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# Two-Fluid Mathematical Models for Blood Flow in Stenosed Arteries: A Comparative Study

*Boundary Value Problems*
**volumeÂ 2009**, ArticleÂ number:Â 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 [1â€“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 fluidâ€”viscosity, yield stress and power lawâ€”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 130â€“1300â€‰, whereas Herschel-Bulkley fluid model could be used in tubes of diameter 20â€“100â€‰.

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; *Î²* 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 .

### 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)â€“(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)â€“(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)â€“(2.25) and (2.29)â€“(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)â€“(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.

### 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.

### 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.

### 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.

### 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.

### 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 0.025â€“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 5.45â€“42.16 and 7.43â€“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.

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 0.025â€“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 4.16â€“25.10 and 5.16â€“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.

## 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