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This paper demonstrates the implications of heat transfer and chemical reactions by considering the flow of a CarreauYasuda fluid through the diseased tapered inclined artery. The analysis has performed on the artery with mild stenosis. The analytical solution has been obtained using the perturbation method due to its low Weissenberg number to get the blood velocity, the wall shear stress distribution, the temperature, the concentration, the resistance impedance and the stream function. The critical results in this study have depicted that the blood velocity for dilatant fluid is higher than that in viscous and pseudoplastic fluids in the core region of the artery. The value of blood velocity through a Newtonian fluid is extraordinarily lower than that through a CarreauYasuda fluid at the core region of the artery. The magnitude of blood velocity across the vertical artery is higher than those in the inclined and horizontal arteries inside the core region of the artery. The wall shear stress distributions are higher in the case of stenosis (diseased artery) than in no stenosis (healthy artery). The temperature transmission for the Carreau fluid is higher than that for the CarreauYasuda fluid. The flow resistance progressively increases as the depth of the stenosis increases. The size of the trapping bolus gradually increases, shifting toward the right at the diverging tapering artery, and it gradually increases, moving toward the left at the converging tapering artery. Finally, the streamlines are noticeably more comprehensive along the vertical artery and are higher than those in the inclined and horizontal arteries.
blood flow, CarreauYasuda fluid, diseased artery, heat transfer, chemical reactions
Arterial stenosis is a lumen narrowing that harasses blood flow to blood vessels. Vascular stenosis may be caused by aneurysms and tumours or linked to arteriosclerosis. Arterial stenosis has motivated biomechanics scientists to execute blood flow awareness and empirical measurement of pressure and velocity in human organs [1].
The CarreauYasuda fluid is a known rheological model utilized to appreciate the steadystate viscosity in nonNewtonian fluid (e.g. pendants, emulsions, and polymer solvents). NonNewtonian CarreauYasuda fluid is the generalized form of Carreau fluid and has numerous industrial implementations like food treatment, drilling, bioengineering processes etc. [2]. CarreauYasuda fluid model predicts the shearthinning/shearthickening behaviours, which display diverse impacts on the thermaltransfer features of fluid [3]. Although various circumstances may appear in these fluids, they have displayed shearthinning behaviour; this refers to the viscosity dwindling due to a plus in the malformation rate implemented to the fluid [4]. The CarreauYasuda model has received widespread emphasis due to its nature being most suitable for depicting shearthinning fluid [5]. A few research attempts to study CarreauYasuda fluid flow through narrowed arteries under certain physical conditions, but it lacks the analytical mathematical procedure [610].
The contemplation of the impacts of heat and chemical reactions on blood flow is impressive to many researchers. Heat and mass transfer survey is the needful theme for researchers due to their biomedical engineering utilizations like tissue delivery and food treatment. So, the effect of rheological features of the fluid with a synchronized scattering of heat and mass transfer is located in enormous indication [11]. The quantitative anticipation of blood flow rate and heat buildup is pivotal for personifying blood circulation disease and gauging blood glucose [12].
Motivated by the studies mentioned above, the unsteady flow of CarreauYasuda fluid across the oblique stenosed catheterized artery in the presence of heat transfer and chemical reactions are considered in this paper. The problem has been modelled, and then the nondimensional governing equations for mild stenosis with the corresponding boundary conditions have been depicted and solved analytically utilizing the perturbation technique. Graphical results and discussions are explained, and some conclusions are included.
An inclined arterial segment with overlapping stenosis can be modelled as a superfine solid cylindrical of finite length L. The nonNewtonian incompressible CarreauYasuda fluid (as a blood model) is injected into the arterial segment. The cylindrical polar coordinate (r, θ, z) is chosen where the zaxis is picked along the arterial segment axis, and all flow parameters are independent of the direction θ. Moreover,, asuming that r=0 is selected as the axis of the symmetry of the arterial segment. The heat and mass transfer phenomena are subjected by afforcing heat T_{0} and concentration C_{0} to the artery's wall, while at the centre of the artery, we are considering symmetry conditions on both temperature and concentration. The geometrical shape of the constricted inclined tapered artery is chosen (see Figure 1) [13].
