Two-Fluid Mathematical Models for Blood Flow in Stenosed Arteries: A Comparative Study
- DS Sankar^{1}Email author and
- AhmadIzaniMd Ismail^{1}
https://doi.org/10.1155/2009/568657
© D. S. Sankar and A. I. Md. Ismail. 2009
Received: 18 December 2008
Accepted: 30 January 2009
Published: 10 February 2009
Abstract
The pulsatile flow of blood through stenosed arteries is analyzed by assuming the blood as a two-fluid model with the suspension of all the erythrocytes in the core region as a non-Newtonian fluid and the plasma in the peripheral layer as a Newtonian fluid. The non-Newtonian fluid in the core region of the artery is assumed as a (i) Herschel-Bulkley fluid and (ii) Casson fluid. Perturbation method is used to solve the resulting system of non-linear partial differential equations. Expressions for various flow quantities are obtained for the two-fluid Casson model. Expressions of the flow quantities obtained by Sankar and Lee (2006) for the two-fluid Herschel-Bulkley model are used to get the data for comparison. It is found that the plug flow velocity and velocity distribution of the two-fluid Casson model are considerably higher than those of the two-fluid Herschel-Bulkley model. It is also observed that the pressure drop, plug core radius, wall shear stress and the resistance to flow are significantly very low for the two-fluid Casson model than those of the two-fluid Herschel-Bulkley model. Hence, the two-fluid Casson model would be more useful than the two-fluid Herschel-Bulkley model to analyze the blood flow through stenosed arteries.
Keywords
1. Introduction
There are many evidences that vascular fluid dynamics plays a major role in the development and progression of arterial stenosis. Arteries are narrowed by the development of atherosclerotic plaques that protrude into the lumen, resulting arterial stenosis. When an obstruction developed in an artery, one of the most serious consequences is the increased resistance and the associated reduction of the blood flow to the particular vascular bed supplied by the artery. Thus, the presence of a stenosis leads to the serious circulatory disorder.
Several theoretical and experimental attempts were made to study the blood flow characteristics in the presence of stenosis [1–8]. The assumption of Newtonian behavior of blood is acceptable for high shear rate flow through larger arteries [9]. But, blood, being a suspension of cells in plasma, exhibits non-Newtonian behavior at low shear rate ( sec) in small diameter arteries [10]. In diseased state, the actual flow is distinctly pulsatile [11, 12]. Many researchers studied the non-Newtonian behavior and pulsatile flow of blood through stenosed arteries [1, 3, 9, 12].
Bugliarello and Sevilla [13] and Cokelet [14] have shown experimentally that for blood flowing through narrow blood vessels, there a peripheral layer of plasma and a core region of suspension of all the erythrocytes. Thus, for a realistic description of the blood flow, it is appropriate to treat blood as a two-fluid model with the suspension of all the erythrocytes in the core region as a non-Newtonian fluid and plasma in the peripheral region as a Newtonian fluid.
Kapur [15] reported that Casson fluid model and Herschel-Bulkley fluid model are the fluid models with nonzero yield stress and they are more suitable for the studies of the blood flow through narrow arteries. It has been reported by Iida [16] that Casson fluid model is simple to apply for blood flow problems, because of the particular form of its constitutive equation, whereas, Herschel-Bulkley fluid model's constitutive equation is not easy to apply because of the form of its empirical relation, since, it contains one more parameter than the Casson fluid model. It has been demonstrated by Scott-Blair [17] and Copley [18] that the parameters appropriate to Casson fluid—viscosity, yield stress and power law—are adequate for the representation of the simple shear behavior of blood. It has been established by Merrill et al. [19] that Casson fluid model holds satisfactorily for blood flowing in tubes of diameter 130–1300 , whereas Herschel-Bulkley fluid model could be used in tubes of diameter 20–100 .
Sankar and Lee [20] have developed a two-fluid model for pulsatile blood flow through arterial stenosis treating the fluid in the core region as Herschel-Bulkley fluid. Thus, in this paper, we extend this study to two-fluid Casson model and compare these models and discuss the advantages of the two-fluid Casson model over the two-fluid Herschel-Bulkley (H-B) model.
2. Mathematical Formulation
2.1. Two-Fluid Casson Model
2.1.1. Governing Equations
2.1.2. Method of Solution
where is the pressure drop. When , the present model reduces to the single fluid Casson model and in such case, the expressions obtained in the present model for velocity , shear stress , wall shear stress , flow rate and plug core radius are in good agreement with those of Chaturani and Samy [12].
