Research Article | Open Access

# Comparative Analysis of Mathematical Models for Blood Flow in Tapered Constricted Arteries

**Academic Editor:**Gaston Mandata N'Guerekata

#### Abstract

Pulsatile flow of blood in narrow tapered arteries with mild overlapping stenosis in the presence of periodic body acceleration is analyzed mathematically, treating it as two-fluid model with the suspension of all the erythrocytes in the core region as non-Newtonian fluid with yield stress and the plasma in the peripheral layer region as Newtonian. The non-Newtonian fluid with yield stress in the core region is assumed as (i) Herschel-Bulkley fluid and (ii) Casson fluid. The expressions for the shear stress, velocity, flow rate, wall shear stress, plug core radius, and longitudinal impedance to flow obtained by Sankar (2010) for two-fluid Herschel-Bulkley model and Sankar and Lee (2011) for two-fluid Casson model are used to compute the data for comparing these fluid models. It is observed that the plug core radius, wall shear stress, and longitudinal impedance to flow are lower for the two-fluid H-B model compared to the corresponding flow quantities of the two-fluid Casson model. It is noted that the plug core radius and longitudinal impedance to flow increases with the increase of the maximum depth of the stenosis. The mean velocity and mean flow rate of two-fluid H-B model are higher than those of the two-fluid Casson model.

#### 1. Introduction

Atherosclerosis is an arterial disease in humans, which leads to the malfunctioning of the cardiovascular system [1]. The intimal thickening of an artery is the initial stage in the progression of atherosclerosis [2–4]. The lumen of the arteries is narrowed by the development of atherosclerotic plaques that protrude into the lumen, resulting in stenosed arteries. The wall of the artery is stiffened by the growth of plaque with a lipid core and a fibromuscular cap and narrowing of lumen of the artery by the deposit of fats, lipids, cholesterol, and so forth [5]. Stenoses in different shapes are formed in the arterial lumen and some of the stenoses shape are axisymmetric, asymmetric, overlapping, and multiple [1, 6–8]. When a stenosis is developed in an artery, its serious consequences are the increased resistance and the associated reduction of blood flow in the downstream [9, 10]. Thus, the development of a stenosis in the lumen of an artery leads to the serious circulatory disorder. Chakravarty et al. [11] pointed out that the blood vessels bifurcate at frequent intervals and although the individual segments of arteries may be treated as uniform between bifurcations, the diameter of the artery reduces considerably at each bifurcation. How and Black [12] pronounced that the study of blood flow in tapered arteries is useful in the design of prosthetic blood vessels as the use of grafts of tapered lumen has the surgical advantage. Hence, it is important to mathematically analyze the blood flow in tapered arteries with stenosis.

In many situations of our routine life such as traveling in vehicles, aircrafts, ships, swinging in a cradle, subjecting to vibration therapy as a treatment for some disease, sudden movements of body in sports activities, our body is exposed to body accelerations or vibrations [8, 13–15]. In some situations like traveling in a bus/train, the whole of the body is subjected to vibrations, while in some other occasions such as when operating jack hammer or lathe machine, driving a car, applying vibration therapy as a medical treatment, some specific part of our body is forced to vibrations [16, 17]. Exposure of our body to high level unintended external body accelerations for a long period causes serious health hazards due to the abnormal functioning of the cardiovascular system [18], and this leads to serious cardiovascular diseases which show some symptoms like headache, abdominal pain, increase in pulse rate, venous pooling of blood in the extremities, loss of vision, and hemorrhage in the face, neck, eye sockets, lungs, and brain [16, 18–20]. Thus, it is useful to investigate the effect of periodic body accelerations on the physiologically important flow measurements of blood flow in arteries of different diameters.

Blood exhibits anomalous viscous properties. Blood, when it flows in larger diameter arteries at high shear rates, it behaves like Newtonian fluid, but when it flows through narrow diameter arteries at low shear rates, it shows notable non-Newtonian behavior [21]. Several researchers investigated blood flow properties in constricted narrow arteries in the absence and presence of externally imposed periodic body accelerations [22–27].

