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Determination of Rheological Properties of Whole Blood with A Journal of Mechanical Science and Technology Journal of Mechanical Science and Technology 23 (2009) 1718~1726 www.springerlink.com/content/1738-494x DOI 10.1007/s12206-009-0420-6 Determination of rheological properties of whole blood with a scanning † capillary-tube rheometer using constitutive models Sangho Kim1, Bumseok Namgung1, Peng Kai Ong1, Young I. Cho2, Keyoung Jin Chun3 and Dohyung Lim3,* 1Division of Bioengineering and Department of Surgery, National University of Singapore, Singapore 117574, Singapore 2Department of Mechanical Engineering & Mechanics, Drexel University, Philadelphia, PA 19104, USA 3Gerontechnology Center, Korea Institute of Industrial Technology, Chungnam, 330-825, Korea (Manuscript Received December 9, 2008; Revised February 5, 2009; Accepted March 12, 2009) -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Abstract We examine the applicability of three different non-Newtonian constitutive models (power-law, Casson, and Herschel-Bulkley models) to the determination of blood viscosity and yield stress with a scanning capillary-tube rheometer. For a Newtonian fluid (distilled water), all three models produced excellent viscosity results, and the meas- ured values of the yield stress with each model were zero. For unadulterated human blood, the Casson and Herschel- Bulkley models produced much stronger shear-thinning viscosity results than the power-law model. The yield stress values for the human blood obtained with the Casson and Herschel-Bulkley models were 13.8 and 17.5 mPa, respec- tively. The two models showed a small discrepancy in the yield stress values, but with the current data analysis method for the scanning capillary-tube rheometer, the Casson model seemed to be more suitable in determining the yield stress of blood than the Herschel-Bulkley model. Keywords: Blood viscosity; Yield stress; Power-law; Casson; Herschel-Bulkley -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Recently, a new U-shaped scanning capillary-tube 1. Introduction rheometer (SCTR) has been developed from the con- The flow properties of the blood are affected by a cept of the conventional capillary-tube viscometer [1- number of factors, which include hematocrit (volume 4]. The rheometer is capable of measuring whole fraction of red blood cells in whole blood), plasma blood viscosity continuously over a wide range of viscosity, red blood cell aggregation, and deform- shear rates without adding any anticoagulants. How- ability. Furthermore, the viscosity of blood is non- ever, there are some limitations in using the SCTR for Newtonian, which increases as shear rate (velocity viscosity measurements of fluids. Since the device gradient in a flowing material) decreases. Hematocrit utilizes charge-coupled devices and light-emitting is one of the major factors determining the blood diode arrays to determine liquid-height changes over viscosity at high shear rates while red blood cell ag- time, viscosity of transparent liquids cannot be meas- gregation plays a significant role at low shear rates. ured [1]. Another major limitation of using this new At normal hematocrit concentrations (35-45%), red device for liquid viscosity measurements lies in the blood cell deformability may affect blood viscosity selection of constitutive models. The SCTR was de- and becomes a major factor in the microcirculation. veloped specially for blood viscosity and yield stress †This paper was recommended for publication in revised form by Associate measurements using the Casson model [2]. However, Editor Gihun Son many earlier studies report that not only the Casson *Corresponding author. Tel.: +82 41 589 8428, Fax.: +82 41 589 8420 E-mail address: [email protected] model but also other non-Newtonian constitutive © KSME & Springer 2009 models effectively describe blood flows in biological S. Kim et al. / Journal of Mechanical Science and Technology 23 (2009) 1718~1726 1719 systems. For instance, Bingham plastic, Casson, and Table 1. Various physiological studies with non-Newtonian constitutive models. Herschel-Bulkley (H-B) models were used to de- scribe the rheological behavior of yield-stress fluids Herschel- Researchers Power-law Casson [5]. Picart et al. [6] used the value of the shear stress Bulkley at a shear rate of 10-3 s-1 as a measure of the yield Chakravarty and x x stress in their report. In a study by Barnes [7], the H- Datta [8] B model described reasonably well the data reported Siauw et al. [9] x x by Picart and coworkers. Tu and Deville [10] x x Table 1 shows three non-Newtonian