$R(z)=\left\{\begin{array}{lc}h(z)\left(1\frac{\delta \cos \phi(zd)}{R_o L_o}\left\{11\frac{94(zd)}{3 L_o}+\frac{32(zd)^2}{L_o^2}\frac{32(zd)^3}{3 L_o^3}\right\}\right) & d \leq z \leq d+\frac{3 L_o}{2} \\ h(z) & \text { otherwise }\end{array}\right.$ (1)
with
$h(z)=m z+R_o$ (2)
where, h(z) is the radius of the tapering arterial segment in the narrowed region, R_{0} is the radius of the nontapering artery in the nonnarrowed region, m=tanφ symbolize the slope of the tapering vessel, φ is the angle of tapering where the possibility of diverse forms of the artery viz, the converging tapering (φ<0), nontapered artery (φ=0) and the diverging tapering (φ>0) [14]. 3L_{0}/2 is the extent of overlapping stenosis, d is the site of the narrowing. Here, δ is the maximum altitude of the stenosis, such that the ratio δ/R_{0}<<1, appears at two specific locations, i.e., z=d+L_{0}/2 and z=d+L_{0}. The height of the stenosis at a distance of z=d+3L_{0}/4 from the origin is 3δ/4 and δcosφ is taken to be the maximum altitude of overlapping stenosis.
Figure 1. Graphical scheme of inclined tapered overlapping narrowed artery
The following governing equations may list the mathematical description of the problem:
Continuity:
$\frac{\partial V_r}{\partial r}+\frac{V_r}{r}+\frac{\partial V_z}{\partial z}=0$ (3)
Momentum:
$\rho\left(\frac{\partial V_r}{\partial t}+V_r \frac{\partial V_r}{\partial r}+V_z \frac{\partial V_r}{\partial z}\right)=\frac{\partial p}{\partial r}+\frac{1}{r} \frac{\partial}{\partial r}\left(r S_n\right)+\frac{\partial S_{r z}}{\partial z}\frac{S_{\theta \theta}}{r}\rho g \cos \alpha$ (4)
$\rho\left(\frac{\partial V_z}{\partial t}+V_r \frac{\partial V_z}{\partial r}+V_z \frac{\partial V_z}{\partial z}\right)=\frac{\partial p}{\partial z}+\frac{1}{r} \frac{\partial}{\partial r}\left(r S_{r z}\right)+\frac{\partial S_{z z}}{\partial z}+\rho g \sin \alpha$ (5)
Energy equation:
$\rho c_p\left(\frac{\partial T}{\partial t}+V_r \frac{\partial T}{\partial r}+V_z \frac{\partial T}{\partial z}\right)=S_r \frac{\partial V_r}{\partial r}+S_{r z} \frac{\partial V_z}{\partial r}+S_{z r} \frac{\partial V_r}{\partial z}+S_{z z} \frac{\partial V_z}{\partial z}+K\left(\frac{\partial^2 T}{\partial r^2}+\frac{1}{r} \frac{\partial T}{\partial r}+\frac{\partial^2 T}{\partial z^2}\right)$ (6)
Concentration equation:
$\frac{\partial C}{\partial t}+V_r \frac{\partial C}{\partial r}+V_z \frac{\partial C}{\partial z}=D\left(\frac{\partial^2 C}{\partial r^2}+\frac{1}{r} \frac{\partial C}{\partial r}+\frac{\partial^2 C}{\partial z^2}\right)+\frac{D K_T}{T_m}\left(\frac{\partial^2 T}{\partial r^2}+\frac{1}{r} \frac{\partial T}{\partial r}+\frac{\partial^2 T}{\partial z^2}\right)$ (7)
where, p represents the pressure of the fluid, ρ is the density of the fluid, V_{r} and V_{z} are the components of the velocity in radial and axial directions respectively, g represent gravitational acceleration, α is tilt angle, T is the temperature, c_{p} is the specific heat at fixed pressure, K denotes the thermal conductivity, C is the concentration of fluid, D is diffusion coefficient, K_{T} is the thermaldiffusion ratio, and T_{m} is the temperature of the medium. The supplementary stress tensor for CarreauYasuda fluid is identificated by [2, 3, 1521]:
$S=\mu_o\left[\frac{\mu_{\infty}}{\mu_o}+\left(1\frac{\mu_{\infty}}{\mu_0}\right)\left(1+(\Gamma \dot{\gamma})^q\right)^{\frac{n1}{q}}\right] A_1$ (8)
with
$A_1=\nabla V+(\nabla V)^T, \dot{\gamma}=\sqrt{\frac{1}{2} \operatorname{tr}\left(A_1\right)^2}$ (9)
where, μ_{∞} is viscosity at infinite shear rate, μ_{0} is viscosity at zero shear rate, Γ is time constant, $\dot{\gamma}$ is shear rate, q is CarreauYasuda parameter, n is power low index number, A_{1} is first Rivlin Ericksen tensor and V=(V_{r}, 0, V_{z}) is velocity vector.
