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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 950323, 27 pages
http://dx.doi.org/10.1155/2012/950323
Research Article

Nonlinear Fluid Models for Biofluid Flow in Constricted Blood Vessels under Body Accelerations: A Comparative Study

1School of Mathematical Sciences, University Science Malaysia, 11800 Penang, Malaysia
2Centre for Applicable Mathematics and Systems Science, Department of Computer Science, Liverpool Hope University, Hope Park, Liverpool L16 9JD, UK

Received 3 January 2012; Accepted 12 February 2012

Academic Editor: M. F. El-Amin

Copyright © 2012 D. S. Sankar and Atulya K. Nagar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Pulsatile flow of blood in constricted narrow arteries under periodic body acceleration is analyzed, modeling blood as non-Newtonian fluid models with yield stress such as (i) Herschel-Bulkley fluid model and (ii) Casson fluid model. The expressions for various flow quantities obtained by Sankar and Ismail (2010) for Herschel-Bulkley fluid model and Nagarani and Sarojamma (2008), in an improved form, for Casson fluid model are used to compute the data for comparing these fluid models. It is found that the plug core radius and wall shear stress are lower for H-B fluid model than those of the Casson fluid model. It is also noted that the plug flow velocity and flow rate are considerably higher for H-B fluid than those of the Casson fluid model. The estimates of the mean velocity and mean flow rate are considerably higher for H-B fluid model than those of the Casson fluid model.

1. Introduction

Atherosclerosis is an arterial disease in large and medium size blood vessels which involve in the complex interactions between the artery wall and blood flow and is caused by intravascular plaques leading to malfunctions of the cardiovascular system [1]. The intimal thickening of an artery is the initial process in the development of atherosclerosis and one of the most wide spread diseases in humans [2]. In atherosclerotic arteries, the lumen is typically narrowed and the wall is stiffened by the buildup of plaque with a lipid core and a fibromuscular cap, and the narrowing of lumen of the artery by the deposit of fats, lipids, cholesterol, and so forth is medically termed as stenosis formation [3]. Different shapes of stenoses are formed in arteries like axisymmetric, asymmetric, overlapping, and multiple and even sometimes it may be arbitrary in shape [47]. Once stenosis develops in an artery, its most serious consequences are the increased resistance and the associated reduction of blood flow to the vascular bed supplied by the artery [8, 9]. Thus, the presence of a stenosis leads to the serious circulatory disorder. Hence, it is very useful to mathematically analyze the blood flow in stenosed arteries.

In many situations of our day to day life, we are exposed to body accelerations or vibrations, like swinging of kids in a cradle, vibration therapy applied to a patient with heart disease, travel of passengers in road vehicles, ships and flights, sudden movement of body in sports activities, and so forth [10, 11]. Sometime, our whole body may be subjected to vibrations, like a passenger sitting in a bus/train, and so forth, while in some other occasions, specific part of our body might be subjected to vibrations, for example, in the operation of jack hammer or lathe machine, driver of a car, and so forth [1214]. Prolonged exposure of our body to high level unintended external body accelerations causes serious health hazards due to the abnormal blood circulation [1517]. Some of the symptoms which result from prolonged exposure of body acceleration are headache, abdominal pain, increase in pulse rate, venous pooling of blood in the extremities, loss of vision, hemorrhage in the face, neck, eye-sockets, lungs, and brain [1820]. Thus, an adequate knowledge in this field is essential to the diagnosis and therapeutic treatment of some health problems, like vision loss, joint pain, and vascular disorder, and so forth, and also in the design of protective pads and machines. Hence, it is important to mathematically analyze and also to quantify the effects of periodic body accelerations in arteries of different diameters.

Due to the rheological importance of the body accelerations and the arterial stenosis, several theoretical studies were performed to understand their effects on the physiologically important flow quantities and also their consequences [1520]. Blood shows anomalous viscous properties. Blood, when it flows through larger diameter arteries at high shear rates, it shows Newtonian character; whereas, when it flows in narrow diameter arteries at low shear rates, it exhibits remarkable non-Newtonian behavior [21, 22]. Many studies pertaining to blood flow analysis treated it as Newtonian fluid [4, 15, 23]. Several researchers used non-Newtonian fluids models for mathematical analysis of blood flow through narrow arteries with different shapes of stenosis under periodic body accelerations [2427]. Casson and Herschel-Bulkley (H-B) fluid models are some of the non-Newtonian fluid models with yield stress and are widely used in the theoretical analysis of blood flow in narrow arteries [28, 29]. The advantages of using H-B fluid model rather than Casson fluid model for modeling of blood flow in narrow arteries are mentioned below.

Chaturani and Samy [8] emphasized the use of H-B fluid model for blood flow modeling with the argument that when blood flows in arteries of diameter 0.095 mm, it behaves like H-B fluid rather than other non-Newtonian fluids. Tu and Deville [21] pronounced that blood obeys Casson fluid’s constitutive equation only at moderate shear rates, whereas H-B fluid model can be used still at low shear rates and represents fairly closely what is occurring in blood. Iida [30] reports “the velocity profiles of blood when it flows in the arterioles having diameter less than 0.1 mm are generally explained fairly by 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 model, but, can still be explained by H-B fluid model.” 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 “n” and thus one can obtain more detailed information about blood flow characteristics by using the H-B fluid model rather than Casson fluid model [31]. Hence, it is appropriate to treat blood as H-B fluid model rather than Casson fluid model when it flows through narrow arteries.

Sankar and Ismail [32] investigated the effects of periodic body accelerations in blood flow through narrow arteries with axisymmetric stenosis, treating blood as H-B fluid model. Nagarani and Sarojamma [33] mathematically analyzed the pulsatile flow of Casson fluid for blood flow through stenosed narrow arteries under body acceleration. The pulsatile flow of H-B fluid model and Casson fluid model for blood flow through narrow arteries with asymmetric stenosis under periodic body acceleration has not been studied so far, to the knowledge of the authors. Hence, in the present study, a comparative study is performed for the pulsatile flow H-B and Casson fluid models for blood flow in narrow arteries with asymmetric shapes of stenoses under periodic body acceleration. The expressions obtained in Sankar and Ismail [32] for shear stress, velocity distribution, wall shear stress, and flow rate are used to compute data for the present comparative study. The aforesaid flow quantities obtained by Nagarani and Sarojamma [33] for Casson fluid model in the corrected form are used in this study to compute data for performing the present comparative study. The layout of the paper is as follows.

