Entropy and stability analysis on blood flow with nanoparticles through a stenosed artery having permeable walls

Abstract

In this research, the electro-osmotic effects are highlighted for a blood-based hybrid nanofluid flow across an artery infected with multiple stenosis. The artery has permeable walls together with slip boundary effects. The slip and permeable boundary conditions model the more realistic blood flow problems. The governing equations of the problem are converted into non-dimensional form by introducing adequate dimensionless variables and acquired the exact solutions. The detailed study of heat transfer is given by Joule heating and viscous dissipation effects. The disorder of fluid flow is investigated by the mathematical study of entropy generation. Analytically attained solutions are examined graphically for both symmetric and non-symmetric shapes of stenosis. Streamlines are analyzed for varying values of flow rate Q and electro-osmotic parameter m. The flow velocity has smallest values on the axis of channel and gets higher value near the boundary walls. The temperature profile delineates opposite behavior to the velocity, and it is parabolic in nature. The velocity reduces towards the non-uniform stenosis except for electroosmotic parameter m. The temperature has larger magnitude in the case of anti-symmetric stenosis. Moreover, the stability of velocity solution is also analyzed.

Introduction

Blood flow is restricted across stenosed arteries that are damaged. This stenosis develops as a result of plaque buildup on the artery walls caused by fats and oils. An artery affected by stenosis terribly may have multiple stenosis. Due to the existence of multiple stenosis the blood artery narrows, and it limits the blood flow across it. The analysis of this kind of stenotic artery has recently picked the curiosity of many researchers. In large vessels, blood depicts Newtonian behavior as well as it shows non-Newtonian behavior while flowing through small vessels.^1–5 The examination of blood flow across a stenotic vessel is crucial due to the characteristics of blood circulation and mechanical properties of the walls of the vessel. The blood flow across a stenotic artery enables us to understand the flow behavior via diseased blood vessel and locate the position of stenosis on the vessels wall. Ponalagusamy ⁶ was the first who had described the behavior of blood flowing through an artery affected by stenosis in his doctoral dissertation. The transient blood flow across tapering tubes is described by using power law model. ⁷ The mathematical analysis of non-Newtonian fluid through a channel is given by Varshney et al. ⁸ The study of carbon nanotubes through an artery damaged by several stenosis, seeing the case of changing viscosity is presented by Nadeem et al. ⁹ Saleem et al. ¹⁰ studied blood flowing via an artery damaged by stenosis having catheter and blood clot in the middle of channel. The transient blood flow across a stenosed artery is examined by Hisham et al. ¹¹ Kadhim et al. studied the influence of non-Newtonian model on properties of blood flow through a left coronary artery with presence of different double stenosis. ¹² Jamil et al. described the effects of magnetic Casson blood flow in an inclined multi-stenosed artery by using Caputo-Fabrizio fractional derivative. ¹³

The nanofluid is the kind of fluids that consists of nanometer sized particles (known as nanoparticle) in the base fluids. The base fluid containing two or more nanoparticles is called hybrid nanofluid. These nanoparticles are scattered in base fluid for improving its thermal efficiency. Turcu et al. ¹⁴ disclosed the production of hybrid nano-composite particles. Akbar et al. ¹⁵ evaluated the CNT flow inside a channel with a sinusoidally vibrating wall. Theoretical investigation of Jeffrey nanofluid flow across an oscillating stretching sheet with an unsteady oblique stagnation point is given by Awan et al. ¹⁶ Heat and mass transfer examination of CNT-based nanofluid and hybrid nanofluid over various geometries is given by Upreti et al.^17,18 Rathore and Srikanth provided the mathematical study of transport phenomena of blood nanofluid in a diseased artery subjected to catheterization. ¹⁹ Karami and Nadooshan examined the power law nanofluid through tapered artery based on consistent couple stress theory. ²⁰

The electroosmotic flow is a flow caused on account of an electric field applied over a channel and electric double layer (EDL) existence at boundary walls. Reuss ²¹ first reported on the electroosmotic flow study, and he demonstrated the flow of water across a plug of clay by applying a proper electric voltage. The electroosmotic flow is used for DNA analysis, and it has a lot of applications in medicine for cure of illnesses like sickle cells, drugs delivery, cellular anomalies. ²² It is also applied for separation techniques like separation of plasma from the blood and utilized in biological agent detectors. Using the Debye-Hunkel assumption, Afonso et al. ²³ investigated the electrokinetic and pressure driven flow of fluid having viscous and elastic properties. Zhao and Yang ²⁴ studied electro-osmotic flow on a nanoscale, where they investigated electro-osmotic influences on a power law fluid across a micro channel. Ferras et al. ²⁵ examined viscous fluid flow through an annular region with electro-osmotic effects. Nadeem et al. ²⁶ probed the electro-osmotic flow of the Bingham plastic fluid across a channel with very small length. Awan et al. ²⁷ analyzed the flow of an Oldroyd-B fluid flowing among parallel plates. Salman et al. ²⁸ studied non-Newtonian flow behavior of blood through arteries affected by several stenosis and electroosmotic effects. Akram et al. analyzed the electroosmosis augmented MHD peristaltic transport of SWCNTs suspension in aqueous media. ²⁹ Sridhar and Ramesh studied peristaltic activity of thermally radiative magneto-nanofluid with electroosmosis and entropy analysis. ³⁰

For describing the disorder of whole system, many researchers have been studying the entropy analysis mathematically. Akbar et al. ³¹ presented peristaltic flow with thermal conductivity of nanofluid and gave brief entropy analysis. Zhang et al. ³² explained entropy generation of blood flowing across tapered arteries filled with nanoparticles. The hybrid nanofluid flow through sinusoidally deforming channel is mathematically studied by Hayat et al. ³³ Riaz et al. presented the entropy analysis for cilia-generated motion of Cu-blood flow of nanofluid in an annulus. ³⁴ Abbas et al. gave the analysis of entropy generation on peristaltic phenomena of MHD slip flow of viscous fluid in diverging tube. ³⁵

The current research is about the study of more realistic problem of the blood flow across a diseased artery. We have studied the flow nature of blood-based hybrid nano fluid through a diseased artery with permeable boundary wall and harmed by multiple stenosis. The fluid flows through the conduit due to the application of electro-osmotic effect and subjected with slip effects. The single-wall and multi-wall carbon nanotubes are taken as nanoparticles in the blood. The transfer of heat is studied in detail by considering the Joule heating effect. Exact solutions are found for mathematical equations by using Mathematica. Moreover, these results are explained in detail graphically. In order to observe the disorder of system, the entropy generation is studied. The flow behavior is also explained by plotting streamlines.