Assume μ_{∞}/μ_{0} to be very small. Eq. (8) becomes
$S=\mu_o\left[\left(1+(\Gamma \dot{\gamma})^q\right)^{\frac{n1}{q}}\right] A_1$ (10)
and for the case Γ $\dot{\gamma}$<<1, Eq. (10) is reduced to
$S=\mu_0\left[1+\frac{n1}{q}(\Gamma \dot{\gamma})^q\right] A_1$ (11)
The specified status of viscous fluid is kept at (n=1). Also, there are two various issues i.e., (n>1) leads to the dilatant fluid and (n<1) drives to pseudoplastic fluid. The original Carreau fluid model can also be retained by substituting (q=2).
The boundary conditions are:
$V_r=0, \frac{\partial V_z}{\partial r}=0, \frac{\partial T}{\partial r}=0, \frac{\partial C}{\partial r}=0$ on $r=0$ (12)
$V_r=0, V_z=0, T=T_o, C=C_o$ on $r=R$ (13)
Using the following dimensionless variables:
$r=R_o r^{\prime}, z=L_o z^{\prime}, V_r=\frac{\delta u_o}{L_o} V_r^{\prime}, V_z=u_o V_z^{\prime}$,
$R=R_o R^{\prime}, t=\frac{L_o}{u_o} t^{\prime}, p=\frac{u_o L_o \mu_o}{R_o^2} p^{\prime}, \theta=\frac{TT_o}{T_o}$,
$\sigma=\frac{CC_o}{C_o}, S_{r r}=\frac{u_o \mu_o}{L_o} S_{r r}{ }^{\prime}, S_{r z}=\frac{\mu_o u_o}{R_o} S_{r z}{ }^{\prime}$,
$S_{z r}=\frac{\mu_o u_o}{R_o} S_{z r}{ }^{\prime}, S_{z z}=\frac{\mu_o u_o}{L_o} S_{z z}{ }^{\prime}, S_{\theta \theta}=\frac{\mu_o u_o}{L_o} S_{\theta \theta}{ }^{\prime}$,
$\dot{\gamma}=\frac{u_o}{R_o} \dot{\gamma}^{\prime}, R_e=\frac{\rho u_o R_o}{\mu_o}, F_r=\frac{u_o^2}{g R_o}, P_r=\frac{c_p \mu_o}{K}$,
$B_r=\frac{\mu_o u_o^2}{K T_o}, S_c=\frac{\mu_o}{\rho D}, S_r=\frac{\rho D K_T T_o}{\mu_o T_m C_o}, W_e=\frac{\Gamma u_o}{R_o}$, (14)
where, u_{0} is the average velocity of the fluid over arterial segment, R_{e} is Reynolds number, F_{r} is the Froud number, P_{r} is Prandtl number, B_{r} is Brickmann number, S_{c} is Schmidt number and S_{r} is Soret number, W_{e} is Weissenberg number. Let us facilitate the governing equations for mild stenosis by adoptiving the two conditions $\left(\delta^*=\frac{\delta}{R_o}<<1\right)$ and $\left(\varepsilon=\frac{R_o}{L_o} \sim o(1)\right)$ [22].