Section 2 mathematically formulates the H-B and Casson fluid models for blood flow and applies the perturbation method of solution. In Section 3, the results of H-B fluid model and Casson fluid model for blood flow in axisymmetric and asymmetrically stenosed narrow arteries 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 (𝑧) direction through a circular narrow artery with constriction. The constriction in the artery is assumed as due to the formation of stenosis in the lumen of the artery and is considered as mild. In this study, we consider the shape of the stenosis as asymmetric. The geometry of segment of a narrow artery with asymmetric shape of mild stenosis is shown in Figure 1(a). For different values of the stenosis shape parameter m, the asymmetric shapes of the stenoses are sketched in Figure 1(b). In Figure 1(b), one can notice the axisymmetric shape of stenosis when the stenosis shape parameter m = 2. 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. Due to the presence of the stenosis in the lumen of the segment of the 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 and blood is modeled as non-Newtonian fluid model with yield stress. In this study, we use two different non-Newtonian fluid models with yield stress for blood flow simulations such as (i) Herschel-Bulkley (H-B) fluid and (ii) Casson fluid. Note that for particular values of the parameters, H-B fluid model’s constitutive equation reduces to the constitutive equations of Newtonian fluid, power law fluid, and Bingham fluid. Also it is to be noted that Casson fluid model’s constitutive equation reduces to the constitutive equation of Newtonian fluid when the yield stress parameter becomes zero. The cylindrical polar coordinate system (𝑟,𝜓,𝑧) has been used to analyze the blood flow.

fig1
Figure 1: Pictorial description of segment of the artery with asymmetric stenosis.
2.1. Herschel-Bulkley Fluid Model
2.1.1. Governing Equations and Boundary Conditions

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 [32]𝜌𝐻𝜕𝑢𝐻𝜕𝑡𝜕=𝑝𝜕𝑧1𝑟𝜕𝜕𝑟𝑟𝜏𝐻+𝐹𝑡,𝜕(2.1)0=𝑝𝜕𝑟,(2.2) where 𝜌𝐻, 𝑢𝐻 are the density and axial component of the velocity of the H-B fluid, respectively; 𝑝 is the pressure; 𝑡 is the time; 𝜏𝐻=|𝜏𝑟𝑧|=𝜏𝑟𝑧 is the shear stress of the H-B fluid; 𝐹(𝑡) is the term which represents the effect of body acceleration and is given by𝐹𝑡=𝑎0cos𝜔𝑏𝑡+𝜙,(2.3) where 𝑎0 is the amplitude of the body acceleration, 𝜔𝑏=2𝜋𝑓𝑏, 𝑓𝑏 is the frequency in Hz and is assumed to be small so that the wave effect can be neglected [14],𝜙 is the lead angle of 𝐹(𝑡) with respect to the heart action. Since, the blood flow is assumed as pulsatile, it is appropriate to assume the pressure gradient as a periodic function as given below [25]:𝜕𝑝𝜕𝑧𝑧,𝑡=𝐴0+𝐴1cos𝜔𝑝𝑡,(2.4) where 𝐴0 is the steady component of the pressure gradient, 𝐴1 is the amplitude of the pulsatile component of the pressure gradient, and 𝜔𝑝=2𝜋𝑓𝑝, 𝑓𝑝 is the pulse frequency in Hz [23]. The constitutive equation of the H-B fluid (which represents blood) is given by𝜏𝐻=𝜇𝐻1/𝑛𝜕𝑢𝐻𝜕𝑟1/𝑛+𝜏𝑦if𝜏𝐻𝜏𝑦,𝜕𝑢𝐻𝜕𝑟=0if𝜏𝐻𝜏𝑦,(2.5) where, 𝜏𝑦 is the yield stress of the H-B fluid and 𝜇𝐻 is the coefficient of viscosity of H-B fluid with dimension (𝑀𝐿1𝑇2)𝑛𝑇. The geometry of the asymmetric shape of stenosis in the arterial segment is mathematically represented by the following equation [34]:𝑅𝑧𝑅0=1𝐺𝐿0𝑚1𝑧𝑑𝑧𝑑𝑚if𝑑𝑧𝑑+𝐿0,1otherwise,(2.6) where 𝐺=(𝛿/𝑅0𝐿0)𝑚(𝑚/𝑚1); 𝛿 denotes the maximum height of the stenosis at 𝑧=𝑑+(𝐿0/𝑚(𝑚/𝑚1)) such that 𝛿/𝑅01; 𝐿0 is the length of the stenosis; 𝑑 denotes its location; 𝑅(𝑧) is the radius of the artery in the stenosed region; 𝑅0 is the radius of the normal artery. It is to be noted that (2.6) also represents the geometry of segment of the artery with axisymmetric stenosis when the stenosis shape parameter m = 2. We make use of the following boundary conditions to solve the system of momentum and constitutive equations for the unknown velocity and shear stress:𝜏𝐻isniteat𝑟=0,𝑢𝐻=0at𝑟=𝑅𝑧.(2.7)

2.1.2. Nondimensionalization

Let us introduce the following nondimensional variables:𝑧=𝑧𝑅0,𝑅(𝑧)=𝑅𝑧𝑅0,𝑟=𝑟𝑅0,𝑡=𝑡𝜔,𝜔=𝜔𝑏𝜔𝑝,𝛿=𝛿𝑅0,𝑢𝐻=𝑢𝐻𝐴0𝑅02/4𝜇0,𝜏𝐻=𝜏𝐻𝐴0𝑅02/2,𝜃=𝜏𝑦𝐴0𝑅0,𝛼2𝐻=𝑅02𝜔𝜌𝐻𝜇0,𝑒=𝐴1𝐴0,𝐵=𝑎0𝐴0,(2.8) where 𝜇0=𝜇𝐻(2/𝑅0𝐴0)𝑛1 having dimension as that of Newtonian fluid’s viscosity [22, 34]; 𝛼𝐻 is the generalized Wormersly frequency parameter or pulsatile Reynolds number, and when 𝑛=1, it reduces to the Newtonian fluid’s pulsatile Reynolds number. Using nondimensional variables defined in (2.8), the momentum and constitutive equations (2.1) and (2.5) can be simplified to the following equations:𝛼2𝐻𝜕𝑢𝐻2𝜕𝑡=4(1+𝑒cos𝑡)+4𝐵cos(𝜔𝑡+𝜙)𝑟𝜕𝜕𝑟𝑟𝜏𝐻,𝜏(2.9)𝐻=12𝜕𝑢𝐻𝜕𝑟1/𝑛+𝜃if𝜏𝐻𝜃,(2.10)𝜕𝑢𝐻𝜕𝑟=0if𝜏𝐻𝜃.(2.11) The geometry of the asymmetric shape of the stenosis in the arterial segment in the nondimensional form reduces to the following equation:𝐿𝑅(𝑧)=1𝐺0𝑚1(𝑧𝑑)(𝑧𝑑)𝑚if𝑑𝑧𝑑+𝐿0,1otherwise,(2.12) where 𝐺=(𝛿/𝑅0𝐿0)𝑚(𝑚/𝑚1). The boundary conditions in the nondimensional form are𝜏𝐻𝑢isniteat𝑟=0,𝐻=0at𝑟=𝑅.(2.13) The volume flow rate in the nondimensional is given by𝑄(𝑧,𝑡)=40𝑅(𝑧)𝑢𝐻(𝑧,𝑟,𝑡)𝑟𝑑𝑟,(2.14) where 𝑄(𝑧,𝑡)=𝑄(𝑧,𝑡)/[𝜋𝑅40𝐴0/8𝜇0], 𝑄 is the volumetric flow rate.

