# An Analytical Approach to Study the Blood Flow over a Nonlinear Tapering Stenosed Artery in Flow of Carreau Fluid Model

Authors: Riaz Ahmad; Asma Farooqi; Rashada Farooqi; Nawaf N. Hamadneh; Md Fayz-Al-Asad; Ilyas Khan; Muhammad Sajid; Ghulam Bary; Muhammad Farooq Saleem Khan
Journal: Complexity
 (2021)

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## Abstract

The current study provides an analytical approach to analyze the blood flow through a stenosed artery by using the Carreau fluid model. The flow governing equations are derived under the consideration of mild stenosis. Mathematical analysis has been carried out by considering the blood as non-Newtonian nature. Then, the analytical solution has been investigated by using the regular perturbation technique. The solutions obtained by this perturbation are up to the second-order in dimensionless Weissenberg numberWe. The performed computations of various parameter values such as velocity, wall shear stress, shear stress, and resistance impedance at the stenotic throat are discussed in detail for different values of Weissenberg number We. The obtained results demonstrate that for shear-thinning fluid, the fluid velocity increases with the increasing parameter m while opposite behavior is observed with the increase in We. Hence, the presented numerical analysis reveals many aspects of the flow by considering the blood as a non-Newtonian Carreau fluid model, and the presented model can be equally applicable to other bio-mathematical studies.


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## Body

## 1. Introduction

The study of artery constriction due to the development of stenosis has attained prime importance in fluid dynamics [1–4]. The blood flow in the vessels is a result of the delicate relationship between pressure and area of the fluid. The size of the stenosis determines the flow type. Three types of flow have been studied: mild stenosis as the flow is laminar, moderate stenosis as the flow is a combination of turbulent and laminar, and thirdly the flow depicts turbulent nature when the size of the stenosis is increased. The characteristics of the blood flow depend on the shape and size of the stenosis. Many researchers have analyzed this biomechanical aspect of the flow theoretically and experimentally in recent years. Tang et al. [5] propounded that when the blood pressure is low, stenotic vessels get collapsed and the perfusion of the area beyond stenosis is highly compromised leading to ischemia/infarct. Applications of micropolar fluids in biomedical sciences have received attention for blood flow in arteries [6–8].In the literature, numerous analytical studies related to blood flow through stenosed arteries have been extensively performed [9–13]. In most of these studies, the flow is considered as laminar with mild stenosis [14]. Considerable work has been done for non-Newtonian fluid models. The Carreau fluid model is one of the generalized Newtonian fluid models which is also considered as a viscosity model. This model helps us to explain the behavior of fluid flow in high shear regions and modeling blood at narrow arteries with low shear rates. Firstly, this model was presented by Carreau [15] in his molecular network theories. Then, Siska et al. [16] have proposed a procedure for the terminal velocity of nonspherical particles by using Carreau fluid in transient flow regions. Such regions are responsible for the development of stenosis in the arteries that could lead to stroke, nausea, back pain, etc., specifically with regard to malfunction of the cardiovascular structures. Therefore, the study of blood vessels especially in stenotic arteries has also attained importance in the fluid dynamics field [17].The shear-thinning and shear-thickening processes of the fluid are well explained by the Carreau fluid model [18, 19]. This method is a mixture of power-law and Newtonian fluid models. Chhabra and Uhlherr [20] have analyzed the Carreau viscosity equations for shear-thinning elastic liquids. Theoretical analysis of the Carreau fluid model is studied by Bush and Phan-Thein [21]. Later on, Lee [22] discussed the Carreau generalized Newtonian model for error estimations. Tabakova et al. [23] analyzed the flow dynamics in blood vessels by using the Carreau model. They have studied the oscillatory and steady flows and approximated their numerical solutions. Liu and Liu [24] investigated the quantitative analysis of blood flow in tapered stenosed arteries. Their main concern is the heat and mass transfer effect on the fluid. Irfan et al. [25] gave a numerical analysis of the unsteady Carreau fluid model. The flow of blood in the arteries is highly pulsatile with unequal velocity distribution which is the cause of many cardiovascular diseases [26]. Ismail et al. [9] constructed a mathematical model to study the generalized Newtonian blood flow through a tapered artery. Liu et al. [27] considered the effects of tapering and stenosis over the blood flow. Moreover, a numerous range of blood flow phenomena, such as hemodynamics behavior [28], axisymmetric micropolar model [10], heat and mass transfer effects [29], Eyring–Prandtl fluid model [30], and the micropolar fluid model for composite stenosis behavior [31], have been analyzed comprehensively. Flow of blood-based nanofluids by using the generalized differential quadrature method and other epidemiological models is carried out by many researchers [32–40].This study aims to investigate the blood flow through a stenosed artery by using the Carreau fluid model. The analytical solution of the governing equations with boundary conditions of the stenotic artery is derived. The perturbation solution is obtained by removing the nonlinearity in the governing equation. Then, the obtained results of the velocity profile, resistance impedance, shear stress, and shear wall stress are shown with rheological parameters of the Carreau fluid model, i.e.,We and m. Eventually, the graphical results are presented with different values of parameters of interest.