2.2. Two-Fluid Herschel-Bulkley Model
The boundary conditions (in dimensionless form) of this model are similar to the boundary conditions of the two-fluid Casson model given in (2.7). Equations (2.38)–(2.42) are also solved using perturbation method with the help of the appropriate boundary conditions as in the case of the two-fluid Casson model. The details of the derivation of the expressions for shear stress, velocity, flow rate, plug core radius, wall shear stress and resistance to flow are given in Sankar and Lee [20].
3. Results and Discussion
3.1. Plug Flow Velocity
3.2. Wall Shear Stress
3.3. Velocity Distribution
3.4. Resistance to Flow
3.5. Quantification of the Wall Shear Stress and Resistance to Flow
Estimates of the wall shear stress | Estimates of the percentage of increase in wall shear stress | |||
---|---|---|---|---|
Two-fluid Casson model | Two-fluid Casson model | |||
0.025 | 1.677 | 3.0057 | 5.45 | 7.43 |
0.050 | 1.8058 | 3.1852 | 11.42 | 15.70 |
0.075 | 1.9495 | 3.3826 | 17.99 | 24.93 |
0.100 | 2.1102 | 3.6005 | 25.24 | 35.25 |
0.125 | 2.2907 | 3.8416 | 33.26 | 46.84 |
0.150 | 2.4939 | 4.1093 | 42.16 | 59.89 |
Stenosis height δ_{p} | Estimates of the resistance | Estimates of the percentage of increase in resistance | ||
---|---|---|---|---|
Two-fluid Casson model | Two-fluid H-B model with n = 0.95 | Two-fluid Casson model | Two-fluid H-B model with n = 0.95 | |
0.025 | 2.4795 | 2.9371 | 4.16 | 5.16 |
0.050 | 2.6135 | 3.0650 | 8.69 | 10.843 |
0.075 | 2.7616 | 3.2049 | 13.66 | 17.12 |
0.100 | 2.9258 | 3.3584 | 19.10 | 24.09 |
0.125 | 3.10868 | 3.5275 | 25.10 | 31.85 |
0.150 | 3.3131 | 3.7143 | 31.72 | 40.52 |
4. Conclusion
The pulsatile flow of blood through stenosed arteries is analyzed by assuming blood as a (i) two-fluid Casson model and (ii) two-fluid Herschel-Bulkley model. It is observed that, for a given set of values of the parameters, the velocity distribution of the two-fluid Casson model is considerably higher than that of the two-fluid Herschel-Bulkley fluid model. Further, it is noticed that the pressure drop, plug core radius, wall shear stress, and the resistance to flow of the two-fluid Casson model are significantly much lower than those of the two-fluid Herschel-Bulkley model.
It is of interest to note that the estimates of the wall shear stress and resistance to flow of the two-fluid Casson model are considerably lower than those of the two-fluid Herschel-Bulkley model. It is also worthy to note that the estimates of the percentage of increase in the wall shear stress and the percentage of increase in the resistance to flow of the two-fluid Casson model are considerably lower than those of the two-fluid Herschel-Bulkley model. Further, it is observed that the difference between the estimates of the wall shear stress, resistance to flow, percentage of increase in the estimates of the wall shear stress, and resistance to flow of the two-fluid Casson model and two-fluid Herschel-bulkley model is substantial. Hence, the two-fluid Casson model would be more useful in the mathematical analysis of the diseased arterial system.