2.1.3. Perturbation Method of Solution

Since, (2.9) and (2.10) form the system of nonlinear partial differential equations, it is not possible to get an exact solution to them. Thus, perturbation method is used to solve this system of nonlinear partial differential equations. Since, the present study deals with slow flow of blood (low Reynolds number flow) where the effect of pulsatile Reynolds number 𝛼𝐻 is negligibly small and also it occurs naturally in the nondimensional form of the momentum equation, it is more appropriate to expand the unknowns 𝑢𝐻 and 𝜏𝐻 in (2.9) and (2.10) in the perturbation series about 𝛼2𝐻. Let us expand the velocity 𝑢𝐻 in the perturbation series about the square of the pulsatile Reynolds number 𝛼2𝐻 as below (where 𝛼2𝐻1):𝑢𝐻(𝑟,𝑧,𝑡)=𝑢𝐻0(𝑟,𝑧,𝑡)+𝛼2𝐻𝑢𝐻1(𝑟,𝑧,𝑡)+.(2.15) Similarly, one can expand the shear stress 𝜏𝐻(𝑟,𝑧,𝑡), the plug core radius 𝑅𝑝(𝑧,𝑡), the plug core velocity 𝑢𝑝(𝑧,𝑡), and the plug core shear stress 𝜏𝑝(𝑧,𝑡) in terms of 𝛼2𝐻. Substituting the perturbation series expansions of 𝑢𝐻 and 𝜏𝐻 in (2.9) and then equating the constant term and 𝛼2𝐻 term, we get𝜕𝜕𝑟𝑟𝜏𝐻0[],=2𝑟(1+𝑒cos𝑡)+𝐵cos(𝜔𝑡+𝜙)𝜕𝑢𝐻02𝜕𝑡=𝑟𝜕𝜕𝑟𝑟𝜏𝐻1.(2.16) Using the binomial series approximation in (2.10) (assuming (𝜃/𝜏)21) and then applying the perturbation series expansions of 𝑢𝐻 and 𝜏𝐻 in the resulting equation and then equating the constant term and 𝛼2𝐻 term, one can obtain𝜕𝑢𝐻0𝜕𝑟=2𝜏𝑛1𝐻0𝜏𝐻0,𝑛𝜃𝜕𝑢𝐻1𝜕𝑟=2𝑛𝜏𝑛2𝐻0𝜏𝐻1𝜏𝐻0.(𝑛1)𝜃(2.17) Applying the perturbation series expansions of 𝑢𝐻 and 𝜏𝐻 in the boundary conditions (2.13), we obtain𝜏𝐻0,𝜏𝐻1𝑢areniteat𝑟=0,𝐻0=0,𝑢𝐻1=0at𝑟=0.(2.18) Solving (2.16)–(2.17) with the help of the boundary conditions (2.18) for the unknowns 𝜏𝑃0,𝜏𝑃1,𝜏𝐻0,𝜏𝐻1,𝑢𝑃0,𝑢𝑃1,𝑢𝐻0, and 𝑢𝐻1, one can get the following expressions (detail of obtaining these expressions is given in [32]):𝜏𝑃0=𝑔(𝑡)𝑅0𝑝,𝜏𝐻0𝑢=𝑔(𝑡)𝑟,𝐻0[]=2𝑔(𝑡)𝑅𝑛𝑅1𝑟(𝑛+1)1𝑅𝑛+1𝑞2𝑅𝑟1𝑅𝑛,𝑢0𝑝[]=2𝑔(𝑡)𝑅𝑛𝑅1𝑞(𝑛+1)12𝑅𝑛+1𝑞2𝑅𝑞12𝑅𝑛,𝜏𝑃1[]=𝑔(𝑡)𝑅𝑛𝐷𝑅2𝑛2𝑞(𝑛+1)2𝑅(𝑛1)2𝑞2𝑅2𝑛2𝑞(𝑛+1)2𝑅𝑛+2,𝜏𝐻1[]=𝑔(𝑧)𝑅𝑛𝐷𝑅2𝑛(𝑛+1)(𝑛+3)𝑛+32𝑟𝑅𝑟𝑅𝑛+2(𝑛1)𝑞(𝑛+2)2𝑅𝑛+22𝑟𝑅𝑟𝑅𝑛+13𝑛2+2𝑛2𝑞2(𝑛+2)(𝑛+3)2𝑅𝑛+3𝑅𝑟,𝑢𝐻1[]=2𝑛𝑔(𝑡)𝑅2𝑛1𝐷𝑅3𝑛2(𝑛+1)2𝑟(𝑛+3)(𝑛+2)(𝑛+3)𝑅𝑛+1+𝑟𝑅2𝑛+2+(𝑛1)𝑞2(𝑛+1)(𝑛+2)(𝑛+3)(2𝑛+1)2𝑅×𝑟(𝑛+2)(𝑛+3)(2𝑛+1)𝑅𝑛+𝑟𝑅𝑛+122𝑛3+9𝑛2++11𝑛+32𝑛2𝑟+6𝑛+3𝑅2𝑛+1+(𝑛1)2𝑞2𝑛(𝑛+2)2𝑅2𝑟(𝑛+1)(𝑛+2)𝑅𝑛+𝑟𝑅2𝑛+3𝑛2+2𝑛2𝑞2(𝑛1)(𝑛+2)(𝑛+3)2𝑅𝑛+3𝑟𝑅𝑛1+3𝑛12+2𝑛2(𝑛1)𝑞2(𝑛2)(𝑛+2)(𝑛+3)2𝑅𝑛+4𝑟1𝑅𝑛2,𝑢𝑃1[]=2𝑛𝑔(𝑡)𝑅2𝑛1𝐷𝑅3𝑛2(𝑛+1)2𝑞(𝑛+3)(𝑛+2)(𝑛+3)2𝑅𝑛+1+𝑞2𝑅2𝑛+2+(𝑛1)𝑞2(𝑛+1)(𝑛+2)(𝑛+3)(2𝑛+1)2𝑅×𝑞(𝑛+2)(𝑛+3)(2𝑛+1)2𝑅𝑛+𝑞2𝑅𝑛+122𝑛3+9𝑛2++11𝑛+32𝑛2𝑞+6𝑛+32𝑅2𝑛+1+(𝑛1)2𝑞2𝑛(𝑛+2)2𝑅2𝑞(𝑛+1)(𝑛+2)2𝑅𝑛+𝑞2𝑅2𝑛+3𝑛2+2𝑛2𝑞2(𝑛1)(𝑛+2)(𝑛+3)2𝑅𝑛+3𝑞2𝑅𝑛1+3𝑛12+2𝑛2(𝑛1)𝑞2(𝑛2)(𝑛+2)(𝑛+3)2𝑅𝑛+4𝑞12𝑅𝑛2,(2.19) where 𝑞2=(𝜃/𝑔(𝑡)), 𝑟|𝜏0𝑝=𝜃=𝑅0𝑝=𝜃/𝑔(𝑡)=𝑞2, 𝑔(𝑡)=(1+𝑒cos𝑡)+𝐵cos(𝜔𝑡+𝜙), and 𝐷=(1/𝑔)(𝑑𝑔/𝑑𝑡). The wall shear stress 𝜏𝑤 is a physiologically important flow quantity which plays an important role in determining the aggregate sites of platelets [3]. The expression for wall shear stress 𝜏𝑤 is given by [32]𝜏𝑤=𝜏𝐻0+𝛼2𝐻𝜏𝐻1𝑟=𝑅=[]×𝑔(𝑡)𝑅1(𝑔(𝑡)𝑅)𝑛1𝛼2𝑅2𝐵×𝑞2(𝑛+2)(𝑛+3)𝑛(𝑛+2)(𝑛1)𝑛(𝑛+3)2𝑅𝑛32𝑞+2𝑛22𝑅𝑛+3.(2.20) The expression for volumetric flow rate 𝑄(𝑧,𝑡) is obtained as below (see [32] for details):𝑄(𝑧,𝑡)=4𝑅0𝑝0𝑟𝑢0𝑝𝑑𝑟+𝑅𝑅0𝑝𝑟𝑢0𝑑𝑟+𝛼2𝑅0𝑝0𝑟𝑢1𝑝𝑑𝑟+𝑅𝑅0𝑝𝑟𝑢1=4[]𝑑𝑟𝑔(𝑡)𝑅𝑛𝑅3𝑞(𝑛+2)(𝑛+3)(𝑛+2)𝑛(𝑛+3)2𝑅+𝑛2𝑞+2𝑛22𝑅𝑛+3𝛼2[]𝑔(𝑡)𝑅𝑛1𝑛𝐷𝑅24×𝑛2𝑛(𝑛1)4𝑛2+12𝑛+5𝑞(2𝑛+1)(2𝑛+3)2𝑅+𝑛(𝑛1)2(𝑛+3)(𝑞𝑛+1)2𝑅2+𝑛32𝑛211𝑛+6(𝑞𝑛+1)2𝑅𝑛+3𝑛(𝑛1)32𝑛211𝑛+6𝑛𝑞2𝑅𝑛+44𝑛5+14𝑛48𝑛345𝑛23𝑛+18𝑞𝑛(𝑛+1)(2𝑛+3)2𝑅2𝑛+4.(2.21) The expression for the plug core radius is obtained as below [32]: 𝑅𝑝=𝑞2+𝛼2[]𝑔(𝑡)𝑅𝑛1𝑛𝐷𝑅3𝑞2(𝑛+1)2𝑅𝑛21𝑛𝑞2𝑅2𝑞2𝑅𝑛+2.(2.22) The longitudinal impedance to flow in the artery is defined asΛ=𝑃(𝑡)𝑄(𝑧,𝑡),(2.23) where[]𝑃(𝑡)=41+𝑒cos(𝑡)(2.24) is the pressure gradient in the nondimensional form.