The mesh diagrams for this interesting arterial geometry are also included by Figure 1(a)–1(c) given as

Figure 1.

Geometry of problem. (a) Mesh diagram of symmetric stenosis. (b) Mesh diagram of non-symmetric stenosis. (c) Mesh diagram displaying corner mesh view.

Mathematical modeling

The electro-osmotic flow of hybrid nanofluid through an artery harmed with several stenosis is considered. The hybrid nanofluid consists of blood as base fluid with single and multi-wall CNT in it. The mathematical expression for multiple stenosis wall geometry in dimensional form is given as ³⁶

$$\eta (\mathrm{z})=\{\begin{array}{cc}R[1-\kappa \{s_l^{n-1}(z-d_l)-{(z-d_l)}^n\}],& d_l\le z\le d_l+s_l\\ R& \mathrm{Otherwise},\end{array}$$ (1)

where

$$\hat{\kappa }={\displaystyle \frac{{\delta }_{\mathrm{l}}^{*}}{{\mathrm{R}\mathrm{s}}_{\mathrm{l}}^{\mathrm{n}}}{\displaystyle \frac{{(\mathrm{n})}^{\mathrm{n}/n-1}}{\mathrm{n}-1}.}}$$ (2)

The incompressible flow is governed by the following equations

$${\displaystyle \frac{\mathrm{\partial }\tilde{u}}{\mathrm{\partial }\tilde{r}}+{\displaystyle \frac{\tilde{u}}{\tilde{r}}+{\displaystyle \frac{\mathrm{\partial }\tilde{w}}{\mathrm{\partial }\tilde{z}}=0,}}}$$ (3)

$${\rho }_{hnf}\left(\tilde{u}{\displaystyle \frac{\mathrm{\partial }\tilde{u}}{\mathrm{\partial }\tilde{r}}+\tilde{w}{\displaystyle \frac{\mathrm{\partial }\tilde{u}}{\mathrm{\partial }\tilde{z}}}}\right)=-{\displaystyle \frac{\mathrm{\partial }\tilde{p}}{\mathrm{\partial }\tilde{r}}+{\mu }_{hnf}\left({\displaystyle \frac{{\mathrm{\partial }}^2\tilde{u}}{\mathrm{\partial }{\tilde{r}}^2}+{\displaystyle \frac{1}{\tilde{r}}{\displaystyle \frac{\mathrm{\partial }\tilde{u}}{\mathrm{\partial }\tilde{r}}+{\displaystyle \frac{{\mathrm{\partial }}^2\tilde{u}}{\mathrm{\partial }{\tilde{z}}^2}-{\displaystyle \frac{\tilde{u}}{{\tilde{r}}^2}}}}}}\right),}$$ (4)

$${\rho }_{hnf}\left(\tilde{u}{\displaystyle \frac{\mathrm{\partial }\tilde{\mathrm{w}}}{\mathrm{\partial }\tilde{\mathrm{r}}}+\overline{w}{\displaystyle \frac{\mathrm{\partial }\tilde{\mathrm{w}}}{\mathrm{\partial }\tilde{z}}}}\right)=-{\displaystyle \frac{\mathrm{\partial }\tilde{p}}{\mathrm{\partial }\tilde{z}}+{\mu }_{hnf}\left({\displaystyle \frac{{\mathrm{\partial }}^2\tilde{\mathrm{w}}}{\mathrm{\partial }{\tilde{r}}^2}+{\displaystyle \frac{1}{\tilde{r}}{\displaystyle \frac{\mathrm{\partial }\tilde{\mathrm{w}}}{\mathrm{\partial }\tilde{r}}+{\displaystyle \frac{{\mathrm{\partial }}^2\tilde{\mathrm{w}}}{\mathrm{\partial }{\tilde{z}}^2}}}}}\right)+{\rho }_e{E}_z,}$$ (5)

$$\begin{array}{rl}(\rho C_p{)}_{hnf}\left(\tilde{u}{\displaystyle \frac{\mathrm{\partial }\tilde{T}}{\mathrm{\partial }\tilde{r}}+\tilde{\mathrm{w}}{\displaystyle \frac{\mathrm{\partial }\tilde{T}}{\mathrm{\partial }\tilde{z}}}}\right)=& k_{hnf}\left({\displaystyle \frac{{\mathrm{\partial }}^2\tilde{T}}{\mathrm{\partial }{\tilde{r}}^2}+{\displaystyle \frac{1}{\tilde{r}}{\displaystyle \frac{\mathrm{\partial }\tilde{T}}{\mathrm{\partial }\tilde{r}}+{\displaystyle \frac{{\mathrm{\partial }}^2\tilde{T}}{\mathrm{\partial }{\tilde{z}}^2}}}}}\right)\\ & +{\mu }_{hnf}\left[2\left\{{\left({\displaystyle \frac{\mathrm{\partial }\tilde{u}}{\mathrm{\partial }\tilde{r}}}\right)}^2+{\left({\displaystyle \frac{\tilde{u}}{\tilde{r}}}\right)}^2+{\left({\displaystyle \frac{\mathrm{\partial }\tilde{w}}{\mathrm{\partial }\tilde{z}}}\right)}^2\right\}+{\left({\displaystyle \frac{\mathrm{\partial }\tilde{\mathrm{w}}}{\mathrm{\partial }\tilde{\mathrm{r}}}+{\displaystyle \frac{\mathrm{\partial }\tilde{u}}{\mathrm{\partial }\tilde{z}}}}\right)}^2\right]+s^{*}.\end{array}$$ (6)

