Electroosmotic and Gyrotactic Microorganisms Effects on MHD Al2O3-Cu/Blood Hybrid Nanofluid Flow through Multi-Stenosed Bifurcated Artery

⁴ Material Science Innovation and Modelling (MaSIM) Research Focus Area, North-West University (Mafikeng Campus), Private Bag X2046, Mmabatho 2735, South Africa

^*Correspondence: mmbhatti@sdust.edu.cn; muhammad09@shu.edu.cn; mubashirme@hotmail.com (M.M. Bhatti)

Front. Biosci. (Landmark Ed) 2024, 29(3), 110; https://doi.org/10.31083/j.fbl2903110

Submitted: 9 November 2023 | Revised: 18 December 2023 | Accepted: 3 January 2024 | Published: 19 March 2024

This is an open access article under the CC BY 4.0 license.

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Abstract

Background: The purpose of this study is to investigate the electroosmotic flow of a hybrid nanofluid (Al ${}_{2}$ O ${}_{3}$ -Cu/Blood) with gyrotactic microorganisms through a bifurcated artery with mild stenosis in both parent and daughter arteries. The flow is subjected to a uniform magnetic field, viscous dissipation, and a heat source. Methods: The governing equations undergo the non-dimensional transformation and coordinate conversion to regularize irregular boundaries, then solve the resulting system using the Crank-Nicolson method. Results: In both sections of the bifurcated artery (parent and daughter artery), the wall shear stress (WSS) profile decreases with increasing stenotic depth. Nusselt profile increases with an increase in the heat source parameter. Conclusions: The present endeavour can be beneficial for designing better biomedical devices and gaining insight into the hemodynamic flow for therapeutic applications in the biomedical sciences.

Keywords

bifurcated artery

Joule heating

stenosis

MHD flow

hybrid nanoparticles

1. Introduction

Numerous studies have been conducted to analyse the hemodynamic characteristics of blood flow through channels, pipes and tubes to understand the pathological mechanism that arises in the stenotic artery. The study provides researchers with insights into hemodynamic flow and facilitates the development of more effective preventative treatments for diseases. Arteriosclerosis, also known as stenosis, is a pathological phenomenon characterised by the accumulation of various substances such as lipids, proteins, fatty compounds, calcium, and other cellular debris along the walls of arteries. This accumulation can result in partial occlusion or complete blockage of the affected blood vessel. A mathematical model was developed by Young [1] to investigate the Newtonian flow within a time-dependent stenosed tube. The study’s findings indicate that the occurrence of stenosis within the artery disrupts the physiological processes of the cardiovascular system, ultimately resulting in severe pathological consequences. The study by Akbar et al. [2] delved into the intricacies of the non-Newtonian fluid model, specifically concerning blood flow within a tapered stenosed artery. The authors approached this investigation by considering blood as a Jeffery fluid. Shit and Roy [3] conducted a study on micropolar fluid to investigate the impact of induced magnetic fields on blood flow through the constricted artery. The study’s findings indicate a positive correlation between the Hartman number and stenosis height with an enhancement in microcirculation. Tripathi and Sharma [4] developed a mathematical model to analyse the two-phase hemodynamic flow through a stenosed artery, incorporating chemical reactions and radiation effects. The study illustrated a reduction in blood velocity adjacent to the arterial wall, as evidenced by the distortion of the velocity contour downstream and the shift of the tapering bolus towards the arterial wall. The study conducted by Khanduri and Sharma [5, 6] pertained to the examination of the impact of Hall effects on the flow of magnetohydrodynamic (MHD) fluid through a stenosed artery that has been affected by thrombosis. The study’s findings indicate a decline in the wall shear stress (WSS) profile as the Hartman number and stenotic depth increase. This phenomenon is caused by reduced blood flow near the arterial walls.

The study revealed that atherosclerotic plaque, which obstructs blood flow in arteries, tends to manifest in regions of complex geometry, such as those proximal to bifurcations, junctions, or areas of high curvature. The presence of arterial curvature and variations in the size of the cross-section both had a role in the preferred localization of the plaque at the arterial walls. Tan et al. [7] explored the blood flow through the bifurcated artery under the gravity effect and irregular stenosis at the parent artery. Srinivasacharya and Rao [8] have designed a mathematical model to investigate the hemodynamic behaviour of blood flow containing copper nanoparticles within a constricted bifurcated artery. The study’s findings indicate a notable alteration in the flow rate and impedance near the apex. This phenomenon is attributed to the occurrence of backflow at the junction and the presence of secondary flow in the region proximal to the apex. Moreover, the researchers [9] proceeded with their investigation by examining the behaviour of a couple’s stress fluid within a bifurcated artery. Shahzadi [10] conducted a theoretical analysis to explore the bio-nanofluid containing copper nanoparticles as a therapeutic agent in the bifurcated artery with compliant walls. The non-Newtonian Casson fluid was investigated by Shahzad et al. [11] in the context of a bifurcated channel featuring stenosis and elastic walls.