2.2. Casson Fluid Model
2.2.1. Governing Equations and Boundary Conditions

The momentum equations governing the blood flow in the axial and radial directions simplify, respectively, to [33]𝜌𝐶𝜕𝑢𝐶𝜕𝑡𝜕=𝑝𝜕𝑧1𝑟𝜕𝜕𝑟𝑟𝜏𝐶+𝐹𝑡,𝜕(2.25)0=𝑝𝜕𝑟,(2.26) where 𝑢𝐶 and 𝜌𝐶 are the axial component of the velocity and density of Casson fluid; 𝑝 is the pressure; 𝑡 is the time; 𝜏𝐶=|𝜏𝑟𝑧|=𝜏𝑟𝑧 is the shear stress of Casson fluid. Equations (2.3) and (2.4) which define mathematically the body acceleration term 𝐹(𝑡) and pressure gradient (𝜕𝑝/𝜕𝑧) are assumed in this subsection. Similarly, (2.6) which mathematically describes the geometry of the axisymmetric shape of stenosis and asymmetric shape of stenosis in the segment of the stenosed artery is also assumed in this subsection (the details of these assumptions can be found in Section 2.1.1) The constitutive equation of the Casson fluid model (which models blood) is defined as below:𝜏𝐶=𝜇𝐶𝜕𝑢𝐶𝜕𝑟+𝜏𝑦if𝜏𝐶𝜏𝑦,𝜕(2.27)𝑢𝐶𝜕𝑟=0if𝜏𝐶𝜏𝑦,(2.28) where 𝜏𝑦 is the yield stress of Casson fluid and 𝜇𝐶 is the coefficient of viscosity of Casson fluid with dimension 𝑀𝐿1𝑇1. The appropriate boundary conditions to solve the system of momentum and constitutive equations (2.25), (2.27), and (2.28) for the unknown velocity and shear stress are𝜏𝐶isniteat𝑟=0,𝑢𝐶=0at𝑟=𝑅𝑧.(2.29)

2.2.2. Nondimensionalization

Similar to (2.8), let us introduce the following nondimensional variables for the Casson fluid flow modeling as follows:𝑧=𝑧𝑅0,𝑅(𝑧)=𝑅𝑧𝑅0,𝑟=𝑟𝑅0,𝑡=𝑡𝜔,𝜔=𝜔𝑏𝜔𝑝,𝛿=𝛿𝑅0,𝑢𝐶=𝑢𝐶𝐴0𝑅20/4𝜇𝐶,𝜏𝐶=𝜏𝐶𝐴0𝑅02/2,𝜃=𝜏𝑦𝐴0𝑅0,𝛼2𝐶=𝑅20𝜔𝜌𝐶𝜇𝐶,𝑒=𝐴1𝐴0,𝐵=𝑎0𝐴0,(2.30) where 𝛼𝐶 is the Wormersly frequency parameter or pulsatile Reynolds number of Casson fluid model. Use of the above nondimensional variables reduces the momentum and constitutive equations (2.25), (2.27), and (2.28), respectively, to the following equations:𝛼2𝐶𝜕𝑢𝐶2𝜕𝑡=4(1+𝑒cos𝑡)+4𝐵cos(𝜔𝑡+𝜙)𝑟𝜕𝜕𝑟𝑟𝜏𝐶,(2.31)𝜏𝐶=12𝜕𝑢𝐶+𝜕𝑟𝜃if𝜏𝐶𝜃,(2.32)𝜕𝑢𝐶𝜕𝑟=0if𝜏𝐶𝜃.(2.33) Equation (2.12) which mathematically defines the nondimensional form of the geometry of the asymmetric shapes of stenosis in the arterial segment is assumed in this sub-section. The boundary conditions in the nondimensional form are𝜏𝐶𝑢isniteat𝑟=0,𝐶=0at𝑟=𝑅.(2.34) The volume flow rate in the nondimensional is given by𝑄=40𝑅(𝑧)𝑢𝐶(𝑧,𝑟,𝑡)𝑟𝑑𝑟,(2.35) where 𝑄=𝑄/[𝜋𝑅40𝐴0/8𝜇𝐶], 𝑄 is the volumetric flow rate.