The value of $s*$ given in equation (6) is ³⁷

$$s*=i_e^2\sigma ,i_e={\displaystyle \frac{E_z}{\sigma }.}$$

An electrolyte mixture $(N{a}^{+}C{l}^{-})$ is uniformly considered and mathematical expression for its electrical potential dispersion is given by Poisson-Boltzmann equation

$${\mathrm{\nabla }}^2\tilde{\varphi }=-{\displaystyle \frac{{\rho }_{\mathrm{e}}}{\mathrm{E}},}$$ (7)

where, ${\rho }_{\mathrm{e}}=\mathrm{ez}*(c^{+}-c^{-})$ is the density of ionic energy, when “no electric double layer overlap is considered”

$$c^{\pm }=c_0{e}^{\pm (\mathrm{ez}*\varphi /K_{B}T*)}.$$ (8)

Inserting the values of ${\rho }_{\mathrm{e}}$ and $c^{\pm }$ in equation (7), we obtain

$${\displaystyle \frac{1}{\tilde{\mathrm{r}}}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }\tilde{\mathrm{r}}}\left(\overline{\mathrm{r}}{\displaystyle \frac{\mathrm{\partial }\tilde{\varphi }}{\mathrm{\partial }\tilde{\mathrm{r}}}}\right)={\displaystyle \frac{2{\mathrm{c}}_0\mathrm{ez}*\sinh (\mathrm{ez}*\tilde{\varphi }/K_{B}T*)}{\mathrm{E}}.}}}$$ (9)

By Debye–Hückel approximation, we have $\sinh (\mathrm{ez}*\tilde{\varphi }/K_{B}T*)\approx (\mathrm{ez}*\tilde{\varphi }/K_{B}T*)$ . Also, by using dimensionless variables $\varphi =\tilde{\varphi }/\zeta$ and $\mathrm{r}=\tilde{\mathrm{r}}/\mathrm{R}$ in equation (9), we obtain

$${\displaystyle \frac{1}{r}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }\mathrm{r}}\left(\mathrm{r}{\displaystyle \frac{\mathrm{\partial }\varphi }{\mathrm{\partial }r}}\right)={\mathrm{m}}^2\varphi .}}$$ (10)

By utilizing the conditions $\mathrm{\partial }\varphi /\mathrm{\partial }\mathrm{r}=0$ at $r=0$ and $\varphi =1$ at $r=\eta (z)$ , we obtain the exact solution of equation (10)

$$\varphi ={\displaystyle \frac{I_0(mr)}{I_0(m\eta )}.}$$ (11)

Now introducing dimensionless variables

$$\begin{array}{rl}& r={\displaystyle \frac{\tilde{r}}{R},z={\displaystyle \frac{\tilde{z}}{s_l},w={\displaystyle \frac{\tilde{w}}{u_0},u={\displaystyle \frac{L\tilde{u}}{u_0{\delta }_l^{*}},p={\displaystyle \frac{R^2\tilde{p}}{s_l{u}_0{\mu }_f},\theta ={\displaystyle \frac{\tilde{T}-{\tilde{T}}_0}{{\tilde{T}}_0},{\tilde{h}}_l={\displaystyle \frac{d_l}{s_l},}}}}}}}\\ & \eta (z)={\displaystyle \frac{\overline{\eta }(z)}{R},{\delta }_l={\displaystyle \frac{{\delta }_l^{*}}{R},S={\displaystyle \frac{E_z^2{R}^2}{\sigma {\tilde{T}}_{0}k},B_r={\displaystyle \frac{{\mu }_f{u}_0^2}{k_f{\tilde{T}}_0},U_{HS}={\displaystyle \frac{\zeta E_{z}E}{\mu u_0},}}}}}\\ & m=\mathrm{ez}*\mathrm{R}\sqrt{{\displaystyle \frac{2{\mathrm{c}}_0}{K_{B}T*E}}}={\displaystyle \frac{R}{{\lambda }_d},\gamma ={\displaystyle \frac{{\tilde{T}}_0}{{\tilde{\theta }}_0},\mu ={\displaystyle \frac{{\mu }_{hnf}}{{\mu }_f},K={\displaystyle \frac{k_{hnf}}{k_f},S_{G_0}={\displaystyle \frac{k_f{\tilde{T}}_0^2}{R^2{\tilde{\theta }}_0^2}.}}}}}\end{array}$$ (12)

By considering the moderate case of multiple stenosis, following assumptions are accounted in this study

$${\delta }_l={\displaystyle \frac{{\delta }_l^{*}}{R}\ll 1,{\displaystyle \frac{R^{*}{n}^{(1/(n-1))}}{s_l}{\sim }^{\circ }1.}}$$ (13)

By utilizing the non-dimensional variables specified in (12), and the assumptions in (13), equations (3)–(6) take the form

$${\displaystyle \frac{\mathrm{\partial }\mathrm{p}}{\mathrm{\partial }\mathrm{r}}=0,}$$ (14)

$${\displaystyle \frac{\mathrm{\partial }p}{\mathrm{\partial }z}=\mu \left({\displaystyle \frac{{\mathrm{\partial }}^{2}w}{\mathrm{\partial }{r}^2}+{\displaystyle \frac{1}{r}{\displaystyle \frac{\mathrm{\partial }w}{\mathrm{\partial }r}}}}\right)+m^2{U}_{HS}{\displaystyle \frac{I_0(mr)}{I_0(m\eta )},}}$$ (15)

$$\mathrm{K}\left({\displaystyle \frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{\mathrm{r}}^2}+{\displaystyle \frac{1}{\mathrm{r}}{\displaystyle \frac{\mathrm{\partial }\theta }{\mathrm{\partial }\mathrm{r}}}}}\right)+\mu B_r{\left({\displaystyle \frac{\mathrm{\partial }w}{\mathrm{\partial }r}}\right)}^2+S=0.$$ (16)