The study of blood rheology is affected by the application of external magnetic and electric fields, resulting in the reduction of the fluid flow, and such type of flow is commonly referred to as electro-magneto hydrodynamics (EMHD) flow. Kolin [12] introduced the concept of MHD in his medical research. The experiments demonstrate that applying a transverse magnetic field to an electrical field decelerates fluid motion. The empirical finding indicated a decrease of $30\%$ in the volumetric blood flux within the duct subject to a high magnetic field of 10 Tesla. Ahmed and Nadeem [13] have constructed a mathematical model encompassing six distinct types of stenosis and have subsequently conducted an investigation on the effect of MHD and hybrid nanoparticles on micropolar fluid. The study revealed an increase in velocity and WSS with an enhancement in nanoparticle concentrations, while an inverse trend was observed for the impedance profile. The MHD fluid flow in an artery was explored numerically by Joshua et al. [14] by employing a non-Newtonian Cross-rheological model. The researchers have considered the presence of multiple stenoses along the arterial walls and have conducted a comprehensive investigation by varying the magnetic parameter, Reynolds number, and stenosis height throughout the entire length of the artery. Kumar et al. [15] analysed MHD fluid flow in a bifurcated artery with permeability, considering the effect of heat source and chemical reaction. The EMHD flow of Au-blood through the inclined constricted artery under periodic body acceleration was investigated by Manchi and Ponalagusamy [16]. The researchers also emphasised the shape of nanoparticles and concluded that the spherical nanoparticles exhibit a greater heat flux at the arterial walls than other nanoparticle shapes. Sharma et al. [17] formulated the MHD two-phase blood flow model by considering the variable viscosity in a curved artery. The study’s findings suggest that atherosclerosis formation positively correlates with curvature and permeability, whereas the heat source parameter reduces the risk of atherosclerosis formation. The study conducted by Mishra et al. [18] delved into the characteristics of magnetohydrodynamic (MHD) nanofluid as it flows through a constricted artery while considering the Soret and Dufour effects. The optimisation of heat transfer in nanofluid blood flow through the stenosed artery was discussed by Sharma et al. [19]. The authors examined the hematocrit-dependent viscosity model and demonstrated that increased hematocrit level and stenosis depth lead to reduced fluid velocity.

Electrokinetics refers to the phenomenon wherein particles are propelled in response to electrical potential differences. Electroosmosis is an electrokinetic phenomenon that arises from applying an external electric field to a charged surface. The flow of an electrically conductive fluid within the blood vessels establishes a net charge at the vessel walls. This, in turn, leads to the development of an opposite charge due to the principle of electro-neutrality within the electrical double layer in close proximity to the walls. The investigation by Mekheimer et al. [20] focused on analysing the impact of electroosmotic and bifurcation effects on the hemodynamic flow in a bifurcated artery with stenosis along the parent artery. Hybrid nanofluid flow through a diseased artery with aneurysmal and stenosed segments at the walls was discussed by Abdelsalam et al. [21]. The study’s findings indicate a correlation between the nanoparticle shape factor and the fluid velocity profile. The study suggests that this information can be applied to improve drug delivery systems. Akhtar et al. [22] elucidated the electroosmotic modulated flow through an artery with multiple stenoses. The dependence of trapping symmetry on the symmetry of multiple stenoses and regulating fluid velocity, temperature, and velocity through electroosmosis are notable findings in this study. In their research, Akram et al. [23] compared the Tiwari-Das model and the modified Buongiorno model to investigate the electroosmotic nanofluid flow under peristaltic pumping. In their study, Khanduri et al. [24] conducted a sensitivity analysis on the MHD fluid flow through a curved artery with stenosis at the wall and thrombosis at the catheter. The WSS profile negatively correlates with the Debye-Huckel parameter and Hartmann number, whereas the impedance profile displays an opposite trend. The EMHD micropolar fluid was analysed by Manchi et al. [25] in the context of a bifurcated artery, considering the effects of Joule heating and body acceleration. The topic of discussion by Zaher et al. [26] pertained to the flow of non-Newtonian fluid with microorganisms in the presence of electroosmotic flow within the boundary layer. In non-Darcian fluid, the velocity is observed to be lower when compared to that of Darcian fluid.