$\frac{\partial V_z}{\partial z}=0$ (15)
$\frac{\partial p}{\partial r}=0$ (16)
$\frac{\partial p}{\partial z}=\frac{1}{r} \frac{\partial}{\partial r}\left(r\left\{\left(\frac{\partial V_z}{\partial r}\right)+\frac{n1}{q}\left(W_e\right)^q\left(\frac{\partial V_z}{\partial r}\right)^{q+1}\right\}\right)+\frac{R_e \sin \alpha}{F_r}$ (17)
$\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial \theta}{\partial r}\right)=B_r\left(\left(\frac{\partial V_z}{\partial r}\right)^2+\frac{n1}{q}\left(W_e\right)^q\left(\frac{\partial V_z}{\partial r}\right)^{q+2}\right)$ (18)
$\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial \sigma}{\partial r}\right)=S_r S_c \frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial \theta}{\partial r}\right)$ (19)
The corresponding boundary conditions (dropping dashes):
$\frac{\partial V_z}{\partial r}=0, \frac{\partial \theta}{\partial r}=0, \frac{\partial \sigma}{\partial r}=0$ on $r=0$ (20)
$V_z=0, \theta=0, \sigma=0$ on $r=R$ (21)
$R(z)= \begin{cases}h^*(z)\left(1\delta^* \cos \phi\left(zd^*\right)\left\{11\frac{94\left(zd^*\right)}{3}+32\left(zd^*\right)^2\frac{32\left(zd^*\right)^3}{3}\right\}\right) & d^* \leq z \leq d^*+\frac{3}{2} \\ h^*(z) & \text { otherwise }\end{cases}$ (22)
where, h^{*}(z)=m^{*}z+1, d^{*}=d/L_{0} and m^{*}= L_{0}m/R_{0} is the corresponding slope of the tapered vessel.
We extend the dependent variables as:
$V_z=V_{z o}+\left(W_e\right)^q V_{z 1}+O\left(W_e\right)^{2 q}$
$\frac{d p}{d z}=\frac{d p_o}{d z}+\left(W_e\right)^q \frac{d p_1}{d z}+O\left(W_e\right)^{2 q}$
$F=F_o+\left(W_e\right)^q F_1+O\left(W_e\right)^{2 q}$
$\theta=\theta_o+\left(W_e\right)^q \theta_1+O\left(W_e\right)^{2 q}$
$\sigma=\sigma_o+\left(W_e\right)^q \sigma_1+O\left(W_e\right)^{2 q}$ (23)
The solutions of pressure gradient, axial velocity, temperature and concentration with the corresponding boundary conditions are
$\frac{d p}{d z}=\frac{16 F}{R^4}+\frac{R_e \sin \alpha}{F_r}+\frac{\left(W_e\right)^q(n1)}{2^{q2} q(q+4)} \times\left(\frac{d p_o}{d z}\frac{R_e \sin \alpha}{F_r}\right)^{q+1} R^q$ (24)
$V_z=\frac{1}{4}\left(\frac{d p}{d z}\frac{R_e \sin \alpha}{F_r}\right)\left(r^2R^2\right)\frac{\left(W_e\right)^q(n1)}{2^{q+1} q(q+2)}\left(\frac{d p_o}{d z}\frac{R_e \sin \alpha}{F_r}\right)^{q+1} \times\left(r^{q+2}R^{q+2}\right)$ (25)
$\theta=2 B_r\left(\frac{d p_o}{d z}\frac{R_e \sin \alpha}{F_r}\right) \times\left[\frac{1}{64}\left(\frac{d p}{d z}\frac{1}{2}\left(\frac{d p_o}{d z}+\frac{R_e \sin \alpha}{F_r}\right)\right)\left(r^4R^4\right)\frac{\left(W_e\right)^q(n1)}{2^{q+3} q(q+4)^2}\left(\frac{d p_o}{d z}\frac{R_e \sin \alpha}{F_r}\right)^{q+1} \times\left(r^{q+4}R^{q+4}\right)\right]$ (26)
$\sigma=2 B_r S_r S_c\left(\frac{d p_o}{d z}\frac{R_e \sin \alpha}{F_r}\right) \times\left[\frac{1}{64}\left(\frac{d p}{d z}\frac{1}{2}\left(\frac{d p_o}{d z}+\frac{R_e \sin \alpha}{F_r}\right)\right)\left(r^4R^4\right)\frac{\left(W_e\right)^q(n1)}{2^{q+3} q(q+4)^2}\left(\frac{d p_o}{d z}\frac{R_e \sin \alpha}{F_r}\right)^{q+1} \times\left(r^{q+4}R^{q+4}\right)\right]$ (27)
Using Eq. (25), the wall shear stress can be listed as
$\tau_R=\frac{1}{2}\left(\frac{d p}{d z}\frac{R_e \sin \alpha}{F_r}\right) R\frac{\left(W_e\right)^q(n1)}{2^{q+1} q} \times\left(\frac{d p_o}{d z}\frac{R_e \sin \alpha}{F_r}\right)^{q+1} R^{q+1}$ (28)
The stream function $\left(V_z=\frac{1}{r} \frac{\partial \psi}{\partial r}\right.$ with $\psi=0$ at $\left.r=0\right)$ is