The relevant boundary conditions in dimensionless form are ³⁸

$${\displaystyle \frac{\mathrm{\partial }\mathrm{w}}{\mathrm{\partial }\mathrm{r}}=0\mathrm{at}r=0\mathrm{and}w={\displaystyle \frac{\sqrt{D_a}}{\alpha }{\displaystyle \frac{\mathrm{\partial }w}{\mathrm{\partial }r}-D_a{\displaystyle \frac{dp}{dz}\mathrm{at}r=\eta ,}}}}$$ (17)

$${\displaystyle \frac{\mathrm{\partial }\theta }{\mathrm{\partial }\mathrm{r}}=0\mathrm{at}r=0\mathrm{and}\theta =0\mathrm{at}r=\eta .}$$ (18)

The non-dimensional mathematical expression of stenosed wall harmed by multiple stenosis is

$$\eta (\mathrm{z})=\{\begin{array}{cc}1-{\displaystyle \frac{{\delta }_l{(n)}^{(n/n-1)}}{n-1}[(z-{\tilde{h}}_l)-{(z-{\tilde{h}}_l)}^n],}& {\tilde{h}}_l\le z\le {\tilde{h}}_l+1\\ 1& \mathrm{Otherwise}.\end{array}$$ (19)

Exact solution

The axial velocity of flow can be expressed mathematically as

$$\begin{array}{rl}& w(r,z)={\displaystyle \frac{1}{4\alpha \mu I_0(m\eta )}}\\ & [-4\alpha U_{HS}{I}_0(mr)+(4\alpha U_{HS}+2\sqrt{D_a}{\displaystyle \frac{dp}{dz}\eta +}\\ & \alpha {\displaystyle \frac{dp}{dz}(r^2-{\eta }^2-4D_a\mu )})I_0(m\eta )-4m{U}_{HS}\sqrt{D_a}{I}_1(m\eta )],\end{array}$$ (20)

where $\eta$ is given in equation (19).

The volume flow rate is evaluated by using

$$Q=2\pi \underset{0}{\overset{\eta }{\int }}rwdr.$$ (21)

Mathematically, pressure gradient is given as

$${\displaystyle \frac{\mathrm{dp}}{\mathrm{dz}}={\displaystyle \frac{4\left[2\alpha ({\mathrm{U}}_{\mathrm{HS}}-(\mathrm{Q}\mu /\pi {\eta }^2))-(2{\mathrm{U}}_{\mathrm{HS}}(2\alpha +\sqrt{{\mathrm{D}}_{\mathrm{a}}}{\mathrm{m}}^2\eta ){\mathrm{I}}_1(\mathrm{m}\eta )/\mathrm{m}\eta {\mathrm{I}}_0(\mathrm{m}\eta ))\right]}{-4\sqrt{{\mathrm{D}}_{\mathrm{a}}}\eta +\alpha {\eta }^2+8\alpha {\mathrm{D}}_{\mathrm{a}}\mu }.}}$$ (22)

The shear stress at stenosed wall is

$${\tau }_{\mathrm{w}}={-{\displaystyle \frac{\mathrm{\partial }\mathrm{w}}{\mathrm{\partial }\mathrm{r}}}|}_{\mathrm{r}=\eta }={\displaystyle \frac{2\alpha (\mathrm{dp}/\mathrm{dz})\eta {\mathrm{I}}_0(\mathrm{m}\eta )-4\mathrm{m}\alpha {\mathrm{U}}_{\mathrm{HS}}{\mathrm{I}}_1(\mathrm{m}\eta )}{4\alpha \mu {\mathrm{I}}_0(\mathrm{m}\eta )}.}$$ (23)

The exact solution for temperature profile is

$$\begin{array}{rl}\theta (\mathrm{r},\mathrm{z})=& -{\displaystyle \frac{r^{2}S}{4K}+{\displaystyle \frac{S{\eta }^2}{4K}-{\displaystyle \frac{B_r{(dp/dz)}^2{r}^4}{64K\mu }+{\displaystyle \frac{2B_r(dp/dz)U_{HS}}{K{m}^2\mu }-{\displaystyle \frac{B_r{m}^2{\mathrm{U}}_{\mathrm{HS}}^2{\eta }^2}{4K\mu }}}}}}\\ & +{\displaystyle \frac{B_r{(dp/dz)}^2{\eta }^4}{64K\mu }-{\displaystyle \frac{B_r{\mathrm{U}}_{\mathrm{HS}}^2(-2+m^2{\eta }^2)}{4K\mu }}}\\ & +{\displaystyle \frac{((B_r{m}^2{\mathrm{U}}_{\mathrm{HS}}^2{r}^2/4K\mu )+(B_r(-2+m^2{r}^2){\mathrm{U}}_{\mathrm{HS}}^2/4K\mu )){\mathrm{I}}_0{(\mathrm{mr})}^2}{{\mathrm{I}}_0{(\mathrm{m}\eta )}^2}}\\ & +{\mathrm{I}}_0(\mathrm{mr})\left(-{\displaystyle \frac{2B_r(dp/dz)U_{HS}}{K{m}^2\mu {\mathrm{I}}_0(\mathrm{m}\eta )}-{\displaystyle \frac{B_r{\mathrm{rmU}}_{\mathrm{HS}}^2{\mathrm{I}}_1(\mathrm{mr})}{2K\mu {\mathrm{I}}_0{(\mathrm{m}\eta )}^2}}}\right)+{\displaystyle \frac{1}{{\mathrm{I}}_0(\mathrm{m}\eta )}}\\ & \left({\displaystyle \frac{B_r(dp/dz)r{U}_{HS}{\mathrm{I}}_1(\mathrm{mr})}{Km\mu }+{\displaystyle \frac{B_r(-2(dp/dz)U_{HS}+m^2{U}^2)\eta {\mathrm{I}}_1(\mathrm{m}\eta )}{2Km\mu }}}\right)\\ & +{\displaystyle \frac{B_r{m}^2{\mathrm{U}}_{\mathrm{HS}}^2{\eta }^2{\mathrm{I}}_1{(\mathrm{m}\eta )}^2-B_r{m}^2{r}^2{\mathrm{U}}_{\mathrm{HS}}^2{\mathrm{I}}_1{(\mathrm{mr})}^2}{2K\mu {\mathrm{I}}_0{(\mathrm{m}\eta )}^2}.}\end{array}$$ (24)

Tables 1 and 2 appear for numerical values and experimental formulas for hybrid nanofluid thermo-physical characteristics respectively.