Both theoretical and experimental results have underscored the significance of nanoparticles in the biomedical domain, as they have been shown to augment the efficacy of delivering diagnostic and therapeutic agents. Numerous investigations have been undertaken to examine the novel potential of nanoparticles at the molecular scale within the realm of life sciences. The successful delivery of nanoparticles into the artery is primarily determined by their physical characteristics, including shape, size, and surface absorption properties. Nanofluid is a suspension of nanoparticles in a base fluid, whereas hybrid nanofluid is a suspension of two or more types of nanoparticles in the base fluid. Synthesis of hybrid nanofluids offers the advantage of incorporating diverse materials’ physical and thermal properties into a singular, homogeneous phase. This results in remarkable physicochemical properties in the resulting synthetic hybrid nanofluid. The hemodynamic flow through permeable walls was investigated by Ellahi et al. [27], employing the homotopy analysis method. The hybrid nanofluid flow through a stenosed artery was analysed by Gandhi et al. [28] in the presence of Joule heating and viscous dissipation. The findings of their research indicate that an increase in the Darcy number results in an enhancement of the velocity profile. This can be attributed to the lower resistance offered by the medium permeability. The researcher, Basha [29], analysed the inclined, uneven, and stenosed artery to investigate the biomagnetic blood flow of Au-Cu. The study conducted by Rishu [30] involved an investigation of the behaviour of Au-Cu hybrid nanoparticles in the context of blood flow through an artery with overlapping stenosis at the walls. The study revealed that an augmentation in the Casson fluid parameter enhances both velocity and temperature. The researchers suggest their findings may have potential applications in nano-pharmacology and biomedical sciences. Kumawat et al. [31] investigated the behaviour of a two-phase power-law nanofluid within a stenosed artery characterized by curvature. Their findings indicate that the presence of arterial curvature augments the likelihood of atherosclerosis deposition. In this investigation, we have opted for Al ${}_{2}$ O ${}_{3}$ -Cu nanoparticles due to their exceptional anti-bacterial and anti-viral properties, offering potential applications in the biomedical domain. The chemical stability and convenient accessibility of these nanoparticles render them an ideal selection for our research.

Bioconvection refers to the phenomenon whereby the macroscopic motion of microorganisms occurs due to spatial variations in density. Microorganisms exhibit self-propulsion, whereas nanoparticles lack this capability. The phenomenon of bioconvection can be observed under conditions where the concentration of nanoparticles is relatively low. The phenomenon of bioconvection is observed due to the instability caused by spatial variation, which leads to the upward movement of microorganisms and the formation of a dense layer at the surface. This layer becomes unstable and results in the crumbling of microorganisms, further enhancing the bioconvection process. Bhatti et al. [32] have discussed the peristaltic motion of Jeffery nanofluid in the presence of microorganisms and a variable magnetic field. The bioconvection movement of microorganisms in a hybrid nanofluid through a porous stretching sheet was investigated by Alharbi et al. [33]. The study conducted by Sharma et al. [34] delved into the dynamics of magnetohydrodynamic (MHD) fluid flow in the presence of microorganisms over an inclined stretching sheet. Mekheimer et al. [35] conducted a study on the delivery of drugs via nanoparticles in the presence of hemodynamic flow within diseased organs. The study conducted by Mostapha et al. [36] was a theoretical investigation of the flow of peristaltic-induced nanofluid, wherein motile gyrotactic microorganisms are observed to move through an endoscope. The study considered the effects of radiation and chemical interaction while incorporating the Soret and Dufour scheme. Khan [37] applied the Homotopy Analysis Method (HAM) to explore the bioconvection phenomena in a hybrid nanofluid with gyrotactic microorganisms. Furthermore, Khan et al. [38] conducted a study on entropy generation analysis for the hybrid nanofluid (Ag-Al ${}_{2}$ O ${}_{3}$ /H ${}_{2}$ O) past a semi-infinite plate. The study mentioned above indicates that no research has been conducted on the electroosmotic flow with gyrotactic microorganisms through the bifurcated artery.

In the aforementioned studies, a notable research gap becomes evident due to the need for more investigations into the behaviour of Al ${}_{2}$ O ${}_{3}$ -Cu/Blood hybrid nanofluids containing gyrotactic microorganisms. Specifically, there is a lack of examination of the hybrid nanofluid flow through multi-stenosed bifurcated arteries with the electroosmotic effect. Filling this research gap is crucial for comprehensively understanding the flow characteristics, thermal characteristics, and biological interactions within such complex vascular conditions. The previously mentioned study has inspired and formed the basis for our ongoing research on the hybrid nanofluids flow in the bifurcated arteries with mild stenosis in both parent and daughter arteries. This study introduces a model that examines the blood flow behavior containing suspended nanoparticles of Al ${}_{2}$ O ${}_{3}$ -Cu, in the presence of gyrotactic microorganisms and electroosmotic force. The governing equations are made non-dimensional and subsequently solved using the Crank-Nicolson scheme. The resulting outcomes are analyzed, and the effects of various parameters are clarified by plotting velocity, temperature, concentration, microorganisms, flow rate, and WSS profiles.