## 2. Problem Formulation

The fundamental equations used in the derivation of the governing equations for the problem considered are(1)∂u∂r+ur+∂w∂z=0,(2)ρu∂∂r+w∂∂zu=−∂p∂r+1r∂∂rrSrr+∂∂zSrz−Sθθr,(3)ρu∂∂r+w∂∂zw=−∂p∂z+1r∂∂rrSrz+∂∂zSzz.It is assumed that the properties of blood flowing the cylindrical tube are described by the constitute relation Carreau model given as [41](4)S=μ1+Γγ˙2m−1/2γ˙.AssumingΓγ˙≪1, we can write(5)S=μ1+m−12Γγ˙2γ˙,where Γ is a time constant of the fluid. The geometrical representation of the constricted portion is shown in Figure 1 and defined as(6)hz=dz1−ηbn−1z−1−z−an,a≤z≤a+b=dz,otherwise,with(7)dz=d0+ξ∗Z.Figure 1
Diagram of axially nonsymmetric stenosis in the artery.In the above equation,ξ∗=tanϕ is called the tapering parameter; a is the length of the nonstenotic part, and b is the length of the stenotic section; the radius of the nontapered artery in the nonstenotic section is d0; and shape parameter is n which defined the constriction shape, n=2 gives the stenosis symmetric behavior. The parameter η is written as(8)η=δnn/n−1d0bnn−1,where δ is the maximum height of the stenosis defined as(9)z=a+b1/nn−1.The detailed derivation of the presenting problem is given here(10)V=ur,z,0,wr,z.Thus(11)L=∇V=∂u∂r0∂u∂z0ur0∂w∂r0∂w∂z,LT=∂u∂r0∂w∂r0ur0∂u∂z0∂w∂z.Here we defined the rate of strain tensor as given(12)γ˙=L+LT=ur0uz0ur0wr0wz+ur0wr0ur0uz0wz,(13)orγ˙=2ur0uz+wr02ur0wr+uz02wz,now(14)γ˙=γ˙=12∑i∑jγ˙ijγ˙ji.Expanding the double sum, we get(15)γ˙=12γ˙11γ˙11+γ˙12γ˙12+γ˙13γ˙13+γ˙21γ˙21+γ˙22γ˙22+γ˙23γ˙23+γ˙31γ˙31+γ˙32γ˙32+γ˙33γ˙331/2.By choosing the values of the component ofγ˙ from equation (13), we get(16)γ˙=124urur+uz+wruz+wr+4urur+wr+uzwr+uz+4wzwz1/2.Simplification of the above expression yields(17)γ˙=124ur2+2uz+wr2+4u2r2+4wz21/2,(18)orγ˙=2ur2+uz+wr2+2u2r2+2wz21/2,by assuming(19)M=Γγ˙2,we get the following components of the extra stress tensor for the Carreau model:(20)Srr=2μ1+m−12MurSrz=μ1+m−12Muz+wz=SzrSθθ=2μ1+m−12Mur.Applying the mild stenotic conditions, we get(21)M=Γ2wr2,Srr=0,Sθθ=0,Szz=0,Srz=μ1+m−12Γ2wr2wr=Szr.Given the above results, the governing equations (1)–(3) can be rewritten as(22)∂p∂r=0,∂u∂r+ur+∂w∂z=0,∂p∂z=μr∂∂rr1+m−12Γ2wr2wr.Dimensionless variables are defined as(23)r¯=rd0,z¯=zb,w¯=wu0,h¯=hd0,u¯=buu0δp¯=d0m+1pu0mbμRe=ρd0m+1u0m−2μb,S¯rr=bSrru0mμS¯rz=d0mSrzu0mμ,h¯=hd0,S¯zz=bSzzu0mμ,S¯θθ=bSθθu0mμ,and get the following dimensionless governing equation after dropping the bars for simplicity(24)∂p∂r=0,(25)∂u∂r+ur+∂w∂z=0,(26)∂p∂z=1r∂∂rr∂w∂r+m−12We2∂w∂r3,where We=Γu0/d0 is the dimensionless Weissenberg number.


