Modeling and analysis of hybrid-blood nanofluid flow in stenotic artery

Current communication deals with the flow impact of blood inside cosine shape stenotic artery. The under consideration blood flow is treated as Newtonian fluid and flow is assumed to be two dimensional. The governing equation are modelled and solved by adopting similarity transformation under the stenosis assumptions. The important quantities like Prandtl number, flow parameter, blood flow rate and skin friction are attained to analyze the blood flow phenomena in stenosis. The variations of different parameters have been shown graphically. It is of interest to note that velocity increases due to change in flow parameter gamma and temperature of blood decreases by increasing nanoparticles volume fraction and Prandtl number. In the area of medicine, the most interesting nanotechnology approach is the nanoparticles applications in chemotherapy. This study provides further motivation to include more convincing consequences in the present model to represent the blood rheology.

www.nature.com/scientificreports/approach enables a comprehensive examination of the intricate physiological processes involved, utilizing mathematical equations to quantify the effects of stenosis on blood flow.For future research, there is an opportunity to enhance existing mathematical models by incorporating more realistic parameters.This could involve considering patient specific characteristics, accounting for dynamic changes in stenosis severity over time, or incorporating variations in blood viscosity.The practical insights gained from improved mathematical models and simulations offer a promising avenue for addressing the complexities of blood flow in stenotic conditions, making the research appealing to readers interested in both theoretical advancements and practical applications in the field.We analyzed the flow of blood in the narrow artery with addition of nanoparticles by considering non-Newtonian nature of blood.The obtained PDE's are transformed into ODE's with the use of similarity transformations.Numerical solution has been calculated for temperature and velocity of blood by using MAT-LAB bvp4c.Obtained results are shown graphically and also in tabular form.This innovative approach not only contributes to deeper scientific understanding of blood flow issues associated with stenosis but also paves the way for developing more precise medical interventions and personalized treatment strategies.

Flow geometry and coordinate system
The following relevant assumptions are made: i.We considered the flow of blood through stenotic artery of cosine shape constriction.
ii. Blood acts like steady, two-dimensional, incompressible Newtonian fluid.iii.The length of stenosis is L 0 2 , width of unblocked region 2R 0 , radius of the artery is R(x) and the maximum values of height is represented by λ. iv.Blood flow along x − axis and r − axis is perpendicular to the flow.v.The region of stenosis is chosen as

Problem formulation and method of solution
The governing steady boundary layer equations of motion, momentum and energy for Newtonian hybrid nanofluid are defined respectively: Boundary conditions can be specified as: Physical properties of nanofluids are defined as follows 13 : where ρ hnf , µ hnf k f and k hnf are the density, viscosity and thermal conductivity of hybrid nanofluid of Cu-Al 2 O 3 nanoparticles and blood, the heat capacity of fluid is (ρC p ) hnf and values of all these properties are defined in Table 1.The value of ψ for u and v is presented in Eq. ( 8), the continuity Eq. ( 2) is satisfied. (1) (5)
where x = x L 0 and after the successful implementation of useful similarity transformation the equations (9 − 10) finally becomes: where where Dimensionless form of Eq. ( 1) is where f = R(x) R 0 and ǫ = R 0 is the non-dimensional measure of stenosis in reference artery.Boundary conditions in dimensionless form are The dimensionless quantities in Eqs. ( 12) and ( 13) are flow parameter γ = ν f L 0 /u 0 R 2 , Prandtl number Pr = k f /(µC p ) f and Cu-Al 2 O 3 nanoparticles concentration are shown by φ 1 and φ 2 .

Physical quantities
The physical quantities of flow field i.e., coefficient of Skin friction C f and heat transfer Nu x are described as: Expression for shear stress τ w and heat flux q w can be find as Non-dimensional form of Eqs. ( 18), ( 19) becomes where Re shows the Reynolds number.

Numerical solution
Numerical solution of Eqs. ( 12) and ( 14) is obtained by using MATLAB bvp4c technique.MATLAB bcp4c solve the boundary values problems for ordinary differential equations.The results for temperature and velocity profiles are obtained and presented graphically.

Graphical results and explanation
Blood flow problem through stenotic artery with addition of hybrid nanoparticles is studied.Results of various parameters on stenotic artery are investigated.Figure 1 shows the geometry of stenotic artery.Figure 2 describes the consequences of temperature for Pr .Graphical results shows that by rising the Pr = 2.0, 3.0, 4.0, 5.0 temperature decreases.Basically Pr is a ratio of diffusivity of momentum to thermal.This implies that there is a inverse relation to heat transfer from the wall of artery, for smaller value of Pr the heat diffusion is greater than momentum.Figure 3 depicts the impact of nanoparticles on temperature field and the curve decreases by increasing φ = 0.01, 0.05, 0.1, 0.2 .Figure 4 depicts the consequences of γ on blood temperature and the curve shows increasing behavior by increasing the value of γ = 0.1, 1.8, 2.5, 3.4 .Impact of nanoparticles on velocity distribution is shown in Fig. 5.By increasing nanoparticles volume fraction values, velocity curve of blood bending down.This is acceptable with the physical presentation that when the nanoparticles volume fraction rises then due to nanoparticles stacked up in blood, the flow velocity decreases.Figure 6 draws the results of γ on velocity profile F′(η ).Velocity of blood increases by increasing γ = 1.0, 1.4, 2.2, 3.7.Figure 7 shows the results of γ on velocity profile F(η ).Velocity of blood increases by increasing γ = 1.0, 1.4, 2.2, 3.7.Figure 8 presents skin friction variations due to change in nanoparticles volume fraction and flow parameter.Skin friction profile goes down by rising γ values.Figure 9 presents consequences of heat transfer coefficient and the curve shows decreasing behavior.Table 1 describes the values for (blood) base fluid and solid hybrid nanoparticles.Table 2 describes the impact of Pr and γ on Nusselt number.Results exhibits that for Pr values coefficient of heat transfer rises by increase in γ while goes down.Table 3 shown the results of γ and φ on skin friction.We can conclude that by rising the flow parameter γ , the values of Skin friction coefficient also rises and by increasing φ , the coefficient of Skin friction decreases.Remarkable properties are show by hybrid nanoparticles which cannot attained in individual state by any component.Mainly the applications of nano drug delivery in biofluid dynamics, for arterial diseases treatments, a detail computational study is presented for hybrid nanoparticles and heat transfer through stenotic artery.Results of present study bay be fruitful during operation procedures in tuning the blood flow.

Conclusion
Flow of blood through stenotic artery with addition of hybrid nanoparticles is studied.The mathematical study of blood flow in stenosis involves modeling complex fluid dynamics, vessels geometry and rheological properties to understand the impact of narrowed passages on flow patterns.A numerical method has been used to attain

Figure 9 .
Figure 9. Consequences of Pr and γ on heat transfer coefficient.

Table 2 .
Nusselt number variations with respect to Pr and γ.

Table 3 .
Numerical values for skin friction coefficient with respect to φ and γ .