Several researchers [11, 28, 29] mentioned that when blood flows in smaller diameter blood vessels at low shear rates, there is erythrocyte-free plasma layer adjacent to the vessel wall and core layer of suspension of all erythrocytes and thus it is not realistic to model blood as simply a single fluid non-Newtonian model. Hence, it is appropriate to model blood as a two-fluid model when it flows through narrow diameter arteries at low shear rates (diameter up to 1300 m) [30], treating the suspension of all the erythrocytes in the core region as a non-Newtonian fluid and the cell free plasma in the peripheral layer region as Newtonian fluid. Herschel-Bulkley (H-B) fluid model and Casson fluid model are some of the non-Newtonian fluid models with yield stress which are commonly used as the non-Newtonian fluids to represent the suspension of all the erythrocytes in the core region of blood flow in narrow arteries [21, 28]. Some advantages of using H-B fluid rather than Casson fluid to model the suspension of all the erythrocytes in the core region of the two-fluid flow modeling of blood in narrow arteries are mentioned below.

Iida [31] reports “the velocity profiles of blood when it flows in the arterioles having diameter less than 0.1 *mm* are generally explained fairly by both Casson and H-B fluid models. However, the velocity profiles of blood flow in the arterioles whose diameters are less than 0.065 *mm* do not conform to the Casson fluid, but can still be explained by H-B fluid.” Tu and Deville [22] reported that blood obeys Casson fluid’s constitutive equation only at moderate shear rates, whereas H-B fluid’s constitutive equation can be used still at low shear rates and represents fairly closely what is occurring in blood. Chaturani and Palanisamy [6] propounded that when blood flows in arteries of diameter 0.095 *mm*, it behaves like H-B fluid rather than other non-Newtonian fluids. Moreover, Casson fluid’s constitutive equation has only one parameter namely the yield stress, whereas the H-B fluid’s constitutive equation has one more parameter, namely, the power law index “”, and thus one can obtain more detailed information about the blood flow characteristics by using the H-B fluid model rather than Casson fluid model [32]. Hence, it is appropriate to represent the suspension of all the erythrocytes in the core region of the two-fluid model of blood (when it flows in narrow diameter arteries at low shear rates) by H-B fluid rather than Casson fluid.

Sankar [33] and Sankar and Lee [34] studied the two-fluid H-B model and two-fluid Casson model, respectively, for blood flow in a narrow artery with mild axisymmetric stenosis under body accelerations. The pulsatile flow of two-fluid H-B fluid model and two-fluid Casson fluid model for blood flow through narrow tapered arteries with mild overlapping stenosis under periodic body acceleration has not been studied so far, to the knowledge of the authors. Hence, in this study, a comparative study is performed for the pulsatile flow of two-fluid H-B and Casson models for blood flow in narrow tapered arteries with mild overlapping stenoses in the presence of periodic body acceleration. For the two-fluid H-B model, the expressions obtained in Sankar [33] for shear stress, velocity distribution, wall shear stress, and flow rate are used to compute the data for the present comparative study. The aforesaid flow quantities obtained by Sankar and Lee [34] for two-fluid Casson model are also used to compute the data for this comparative study. The layout of the paper is as follows.

Section 2 mathematically formulates the two-fluid H-B and Casson models for blood flow and applies the perturbation method of solution. In Section 3, the results of two-fluid H-B model and two-fluid Casson model for blood flow in narrow tapered arteries with mild overlapping stenosis are compared. Some possible clinical applications to the present study are also given in Section 3. The main results are summarized in the concluding Section 4.

#### 2. Mathematical Formulation

Consider an axially symmetric, laminar, pulsatile, and fully developed flow of blood (assumed to be incompressible) in the axial direction through a narrow tapered artery with mild overlapping stenosis. Geometry of the segment of a narrow artery with mild overlapping stenosis is shown in Figure 1(a). For different angles of tapering, the geometry of the stenosed artery is depicted in Figure 1(b). The geometry of the stenosed tapered artery at a cross-section in a time cycle is sketched in Figure 1(c). The segment of the artery under study is considered to be long enough so that the entrance, end, and special wall effects can be neglected. Since, the stenosis developed in the lumen of the segment of artery, it is appropriate to treat the segment of the stenosed artery under study as rigid walled. Assume that there is periodical body acceleration in the region of blood flow. Blood is modeled as a two-fluid model, treating the suspension of all the erythrocytes in the core region as non-Newtonian fluid with yield stress and the plasma in the peripheral layer region as Newtonian fluid. The non-Newtonian fluid in the core region is represented by (i) Herschel-Bulkley (H-B) fluid model and (ii) Casson fluid model. Cylindrical polar coordinate system is used to analyze the blood flow.