constitutive Liepsch and x models (power-law, Casson, and H-B models) that Moravec [11] have been used by a number of researchers for inves- Rohlf and Tenti [12] x tigating blood rheology and flows. Siauw et al. [9] Misra et al. [13, 14] x performed a comparative study of the power-law and Das and Batra [15] x Casson models for prediction of unsteady stenosis Dash et al. [16] x flows, whereas Tu and Deville [10] used H-B, Bing- Misra and Kar [17] x ham, and power-law models for their study on steno- Chakravarty and sis flows. It has been pointed out that the selection of x Datta [18, 19] constitutive model may significantly affect the analy- sis of blood flows [9, 10]. The main purpose of the present study is to exam- ine whether the results of blood rheology and flow A obtained with the current data analysis method for the SCTR are significantly altered by the choice of con- stitutive model. Hence, the study investigated the effect of constitutive model selection on the blood viscosity and yield stress measurements and predic- tion of blood flow patterns using the SCTR. It is well known that human blood has both shear-thinning characteristics and yield stress, which make the blood viscosity increase with decreasing shear rate. We examined the capability of three constitutive models in handling data obtained with the SCTR for determi- nations of blood viscosity and yield stress. The B power-law model was chosen to address the shear- thinning behavior of blood. The Casson and H-B models were selected to address both the shear- thinning viscosity and yield stress of blood. 2. Materials and methods The experimental preparation and data acquisition methods have been described previously [2, 3], and the reader is referred to those studies for a complete description. Viscosity values of distilled water and Fig. 1. Liquid-solid interface condition for each vertical tube of a U-shaped scanning capillary tube rheometer. unadulterated human blood were determined with the scanning capillary-tube rheometer (SCTR) over a wide range of shear rates. Human blood was obtained viscosity and yield stress determinations. The length from a healthy male donor aged 29. Venous blood and inner diameter of the capillary tube used in this sample ~3 ml was withdrawn and directly transferred study were 100 and 0.797 mm, respectively. Fluid into a disposable capillary-tube set in the SCTR for height changes with time in the riser tubes shown in 1720 S. Kim et al. / Journal of Mechanical Science and Technology 23 (2009) 1718~1726 Fig. 1 were measured with an accuracy of 0.083 mm. the more pronounced the non-Newtonian properties Human blood was examined at a body temperature of of the fluid are. The mathematical details of curve- 37°C, while tests with distilled water were performed fitting procedure to determine the two unknowns, m at a room temperature of 25°C. and n , in the SCTR were reported in our previous study [3]. The velocity profile in the capillary tube 2.1 Surface tension and yield stress (Vc ) can be expressed by the following equation: Fig. 1A shows that the fluid level in the right-side 1 n nn++11 vertical tube (riser tube 1) decreases while that in the ⎛⎞n ⎛⎞ρght12()−−∆ h () t ht=∞ ⎛⎞ []nn Vtrc (, )=⋅⎜⎟⎜⎟ ⋅−⎜⎟ Rc r left-side vertical tube (riser tube 2) increases. Even ⎝⎠nmL+12⎝⎠c ⎝⎠ when time goes to infinity (i.e., t ~ 3 minutes), the (3) two fluid levels do not become equal due to any dif- ferences in surface tension and yield stress at the two where Rc and Lc are the radius and length of the riser tubes (see ∆ht=∞ in Fig. 1B). As shown in Fig. capillary tube, respectively. Note that when the 1A, the riser tube 1 always has a fully wet surface power-law index, n , becomes 1, the above equation condition, whereas the riser tube 2 has an almost per- yields to the Newtonian velocity profile. fectly dry surface condition during the entire test. Therefore, the surface tension at the riser tube 1 is 2.3 Casson model consistently greater than that at the riser tube 2 [2]. The Casson model mathematically describes the The time-dependent flow in the SCTR is analyzed shear-thinning characteristics of blood viscosity with to determine the viscosities of distilled water and a yield stress term: whole blood. Assuming a quasi-steady flow, the gov- erning equation can be simplified as follows: ττ=+y k γ& when ττ≥ y (4a) γ& = 0 when ττ≤ y (4b) ∆=Ptcst()ρ ght12 () − h () t −∆ h (1) [ ] where τ y is a constant that is interpreted as the yield where ∆Ptc () is the pressure drop across the capil- stress, and k is a Casson model constant. lary tube, ρ is the density of fluid, g is gravita-
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