The novelty of the present work:

• To investigate the influence of hybrid nanofluid (Al ${}_{2}$ O ${}_{3}$ -Cu/Blood) flow with gyrotactic microorganism through the stenosed bifurcated artery.

• Examine the blood flow characteristics with combined effect of Joule heating, electroosmosis, heat source and viscous dissipation.

2. Model Formulation

Consider a fully developed, unsteady, laminar, incompressible two-dimensional MHD blood flow passing through the stenosed bifurcated artery. The arteries are assumed to be straight, circular cylinders passing the centre line of the parent artery. The bifurcated artery has overlapping stenosis at the parent artery, while the irregular stenosis at the daughter arteries is shown in Fig. 1. Let’s assume the cylindrical coordinate system $r_{1}^{*},\theta_{1}^{*}$ and $z_{1}^{*}$ to represent the material point in which $z_{1}^{*}$ is along the centerline of the parent artery and $r_{1}^{*}$ and $\theta_{1}^{*}$ are assumed to be radial and circumferential directions, respectively. The flow is assumed to be antisymmetric, so all the variables are independent of $\theta$ . A uniform magnetic field denoted by $\textbf{B}=(B_{0},0,0)$ and an electric field denoted by $\textbf{E}=(0,0,E_{0})$ are applied to the flow of blood, where $B_{0}$ is constant. The gravitational force acts in the downward direction and the induced magnetic field is assumed to be negligible due to low magnetic Reynold’s number assumption. The potential for a flow separation zone is eliminated by incorporating curvature at the lateral junction and bifurcation’s apex.

Fig. 1.

Representation of bifurcated stenosed artery.

The geometry of the bifurcated artery with multi-stenosis in the parent artery and an overlapping stenosis in the daughter artery is expressed as follows [10]:

(1) $R_{1}^{*}(z_{1}^{*})=\begin{cases}R_{0},&0<z_{1}^{*}\leq d\\ R_{0}-\frac{6}{5L_{0}^{4}}\delta(11(z_{1}^{*}-d)L_{0}^{3}-47(z_{1}^{*}-d)^{2}L% _{0}^{2}+72(z_{1}^{*}-d)^{3}L_{0}-36(z_{1}^{*}-d)^{4}),&d<z_{1}^{*}\leq d+L_{0% }\\ R_{0},&d+L_{0}<z_{1}^{*}\leq z_{1}\\ R_{0}+r_{1}-\sqrt{\bigg{(}r_{1}^{2}-\frac{R_{0}^{2}}{L_{0}^{2}}(z_{1}^{*}-z_{1% })^{2}\bigg{)}}&z_{1}<z_{1}^{*}\leq z_{2}\\ 2r_{0}\sec\eta+\frac{R_{0}}{L_{0}}\biggl{(}z_{1}^{*}-z_{2}\biggr{)}\tan\eta&z_% {2}<z_{1}^{*}\leq z_{4}+0.5L_{0}\\ 2r_{0}\sec\eta+\frac{R_{0}}{L_{0}}\biggl{(}z_{1}^{*}-z_{2}\biggr{)}\tan\eta-A^% {*}&z_{4}+0.5L_{0}<z_{1}^{*}\leq z_{4}+2.5L_{0},\\ 2r_{0}\sec\eta++\frac{R_{0}}{L_{0}}\biggl{(}z_{1}^{*}-z_{2}\biggr{)}\tan\eta,&% z_{4}+2.5L_{0}<z_{1}^{*}\leq z_{max},\end{cases}$

where $A^{*}=2\delta_{1}\bigg{\{}\cos\bigg{(}\frac{2\pi}{L_{0}}\bigg{(}\frac{z_{1}^{*% }-d}{4}\bigg{)}-0.25\bigg{)}-0.07\cos\bigg{(}\frac{128\pi}{L_{0}}\bigg{(}z_{1}% ^{*}-d-0.5L_{0}\bigg{)}\bigg{)}\bigg{\}}\tan(\eta)$ , $R_{0}$ represents the radius of the normal artery, $L_{0}$ is the length of the diseased segment and Kroncecker delta function is defined as:

(2) $\delta_{ij}=\begin{cases}0~{}~{}~{}~{}~{}~{}\text{for}~{}~{}~{}~{}0\leq z_{1}^% {*}\leq z_{3},\\ 1~{}~{}~{}~{}~{}~{}\text{for}~{}~{}~{}~{}z_{3}\leq z_{1}^{*}\leq z_{max},\end{cases}$

The inner wall is represent as:

