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

Faculty of Science, Yibin University, Yibin 644000, Sichuan, China Wah Medical College POF Hospital, Wah Cantt 47040, Pakistan Department of Basic Sciences, College of Science and ,eoretical Studies, Saudi Electronic University, Riyadh 11673, Saudi Arabia Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka 1000, Bangladesh Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah, Majmaah 11952, Saudi Arabia Faculty of Materials and Chemical Engineering, Yibin University, Yibin 644000, Sichuan, China Faculty of International Applied Technology, Yibin University, Yibin 644000, Sichuan, China School of Environmental Science and Technology, Tsinghua University, Haidian, Beijing, China


Introduction
e study of artery constriction due to the development of stenosis has attained prime importance in fluid dynamics [1][2][3][4]. e blood flow in the vessels is a result of the delicate relationship between pressure and area of the fluid. e size of the stenosis determines the flow type. ree 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. e 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][7][8].
In the literature, numerous analytical studies related to blood flow through stenosed arteries have been extensively performed [9][10][11][12][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. e Carreau fluid model is one of the generalized Newtonian fluid models which is also considered as a viscosity model.
is 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. en, 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. erefore, the study of blood vessels especially in stenotic arteries has also attained importance in the fluid dynamics field [17]. e shear-thinning and shear-thickening processes of the fluid are well explained by the Carreau fluid model [18,19].
is 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. eoretical analysis of the Carreau fluid model is studied by Bush and Phan-ein [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. ey 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. eir 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. e 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 bloodbased nanofluids by using the generalized differential quadrature method and other epidemiological models is carried out by many researchers [32][33][34][35][36][37][38][39][40].
is study aims to investigate the blood flow through a stenosed artery by using the Carreau fluid model. e analytical solution of the governing equations with boundary conditions of the stenotic artery is derived. e perturbation solution is obtained by removing the nonlinearity in the governing equation. en, 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.

Problem Formulation
e fundamental equations used in the derivation of the governing equations for the problem considered are It is assumed that the properties of blood flowing the cylindrical tube are described by the constitute relation Carreau model given as [41] Assuming Γ _ c ≪ 1, we can write where Γ is a time constant of the fluid. e geometrical representation of the constricted portion is shown in Figure 1 and defined as In the above equation, ξ * � tan ϕ 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 d 0 ; and shape parameter is n which defined the constriction shape, n � 2 gives the stenosis symmetric behavior. e parameter η is written as where δ is the maximum height of the stenosis defined as e detailed derivation of the presenting problem is given here us 2 Complexity Here we defined the rate of strain tensor as given Expanding the double sum, we get By choosing the values of the component of _ c from equation (13), we get Simplification of the above expression yields by assuming we get the following components of the extra stress tensor for the Carreau model:

Complexity 3
Applying the mild stenotic conditions, we get Given the above results, the governing equations (1)-(3) can be rewritten as Dimensionless variables are defined as and get the following dimensionless governing equation after dropping the bars for simplicity where We � Γu 0 /d 0 is the dimensionless Weissenberg number.

Boundary
Conditions. e following boundary conditions are applied along with geometrical interpretation: where h(z) is already defined in equation (6).

Employing Perturbation Approach
To get the perturbation solution for the above defined mathematical model, equation (26) can be rearranged as and by integrating equation (29), we get e symmetry condition at the center line r � 0 yields C 1 � 0, and thus we can write equation (30) as Due to nonlinearity arising in equation (31), we opt perturbation technique to solve it. We assume Substituting equations (32) and (33) Now comparing the coefficients of various powers of We in equations (35) and (36), the following systems can be obtained.
System of order We 0 : System of order We 2 : Solving these system results in the following values of w 0 and w 1 : e definition of flow rate allows us to write inserting the value w 0 from equation (39) leads to the following expression: similarly, we can write and thus, We summarize the results of the perturbation series through an order We 2 . By substituting equations (39)-(44), we get Complexity 5 If we define On substituting these expressions into equations (45) and (46) and retaining only terms up to O(We 2 ), we obtain Now the pressure drop is and on substituting the value of (dp/dz) from equation (48), we derive and similarly, the resistance impedance is defined by the given expression: and the nondimensional shear stress for the Carreau law model is given as since Complexity we obtain and thus equation (53) gives

Complexity 7
Finally, at maximum height, i.e., h � 1 − δ, the wall shear stress of the stenosis can be written as

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. e 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. In Figure 3, 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. e behavior of shear stress for We 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. e opposite behavior is observed when m is increased as seen in Figure 5. Complexity e effect of the severity of stenosis on the velocity profile w is shown in Figures 6 and 7. In Figure 6, the velocity profile increases at the center of the channel by increasing m.
And in Figure 7, 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 Figure 8, 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.

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. e 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 number (We) and power-law index m. From the analysis of the related facts and figures, the following results are concluded: (1) e velocity profile increases at the center of the channel by increasing m. However, an increase in We decreases the velocity at the channel center. Conclusively, the present work may be an improvement in the analysis of pulsatile blood flow through a mild stenotic tapering artery.

Nomenclature
We: Weissenberg number Γ: Time constant of the fluid ξ * : Tapering parameter a: Length of the nonstenotic part b: Length of the stenotic section d 0 : Radius of the nontapered artery in the nonstenotic section n: Shape parameter δ: Maximum height of the stenosis τ s : Wall shear stress Δp: Pressure drop λ: Resistance impedance m: Power-law index w: Velocity profile F: Flow rate S rz : Shear stress S: Extra stress tensor θ: Tapering angle.

Data Availability
No data were required to perform this research.

Conflicts of Interest
e authors declare that there are no conflicts of interest.