### 2.1. Boundary Conditions

The following boundary conditions are applied along with geometrical interpretation:(27)w=0,atr=hz,(28)∂w∂r=0,atr=0,where hz is already defined in equation (6).


## 2.1. Boundary Conditions

The following boundary conditions are applied along with geometrical interpretation:(27)w=0,atr=hz,(28)∂w∂r=0,atr=0,where hz is already defined in equation (6).


## 3. Employing Perturbation Approach

To get the perturbation solution for the above defined mathematical model, equation (26) can be rearranged as(29)∂p∂z=1r∂∂rr1+We2m−12∂w∂r2∂w∂r,and by integrating equation (29), we get(30)r22∂p∂z+C1=r1+m−12We2∂w∂r2∂w∂r.The symmetry condition at the center liner=0 yields C1=0, and thus we can write equation (30) as(31)r2∂p∂z=1+m−12We2∂w∂r2∂w∂r.Due to nonlinearity arising in equation (31), we opt perturbation technique to solve it. We assume(32)w=w0+We2w1+⋯,(33)p=p0+We2p1+⋯,(34)F=F0+We2F1+⋯.Substituting equations (32) and (33) into equations (27) and (34), we find that(35)r2∂p0∂z+We2∂p1∂z=∂w0∂r+We2∂w1∂r+m−12∂w0∂r3+We4m−12∂w0∂r2∂w1∂r+2m−12∂w1∂r∂w0∂r2+We6m−12∂w1∂r4∂w0∂r+2m−12∂w0∂r∂w1∂r2+We8m−12∂w1∂r5,(36)w0+We2w1=0,atr=hz.Now comparing the coefficients of various powers ofWe in equations (35) and (36), the following systems can be obtained.System of orderWe0:(37)r2∂p0∂z=∂w0∂r,w0=0,atr=hz.System of orderWe2:(38)r2∂p0∂z=∂w1∂r+m−12∂w0∂r3,w1=0,atr=hz.Solving these system results in the following values ofw0 and w1:(39)w0=14∂p0∂zr2−h2,(40)w1=14∂p1∂zr2−h2+m−164∂p0∂z3h4−r4.The definition of flow rate allows us to write(41)F0=2π∫0hrw0dr,inserting the value w0 from equation (39) leads to the following expression:(42)∂p0∂z=−8πh4F0,similarly, we can write(43)F1=2π∫0hrw1dr,and thus,(44)∂p1∂z=−8F1h4π−m−116×83π3h10F03.We summarize the results of the perturbation series through an orderWe2. By substituting equations (39)–(44), we get(45)w=−2πh4F0r2−h2+We24r2−h2−8F1h4π−m−11283π3h10F03−8We2π3h12m−1h4−r4F03,(46)∂p∂z=−8πh4F0−We28F1h4π+m−116×83π3h10F03.If we define(47)F2=F0+We2F1,F0=F2−We2F1.On substituting these expressions into equations (45) and (46) and retaining only terms up to OWe2, we obtain(48)∂p∂z=−8Fπh4−We2m−116×83π3h10F3,(49)w=−2Fπh4r2−h2−32We23π3h10F3r2−h2m−1−8We2π3h12F3m−1h4−r4.Now the pressure drop is(50)Δp=∫0L/b−dpdzdz,and on substituting the value of dp/dz from equation (48), we derive(51)Δp=∫0L/b8Fπh4+We2m−11283π3h10F3dz,and similarly, the resistance impedance is defined by the given expression:(52)λ=ΔpF=∫0a/bRz|h=1+ξzdz+∫a/ba/b+1Rzdz+∫a/b+1L/bRz|h=1+ξzdz,and the nondimensional shear stress for the Carreau law model is given as(53)Srz=∂w∂r+We2m−12∂w∂r3,since(54)w=−2Fπh4r2−h2−32We23π3h10F3r2−h2m−1−8We2π3h12F3m−1h4−r4,we obtain(55)∂w∂r=−4Frπh4−64We2F3m−1r3π3h10+32r3We2π3h12F3m−1,and thus equation (53) gives(56)Srz=−4Frπh4−64We2F3m−1r3π3h10+32r3We2π3h12F3m−1+We2m−12−4Frπh4−64We2F3m−1r3π3h10+32r3We2π3h12F3m−13.Finally, at maximum height, i.e.,h=1−δ, the wall shear stress of the stenosis can be written as(57)τs=Srz|h=1−δ=4π1−δ3−F−16We23π21−δ6m−1F3+8We2m−1π21−δ8F3.


