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 h t= 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
H-4. A typical experimental data set of human blood obtained by a scanning capillary-tube rheometer with plastic riser tubes...218 ix
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.