**(a) Geometry of the segment of an artery with overlapping stenosis**

**(b) Shapes of the overlapping stenosis in the peripheral layer and core regions for different angles of tapering at**

**(c) Changes in the shape of the arterial geometry in a time cycle at and**

##### 2.1. Two-Fluid Herschel-Bulkley (H-B) Model

###### 2.1.1. Governing Equations and Boundary Conditions

The geometry of the artery as shown in Figure 1 is mathematically defined as follows [29, 35]: where where are the radius of the tapered stenosed arterial segment in the peripheral layer region and core region, respectively; is the radius of the artery in the normal region; and are the angle of tapering and slope of the tapered vessel respectively; is the location of the stenosis; is the length of the stenosis; , are the critical heights of the overlapping stenosis in the peripheral layer region and core region, respectively; is the time variant parameter; is a constant; is the angular frequency with as the pulse frequency. Length of the arterial segment is taken to be of finite length . It has been reported that the radial velocity is negligibly small and can be neglected for a low Reynolds number flow in a narrow artery with mild stenosis. The momentum equations governing the blood flow in the axial and radial directions simplify respectively to [33] as follows: where the shear stress . The constitutive equations of the fluids in motion in the core region (for H-B fluid) and in the peripheral region (for Newtonian fluid) are given by where are the axial component of the fluid’s velocity in the core region and peripheral region; are the shear stress of the fluid in the core region (H-B fluid) and peripheral layer region (Newtonian fluid); are the viscosities of the H-B fluid and Newtonian fluid with respective dimensions and are the densities of the H-B fluid and Newtonian fluid; is the pressure; is the time; is the yield stress of the fluid in the core region. From (2.6), it is clear that the velocity gradient vanishes in the region where the shear stress is less than the yield stress which implies a plug flow whenever and normal flow otherwise. The boundary conditions are Since the blood flow in arteries is due to the applied pressure gradient (due to the pumping action of the heart) and is highly pulsatile, it is appropriate to assume the pressure gradient as the following periodic function of and [16, 20]. where is the steady component of the pressure gradient, is the amplitude of the pulsatile component of the pressure gradient, and is the pulse frequency in Hz. Both and are functions of [16]. The periodic body acceleration in the axial direction is given by where is the amplitude, is the frequency in Hz and is assumed to be small so that the wave effect can be neglected [20], and is the lead angle of with respect to the heart action.

###### 2.1.2. Nondimensionalization

Let us introduce the following nondimensional variables: where , which has the dimension as that of the Newtonian fluid’s viscosity, is the pulsatile Reynolds number or generalized Wormersly frequency parameter, and when , we get the Wormersly frequency parameter of the Newtonian fluid. Applying (2.11) into (2.1)-(2.2), one can get the nondimensional form of the equations for the geometry of the tapered stenosed arterial segment as follows: where Using the above nondimensional variables in (2.3) and (2.5)–(2.7), we obtain The boundary conditions (in dimensionless form) are

The volumetric flow rate (in nondimensional) is given by where and is the volume flow rate.