Table 1.

Thermo-physical characteristics of base fluid and nanoparticles.

Physical parameters	Base Fluid (f)	Nanoparticles
	Blood	Single-wall CNT (s1)	Multi-wall CNT (s2)
$\rho$	1063	2600	1600
$k$	0.492	6600	3000
${\mathrm{C}}_{\mathrm{p}}$	3617	425	796

Table 2.

Hybrid nanofluid Model. ³⁹

Viscosity	${\mu }_{\mathrm{hnf}}=\frac{{\mu }_{\mathrm{f}}}{{[(1-{\phi }_1)(1-{\phi }_2)]}^{2.5}}$
Density	${\rho }_{\mathrm{hnf}}=[(1-{\phi }_2)\{(1-{\phi }_1){\rho }_{\mathrm{f}}+{\phi }_1{\rho }_{\mathrm{s}1}\}]$
Heat capacity	$(\rho {\mathrm{C}}_{\mathrm{p}}{)}_{\mathrm{hnf}}=[(1-{\phi }_2)\{(1-{\phi }_1)(\rho {\mathrm{C}}_{\mathrm{p}}{)}_{\mathrm{f}}+{\phi }_1(\rho {\mathrm{C}}_{\mathrm{p}}{)}_{\mathrm{s}1}\}]+{\phi }_2(\rho {\mathrm{C}}_{\mathrm{p}}{)}_{\mathrm{s}2}$
Thermal conductivity	$\begin{array}{rl}{\mathrm{k}}_{\mathrm{hnf}}=& {\mathrm{k}}_{\mathrm{bf}}\left[\frac{{\mathrm{k}}_{\mathrm{s}2}+(\mathrm{n}-1)({\mathrm{k}}_{\mathrm{bf}}-{\phi }_2({\mathrm{k}}_{\mathrm{bf}}-{\mathrm{k}}_{\mathrm{s}2}))}{{\mathrm{k}}_{\mathrm{s}2}+(\mathrm{n}-1){\mathrm{k}}_{\mathrm{bf}}+{\phi }_2({\mathrm{k}}_{\mathrm{bf}}-{\mathrm{k}}_{\mathrm{s}2})}\right]\\ {\mathrm{k}}_{\mathrm{bf}}=& {\mathrm{k}}_{\mathrm{f}}\left[\frac{{\mathrm{k}}_{\mathrm{s}1}+(\mathrm{n}-1)({\mathrm{k}}_{\mathrm{f}}-{\phi }_1({\mathrm{k}}_{\mathrm{f}}-{\mathrm{k}}_{\mathrm{s}1}))}{{\mathrm{k}}_{\mathrm{s}1}+(\mathrm{n}-1){\mathrm{k}}_{\mathrm{f}}+{\phi }_1({\mathrm{k}}_{\mathrm{f}}-{\mathrm{k}}_{\mathrm{s}1})}\right]\end{array}$

Entropy analysis

The mathematical expression in dimensional form is given as ⁴⁰

$${\mathrm{S}}_{\mathrm{G}}={\displaystyle \frac{{\mathrm{k}}_{\mathrm{hnf}}}{{\tilde{\theta }}_0^2}\left[{\left({\displaystyle \frac{\mathrm{\partial }\tilde{\mathrm{T}}}{\mathrm{\partial }\tilde{\mathrm{r}}}}\right)}^2+{\left({\displaystyle \frac{\mathrm{\partial }\tilde{\mathrm{T}}}{\mathrm{\partial }\tilde{z}}}\right)}^2\right]+{\displaystyle \frac{{\mu }_{hnf}}{{\tilde{\theta }}_0}\left[2\left\{{\left({\displaystyle \frac{\mathrm{\partial }\tilde{u}}{\mathrm{\partial }\tilde{\mathrm{r}}}}\right)}^2+{\left({\displaystyle \frac{\tilde{u}}{\tilde{\mathrm{r}}}}\right)}^2+{\left({\displaystyle \frac{\mathrm{\partial }\tilde{\mathrm{w}}}{\mathrm{\partial }\tilde{z}}}\right)}^2\right\}+{\left({\displaystyle \frac{\mathrm{\partial }\tilde{\mathrm{w}}}{\mathrm{\partial }\tilde{\mathrm{r}}}+{\displaystyle \frac{\mathrm{\partial }\tilde{u}}{\mathrm{\partial }\tilde{z}}}}\right)}^2\right].}}$$ (25)

By employing the non-dimensional variables provided in (12), equation (25) becomes

$${\mathrm{N}}_{\mathrm{S}}={\displaystyle \frac{{\mathrm{S}}_{\mathrm{G}}}{{\mathrm{S}}_{{\mathrm{G}}_0}}={\displaystyle \frac{k_{hnf}}{k_f}{\left({\displaystyle \frac{\mathrm{\partial }\theta }{\mathrm{\partial }r}}\right)}^2+{\displaystyle \frac{{\mu }_{hnf}}{{\mu }_f}{\displaystyle \frac{B_r}{\gamma }{\left({\displaystyle \frac{\mathrm{\partial }w}{\mathrm{\partial }r}}\right)}^2.}}}}$$ (26)

There are two parts of equation (26), entropy generation because of conduction $({\mathrm{N}}_s{)}_{cond}$ and entropy generation on account of viscous effects $({\mathrm{N}}_s{)}_{visc}$ .