Authors’ Affiliations
References
- Mandal PK: An unsteady analysis of non-Newtonian blood flow through tapered arteries with a stenosis. International Journal of Non-Linear Mechanics 2005, 40(1):151–164. 10.1016/j.ijnonlinmec.2004.07.007MATHView ArticleGoogle Scholar
- Marshall I, Zhao S, Papathanasopoulou P, Hoskins P, Xu XY: MRI and CFD studies of pulsatile flow in healthy and stenosed carotid bifurcation models. Journal of Biomechanics 2004, 37(5):679–687. 10.1016/j.jbiomech.2003.09.032View ArticleGoogle Scholar
- Sankar DS, Hemalatha K: Pulsatile flow of Herschel-Bulkey fluid through stenosed arteries—a mathematical model. International Journal of Non-Linear Mechanics 2006, 41(8):979–990. 10.1016/j.ijnonlinmec.2006.02.007MATHView ArticleGoogle Scholar
- Moayeri MS, Zendehbudi GR: Effects of elastic property of the wall on flow characteristics through arterial stenoses. Journal of Biomechanics 2003, 36(4):525–535. 10.1016/S0021-9290(02)00421-9View ArticleGoogle Scholar
- Chakravarty S, Mandal PK: Two-dimensional blood flow through tapered arteries under stenotic conditions. International Journal of Non-Linear Mechanics 2000, 35(5):779–793. 10.1016/S0020-7462(99)00059-1MATHView ArticleGoogle Scholar
- Liu G-T, Wang X-J, Ai B-Q, Liu L-G: Numerical study of pulsating flow through a tapered artery with stenosis. Chinese Journal of Physics 2004, 42(4):401–409.Google Scholar
- Long Q, Xu XY, Ramnarine KV, Hoskins P: Numerical investigation of physiologically realistic pulsatile flow through arterial stenosis. Journal of Biomechanics 2001, 34(10):1229–1242. 10.1016/S0021-9290(01)00100-2View ArticleGoogle Scholar
- Dash RK, Jayaraman G, Mehta KN: Flow in a catheterized curved artery with stenosis. Journal of Biomechanics 1999, 32(1):49–61. 10.1016/S0021-9290(98)00142-0View ArticleGoogle Scholar
- Tu C, Deville M: Pulsatile flow of non-Newtonian fluids through arterial stenoses. Journal of Biomechanics 1996, 29(7):899–908. 10.1016/0021-9290(95)00151-4View ArticleGoogle Scholar
- Chien S: Hemorheology in clinical medicine. Recent Advances in Cardiovascular Diseases 1981, 2: 21–26.MathSciNetGoogle Scholar
- Srivastava VP, Saxena M: Two-layered model of Casson fluid flow through stenotic blood vessels: applications to the cardiovascular system. Journal of Biomechanics 1994, 27(7):921–928. 10.1016/0021-9290(94)90264-XView ArticleGoogle Scholar
- Chaturani P, Samy RP: Pulsatile flow of Casson's fluid through stenosed arteries with applications to blood flow. Biorheology 1986, 23(5):499–511.Google Scholar
- Bugliarello G, Sevilla J: Velocity distribution and other characteristics of steady and pulsatile blood flow in fine glass tubes. Biorheology 1970, 7(2):85–107.Google Scholar
- Cokelet GR: The Rheology of Human Blood. Prentice-Hall, Englewood Cliffs, NJ, USA; 1972.Google Scholar
- Kapur JN: Mathematical Models in Biology and Medicine. Affiliated East West Press, New Delhi, India; 1992.Google Scholar
- Iida N: Influence of plasma layer on steady blood flow in micro vessels. Japanese Journal of Applied Physics 1978, 17(1):203–214. 10.1143/JJAP.17.203View ArticleGoogle Scholar
- Scott-Blair GW: An equation for the flow of blood, plasma and serum through glass capillaries. Nature 1959, 183(4661):613–614. 10.1038/183613a0View ArticleGoogle Scholar
- Copley AL: Apparent viscosity and wall adherence of blood systems. In Flow Properties of Blood and Other Biological Systems. Edited by: Copley AL, Stainsby G. Pergamon Press, Oxford, UK; 1960.Google Scholar
- Merrill EW, Benis AM, Gilliland ER, Sherwood TK, Salzman EW: Pressure-flow relations of human blood in hollow fibers at low flow rates. Journal of Applied Physiology 1965, 20(5):954–967.Google Scholar
- Sankar DS, Lee U: Two-phase non-linear model for the flow through stenosed blood vessels. Journal of Mechanical Science and Technology 2007, 21(4):678–689. 10.1007/BF03026973View ArticleGoogle Scholar
- Shukla JB, Parihar RS, Gupta SP: Effects of peripheral layer viscosity on blood flow through the artery with mild stenosis. Bulletin of Mathematical Biology 1980, 42(6):797–805.MATHView ArticleGoogle Scholar
- Karino T, Goldsmith HL: Flow behavior of blood cells and rigid spheres in annular vortex. Philosophical Transactions of the Royal Society of London. Series B 1977, 279(967):413–445. 10.1098/rstb.1977.0095View ArticleGoogle Scholar
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