(3) $R_{2}^{*}(z_{1}^{*})=\begin{cases}0,&0<z_{1}^{*}\leq z_{3},\\ \sqrt{\tilde{r}_{1}^{2}-\bigg{(}R_{0}\bigg{(}\frac{z_{1}^{*}-z_{3}}{L_{0}}% \bigg{)}-\tilde{r}_{1}\bigg{)}^{2}},&z_{3}<z_{1}^{*}\leq z_{4},\\ \frac{R_{0}}{L_{0}}(z_{1}^{*}-z_{2})\tan\eta,&z_{4}<z_{1}^{*}\leq z_{max}.\end% {cases}$

Where, d represent location of stenosis, $z_{1}$ denotes the location of insert lateral junction, $z_{max}$ signifies the maximum length of stenosis; The radii of curvature at the flow divider and at lateral junction is represented by:

(4) $\tilde{r}_{1}=\frac{(z_{3}-z_{2})R_{0}\sin\eta}{L_{0}(1-\sin\eta)},~{}~{}r_{1}% =\frac{R_{0}-2r_{0}\sec\eta}{\cos\eta-1},~{}$

The positional of the lateral junction offset, apex, and curvature offset at the inner wall are presented as follows:

(5)

$\displaystyle z_{2}=z_{1}+r_{1}\bigg{(}\frac{L_{0}}{R_{0}}\bigg{)}\sin\eta,$

(6)

$\displaystyle z_{3}=z_{2}+q_{1}L_{0},$

(7)

$\displaystyle z_{4}=z_{3}+\tilde{r}_{1}\bigg{(}\frac{L_{0}}{R_{0}}\bigg{)}(1-% \sin\eta).$

Where $\eta$ is half of the bifurcation angle and value of $q_{1}$ lies between 0.1 and 0.5, $z_{max}$ represents the finite length of the bifurcated artery.

2.1 Governing Equations

Assuming the aforementioned conditions and utilising the Boussinesq approximation, the equations dictating the flow can be expressed as follows [30]:

Continuity Equation:

(8) $\frac{\partial u_{1}^{*}}{\partial r_{1}^{*}}+\frac{u_{1}^{*}}{r_{1}^{*}}+% \frac{\partial w_{1}^{*}}{\partial z_{1}^{*}}=0,$

Momentum Equation: $r_{1}^{*}$ -direction:

(9) $\displaystyle\rho_{hnf}\bigg{[}\frac{\partial u_{1}^{*}}{\partial t_{1}^{*}}+u% _{1}^{*}\frac{\partial u_{1}^{*}}{\partial r_{1}^{*}}+w_{1}^{*}\frac{\partial u% _{1}^{*}}{\partial z_{1}^{*}}\bigg{]}=-\frac{\partial p_{1}^{*}}{\partial r_{1% }^{*}}+\frac{1}{r_{1}^{*}}\frac{\partial}{\partial r_{1}^{*}}\bigg{[}\mu_{hnf}% r_{1}^{*}\frac{\partial u_{1}^{*}}{\partial r_{1}^{*}}\bigg{]}+\frac{1}{2}% \frac{\partial}{\partial z_{1}^{*}}\bigg{[}\mu_{hnf}\bigg{(}\frac{\partial w_{% 1}^{*}}{\partial r_{1}^{*}}+\frac{\partial u_{1}^{*}}{\partial z_{1}^{*}}\bigg% {)}\bigg{]}-\mu_{hnf}\frac{u_{1}^{*}}{r_{1}^{*2}},$

$z_{1}^{*}$ -direction:

(10) $\displaystyle\rho_{hnf}\bigg{[}\frac{\partial w_{1}^{*}}{\partial t_{1}^{*}}+u% _{1}^{*}\frac{\partial w_{1}^{*}}{\partial r_{1}^{*}}+w_{1}^{*}\frac{\partial w% _{1}^{*}}{\partial z_{1}^{*}}\bigg{]}=-\frac{\partial p_{1}^{*}}{\partial z_{1% }^{*}}+\frac{1}{2}\frac{1}{r_{1}^{*}}\frac{\partial}{\partial r_{1}^{*}}\bigg{% [}\mu_{hnf}r_{1}^{*}\bigg{(}\frac{\partial u_{1}^{*}}{\partial z_{1}^{*}}+% \frac{\partial w_{1}^{*}}{\partial r_{1}^{*}}\bigg{)}\bigg{]}+\frac{\partial}{% \partial z_{1}^{*}}\bigg{[}\mu_{hnf}\frac{\partial w_{1}^{*}}{\partial z_{1}^{% *}}\bigg{]}\\ \displaystyle+(\rho\gamma)_{hnf}g\biggl{[}(\tilde{T}^{*}-\tilde{T}^{*}_{1})+(% \tilde{C}^{*}-\tilde{C}^{*}_{1})-(\tilde{n}^{*}-\tilde{n}^{*}_{1})\biggr{]}% \cos(\eta\delta_{ij})+G(t_{1}^{*})+\rho_{e}E_{0}-\sigma_{hnf}B_{0}^{2}w_{1}^{*},$