## 4. Graphical Results and Discussion

In this section, the effect of rheological parameters of the Carreau fluid model, i.e.,We and m on velocity profile, resistance impedance, wall shear stress, and shear stress at the stenotic throat, is graphically performed and discussed. The Weissenberg number is the ratio of the relaxation time of the fluid and a specific process time. Relaxation time increases when we increase the Weissenberg number and velocity field easily increases and skin friction decreases. Another aspect of increasing the Weissenberg number is that it reduces the magnitude of the fluid velocity for shear-thinning fluid, while it arises for the shear-thickening fluid. Figure 2 shows that resistance impedance is an increasing function of the Weissenberg number We.Figure 2
Variation of resistance for the Carreau model form=2,n=2,L=1,andF=0.6.In Figure3, it is observed that the resistance impedance increases as there is an increase in power-law index m. It explains the the fluid shear-thinning behavior m<1 slowly compared to Newtonian m=1 and shear-thickening m>1 fluids.Figure 3
Variation of resistance for the Carreau model forWe=0.2,n=2,L=1,andF=0.6.The behavior of shear stress forWe and m is shown in Figures 4 and 5. It is observed that the magnitude of wall shear stress increases by increasing We as shown in Figure 4. The opposite behavior is observed when m is increased as seen in Figure 5.Figure 4
Wall shear stress variation for the Carreau model forn=2,L=1,F=1,andm=0.8.Figure 5
Wall shear stress variation for the Carreau model forn=2,L=1,F=1,and We=0.2.The effect of the severity of stenosis on the velocity profilew is shown in Figures 6 and 7. In Figure 6, the velocity profile increases at the center of the channel by increasing m.Figure 6
Velocity variation for the Carreau model forn=2,L=1,F=3,andm=0.5.Figure 7
Velocity variation for the Carreau model forn=3,L=1,F=3,and We=0.1.And in Figure7, an increase in We decreases the velocity at the channel center. It can be seen that for a fixed value of prescribed flux F, the velocity profile w increases near the center, while it decreases near the wall with an increase in the severity of stenosis.In Figure8, we noticed that the shear stress at the maximum height of stenosis behaves differently by increasing We and m. It is analyzed that shear stress by fixing Weissenberg number We and flow rate F at the stenosis throat increases with an increase in m. While shear stress decreases with an increase in We with defined values of power-law index m and flow rate F.Figure 8
Variation of shear stress at the stenotic throat.


## 5. Conclusion

In this study, we have analyzed the numerical solution of the blood flow with its non-Newtonian nature. Analytical solutions are derived from the given governing equations. The presented computation results of various parameter values, namely, velocity, wall shear stress, shear stress, and resistance impedance at the stenotic throat, are studied in detail for different values of Weissenberg numberWe and power-law index m. From the analysis of the related facts and figures, the following results are concluded:(1)
The velocity profile increases at the center of the channel by increasingm. However, an increase in We decreases the velocity at the channel center.(2)
Resistance impedance is an increasing function ofWe and m.(3)
The magnitude of shear stress increases by increasingWe, while it shows the opposite behavior when m is increased.(4)
Shear stress at the maximum height of stenosis behaves differently by increasingWe and m.Conclusively, the present work may be an improvement in the analysis of pulsatile blood flow through a mild stenotic tapering artery.



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