Furthermore, the ratio of entropy because of conduction to the total entropy is known as Bejan number, ⁴¹

$$\mathrm{Be}={\displaystyle \frac{{({\mathrm{N}}_s)}_{cond}}{{({\mathrm{N}}_s)}_{cond}+{({\mathrm{N}}_s)}_{visc}}.}$$ (27)

Results and discussion

This section includes the graphical explanation of analytical results attained in above portion. The influences of different physical parameters on the behavior of flow are inspected. The changes in behavior of flow for symmetric shape of multiple stenosis (i.e. n = 2) and anti-symmetric shape of multiple stenosis (i.e. n = 6) are also discussed. The graphs are plotted by computer coding on Mathematica 12. In Figures 2–7, the graphs are plotted for velocity $w(r,z)$ versus r (radial coordinate) for different rising values of physical parameters. Figure 2 explains the relationship among the velocity $w(r,z)$ and flow rate Q. It depicts that velocity of fluid has lower value at the mean of channel while it increases with the advancing value of Q and gets its higher value near the walls of channel for both cases of symmetric as well as anti-symmetric multiple stenosis. It also clarifies that higher values of Q results the rapid motion of fluid. Moreover, magnitude of velocity declines toward non-symmetric multiple stenosis. The velocity diminishes for enhancing values of electro-osmotic parameter m at the center of channel, its behavior totally changes near the walls due to decrease in Debye length and it shows more descending behavior towards the shape with $n=6$ as Figure 3 displays. Figure 4 illustrates that velocity profile decreases for improving values of ${\mathrm{U}}_{\mathrm{HS}}$ for both shapes of multiple stenosis and it also declines towards non-symmetric multiple stenosis. Figure 5 talks about the impact of height of stenosis ${\delta }_l$ on flow behavior. It indicates the improvement in the velocity graph for incrementing values of ${\delta }_l$ in both shapes of stenosis. The magnitude of fluid velocity raised from its lower value at mid of the channel and gets maximum magnitude near the walls for increasing height of stenosis. It also describes that non-symmetric stenosis has higher velocity values than symmetric stenosis. Figure 6 gives us information about the effects of advancing values of Darcy number $D_a$ . It displays that velocity decreases close to the mid of channel for higher values of Darcy number, but its behavior reverses while moving near to the wall because of increase in permeability for both shapes of stenosis. Figure 7 provides the belongings of slip parameter α with velocity profile. It reveals that as we give increment to the values of slip parameter, it reduces the motion of fluid near the axis of channel, but its behavior also reverses near the walls due to slip effect whether there is symmetric or non-symmetric shape of stenosis. Also, velocity graph for non-symmetric shape is down in comparison with symmetric shape. The graphical analysis of velocity profile for various involved parameters shows its axially symmetric behavior.

Figure 2.

Velocity profile for Q.

Figure 3.

Velocity profile for m.

Figure 4.

Velocity profile for $U_{HS}$ .

Figure 5.

Velocity profile for ${\delta }_l$ .

Figure 6.

Velocity profile for $D_a$ .

Figure 7.

Velocity profile for $\alpha$ .

Figures 8–13 provide the consequences of dependence of the wall shear stress on different physical parameters. It makes clear from Figure 8 that the wall shear stress ${\tau }_{\mathrm{w}}$ gains higher magnitude for quick flow rate. Figure 9 and Figure 10 explain the dependence of shear stress ${\tau }_{\mathrm{w}}$ on ${\mathrm{U}}_{\mathrm{HS}}$ and m respectively. The graph of shear stress increases for the rising values of both ${\mathrm{U}}_{\mathrm{HS}}$ and m. Figure 11 describes the effect of height of stenosis on ${\tau }_{\mathrm{w}}$ . The wall shear stress has larger magnitude when artery becomes narrower for higher stenosis. Figure 12 exhibits the rise of shear stress graph for successive values of Darcy number. Figure 13 unveils that the stress values drop for the advancing values of slip parameter α.

Figure 8.

Shear stress for Q.

Figure 9.

Shear stress for ${\mathit{U}}_{\mathit{H}\mathit{S}}$ .

Figure 10.

Shear stress for m.

Figure 11.

Shear stress for ${\delta }_l$ .

Figure 12.

Shear stress for $D_a$ .

Figure 13.

Shear stress for $\alpha$ .

