036 550 8.00 1004 583 10.00 976 609 12.00 946 632 14.00 922 650 16.00 907 666 18.00 894 679 20.00 885 690 22.00 891 700 24.00 882 707 26.00 875 714 28.00 868 720 30.00 863 725 32.00 859 730 34.00 854 734 36.00 851 737 38.00 847 740 40.00 845 742 42.00 842 743 44.00 840 745 46.00 838 747 48.00 836 748 50.00 834 750 52.00 833 751 54.00 832 752 56.00 830 753 58.00 829 754 60.00 828 754
211
Table H-1. Continued.
Time (s) Pixel Number at Riser Tube 1 Pixel Number at Riser Tube 2 62.00 827 755 64.00 826 756 66.00 825 756 68.00 825 757 70.00 824 757 72.00 823 757 74.00 822 758 76.00 822 758 78.00 821 758 80.00 820 759 82.00 820 759 84.00 819 759 86.00 819 759 88.00 818 759 90.00 818 760 92.00 818 760 94.00 817 760 96.00 817 760 98.00 816 760 100.00 816 760 102.00 816 760 104.00 815 760 106.00 815 760 108.00 814 760 110.00 814 760 112.00 814 760 114.00 813 760 116.00 813 760 118.00 813 759 120.00 812 759 122.00 812 759 124.00 812 759 126.00 812 759 128.00 811 759 130.00 811 759
212
Table H-1. Continued.
Time (s) Pixel Number at Riser Tube 1 Pixel Number at Riser Tube 2 132.00 811 759 134.00 811 758 136.00 811 758 138.00 810 758 140.00 810 758 142.00 810 758 144.00 809 758 146.00 809 758 148.00 809 758 150.00 809 758 152.00 809 758 154.00 808 758 156.00 808 758 158.00 808 758 160.00 808 758 162.00 809 759 164.00 808 758 166.00 808 758 168.00 808 758 170.00 808 758 172.00 807 758 174.00 808 758 176.00 807 758 178.00 807 758 180.00 807 758
213
Table H-2. A typical experimental data set of distilled water obtained by a scanning capillary-tube rheometer with plastic riser tubes. One out of 100 data points is selected from an original data set.
Time (s) Pixel Number at Riser Tube 1 Pixel Number at Riser Tube 2 0.00 568 130 2.00 411 288 4.00 292 407 6.00 220 478 8.00 178 522 10.00 152 548 12.00 137 561 14.00 128 571 16.00 122 576 18.00 118 580 20.00 117 583 22.00 115 584 24.00 115 585 26.00 114 585 28.00 112 585 30.00 112 586 32.00 112 586 34.00 112 586 36.00 111 587 38.00 111 587 40.00 111 587 42.00 111 587 44.00 111 587 46.00 111 587 48.00 111 587 50.00 111 587 52.00 111 587 54.00 111 587 56.00 111 587 58.00 111 587 60.00 111 587 62.00 111 587 64.00 111 587 66.00 111 587 68.00 111 587 70.00 111 587
214
Table H-2. Continued.
Time (s) Pixel Number at Riser Tube 1 Pixel Number at Riser Tube 2 72.00 111 587 74.00 111 587 76.00 111 587 78.00 111 587 80.00 111 587 82.00 111 587 84.00 111 587 86.00 111 587 88.00 111 587 90.00 111 587 92.00 111 587 94.00 111 587 96.00 111 587 98.00 111 587 100.00 111 587 102.00 111 587 104.00 111 587 106.00 111 587 108.00 111 587 110.00 111 587 112.00 111 587 114.00 111 587 116.00 111 587 118.00 111 587 120.00 111 587 122.00 111 587 124.00 111 587 126.00 111 587 128.00 111 587 130.00 111 587 132.00 111 587 134.00 111 587 136.00 111 587 138.00 111 587 140.00 111 587 142.00 111 587 144.00 111 587 146.00 111 587 148.00 111 587 150.00 111 587
215
Table H-3. A typical experimental data set of bovine blood obtained by a scanning capillary-tube rheometer with plastic riser tubes. One out of 100 data points is selected from an original data set.
Time (s) Pixel Number at Riser Tube 1 Pixel Number at Riser Tube 2 0 550 115 2 520 152 4 481 182 6 455 210 8 430 235 10 405 259 12 384 280 14 365 299 16 349 316 18 332 331 20 319 345 22 304 359 24 288 370 26 278 381 28 268 390 30 259 399 32 251 408 34 244 415 36 237 422 38 231 428 40 224 435 42 218 440 44 213 445 46 208 450 48 203 455 50 198 458 52 193 462 54 190 465 56 187 469 58 184 472 60 181 475 62 179 477 64 176 480 66 174 482 68 172 485 70 170 487
216
Table H-3. Continued.
Time (s) Pixel Number at Riser Tube 1 Pixel Number at Riser Tube 2 72 168 489 74 166 491 76 164 492 78 162 494 80 161 496 82 159 497 84 158 498 86 156 500 88 155 501 90 154 502 92 153 503 94 152 504 96 150 505 98 150 506 100 149 507 102 148 508 104 147 509 106 146 510 108 146 511 110 145 511 112 144 512 114 144 512 116 143 513 118 143 513 120 142 514 122 142 515 124 141 515 126 141 516 128 140 516 130 140 517 132 139 517 134 139 517 136 139 518 138 138 518 140 138 518 142 138 518 144 138 519 146 137 519 148 137 519 150 137 519
217
Table H-3. Continued.