2.2.3. Perturbation Method of Solution

As described in Section 2.1.3, perturbation method is applied to solve the system of nonlinear partial differential equations (2.31) and (2.32). Let us expand the velocity 𝑢𝐶 in the perturbation series about the square of the pulsatile Reynolds number 𝛼2𝐶 as below (where 𝛼2𝐶1):𝑢𝐶(𝑟,𝑧,𝑡)=𝑢𝐶0(𝑟,𝑧,𝑡)+𝛼2𝐶𝑢𝐶1(𝑟,𝑧,𝑡)+.(2.36) Similarly, one can expand the shear stress 𝜏𝐶(𝑟,𝑧,𝑡), the plug core radius 𝑅𝑝(𝑧,𝑡), the plug core velocity 𝑢𝑝(𝑧,𝑡), and the plug core shear stress 𝜏𝑝(𝑧,𝑡) in terms of 𝛼2𝐶. Substituting the perturbation series expansions of 𝑢𝐶and 𝜏𝐶 in (2.31) and then equating the constant term and 𝛼2𝐶 term, one can obtain𝜕𝜕𝑟𝑟𝜏𝐶0[],=2𝑟(1+𝑒cos𝑡)+𝐵cos(𝜔𝑡+𝜙)𝜕𝑢𝐶02𝜕𝑡=𝑟𝜕𝜕𝑟𝑟𝜏𝐶1.(2.37) Applying the perturbation series expansions of 𝑢𝐶 and 𝜏𝐶 in (2.32) and then equating the constant term and 𝛼2𝐶 term, we get𝜕𝑢𝐶0𝜏𝜕𝑟=2𝐶02𝜃𝜏𝐶0,+𝜃𝜕𝑢𝐶1𝜕𝑟=2𝜏𝐶11𝜃𝜏𝐶0.(2.38) Applying the perturbation series expansions of 𝑢𝐶 and 𝜏𝐶 in the boundary conditions (2.34) and then equating the constant terms and 𝛼2𝐶 terms, one can get𝜏𝐶0,𝜏𝐶1𝑢areniteat𝑟=0,𝐶0=0,𝑢𝐶1=0at𝑟=0.(2.39) Solving (2.37)–(2.38) with the help of the boundary conditions (2.39) for the unknowns 𝜏𝑃0,𝜏𝑃1,𝜏𝐶0,𝜏𝐶1,𝑢𝑃0,𝑢𝑃1,𝑢𝐶0, and 𝑢𝐶1, one can get the following expressions as in [33], but in a corrected form ((2.40)–(2.50)):𝜏𝑃0=𝑔(𝑡)𝑅0𝑝𝜏,(2.40)𝐶0𝑢=𝑔(𝑡)𝑟,(2.41)𝐶0=𝑔(𝑡)𝑅2𝑟1𝑅283𝑞𝑅𝑟1𝑅3/2+2𝑞2𝑅𝑟1𝑅𝑢,(2.42)𝑃0=𝑔(𝑡)𝑅2813𝑞𝑅𝑞+22𝑅13𝑞2𝑅2𝜏,(2.43)𝑃1=𝑔(𝑡)𝐷𝑅5𝑞122𝑅34𝑞2𝑅+𝑞2𝑅2𝜏,(2.44)𝐶1=𝑔(𝑡)𝐷𝑅38×2𝑟𝑅𝑟𝑅3𝑞2𝑅4𝑅𝑟821𝑞2𝑅7𝑟𝑅𝑟4𝑅5/2𝑞32𝑅7/2𝑅𝑟,𝑢(2.45)𝐶1=𝑔(𝑡)𝐷𝑅41𝑟121𝑅213𝑞2𝑅𝑟1𝑅3/21𝑟161𝑅4+53294𝑞2𝑅𝑟1𝑅7/2+49𝑞2𝑅𝑟1𝑅3/28𝑘632𝑅𝑟1𝑅31𝑞282𝑅4𝑟log𝑅+1𝑞142𝑅9/21𝑅𝑟,𝑢(2.46)𝑃1=𝑔(𝑡)𝐷𝑅47+481598𝑞2𝑅20𝑞632𝑅+5𝑞122𝑅249𝑞2𝑅5/2439𝑞70562𝑅4+1𝑞142𝑅9/21𝑞282𝑅4𝑞log2𝑅,(2.47) where 𝑞2=(𝜃/𝑔(𝑡)), 𝑟|𝜏0𝑝=𝜃=𝑅0𝑝=𝜃/𝑔(𝑡)=𝑞2, 𝑔(𝑡)=(1+𝑒cos𝑡)+𝐵cos(𝜔𝑡+𝜙), and 𝐷=(1/𝑔)(𝑑𝑔/𝑑𝑡). Using (2.41) and (2.45), the expression for wall shear stress 𝜏𝑤 is obtained as below:𝜏𝑤=𝜏𝐶0+𝛼2𝐶𝜏𝐶1𝑟=𝑅𝛼=𝑔(𝑡)𝑅12𝐶𝑅2𝐷8817𝑞𝑅+17𝑞2𝑅4.(2.48) The expression for volumetric flow rate 𝑄(𝑧,𝑡) is obtained as below:𝑄(𝑧,𝑡)=4𝑅0𝑃0𝑟𝑢0𝑝𝑑𝑟+𝑅𝑅0𝑃𝑟𝑢0𝑑𝑟+𝛼2𝑅0𝑃0𝑟𝑢1𝑝𝑑𝑟+𝑅𝑅0𝑃𝑟𝑢1𝑑𝑟=𝑔(𝑡)𝑅41167𝑞𝑅+43𝑞2𝑅1𝑞212𝑅4𝛼2𝐶𝑅6×1𝑔(𝑡)𝐷630𝑞77𝑅+8𝑞352𝑅13𝑞2𝑅5/2+1𝑞142𝑅9/241𝑞7702𝑅61𝑞142𝑅6𝑞log2𝑅+1𝑞142𝑅4𝑞12𝑅2.log(𝑞)(2.49) The expression for the plug core radius is obtained as below [33]:𝑅𝑝=𝑞2𝐷𝛼2𝐶𝑅34𝑞2𝑅43𝑞2𝑅3/2+13𝑞2𝑅3.(2.50) The longitudinal impedance to flow in the artery is defined asΛ=𝑃(𝑡)𝑄(𝑧,𝑡).(2.51)