$\psi=\frac{1}{4}\left(\frac{d p}{d z}\frac{R_e \sin \alpha}{F_r}\right)\left(\frac{r^4}{4}\frac{r^2 R^2}{2}\right)\frac{\left(W_e\right)^q(n1)}{2^{q+1} q(q+2)}\left(\frac{d p_o}{d z}\frac{R_e \sin \alpha}{F_r}\right)^{q+1} \times\left(\frac{r^{q+4}}{q+4}\frac{r^2 R^{q+2}}{2}\right)$ (29)
Assuming the flow rate is fixed for all the syllables of artery. The pressure drop along the overlapping stenosis is
$\Delta p=\int_0^{L^*}\left(\frac{d p}{d z}\right) d z=F \int_0^{L^*} \chi(z) d z$ (30)
where,
$\chi(z)=\frac{16}{R^4}\frac{R_e \sin \alpha}{F F_r}\frac{\left(W_e\right)^q(n1)}{2^{q2} q(q+4) F} \times\left(\frac{d p_o}{d z}\frac{R_e \sin \alpha}{F_r}\right)^{q+1} R^q$ (31)
The resistance impedance exposed by the flowing blood in the arterial segment under deem using Eq. (30) may be recognised as
$\lambda=\frac{\Delta p}{F}=\int_0^{d^*} \sum(z) d z+\int_{d^*}^{d^*+\frac{3}{2}} \chi(z) d z+\int_{d^*+\frac{3}{2}}^{L^*} \sum(z) d z$ (32)
where, L^{*}=L/L_{0} and ∑(z)=χ(z)_{m*z+1}.
This paper uses computer simulations of blood flow in diseased tapered inclined arteries subject to heat transfer and chemical reactions. In this part, we exhibit drawing illustrations of some essential features of the flow features, like blood velocity, wall shear stress, temperature, concentration, resistance impedance and the trapping phenomenon for distinct values of power low index parameter n, Weissenberg parameter W_{e}, tilt angle α, Froud parameter F_{r}, angle of tapering φ, maximum constriction altitude δ^{*}, the Brickmann parameter B_{r}, the Soret parameter S_{r} and the length of the artery L. For the goal of numerical calculations, we use the following informations (R_{e}=1, dp_{0}/dz=0015, F=0.3, d^{*}=0.75). The numerical results are simulated in the following figures below to get a deeper insight into the qualitative analysis of the results.
The impacts of power low index number n and Weissenberg parameter W_{e} on the velocity profile of blood are analyzed. Figure 2(a) describes how the blood velocity varies with power low index number. The blood velocity declines with an increment in the exponent number in the two terminal regions and whilst the blood velocity increases with a rise in the exponent number in the core region. Moreover, the curves of dilatant fluid (n>1) are higher than those in the viscous fluid (n=1) and the pseudoplastic fluid (n<1) in the core region. Figure 2(b) shows the behaviour of the axial velocity with the Weissenberg parameter W_{e}. The velocity increases as the Weissenberg parameter increases in the core region this consistent with the results of Khan et al. [2]. At the same time, the opposite happens in the two terminal regions. Furthermore, the values of blood velocity for a Newtonian fluid (W_{e}=0 or n=1) are lower than that for a CarreauYasuda fluid (W_{e}≠0 or n≠1) at the core region. The archaeology of tilt angle α and Froud parameter F_{r} on blood velocity V_{z} are presented in Figures 3(a) and 3(b). The magnitude of the blood velocity across the vertical artery (α=90^{o}) is superior than those across inclined artery (α=30^{o}) and horizontal artery (α=0^{o}) inside the core region, while the inverse occurs at the two terminal ends.
We can also record that blood velocity grows by a rise in the Froud number F_{r} near the two terminal ends. At the same time, it reduces by dwindling the Froud number F_{r} at the core region.