###### 2.1.3. Perturbation Method of Solution

As (2.14)–(2.18) form a system of nonlinear partial differential equations, it is not possible to obtain the exact solution to it. Perturbation method is applied to solve this system of differential equations with the boundary conditions (2.19). Since, the present study deals with the slow flow of blood (low Reynolds number flow) where the effect of pulsatile Reynolds numbers and are negligibly small and also they occur naturally in the nondimensional form of the momentum equation, it is more appropriate to expand the unknowns , and in (2.14) and (2.18) in the perturbation series about and . The plug core velocity and the velocity in the core region are expanded in the perturbation series of powers of (where ) as follows: Similarly, we can expand , and in powers of and and in powers of . Applying the perturbation series expansions of and in (2.14) and then equating the constant terms and terms, we obtain Approximating (2.16) using binomial series and then applying the perturbation series expansions of and in (2.16) and thereafter equating the constant terms and terms, one can get Substituting the perturbation series expansions of and in (2.15) and then equating the constant terms and terms, one can obtain On applying the perturbation series expansions of and in (2.18) and then equating the constant terms and terms, we can easily get Use of the perturbation series expansion of , and in (2.19) and then equating the constant terms and and terms, the boundary conditions decomposes respectively to On solving the system of differential equations (2.22)–(2.25) with the help of boundary conditions (2.26)–(45), one can get the following expressions for the unknowns , and (detail of obtaining these expressions is given in Sankar [33]): where , and . The expression for wall shear stress is obtained as follows (see [33] for detail): The expression for the volume flow rate is obtained as follows (for detail see [33]): The expression for plug core radius is obtained as follows (detail of obtaining this expression is given in [33]): The resistance to flow in the artery is given by when ; the present model reduces to the single-fluid H-B model and in this 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 Sankar and Ismail [14].

##### 2.2. Two-Fluid Casson Fluid Model

###### 2.2.1. Governing Equations and Boundary Conditions

Equations (2.1)-(2.2) which mathematically define the geometry of the tapered artery with overlapping stenosis are assumed in this subsection. The momentum equations governing the flow in the core region and peripheral layer region simplify to [34] where the shear stress ; and are the shear stress of the fluid in the core region (Casson fluid) and peripheral layer region (Newtonian fluid), respectively; and are the axial velocity of the fluid in the core region and peripheral layer region, respectively; and are the densities of the Casson fluid and Newtonian fluid, respectively; is the pressure; is the time. Equations (2.9) and (2.10) which define mathematically the body acceleration term and pressure gradient are assumed in this subsection. The constitutive equations of the fluids in motion in the core region (Casson fluid) and peripheral layer region (Newtonian fluid) are where is the yield stress; is the plug core radius; and are the viscosities of the Casson fluid and Newtonian fluid, respectively. The appropriate boundary conditions of the two-fluid flow are

###### 2.2.2. Nondimensionalization

Let us introduce the following nondimensional variables: where and are the pulsatile Reynolds numbers of the Casson fluid and Newtonian fluid, respectively. Using the nondimensional variables in the momentum equations (2.32) and (2.33) and the constitutive equations (2.35), the simplified form of these equations can be obtained respectively as follows: Using the nondimensional variables, the boundary conditions become Equations (2.12)-(2.13) which mathematically defines the nondimensional form of the geometry of the segment of the tapered artery with overlapping stenosis is assumed in this subsection.

The nondimensional volume flow rate is given by where is the volume flow rate.

###### 2.2.3. Perturbation Method of Solution

As it is not possible to find an exact solution to the system of nonlinear partial differential equations (2.38)–(2.42), perturbation method is applied to obtain the asymptotic solution to the unknowns , and . Since, the present study deals with the slow flow of blood (low Reynolds number flow) where the effect of pulsatile Reynolds numbers and are negligibly small and also they occur naturally in the nondimensional form of the momentum equation, it is appropriate to expand (2.38)–(2.42) in the perturbation series about and . The plug core velocity and the velocity in the core region are expanded in the perturbation series of as follows (where ):

Similarly, one may expand , and the plug core radius in the perturbation series about and , where . Using the perturbation series expansions of and in (2.38) and then equating the constant terms and terms, the momentum equation of the core region decomposes to Applying the perturbation series expansions of and in (2.40) and then equating the constant terms and terms, the constitutive equation of the core region simplifies to Similarly, substituting the perturbation series expansions of and in (2.39) and then equating the constant terms and terms, the momentum equation of the peripheral region decomposes to Applying the perturbation series expansions of and in (2.42) and then equating the constant terms and terms, the constitutive equation of the peripheral region reduces to Using the perturbation series expansions of , and in (2.43) and then equating the constant terms and and terms, one can obtain Solving the system of (2.46)–(2.49) using the boundary conditions (2.50), one can obtain the following expressions for the unknowns , and :