3. Numerical Simulation of the Results

The main objective of the present mathematical analysis is to compare the H-B and Casson fluid models for blood flow in constricted arteries and spell out the advantageous of using H-B fluid model rather than Casson fluid for the mathematical modeling of blood flow in a narrow artery with asymmetric stenosis. It is also aimed to bring out the effect of body acceleration, stenosis shape parameter, yield stress, and pressure gradient on the physiologically important flow quantities such as plug core radius, plug flow velocity, velocity distribution, flow rate, wall shear stress, and longitudinal impedance to flow. The different parameters used in this analysis and their range of values are given below [3235].

Yield stress θ: 0–0.3; power law index n: 0.95–1.05; pressure gradient e: 0-1; body acceleration B: 0–2; frequency parameter ω: 0-1; pulsatile Reynolds numbers 𝛼𝐻 and 𝛼𝐶: 0.2–0.7; lead angle ϕ: 0.2–0.5; asymmetry parameter m: 2–7; stenosis depth δ: 0–0.2.

3.1. Plug Core Radius

The variation of the plug core with axial distance in axisymmetric stenosed artery (m = 2) for different values of the yield stress of H-B and Casson fluid models with δ = 0.15, B = 2, 𝛼𝐻 = 𝛼𝐶 = 0.2, e = ϕ = 0.7 and t = 45° is shown in Figure 2. It is observed that the plug core radius decreases slowly when the axial variable 𝑧 increases from 0 to 4 and then it increases when 𝑧 increases further from 4 to 8. The plug core radius is minimum at the centre of the stenosis (z = 4), since the stenosis is axisymmetric. The plug core radius of the H-B fluid model is slightly lower than that of the Casson fluid model. One can note that the plug core radius increases very significantly when the yield stress of the flowing blood increases. Figure 3 sketches the variation of plug core radius with pressure gradient ratio in asymmetrically stenosed artery (m = 4) for H-B and Casson fluid models and for different values of the body acceleration parameter with θ = δ = 0.1, t = 60°, ϕ = 0.7, m = 4, and z = 4. It is noticed that the plug core radius decreases rapidly with the increase of the pressure gradient ratio 𝑒 from 0 to 0.5 and then it decreases slowly with the increase of the pressure gradient ratio 𝑒 from 0.5 to 1. It is seen that plug core radius increases significantly with the increase of the body acceleration parameter 𝐵. Figures 2 and 3 bring out the influence of the non-Newtonian behavior of blood and the effects of body acceleration and pressure gradient on the plug core radius when blood flows in asymmetrically stenosed artery.

950323.fig.002
Figure 2: Variation of plug core radius with axial distance for H-B and Casson fluid models with different values of yield stress θ and with δ = 0.15, 𝛼𝐻 = 𝛼𝐶 = 0.2, B = 2, e = ϕ = 0.7, and t = 45°.
950323.fig.003
Figure 3: Variation of plug core radius with pressure gradient for H-B and Casson fluids and for different values of body acceleration parameter 𝐵 with θ = δ = 0.1, t = 60°, ϕ = 0.7, m = 4, and z = 4.
3.2. Plug Flow Velocity

Figure 4 shows the variation of the plug flow velocity with yield stress for H-B and Casson fluid models and for different values of the stenosis shape parameter with e = 0.5, ϕ = 0.2, z = 4, t = 60°, ω = 0.5, B = 1, and δ = 0.1. It is noted that for H-B fluid model, the plug flow velocity decreases very slowly with the increase of the yield stress, whereas, in the case of Casson fluid model, it decreases rapidly when the yield stress θ increases from 0 to 0.05 and then it decreases slowly with the increase of the yield stress from 0.05 to 0.3. It is seen that the plug flow velocity is considerably higher for H-B fluid model than that of the Casson fluid model. One can easily observe that the plug flow velocity decreases significantly with the increase of the stenosis shape parameter 𝑚. The variation of plug flow velocity with axial distance for H-B and Casson fluid models and for different values of the body acceleration 𝐵 and pressure gradient ratio 𝑒 with 𝛿 = θ = 0.1, m = 4, t = 60°, ϕ = 0.2, and ω = 0.5 is depicted in Figure 5. It is seen that the plug flow velocity skews more to the right-hand side in the axial direction which is attributed by the skewness of the stenosis. It is clear that the plug flow velocity increases considerably with the increase of the body acceleration parameter 𝐵 and pressure gradient ratio 𝑒. Figures 4 and 5 show the non-Newtonian character of blood and effects of body acceleration, pressure gradient, and asymmetry of the stenosis on the plug flow velocity of blood when it flows through a constricted artery.

950323.fig.004
Figure 4: Variation of plug flow velocity with yield stress for H-B and Casson fluid models and for different values of stenosis shape parameter 𝑚 with e = 0.5, ϕ = 0.2, t = 60°, z = 4, ω = 0.5, B = 1, and δ = 0.1.
950323.fig.005
Figure 5: Variation of plug flow velocity with axial distance for H-B and Casson fluids and for different values of 𝐵 and 𝑒 with δ = θ = 0.1, m = 4, t = 60°, ϕ = 0.2, and ω = 0.5.
3.3. Velocity Distribution

Figure 6 sketches the velocity distribution for H-B and Casson fluid models and for different values of yield stress θ, stenosis depth δ with m = 2, e = 0.2, 𝛼𝐻 = 𝛼𝐶 = 0.5, ϕ = 0.2, ω = 1, t = 60°, and B = 1. It is observed that the velocity of H-B fluid model is considerably higher than that of Casson fluid model. It is also found that the velocity of the blood flow decreases with the increase of the yield stress θ and stenosis depth δ. But the decrease in the velocity is considerable when the stenosis depth δ increases, whereas it decreases significantly with the increase of the yield stress. It is of interest to note that the velocity distribution of H-B fluid with δ = 0.2 and θ = 0.05 and B = 0 is in good agreement with the corresponding plot in Figure 6 of Sankar and Lee [34]. It is also to be noted that the velocity distribution of Casson fluid with δ = 0.2, θ = 0.01, and B = 0 is in good agreement with the corresponding plot in Figure 6 of Siddiqui et al. [35].