Figure 2. Variance of velocity aspects V_{z} versus r for diversified values of n and W_{e} at z=1.2, δ^{*}=0.09, φ=0, q=3, α=15°, F_{r}=0.1 (panels (a) and (b) respectively)
Figure 3. Variance of velocity aspects V_{z} versus r for diversified values of α and F_{r} at z=1.2, δ^{*}=0.09, φ=0, W_{e}=0.3, q=2, n=3 (panels (a) and (b) respectively)
Figure 4. Distribution of wall shear stress τ_{R} versus z for diverse values of φ and δ^{*} at W_{e}=0.1, q=2, n=3, α=30°, F_{r}=0.1 (panels (a) and (b) respectively)
Figure 5. Distribution of wall shear stress τ_{R} versus z for diverse values of n and W_{e} at δ^{*}=0.09, φ=0, q=2, α=30°, F_{r}=0.1 (panels (a) and (b) respectively)
The profiles of wall shear stress τ_{R} in the narrow area (0.75≤z≤2.25) for distinct values of taper angle φ, and maximum constriction altitude δ^{*} are displayed in Figures 4(a) and 4(b), consecutively. Figure 4(a) indicates that the curves via the converging tapering artery (φ=0.05<0) are above those in the nontapering artery (φ=0) and the diverging tapering artery (φ=0.05>0). This is consistent with the results of Padma et al. [23]. Figure 4(b) depicts that the wall shear stress increases from the purely healthy artery (no stenosis) to the diseased artery case because the augmentation of the maximum constriction altitude decelerates the flow and increases wall stress. Hence, the wall shear stress disturbances are higher for stenosis (δ^{*}≠0) (diseased artery) than that for no stenosis (δ^{*}=0) (healthy artery). The wall shear stress distros along the narrow zone for diversified values of the power low index parameter n and the Weissenberg number W_{e} are exhibited in Figures 5(a) and 5(b). It should be observed that the shear wall stress increases as the values of n and W_{e} increase. It is surprisingly enjoyable to note that the exponent number n has a reinforcing impact on the wall shear stress. The consequences in Figure 5(a) show that the curves of wall shear stress for dilatant fluid (n>1) are higher than those in the viscous fluid (n=1) and the pseudoplastic fluid (n<1). It also reveals that from Figure 5(b), the values of wall shear stress for CarreauYasuda fluid (W_{e}≠0) are substantially maximal than that for Newtonian fluid (W_{e}=0). The results of wall shear stress distributed over arterial segment for diversified values of the angle of inclination α and Froud number F_{r} are presented in Figures 6(a) and 6(b). It might be evident that the shear stress distribution along the inclined vessel (α=30°) is superior to those in the horizontal vessel (α=0°) and the vertical vessel (α=90°). Also, the wall shear stress increases by growing the Froud number F_{r}.
Figure 6. Distribution of wall shear stress τ_{R} versus z for diverse values of α and F_{r} at δ^{*}=0.09, φ=0, W_{e}=0.1, q=2, n=4 (panels (a) and (b) respectively)
Figure 7. Variation of tempresure θ versus r for diverse values of n and W_{e} at z=1.2, δ^{*}=0.09, φ=0, q=2, α=15^{o}, F_{r}=0.1, B_{r}=0.3 (panels (a) and (b) respectively)
The variation of temperature θ with radial distance r presented in Figures 7(a) and 7(b) predict the effects of the exponent number n and the Weissenberg parameter W_{e} on the thermal transmission of the fluid while Figures 8(a) and 8(b) help making a comparative study of the mass transfer for various values of the CarreauYasuda parameter q and the Brickmann number B_{r}. All these results suggest that the nondimensional temperature of the fluid increases with the increase of the power low index number n, Weissenberg number W_{e}, and the Brickmann number B_{r} while it decreases by increasing of the CarreauYasuda parameter q. Figure 7(a) depicts that the dimensionless heat transfer rate for the dilatant fluid (n>1) is maximal than that in the viscous fluid (n=1) and the pseudoplastic fluid (n<1). It is observed that the heat values for a Newtonian fluid (W_{e}=0) are much less than that for the CarreauYasuda fluid (W_{e}≠0), as shown in 7(b). Moreover, the transmission of the temperature for Carreau fluid (q=2) is higher than that for CarreauYasuda fluid (q≠2), as described in Figure 8(a). It can be seen in Figure 8(b) that, the higher Brickmann number value, the larger temperature of the fluid rise. This is consistent with the results of Nazir et al. [11] and Farooq et al. [24].