Energy Equation:

(11) $(\rho C_{p})_{hnf}\bigg{[}\frac{\partial\tilde{T}^{*}}{\partial t_{1}^{*}}+u_{% 1}^{*}\frac{\partial\tilde{T}^{*}}{\partial r_{1}^{*}}+w_{1}^{*}\frac{\partial% \tilde{T}^{*}}{\partial z_{1}^{*}}\bigg{]}=k_{hnf}\bigg{[}\frac{\partial^{2}% \tilde{T}^{*}}{\partial r_{1}^{*2}}+\frac{1}{r_{1}^{*}}\frac{\partial\tilde{T}% ^{*}}{\partial r_{1}^{*}}+\frac{\partial^{2}\tilde{T}^{*}}{\partial z_{1}^{*2}% }\bigg{]}+\tilde{Q}_{0}+\sigma_{hnf}B_{0}^{2}w_{1}^{*2}+\sigma_{nf}E_{0}^{2}+% \phi^{*},$

where,

$\phi^{*}=2\mu_{hnf}\bigg{[}\bigg{(}\frac{\partial u_{1}^{*}}{\partial r_{1}^{*% }}\bigg{)}^{2}+\bigg{(}\frac{u_{1}^{*}}{r_{1}^{*}}\bigg{)}^{2}+\bigg{(}\frac{% \partial w_{1}^{*}}{\partial z_{1}^{*}}\bigg{)}^{2}+\frac{1}{2}\bigg{(}\frac{% \partial u_{1}^{*}}{\partial z_{1}^{*}}+\frac{\partial w_{1}^{*}}{\partial r_{% 1}^{*}}\bigg{)}^{2}\bigg{]}.$

Concentration Equation:

(12) $\frac{\partial\tilde{C}^{*}}{\partial t_{1}^{*}}+u_{1}^{*}\frac{\partial\tilde% {C}^{*}}{\partial r_{1}^{*}}+w_{1}^{*}\frac{\partial\tilde{C}^{*}}{\partial z_% {1}^{*}}=D_{b}\biggl{[}\frac{\partial^{2}\tilde{C}^{*}}{\partial r_{1}^{*2}}+% \frac{1}{r_{1}^{*}}\frac{\partial\tilde{C}^{*}}{\partial r_{1}^{*}}+\frac{% \partial^{2}\tilde{C}^{*}}{\partial z_{1}^{*2}}\biggr{]}-R_{c}(\tilde{C}^{*}-% \tilde{C_{1}}^{*})$

Microorganism Equation:

(13) $\frac{\partial\tilde{n}^{*}}{\partial t_{1}^{*}}+u_{1}^{*}\frac{\partial\tilde% {n}^{*}}{\partial r_{1}^{*}}+w_{1}^{*}\frac{\partial\tilde{n}^{*}}{\partial z_% {1}^{*}}+\frac{bW_{c}}{\tilde{C_{0}}^{*}-\tilde{C_{w}}^{*}}\biggl{[}\frac{% \partial}{\partial r_{1}^{*}}\biggl{(}\frac{\tilde{n}^{*}\partial\tilde{C}^{*}% }{\partial r_{1}^{*}}+\frac{\partial}{\partial z_{1}^{*}}\biggl{(}\frac{\tilde% {n}^{*}\partial\tilde{C}^{*}}{\partial z_{1}^{*}}\biggr{)}\biggr{]}=D_{n}% \biggl{[}\frac{\partial^{2}\tilde{n}^{*}}{\partial r_{1}^{*2}}+\frac{1}{r_{1}^% {*}}\frac{\partial\tilde{n}^{*}}{\partial r_{1}^{*}}+\frac{\partial^{2}\tilde{% n}^{*}}{\partial z_{1}^{*2}}\biggr{]}$

The boundary conditions are:

(14) $w_{1}^{*}=0,~{}~{}\tilde{T}^{*}=\tilde{T}^{*}_{w}~{}~{}\tilde{C}^{*}=\tilde{C}% ^{*}_{w}~{}~{}\tilde{n}^{*}=\tilde{n}^{*}_{w}~{}~{}\text{at}~{}~{}r_{1}^{*}=R_% {1}(z_{1}^{*})~{}~{}~{}~{}\text{for all}~{}~{}~{}~{}z_{1}^{*},$