950323.fig.006
Figure 6: Velocity distribution for different fluid models with e = 0.2, 𝛼𝐻 = 𝛼𝐶 = 0.5, ϕ = 0.2, ω = 1, t = 60°, and B = 1.
3.4. Flow Rate

The variation of flow rate with pressure gradient ratio for H-B and Casson fluid models and for different values of the power law index n, body acceleration parameter 𝐵, and stenosis shape parameter 𝑚 with θ = δ = 0.1, 𝛼𝐻 = 𝛼𝐶 = ϕ = 0.2, z = 4, t = 60°, and ω = 1 is shown in Figure 7. It is seen that the flow rate increases with the pressure gradient ratio 𝑒. But the increase in the flow rate is linear for H-B fluid model and almost constant for Casson fluid model. For a given set of values of the parameters, the flow rate for H-B fluid model is considerably higher than that of the Casson fluid model. It is also clear that for a given set of values of 𝑛 and m, the flow rate increases considerably with the increase of the body acceleration parameter 𝐵. One can observe that for fixed values of 𝑛 and B, the flow rate decreases significantly with the increase of the stenosis shape parameter 𝑚. When the power law index 𝑛 increases from 0.95 to 1.05 and all the other parameters were held constant, the flow rate decreases slightly when the range of the pressure gradient ratio 𝑒 is 0–0.5 and this behavior is reversed when the range of the pressure gradient ratio 𝑒 is 0.5 to 1. Figure 7 brings out the effects of body acceleration and stenosis shape on the flow rate of blood when it flows through narrow artery with mild stenosis.

950323.fig.007
Figure 7: Variation of flow rate with pressure gradient for H-B and Casson fluid models and for different values of 𝐵 and 𝑚 with θ = δ = 0.1, 𝛼𝐻 = 𝛼𝐶 = ϕ = 0.2, z = 4, t = 60°, and ω = 1.
3.5. Wall Shear Stress

Figure 8 shows the variation of wall shear stress with frequency ratio for H-B and Casson fluid models and for different values of the ϕ (lead angle), 𝛼𝐻 (pulsatile Reynolds number for H-B fluid model), and 𝛼𝐶 (pulsatile Reynolds number of Casson fluid model) with m = 2, θ = δ = 0.1, e = 0.5, B = 1, z = 4, and t = 60°. It is seen that the wall shear stress decreases slightly nonlinearly with frequency ratio for lower values of the pulsatile Reynolds numbers 𝛼𝐻 and 𝛼𝐶 and lead angle ϕ, and it decreases linearly with frequency ratio for higher values of the pulsatile Reynolds numbers 𝛼𝐻 and 𝛼𝐶 and lead angle ϕ. It is found that for a given set of values of the parameters, the wall shear stress is marginally lower for H-B fluid model than that of the Casson fluid model. Also, one can note that for fixed value of the lead angle ϕ, the wall shear stress decreases significantly with the increase of the pulsatile Reynolds numbers 𝛼𝐻 and 𝛼𝐶. It is also observed that the wall shear stress decreases marginally with the increase of the lead angle ϕ when all the other parameters were kept as invariables. Figure 8 spells out the effects of pulsatility and non-Newtonian character of blood on the wall shear stress when it flows in a narrow artery with mild stenosis.

950323.fig.008
Figure 8: Variation of wall shear stress with frequency ratio for H-B and Casson fluids and for different values of 𝛼𝐻, 𝛼𝐶 and ϕ with θ = δ = 0.1, m = 2, e = 0.5, B = 1, z = 4, and t = 60°.
3.6. Longitudinal Impedance to Flow

The variation of the longitudinal impedance to flow with axial distance for different values of the stenosis shape parameter 𝑚 and body acceleration parameter 𝐵 with θ = δ = 0.1, t = 60°, 𝛼𝐻 = 𝛼𝐶 = ϕ = 0.2, e = 0.5, and ω = 1 is depicted in Figures 9(a) (for H-B fluid model) and 9(b) (Casson fluid model). It is noticed that the longitudinal impedance to flow increases with the increase of the axial variable 𝑧 from 0 to the point where the stenosis depth is maximum and then it decreases as the axial variable 𝑧 increases further from that point to 8. One can see the significant increase in the longitudinal impedance to flow when the stenosis shape parameter 𝑚 increases and marginal increase in the longitudinal impedance to flow when the body acceleration parameter B increases. It is also clear that for the same set of values of the parameters, the longitudinal impedance to flow is significantly lower for H-B fluid model than that of the Casson fluid model. Figures 9(a) and 9(b) bring out the effects of body acceleration and asymmetry of the stenosis shape on the longitudinal impedance to blood flow.

fig9
Figure 9: Variation of longitudinal impedance to flow with axial distance for H-B and Casson fluid models and for different values of 𝑚 and 𝐵 with θ = δ = 0.1, t = 60°, 𝛼𝐻 = 𝛼𝐶 = ϕ = 0.2, e = 0.5, and ω = 1.

The increase in the longitudinal impedance to blood flow due to the asymmetry shape of the stenosis is defined as the ratio between the longitudinal impedance to flow of a fluid model for a given set of values of the parameters in an artery with asymmetric stenosis and the longitudinal impedance of the same fluid model and for the same set of values of the parameters in that artery with axisymmetric stenosis. The estimates of the increase in the longitudinal impedance to flow are computed in Table 1 for different values of the stenosis shape parameter 𝑚 and body acceleration parameter 𝐵 with δ = θ = 0.1, e = 0.5, ω = 1, z = 4, 𝛼𝐻 =𝛼𝐶 = ϕ = 0.2, and t = 60°. It is observed that the estimates of the increase in the longitudinal impedance to flow increase considerably when the stenosis shape parameter 𝑚 increases and they decrease slightly when the body acceleration parameter 𝐵 increases. Hence, the longitudinal impedance to flow is significantly higher in the arteries with asymmetric shape of the stenosis compared to that in the arteries with axisymmetric stenosis. It is also noted that the presence of the body acceleration decreases the longitudinal impedance to blood flow considerably.

tab1
Table 1: Estimates of the increase in the longitudinal impedance to flow due to the increase in the stenosis shape parameter and body acceleration with δ = θ = 0.1, e = 0.5, ω = 1, 𝛼𝐻 = 𝛼𝐶 = ϕ = 0.2, z = 4, and t = 60°.
3.7. Some Possible Clinical Applications

To discuss some possible clinical applications of the present study, the data (for different types of arteries, their corresponding radii, steady and pulsatile pressure gradient values) reported by Chaturani and Wassf Issac [23] are given in Table 2 and are used in this applications part of our study. For these clinical data (given in Table 2), the estimates of the mean velocity of H-B and Casson fluid models for different values of the stenosis shape parameter 𝑚 and different values of the body acceleration parameter 𝐵 with θ = δ = 0.1, t = 60°, ω = 1, z = 4, ϕ = 0.2, 𝛼𝐻 = 𝛼𝐶=0.2, and e = 0.2 are computed in Table 3. It is recorded that the estimates of the mean velocity increase significantly with the increase of the artery radius, except in arterioles. It is also found that the estimates of the mean velocity of H-B fluid model are marginally higher than those of the Casson fluid model. It is noted that the mean velocity increases considerably with the increase of the body acceleration parameter 𝐵 and the reverse behavior is found when the stenosis shape parameter 𝑚 increases.

tab2
Table 2: Physiological data for different arteries.
tab3
Table 3: Estimates of mean velocity of H-B and Casson fluid models for different values of 𝑚 and 𝐵 in arteries with different radii with θ = δ = 0.1, t = 60°, ω = 1, z = 4, e = 0.2, and 𝛼𝐻 = 𝛼𝐶 = ϕ = 0.2.