Figure 8. Variation of tempresure θ versus r for diverse values of q and B_{r} at z=1.2, δ^{*}=0.09, φ=0, W_{e}=0.2, α=15^{o}, n=3, F_{r}=0.1 (panels (a) and (b) respectively)
Figures 9 and 10 are intended to show the disparity of concentration for the power low index parameter n, the Weissenberg number W_{e}, the Brickmann number B_{r}, and the Soret parameter S_{r}. It is observed that the concentration has an adverse behaviour as compared to the temperature. Figures 9(a) and 9(b) depict that the concentration for the dilatant liquid (n>1) is more minimal than that for the viscous liquid (n=1) and the pseudoplastic liquid (n<1). Also, the concentration for a Newtonian fluid (W_{e}=0) is maximal than that for CarreauYasuda fluid (W_{e}≠0). Figures 10(b) and 10(b) explain the difference of concentration profile for the Brickmann parameter B_{r}, and the Soret parameter S_{r} to show the concentration of the fluid decreases as the Brickmann parameter B_{r} and the Soret parameter S_{r} increase. This is consistent with the results of Alam et al. [25].
Figure 9. Variation of concentration profiles σ versus r for diversified values of n and W_{e} at z=1.2, δ^{*}=0.09, φ=0, q=2, α=15^{o}, F_{r}=0.1, B_{r}=0.3, S_{r}=0.1, S_{c}=0.1 (panels (a) and (b) respectively)
Figure 10. Variation of concentration profiles σ versus r for diversified values of B_{r} and S_{r} at z=1.2, δ^{*}=0.09, φ=0, W_{e}=0.5, q=2, n=2, α=15^{o}, F_{r}=0.1, S_{c}=0.2 (panels (a) and (b) respectively)
The diversity of flow impedance λ with the altitude of the stenosis δ^{*} for diverse values of the power low index number n, the Weissenberg number W_{e}, the angle of inclination α and artery length L^{*} has been shown in Figures 11 and 12. The impedance λ piecemealy enhances as the altitude of the contraction grows. Also impedance curves in the dilatant fluid (n>1) are lower than those in the viscous fluid (n=1) and the pseudoplastic fluid (n<1). Furthermore, the values of resistance for a Newtonian fluid (W_{e}=0) is higher than that for a CarreauYasuda fluid (W_{e}≠0). Moreover, the impedance along the vertical artery (α=90°) is higher than that along the horizontal artery (α=0°) and the inclined artery (α=30°). It has been perceived from Figure 11(b) that the impedance diminishes with an increment in the Weissenberg number W_{e}. Also, the impedance grows with enhancement in the artery length L^{*}, as shown in Figure 12(b).
Figure 11. Variance of impedance λ versus δ^{*} for diversified values of n and W_{e} at φ=0, q=2, α=60^{o}, F_{r}=0.1, L^{*}=3 (panels (a) and (b) respectively)
Figure 12. Variance of impedance λ versus δ^{*} for diversified values of α and L^{*} at φ=0, W_{e}=0.3, q=2, n=3, F_{r}=0.1 (panels (a) and (b) respectively)
The taper angle φ effect is elucidated in Figure 13. It is remarked that the size of the trapping bolus gradually increases, shifting toward the right at (φ=0.2>0) (diverging tapering artery), and it gradually increases moving toward the left at (φ=0.2<0) (converging tapering artery). Figure 14 depicts the streamlined patterns for diversified values of exponent number n when all other parametric values are kept fixed. It is observed that the streamlines are converging at the upper and lower walls of the artery for the viscous fluid (n=1). It is shown that the streamlines are diverging at the arterial walls noting that there are many circular trapping boluses near the lower border of the artery for the dilatant fluid (n>1). It is seen that the size and the number of the intermittent trapping bolus increase near the lower border of the artery for the pseudoplastic fluid (n<1). Figure 15 indicates that the streamlines are noticeably wider along the vertical artery (α=90°) is maximal than that in the inclined artery (α=30°) and the horizontal one (α=0°).
Figure 13. Streamlines for diversified values of the tapering angle φ at δ^{*}=0.4, W_{e}=0.3, q=2, n=2, α=45°, F_{r}=0.1
Figure 14. Streamlines for diversified values of the exponent number n at δ^{*}=0.4, φ=0, W_{e}=0.5, q=5, α=30°, F_{r}=0.1
Figure 15. Streamlines for diversified values of tilt angle α at δ^{*}=0.4, φ=0, W_{e}=0.5, q=2, n=3, F_{r}=0.1
An analytical investigation of the flow of a CarreauYasuda fluid through the diseased tapered inclined artery subject to heat transfer and chemical reactions has been conducted. The core findings of the analysis are as follows:
●The blood velocity and the temperature curves for the dilatant fluid (n>1) are higher than those in the viscous fluid (n=1) and the pseudoplastic fluid (n<1) in the core region of the artery.