(15) $\frac{\partial w_{1}^{*}}{\partial r_{1}^{*}}=0,~{}~{}\frac{\partial\tilde{T}^% {*}}{\partial r_{1}^{*}}=0~{}~{}\frac{\partial\tilde{C}^{*}}{\partial r_{1}^{*% }}=0~{}~{}\frac{\partial\tilde{n}^{*}}{\partial r_{1}^{*}}=0~{}~{}\text{at}~{}% ~{}r_{1}^{*}=0~{}~{}~{}~{}~{}~{}\text{for}~{}~{}~{}~{}0\leq z_{1}^{*}\leq z_{3},$

(16) $w_{1}^{*}=0,~{}~{}\tilde{T}^{*}=\tilde{T}^{*}_{w},~{}~{}\tilde{C}^{*}=\tilde{C% }^{*}_{w},~{}~{}\tilde{n}^{*}=\tilde{n}^{*}_{w}~{}~{}\text{at}~{}~{}r_{1}^{*}=% R_{2}(z_{1}^{*})~{}~{}~{}~{}~{}~{}\text{for}~{}~{}~{}~{}z_{3}\leq z_{1}^{*}% \leq z_{max}.$

The initial condition regarding velocity, temperature, concentration and microorganisms are considered as:

(17) $w_{1}^{*}=0,~{}~{}\tilde{T}^{*}=0,~{}~{}\tilde{C}^{*}=0,~{}~{}\tilde{n}^{*}=0~% {}~{}\text{at}~{}~{}t_{1}^{*}=0.$

The body acceleration and pressure gradient terms are given as;

(18)

$\displaystyle G(t_{1}^{*})=\tilde{F}_{b}\cos(\omega_{b}t_{1}^{*}+\psi)$

(19)

$\displaystyle-\frac{\partial p_{1}^{*}}{\partial z_{1}^{*}}=D_{0}+D_{1}\cos(% \omega_{p}t_{1}^{*})$

Here, $\omega_{p}=2\pi\nu_{p}$ , $\nu_{p}$ is pulse frequency, $\tilde{F}_{b}$ is the body acceleration term, $\omega_{b}=2\pi\nu_{p}$ , $\nu_{b}$ is frequency of body acceleration with $\psi$ as a phase angle.

Electrohydrodynamics (EHD)

Blood is a complex physiological fluid consisting of haemoglobin, plasma, white blood cells, and various ionic components. Its unique composition enables it to function as an electrically conductive fluid. When the arterial walls are exposed to an electrolyte solution, a net charge is generated at the arterial walls. This leads blood to take the opposite charge near the arterial walls. The charged ion undergoes movement upon applying an electric field and subsequently induces fluid motion. This phenomenon is commonly referred to as electro-osmotic flow. The Poisson-Boltzmann equation provides the electro-osmotic potential function, as stated in work by Manchi et al. [25]:

(20) $\nabla^{2}\tilde{\Phi}=-\frac{\rho_{e}}{\epsilon},$

where, $\tilde{\Phi}$ is the electro-osmotic function, $\epsilon$ is dielectric constant, and $\rho_{e}$ is given as:

(21) $\rho_{e}=(m^{+}-m^{-})e_{0}z_{0}.$

The Boltzmann distribution can effectively describe the determination of the number density of cations and anions as:

(22) $m^{\pm}=m_{0}\exp\biggl{(}\mp\frac{e_{0}z_{0}\tilde{\Phi}}{k_{B}T_{avg}}\biggr% {)},$

where, $z_{0}$ is the charge balance, $e_{0}$ is electric constant, $k_{B}$ is Boltzmann constant.

Using the Debye-Huckel linearizion, the Poisson equation takes the form:

(23) $\Biggl{(}\frac{\partial^{2}}{\partial r_{1}^{*}{{}^{2}}}+\frac{1}{r_{1}^{*}}% \frac{\partial}{\partial r_{1}^{*}}+\frac{\partial^{2}}{\partial z_{1}^{*}{{}^% {2}}}\Biggr{)}\tilde{\Phi}=\frac{\tilde{\Phi}}{q_{m}^{2}},$

where $q_{m}=\frac{1}{e_{0}z_{0}}\sqrt{\frac{\epsilon k_{B}T_{avg}}{2m_{0}}}$ .