Time (s) Pixel Number at Riser Tube 1 Pixel Number at Riser Tube 2 152 137 520 154 136 520 156 136 520 158 136 520 160 136 521 162 135 521 164 135 521 166 135 522 168 135 522 170 134 522 172 134 522 174 134 523 176 133 523 178 133 523 180 159 555
218
Table H-4. A typical experimental data set of human blood obtained by a scanning capillary-tube rheometer with plastic riser tubes. One out of 100 data points is selected from an original data set.
Time (s) Pixel Number at Riser Tube 1 Pixel Number at Riser Tube 2 0.00 584 133 2.00 535 185 4.00 484 235 6.00 446 273 8.00 411 311 10.00 380 341 12.00 356 366 14.00 335 387 16.00 315 408 18.00 299 423 20.00 283 437 22.00 272 450 24.00 261 460 26.00 252 469 28.00 244 476 30.00 237 482 32.00 230 491 34.00 224 497 36.00 219 502 38.00 214 507 40.00 210 510 42.00 206 513 44.00 201 517 46.00 196 520 48.00 193 522 50.00 192 525 52.00 189 527 54.00 187 529 56.00 186 531 58.00 184 533 60.00 182 534 62.00 181 535 64.00 179 537 66.00 178 538 68.00 177 539 70.00 176 540
219
Table H-4. Continued.
Time (s) Pixel Number at Riser Tube 1 Pixel Number at Riser Tube 2 72.00 176 541 74.00 175 542 76.00 174 543 78.00 173 544 80.00 172 545 82.00 171 545 84.00 171 546 86.00 170 546 88.00 170 547 90.00 169 547 92.00 169 548 94.00 168 548 96.00 168 549 98.00 167 549 100.00 167 550 102.00 167 550 104.00 166 550 106.00 166 551 108.00 166 551 110.00 165 551 112.00 165 551 114.00 165 552 116.00 165 552 118.00 164 552 120.00 164 552 122.00 164 552 124.00 163 552 126.00 163 553 128.00 163 553 130.00 163 553 132.00 162 553 134.00 162 553 136.00 162 553 138.00 162 553 140.00 162 553 142.00 161 553 144.00 161 554 146.00 161 554 148.00 161 554 150.00 160 554
220
Table H-4. Continued.
Time (s) Pixel Number at Riser Tube 1 Pixel Number at Riser Tube 2 152.00 160 554 154.00 160 554 156.00 160 554 158.00 160 554 160.00 160 554 162.00 160 555 164.00 160 555 166.00 160 555 168.00 159 555 170.00 159 555 172.00 159 555 174.00 159 555 176.00 159 555 178.00 159 555 180.00 159 555
221
VITA
Sangho Kim
Education Drexel University Philadelphia, PA Doctor of Philosophy in Mechanical Engineering 12/2002 Master of Science in Mechanical Engineering Kyungpook National University Taegu, Korea Bachelor of Science in Mechanical Engineering 02/1997
Journal Publications • S. Kim, Y.I. Cho, K.R. Kensey, R.O. Pellizzari and P.R. Stark “A scanning dual-capillary-tube viscometer” Review of Scientific Instruments, Vol. 71, No. 8, August 2000, 3188-3192 • S. Kim, Y.I. Cho, A.H. Jeon, B. Hogenaeur and K.R. Kensey “A new method for blood viscosity measurement” J. Non-Newtonian Fluid Mechanics, 94 (2000) 47-56 • S. Kim, Y.I. Cho, W.N. Hogenaeur and K.R. Kensey “A method of isolating surface tension and yield stress effects in a U-shaped scanning capillary-tube viscometer using a Casson model” J. Non-Newtonian Fluid Mechanics, 103 (2002) 205-219 ● S. Kim and Y.I. Cho “The effect of dye concentration on the viscosity of water in a scanning capillary-tube viscometer” J. Non-Newtonian Fluid Mechanics, 2002 (submitted) ● S. Kim, Y.I. Cho, and W.N. Hogenaeuer “Non-Newtonian constitutive models for the viscosity and yield-stress measurements of blood using a scanning capillary-tube rheometer” Biorheology, 2002 (submitted)
Conference Publications • S. Kim and Y.I. Cho “A new method of measuring blood viscosity with a U-shaped scanning capillary-tube viscometer using a Casson model” Proceedings of the IEEE 28th Annual Northeast Bioengineering Conference, April 20-21, 2002, 253-254 • Y.I. Cho and S. Kim “A new scanning capillary tube viscometer for blood viscosity measurement” Proceedings of the First National Congress on Fluids Engineering, September 1-2, Korea, 2000, 5-8
US Patents
• U.S. Patent No. 6,428,488 • U.S. Patent No. 6,412,336 • U.S. Patent No. 6,322,524 • U.S. Patent No. 6,402,703 • U.S. Patent No. 6,450,974