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Two-Fluid Mathematical Models for Blood Flow in Stenosed Arteries: A Comparative Study
Boundary Value Problems volume 2009, Article number: 568657 (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.
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
Consider an axially symmetric, laminar, pulsatile, and fully developed flow of blood (assumed to be incompressible) in the 
direction through a rigid-walled circular artery with an axially symmetric mild stenosis. The geometry of the arterial stenosis is shown in Figure 1. We have used the cylindrical polar coordinates 
. Blood is represented by 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 region as a Newtonian fluid. The non-Newtonian fluid in the core region is represented by (i) Casson fluid model and (ii) Herschel-Bulkley fluid model. The geometry of the stenosis in the peripheral region (in dimensionless form) and core region are, respectively, given by
where 
and 
are the radii of the stenosed artery with the peripheral region and core region, respectively; 
and 
are the radii of the normal artery and core region of the normal artery, respectively; β is the ratio of the central core radius to the normal artery radius; 
is the length of the stenosis; 
indicates the location of the stenosis; 
and 
are the maximum projections of the stenosis in the peripheral region and core region, respectively, such that 
and 
.
Geometry of the two-fluid models with arterial stenosis. Two-fluid Casson modelTwo-fluid H-B model
2.1. Two-Fluid Casson Model
2.1.1. Governing Equations
It can be shown that the radial velocity is negligibly small and can be neglected for a low Reynolds number flow. The basic momentum equations governing the flow are
where the shear stress 
; 
is the pressure; 
are the axial velocities of the fluid in the core region and peripheral region, respectively; 
are the shear stresses of the Casson fluid and Newtonian fluid, respectively; 
are the densities of the Casson fluid and Newtonian fluid, respectively; 
is the time. The relationships between the shear stress and strain rate of the fluids in motion in the core region (Casson fluid) and peripheral region (Newtonian fluid) are given by
where 
are the viscosities of the Casson and Newtonian fluids, respectively; 
is the yield stress; 
is the plug core radius. The boundary conditions are
Since the pressure gradient is a function of 
and 
, we assume
where 
. Since any periodic function can be expanded in a Fourier sine series, it is reasonable to choose 
as a good approximation for 
,where 
and 
are the amplitude and angular frequency of the flow, respectively. We introduce the following nondimensional variables:
where 
is the negative of the pressure gradient in the normal artery; 
and 
are the pulsatile Reynolds numbers of the Casson fluid and Newtonian fluid, respectively. Using the nondimensional variables, (2.1)–(2.4) are simplified to
where
The boundary conditions (in the dimensionless form) are
The geometry of the stenosis in the peripheral region and core region (in the dimensionless form) are given by
The nondimensional volume flow rate 
is given by
where 
; 
is the volume flow rate.
2.1.2. Method of Solution
When we nondimensionalize the constitutive (2.1), (2.2), 
and 
occur naturally and these are time dependent and hence, it is more appropriate to expand (2.10)–(2.13) about 
and 
. Let us expand the plug core velocity 
and the velocity in the core region 
in the perturbation series of 
as follows: (where 
)
Similarly, one can expand 
and 
in powers of 
and 
, where 
. Using the perturbation series in (2.10), (2.11) and then equating the constant terms and 
terms, the differential equations of the core region become
Similarly, using the perturbation series expansions in (2.13) and then equating the constant terms and 
terms, the differential equations of the peripheral region become
Substituting the perturbation series expansions in (2.15) and then equating the constant terms and 
and 
terms, we get
Solving the system of (2.19) and (2.20) using (2.21) for the unknowns 
, 
one can obtain
where 
and 
. The wall shear stress 
can be obtained as follows:
Using (2.23)–(2.25) and (2.29)–(2.31) in (2.17), the volume flow rate is obtained as
The shear stress 
at 
is given by
Using Taylor's series of 
and 
about 
and using 
, we get
Using (2.22), (2.27), and (2.35) in the two term approximated perturbation series of 
, the expression for 
can be obtained as
The resistance to flow is given by
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 basic momentum equations governing the flow and the constitutive equations in the nondimensional form are
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
The objective of the present analysis is to compare and bring out the advantages of the two-fluid Casson model over the two-fluid Herschel-Bulkley model. It is observed that the typical value of the power law index 
for blood flow models is taken as 0.95 [3]. The value 0.1 is used for the nondimensional yield stress 
in this study. Even though the range of the amplitude 
is from 0 to 1, we have used the value 0.5. The value 0.5 is used for the pulsatile Reynolds numbers 
and pulsatile Reynolds number ratio 
of both the two-fluid models [11]. The value of the ratio 
of central core radius 
to the normal artery radius 
in the unobstructed artery is generally taken as 0.95 [15]. Following Shukla et al. [21], relations 
and 
are used to estimate 
and 
. The maximum thickness of the stenosis in the peripheral region 
is taken as 0.1 [11]. The steady flow rate 
value is taken as 1.0 [12]. It is observed that in the expression of the flow rate of the two-fluid Casson model, 
and 
are the knowns, and 
and 
are the unknowns to be determined. A careful analysis of the flow rate expression reveals the fact that 
is the pressure gradient of the steady flow. Thus, if steady flow is assumed, then the expression of the flow rate can be solved for 
[3, 12]. For steady flow, the expression for flow rate of the two-fluid Casson model reduces to
The similar equation of the two-fluid Herschel-Bulkley model is
The variation of pressure drop in a time cycle of the two-fluid Herschel-Bulkley (H-B) and Casson models with 
and 
is shown in Figure 2. It is observed that for both the two-fluid models the pressure drop increases as time 
(in degrees) increases from 
to 
, then it decreases as 
increases from 
to 
, and again the pressure drop increases as 
increases further from 
to 
. The pressure drop is maximum at 
and minimum at 
. It is found that, at any time, the pressure drop of the two-fluid Casson model is considerably much lower than that of the two-fluid H-B model while all the other parameters held constant. Figure 3 depicts the variation of the plug core radius with axial distance of the two-fluid H-B and Casson models with 
and 
. It is noticed that the plug core radius decreases as the axial variable 
increases from 4 to 5 and it increases symmetrically when the axial variable increases from 5 to 6. It is noted that for a given set of values of the parameters, the plug core radius values of the two-fluid Casson model are significantly much lower than that of the two-fluid H-B model.