●The values of axial velocity and heat transfer rate along a Newtonian fluid (W_{e}=0 or n=1) are more minimal than that along a CarreauYasuda fluid (W_{e}≠0 or n≠1) at the core region of the artery.
●The magnitude of blood velocity through the vertical artery is maximal than that in the inclined and horizontal artery inside the core region of the artery.
●The wall shear stress distributions are higher for stenosis (diseased artery) than that for no stenosis (healthy artery).
●The wall shear stress values through the CarreauYasuda fluid (W_{e}≠0 or n≠1) are maximal than that along a Newtonian fluid (W_{e}=0 or n=1).
●The wall shear stress values along the inclined artery are higher than those in the horizontal and vertical arteries.
●The transmission of the temperature for the Carreau fluid (q=2) is higher than that for the CarreauYasuda fluid (q≠2).
●The concentration profile has a reverse behaviour as compared to the temperature profile.
●The impedance gradually increases as the maximum height of stenosis increases.
●The curves of resistance impedance for the dilatant fluid (n>1) are lower than those in the viscous fluid (n=1) and the pseudoplastic fluid (n<1).
●The resistance values for a Newtonian fluid (W_{e}=0 or n=1) is higher than that for a CarreauYasuda fluid (W_{e}≠0 or n≠1).
●The impedance along the vertical artery is higher than horizontal and inclined arteries.
●The size of the trapping bolus gradually increases, shifting toward the right at (φ>0) (diverging tapering artery), and it gradually increases, moving toward the left at (φ<0) (converging tapering artery).
●The streamlines are noticeably more comprehensive along the vertical artery is higher than that in the inclined artery and the horizontal one.
B_{r} 
Brickmann number 
C 
concentration, Kg. m^{2} .s^{2} 
c_{p} 
specific heat capacity, J. Kg^{1}. K^{1} 
C_{0} & C_{1} 
walls concentration, Kg. m^{2} .s^{2} 
D 
diffusion coefficient, m^{2}. s^{1} 
d 
site of the stenosis, m 
d^{*} 
dimensional stenosis site 
dp_{0}/dz 
initial pressure rise, Kg. m^{2}. s^{2} 
F 
flow rate, m^{3}.s^{1} 
F_{r} 
Froud number 
G 
gravitational acceleration, m. s^{2} 
K 
thermal conductivity, W. m^{1}. K^{1} 
K_{T} 
thermaldiffusion ratio, Kg. m^{2} .s^{2} 
3L_{0}/2 
length of overlapping stenosis, m 
L^{*} 
nondimensional length of arterial segment 
m^{*} 
slope of the tapered vessel 
n 
power low index number 
p 
pressure, N. m^{2} 
P_{r} 
Prandtl number 
q 
CarreauYasuda parameter 
R 
radius of the tapered artery, m 
r 
radial direction, m 
R_{e} 
Reynolds number 
R_{0} 
radius of the nontapered artery, m 
S_{c} 
Schmidt number 
S_{r} 
Soret number 
T 
temperature field, K 
t 
time, s 
T_{m} 
temperature of the medium, K 
T_{0} & T_{1} 
walls temperature, K 
u_{0} 
average velocity of the fluid over arterial segment, m. s^{1} 
V_{z} 
axial velocity, m. s^{1} 
V_{r} 
radial velocity, m. s^{1} 
W_{e} 
Weissenberg number 
z 
axial direction, m 
Greek symbols 

α 
angle of inclination 
δ 
height of the stenosis, m 
δ^{*} 
dimensionless height of the stenosis 
θ 
dimensionless temperature 
λ 
resistance impedance, Kg. m^{4}. s^{1} 
μ_{0} & μ_{∞} 
viscosity at zero and infinite shear rates, Kg. m^{1}. s^{1} 
ρ 
fluid density, Kg. m^{3} 
σ 
dimensionless concentration 
τ_{R} 
wall shear stress distribution, N. m^{2} 
ϕ 
angle of tapering 
ψ 
stream function, m^{3}. s^{1} 
Γ 
time constant 
Δp 
pressure drop, Kg. m^{1}. s^{2} 
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