Figures 14–19 give graphical representation of the temperature profile for several involved parameters. Figure 14 provides details about the temperature profile for increasing flow rate. It shows that the temperature enhances for rapid flow of fluid and having maximum altitude at the center of the channel. It begins to descend from center towards walls and vanishes at the boundary. Figure 15 demonstrates that temperature graph gets height for the larger values of $U_{HS}$ . Figure 16 shows the impacts of Joule heating parameter S on the temperature graph. The magnitude of the temperature gains increment by giving increment to the values of S. Figure 17 describes the relationship between temperature and Brinkman number ${\mathrm{B}}_{\mathrm{r}}$ . It unveils that the temperature enhances by giving larger magnitude to Brinkman number. The temperature profile advances for progressive values of Darcy number ${\mathrm{D}}_{\mathrm{a}}$ as Figure 18 reveals. Figure 19 explains the effect of the slip parameter α on the temperature graph. It clears that the temperature profile gets improvement for enlarging values of α. The temperature profile shows more progressive behavior for all discussed parameters in the case of non-uniform shape of stenosis than uniform shape. The graphical representation of entropy generation ${\mathrm{N}}_{\mathrm{S}}$ is appeared in Figures 20–26. These figures depict that ${\mathrm{N}}_{\mathrm{S}}$ attains more increasing values for non-uniform shapes of stenosis than uniform shapes for all discussed parameters except ${\mathrm{D}}_{\mathrm{a}}$ . In the case of increasing value of Darcy number, ${\mathrm{N}}_{\mathrm{S}}$ shows similar behavior for both shapes of stenosis. Figure 20 displays that entropy generation has larger magnitude for speedy flow rate. Figure 21 indicates rising behavior of ${\mathrm{N}}_{\mathrm{S}}$ for the rising values of ${\mathrm{U}}_{\mathrm{HS}}$ . For incrementing values of electro-osmotic parameter, ${\mathrm{N}}_{\mathrm{S}}$ has higher magnitude as observed from Figure 22. Figure 23 exhibits that ${\mathrm{N}}_{\mathrm{S}}$ is an increasing function of Brinkman number ${\mathrm{B}}_{\mathrm{r}}$ . Figure 24 illustrates that entropy graph also rises for improving values of Darcy number. Figures 25 and 26 describe the effect of physical parameters $\alpha$ and γ on the behavior of ${\mathrm{N}}_{\mathrm{S}}$ . An increment in the values of $\alpha$ and γ gives advancing and declining behavior of ${\mathrm{N}}_{\mathrm{S}}$ respectivetly. It is observed from all the figures giving graphical analysis of entropy generation that entropy has lowest magnitude near the axis of the channel. It gradually rises and reaches its highest value near the boundary of the channel for both symmetric and non-symmetric stenosis. Figures 27–33 explain the effect of several constraints on $Be$ . These figures allow to write that the effect of Bejan number $Be$ is exactly same for enlarging values of different parameters whether stenosis has symmetric shape or anti-symmetric shape. $Be$ plot for Q i.e., Figure 27 tells the advancing behavior of $Be$ for higher flow rate. Figure 28 depicts $Be$ as an increasing function of ${\mathrm{U}}_{\mathrm{HS}}$ . Figure 29 shows belongings of m with $Be$ . It is observed that $Be$ graph attains height for larger values of m. Figure 30 gives dependence of $Be$ on ${\mathrm{B}}_{\mathrm{r}}$ . It describes that $Be$ graph moves up by incrementing the magnitude of ${\mathrm{B}}_{\mathrm{r}}$ . Graph of $Be$ is also affected by incrementing value of Darcy number, it shows successive behavior for successive values of ${\mathrm{D}}_{\mathrm{a}}$ as shown in Figure 31. Figures 32 and 33 display the changing behavior of $Be$ for distinct values of α and γ. $Be$ illustrates the ascending behavior for ascending values of both parameters α and γ. Moreover, streamlines are also plotted to analyze the flow behavior for both shapes of stenosis in Figures 34–41. These figures depict that there are vortices at the points on the axis where the walls have symmetric stenosis. But these vortices are descending in size for higher values of Q. Similarly, there are also vortices appearing for non-symmetric stenosis but their size declines for increasing values of Q. Figures 38–41 explain flow behavior for enhancing values of m. There is trapping at the points where the walls have stenosis for both shapes of stenosis. The number of trappings rises for rising values of m in the case of symmetric shape of stenosis but trappings are less in count for larger values of m in the case of non-symmytric shape of stenosis. Furthermore, it is also observed from streamline plotting that the walls and vortices as well are symmetrical shape for multiple stenosis of uniform shape, while the walls and vortices as well are anti-symmetric in shape for non-uniform shape of stenosis.

Figure 14.

Temperature profile for Q.

Figure 15.

Temperature profile for $U_{HS}$ .

Figure 16.

Temperature profile for S.

Figure 17.

Temperature profile for $B_r$ .

Figure 18.

Temperature profile for $D_a$ .

Figure 19.

Temperature profile for $\alpha$ .

Figure 20.

Entropy for Q.

Figure 21.

Entropy for $U_{HS}$ .

Figure 22.

Entropy for m.

Figure 23.

Entropy for $B_r$ .

Figure 24.

Entropy for $D_a$ .

Figure 25.

Entropy for $\alpha$ .

Figure 26.

Entropy for $\gamma$ .

Figure 27.

Be plot for Q.

Figure 28.

Be plot for $U_{HS}$ .

Figure 29.

Be plot for m.

Figure 30.

Be plot for $B_r$ .

Figure 31.

Be plot for $D_a$ .

Figure 32.

Be plot for $\alpha$ .

Figure 33.

Be plot for γ.

Figure 34.

Streamline for Q = 1, n = 2.

Figure 35.

Streamline for Q = 2, n = 2.

Figure 36.

Streamline for Q = 1.3, n = 6.

Figure 37.

Streamline for Q = 1.5, n = 6.

Figure 38.

Streamline for m = 2, n = 2.

Figure 39.

Streamline for m = 3, n = 2.

Figure 40.

Streamline for m = 2, n = 6.

Figure 41.

Streamline for m = 3, n = 6.

Stability analysis

The stability of solution for velocity of blood flow $w(r,z)$ is analyzed in this section. The critical points are found by giving numerical values to the involved parameters, and stability at these points is observed by finding eigen values of Hessian matrix of $w(r,z)$ given as

$$H_w=\left[\begin{array}{cc}{\displaystyle \frac{{\mathrm{\partial }}^{2}w}{\mathrm{\partial }{r}^2}}& {\displaystyle \frac{{\mathrm{\partial }}^{2}w}{\mathrm{\partial }z\mathrm{\partial }r}}\\ {\displaystyle \frac{{\mathrm{\partial }}^{2}w}{\mathrm{\partial }r\mathrm{\partial }z}}& {\displaystyle \frac{{\mathrm{\partial }}^{2}w}{\mathrm{\partial }{z}^2}}\end{array}\right].$$

The eigen values of the Hessian matrix on the axis of the channel (i.e. $r=0$ ) for both shapes of stenosis are real and have opposite signs for different values of z as shown in

Tables 3 and 4 It depicts that $H_w$ is indefinite matrix, therefore flow velocity is not stable at these points. The velocity solution at other numerically determined critical points is also unstable as mentioned in the Tables 3 and 4. The velocity solution is stable at other points. The critical point where the velocity has maximum value, or it behaves as a saddle point is mentioned in the Tables 3 and 4 for uniform and non-uniform shape of stenosis respectively. Moreover, the eigen values that are given in Tables 3 and 4 are all real.