Variation of pressure drop in a time cycle of the two-fluid Casson and H-B models.
Variation of plug core radius with axial distance of the two-fluid Casson and Herschel-Bulkley models.
3.1. Plug Flow Velocity
The variation of the plug flow velocity in a time cycle of the two-fluid Casson and H-B models with 
, 
, 
and 
is depicted in Figure 4. It is seen that the plug flow velocity decreases as time 
(in degrees) increases from 
to 
, then it increases as 
increases from 
to 
, and then again it decreases from 
to 
. The plug flow velocity is minimum at 
and maximum at 
. It is noted that the plug flow velocity of the two-fluid Casson model is considerably higher than that of the two-fluid H-B model.
Variation of plug flow velocity in a time cycle of the two-fluid Casson and two fluid Herschel-Bulkley models.
3.2. Wall Shear Stress
Figure 5 shows the variation of the wall shear stress in a time cycle of the two-fluid Casson and H-B models with 
and 
. The behavior of the wall shear stress is just reversed of the two-fluid models, that we observed in Figure 4 for the plug flow velocity.
Variation of wall shear stress in a cycle of the two-fluid Casson and two fluid Herschel-Bulkley models.
3.3. Velocity Distribution
The velocity distributions of the two-fluid H-B and Casson models with 
, 
and 
are sketched in Figure 6. One can notice the plug flow around the tube axis for both the fluid models. It is further recorded that, for a given set values of the parameters, a significantly high-magnitude velocity profile is found in the two-fluid Casson model than in the two-fluid H-B model.
Velocity distribution of the two-fluid Casson and two-fluid Herschel-Bulkley model.
3.4. Resistance to Flow
The variation of resistance to flow with peripheral layer stenosis height of the two-fluid Casson and H-B models with 
and 
is shown in Figure 7. It is observed that the resistance to flow increases nonlinearly with the increase of the peripheral stenosis height. It is of interest to note that, for any value of the stenosis height, the resistance to flow of the two-fluid Casson model is considerably much lower than that of the H-B model.
Variation of resistance to flow with peripheral layer stenosis height of the two-fluid Casson and two-fluid Herschel-Bulkley models.
3.5. Quantification of the Wall Shear Stress and Resistance to Flow
The wall shear stress 
and resistance to flow 
are physiologically important quantities which play an important role in the formation of platelets [22]. High wall shear stress not only damages the vessel wall and causes intimal thickening but also activates platelets, causes platelet aggregation, and finally results in the formation of thrombus [6]. Estimates of the wall shear stress 
and the percentage of increase in the wall shear stress of the two-fluid Casson model and two-fluid Herschel-Bulkley model with 
for different stenosis heights with 
, 
, 
and 
are computed in Table 1. It is found that for the range 0.025–0.15 of the stenosis height, the corresponding range of the percentage of increase in the estimates of the wall shear stress of the two-fluid Casson model and two-fluid Herschel-Bulkley model with 
are 5.45–42.16 and 7.43–59.89, respectively. One can notice that both the estimates of the wall shear stress and the percentage of increase in the wall shear stress of the two-fluid Casson model are significantly lower than those of the two-fluid Herschel-Bulkley model.
and percentage of increase in the wall shear stress 
of the two-fluid Casson model and two-fluid Herschel-Bulkley model over uniform diameter tube for different stenosis sizes with 
and 
.Estimates of the resistance to flow 
and the percentage of increase in the resistance to flow for the two-fluid Casson model and two-fluid Herschel-Bulkley model with 
for different stenosis heights with 
and 
are given in Table 2. It is observed that, for the range 0.025–0.15 of the stenosis height, the corresponding ranges of the percentage of increase in the estimates of the resistance to flow of the two-fluid Casson model and two-fluid Herschel-Bulkley model are 4.16–25.10 and 5.16–31.85, respectively. It is clear that both the estimates of the wall shear stress and the percentage of increase in the wall shear stress of the two-fluid Casson model are significantly lower than those of the two-fluid Herschel-Bulkley model. Hence, it is clear that the two-fluid Casson model layer is useful in the functioning of the diseased arterial system.
of the two-fluid Casson model and two-fluid Herschel-Bulkley model over uniform diameter tube for different stenosis heights with 
and 
.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.
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Sankar, D., Ismail, A. Two-Fluid Mathematical Models for Blood Flow in Stenosed Arteries: A Comparative Study. Bound Value Probl 2009, 568657 (2009). https://doi.org/10.1155/2009/568657
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DOI: https://doi.org/10.1155/2009/568657