For the clinical data given in Table 2, the estimates of the mean flow rate of H-B and Casson fluid models are computed in Table 4 for different values of the stenosis shape parameter 𝑚 and different values of the body acceleration parameter 𝐵 with θ = δ = 0.1, ω = 1, t = 60°, z = 4, ϕ = 0.2, 𝛼𝐻 = 𝛼𝐶=0.2, and 𝑒=0.2. It is observed that the estimates of the mean flow rate decrease very significantly with the increase of the artery radius. It is also found that the estimates of the mean flow rate of H-B fluid model are considerably higher than those of the Casson fluid model. It is noted that the estimates of the mean flow rate increase significantly with the increase of the body acceleration parameter 𝐵 and the reverse behavior is recorded when the stenosis shape parameter 𝑚 increases.

tab4
Table 4: Estimates of mean flow rate of H-B and Casson fluid models for different values of 𝑚 and 𝐵 in arteries with different radii with θ = δ = 0.1, t = 60°, ω = 1, z = 4, e = 0.2, and 𝛼𝐻 = 𝛼𝐶 = ϕ = 0.2.

4. Conclusions

The present mathematical analysis brings out various interesting rheological properties of blood when it flows through narrow stenosed arteries with body acceleration, treating it as different non-Newtonian fluid models with yield stress such as (i) Herschel-Bulkley fluid model and (ii) Casson fluid model. By the use of appropriate mathematical expression for the geometry of segment of the stenosed artery, both axisymmetric and asymmetric shapes of stenoses are considered to study the effects of stenosis shape and size on the physiologically important quantities. Some major findings of this mathematical analysis are summarized below.(i)The plug core radius, wall shear stress, and longitudinal impedance to flow are marginally lower for H-B fluid model than those of the Casson fluid model.(ii)The plug flow velocity, velocity distribution, and flow rate are considerably higher for H-B fluid model than those of the Casson fluid model.(iii)The plug core radius and longitudinal impedance to flow increase significantly with the increase of the stenosis shape parameter, and the reverse behavior is observed for plug flow velocity, velocity distribution, and flow rate.(iv)The estimates of the mean velocity and mean flow rate are considerably higher for H-B fluid model than those of the Casson fluid model.(v)The estimates of the mean velocity and mean flow rate increase considerably with the increase of the body acceleration, and this behavior is reversed when the stenosis shape parameter increases.

Based on these results, one can note that there is substantial difference between the flow quantities of H-B fluid model and Casson fluid model, and thus it is expected that the use of H-B fluid model for blood flow in diseased artery may provide better results which may be useful to physicians in predicting the effects of body accelerations and different shapes and sizes of stenosis in the artery on the physiologically important flow quantities. Also, it is hoped that this study may provide some useful information to surgeons to take some crucial decisions regarding the treatment of patients, whether the cardiovascular disease can be treated with medicines or should the patient undergo a surgery. Hence, it is concluded that the present study can be treated as an improvement in the mathematical modeling of blood flow in narrow arteries with mild stenosis under the influence of periodic body accelerations.

Nomenclature

𝑟:Radial distance
𝑟:Dimensionless radial distance
𝑧:Axial distance
𝑧:Dimensionless axial distance
𝑛:Power law index
𝑝:Pressure
𝑝:Dimensionless pressure
𝑃:Dimensionless pressure gradient
𝑄:Flow rate
𝑄:Dimensionless flow rate
𝑅0:Radius of the normal artery
𝑅(𝑧):Radius of the artery in the stenosed region
𝑅(𝑧):Dimensionless radius of the artery in the stenosed region
𝐹(𝑡):Body acceleration function
𝑎0:Amplitude of the body acceleration
𝑅𝑃:Plug core radius
𝑅𝑃:Dimensionless plug core radius
𝑢𝐻:Axial velocity of Herschel-Bulkley fluid
𝑢𝐻:Dimensionless axial velocity of Herschel-Bulkley fluid
𝑢𝐶:Axial velocity of Casson fluid
𝑢𝐶:Dimensionless axial velocity of Casson fluid
𝐴0:Steady component of the pressure gradient
𝐴1:Amplitude of the pulsatile component of the pressure gradient
𝐿:Length of the normal artery
𝐿0:Length of the stenosis
𝑚:Stenosis shape parameter
𝐿0:Dimensionless length of the stenosis
𝑑:Location of the stenosis
𝑑:Dimensionless location of the stenosis
𝑡:Time
𝑡:Dimensionless time.
Greek Letters
Λ:Dimensionless longitudinal impedance to flow
𝜙:Azimuthal angle
̇𝛾:Shear rate
𝜏𝑦:Yield stress
𝜃:Dimensionless yield stress
𝜏𝐻:Shear stress of the Herschel-Bulkley fluid
𝜏𝐻:Dimensionless shear stress of Herschel-Bulkley fluid
𝜏𝐶:Shear stress for Casson fluid
𝜏𝐶:Dimensionless shear stress of Casson fluid
𝜏𝑤:Dimensionless wall shear stress
𝜌𝐻:Density of Herschel-Bulkley fluid
𝜌𝐶:Density of Casson fluid
𝜇𝐻:Viscosity of Herschel-Bulkley fluid
𝜇𝐶:Viscosity of the Casson fluid
𝛼𝐻:Pulsatile Reynolds number of Herschel-Bulkley fluid
𝛼𝐶:Pulsatile Reynolds number of Casson fluid
𝛿:Depth of the stenosis
𝛿:Dimensionless depth of the stenosis
𝜔:Angular frequency of the blood flow
𝜙:Lead angle.
Subscripts
𝑤:Wall shear stress (used for τ)
𝐻:Herschel-Bulkley fluid (used for 𝑢,𝑢,𝜏,𝜏)
𝐶:Newtonian fluid (used for 𝑢,𝑢,𝜏,𝜏).

Acknowledgments

This research work was supported by the Research University Grant of Universiti Sains Malaysia, Malaysia (RU Grant ref. no. 1001/PMATHS/811177). The authors thank the reviewers for their valuable comments which helped to improve the technical quality of this research article.

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