Table 3.

Behavior of velocity at critical points for $\alpha =0.1,\mathrm{m}=2,{\delta }_{\mathrm{l}}=0.1,{\mathrm{D}}_{\mathrm{a}}=0.1,{\varphi }_1=0.02,{\varphi }_1=0.03,$ $Q=0.25,h_1=0.5,h_2=2,h_3=3.5,U=1,n=2$ .

Critical point $(r,z)$		Sign of Eigen Values of Hessian Matrix	Classification of Critical Point	Stability
$z$	$r$
0.8	0	Opposite	Saddle	Unstable
	$1.06064007984749$	Same and Negative	Maximum	Unstable
	$-1.06064007984749$	Same and Negative	Maximum	Unstable
1	0	Opposite	Saddle	Unstable
	$1.04887316361867$	Same and Negative	Maximum	Unstable
	$-1.04887316361867$	Same and Negative	Maximum	Unstable
2.6	0	Opposite	Saddle	Unstable
	1.05178837274242	Same and Negative	Maximum	Unstable
	$-1.05178837274242$	Same and Negative	Maximum	Unstable
2.8	0	Opposite	Saddle	Unstable
	1.07568864559133	Same and Negative	Max	Unstable
	−1.07568864559133	Same and Negative	Max	Unstable
4	0	Opposite	Saddle	Unstable
	$1.04886754141470$	Same and Negative	Maximum	Unstable
	$-1.04886754141470$	Same and Negative	Maximum	Unstable

Table 4.

Behavior of velocity at critical points for $\alpha =0.1,m=2,{\delta }_l=0.1,D_a=0.1,{\varphi }_1=0.02,{\varphi }_1=0.03,$ $Q=0.25,h_1=0.5,h_2=2,h_3=3.5,U=1,n=6$ .

Critical point $(r,z)$		Sign of Eigen Values of Hessian Matrix	Classification of Critical Point	Stability
$z$	$r$
0.8	0	Opposite	Saddle	Unstable
	1.08537010970125	Opposite	Saddle	Unstable
	$-1.08537010970125$	Opposite	Saddle	Unstable
1	0	Opposite	Saddle	Unstable
	1.06124894237561	Same and Negative	Max	Unstable
	$-1.06124894237561$	Same and Negative	Max	Unstable
2.6	0	Opposite	Saddle	Unstable
	1.05250878837873	Same and Negative	Max	Unstable
	$-1.05250878837873$	Same and Negative	Max	Unstable
2.8	0	Opposite	Saddle	Unstable
	1.05445628387879	Same and Negative	Max	Unstable
	−1.05445628387879	Same and Negative	Max	Unstable
4	0	Opposite	Saddle	Unstable
	1.06124894237561	Same and Negative	Max	Unstable
	$-1.06124894237561$	Same and Negative	Max	Unstable

Conclusion

The flow behavior of hybrid nanofluid through an artery harmed by several stenosis is examined mathematically. The symmetric and anti-symmetric shapes of stenosis are included in this examination. The main findings of this study are

• The fluid moves quickly through the artery for increasing height of stenosis,

• The fluid flow slows down for higher values of slip parameter, Darcy number and electro-osmotic parameter near the axis, but its behavior reverses near the walls.

• The flow velocity shows axisymmetric behavior.

• Temperature has larger magnitude towards the axis of channel, and it vanishes on the boundary.

• Temperature rises rapidly for the symmetric shape of stenosis than the non-symmetric shape.

• Entropy analysis elucidates the lowest disorder near the mean of channel. It increases along the radial axis and gets higher towards the walls.

Nomenclature

( $\tilde{r}$ , $\tilde{z}$ )

Cylindrical coordinates

( $\tilde{u},\tilde{w}$ )

Radial and axial velocities

R

Non-stenotic radius of artery

$d_l$

Position of stenosis ( $l=1,2,3$ )

$s_l$

Length of stenosis ( $l=1,2,3$ )

$E_z$

Axial electric field

$i_e$

Current density

σ

Electrical resistivity of fluid

${\rho }_e$

Density of total ionic charge

E

Permittivity

e

Electronic charge

${\varphi }_1$

Concentration of single wall CNT

$c^{+},c^{-}$

Cations and anions density

$c_0$

Ions concentration

$U_{HS}$

Helmhotlz-Smoluchowski velocity

$T*$

Average temperature of electrolyte solution

hnf

Hybrid nanofluid

CNT

Carbon nanotubes

$K_B$

Boltzmann constant

$m$

Electro-osmotic parameter

$S$

Joule heating parameter

$k$

Thermal conductivity

$B_r$

Brinkman number

${\lambda }_d$

Debye-length

$\zeta$

Zeta potential

α

Slip parameter

$D_a$

Darcy number

$z^{*}$

Charge balance

${\varphi }_2$

Concentration of multi wall CNT

µ

Dynamic viscosity

$\tilde{\varphi }$

Electro-kinetic potential function

${\delta }_l^{*}$

Maximum height of stenosis in dimensional form

n ≥ 2

Multiple stenosis shape Parameter

Author biographies

Muhammad Hasnain Shahzad is a PhD Scholar at Department of Mathematics, University of the Punjab Lahore. His field of research is applied mathematics and computational fluid dynamics. He is working under the supervision of Dr Aziz Ullah Awan.

Dr Aziz Ullah Awan is an Associate Professor at Department of Mathematics, University of the Punjab Lahore. He has published many quality research articles in the field of applied Mathematics.

Salman Akhtar is a PhD research fellow at Department of Mathematics, Quaid-i-Azam University Islamabad. He has published many research articles under the supervision of Prof. Dr. Sohail Nadeem.

Dr. Sohail Nadeem is Professor of Mathematics at Quaid-i-Azam University Islamabad. His contribution to the field of applied Mathematics is recognized at national and international level. He has produced a good number of PhD students. Also, he is serving as editor of various well reputed international journals.