The boundary conditions for electro-osmotic equation are:

(24) $\begin{cases}\tilde{\Phi}=\tilde{\zeta}_{1},&\text{at}~{}~{}~{}~{}r_{1}^{*}=R_% {1}(z_{1}^{*})~{}~{}~{}~{}\text{for ~{}~{}all}~{}~{}~{}~{}z_{1}^{*},\\ \frac{\partial\tilde{\Phi}}{\partial r_{1}^{*}}=0,&\text{at}~{}~{}~{}~{}r_{1}^% {*}=0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for}~{}~{}~{}~{}0\leq z_{1}^{*}% \leq z_{3},\\ \tilde{\Phi}=\tilde{\zeta}_{2},&\text{at}~{}~{}r_{1}^{*}=R_{2}(z_{1}^{*})~{}~{% }~{}~{}~{}~{}\text{for}~{}~{}~{}~{}z_{3}\leq z_{1}^{*}\leq z_{max}.\end{cases}$

2.2 Non-Dimensionalization

It is necessary to convert the governing equations presented in Eqns. (8)-(13) into dimensionless form in order to obtain a numerical solution. The introduction of non-dimensional variables is performed in the following manner:

(25) $\displaystyle\bar{u}_{1}^{*}=\frac{L_{0}u_{1}^{*}}{\delta^{*}U_{0}},~{}~{}\bar% {t}_{1}^{*}=\frac{U_{0}t_{1}^{*}}{R_{0}},~{}\bar{z}_{1}^{*}=\frac{z_{1}^{*}}{L% _{0}},\bar{p}_{1}^{*}=\frac{R_{0}^{2}p_{1}^{*}}{U_{0}L_{0}\mu_{0}},~{}\bar{r}_% {1}^{*}=\frac{r_{1}^{*}}{R_{0}},\bar{w}_{1}^{*}=\frac{w_{1}^{*}}{U_{0}},~{}% \bar{\theta}=\frac{\tilde{T}^{*}-\tilde{T}^{*}_{1}}{\tilde{T}^{*}_{w}-\tilde{T% }^{*}_{1}},~{}\bar{\phi}=\frac{\tilde{C}^{*}-\tilde{C}^{*}_{1}}{\tilde{C}^{*}_% {w}-\tilde{C}^{*}_{1}},{\chi_{1}}=\frac{\tilde{n}^{*}-\tilde{n}^{*}_{1}}{% \tilde{n}^{*}_{w}-\tilde{n}^{*}_{1}},~{}~{}\Phi=\frac{\tilde{\Phi}}{\tilde{% \zeta}_{1}}\bar{R}_{i}^{*}=\frac{R_{i}^{*}}{R_{0}}(i=1,2),~{}\bar{d}=\frac{d}{% L_{0}},\bar{r}_{0}=\frac{r_{0}}{R_{0}},~{}\bar{z_{i}}=\frac{z_{i}}{L_{0}}(i=1,% ..,4),~{}\bar{r_{1}}=\frac{r_{1}}{R_{0}},\bar{\tilde{r_{1}}}=\frac{\tilde{r_{1% }}}{R_{0}},~{}Re=\frac{U_{0}\rho_{f}R_{0}}{\mu_{f}},~{}M^{2}=\frac{\sigma_{f}B% _{0}^{2}R_{0}^{2}}{\mu_{f}},~{}E_{1}^{*}=\frac{E_{0}}{B_{0}U_{0}},Gr=\frac{% \rho_{f}R_{0}^{2}g\gamma_{f}(\tilde{T}^{*}_{w}-\tilde{T}^{*}_{1})}{\mu_{f}U_{0% }},~{}Gc=\frac{\rho_{f}R_{0}^{2}g\gamma_{f}(\tilde{C}^{*}_{w}-\tilde{C}^{*}_{1% })}{\mu_{f}U_{0}},~{}Rb=\frac{\rho_{f}R_{0}^{2}g\gamma_{f}(\tilde{n}^{*}_{w}-% \tilde{n}^{*}_{1})}{\mu_{f}U_{0}},~{}Ec=\frac{U_{0}^{2}}{C_{p}(\tilde{T}^{*}_{% w}-\tilde{T}^{*}_{1})},~{}Pr=\frac{\mu_{0}C_{p}}{k_{f}},\tilde{Q}=\frac{\tilde% {Q}_{0}R_{0}^{2}}{\kappa_{f}(\tilde{T}^{*}_{w}-\tilde{T}^{*}_{1})},~{}S_{z}=% \frac{\sigma_{f}}{\kappa_{f}}\frac{R_{0}^{2}E_{0}^{2}}{(\tilde{T}_{w}-\tilde{T% }_{0})},~{}q=\frac{q_{m}}{R_{0}},~{}Sc=\frac{\nu}{D_{m}},~{}Pe=\frac{bW_{c}}{D% _{n}},~{}\sigma_{1}=\frac{\tilde{n}^{*}}{(\tilde{n}^{*}_{w}-\tilde{n}^{*}_{1})% },~{}\xi=\frac{R_{b}\rho_{f}R_{0}^{2}}{\mu_{0}},U_{hs}=-\frac{\zeta\epsilon E_% {0}}{\mu_{0}U_{0}}.$

The aforementioned non-dimensional parameters mentioned in Eqn. (25) are inserted into the governing Eqns. (8)-(13) and removed the bars. The mild stenotic hypotheses are applied, i.e., $\delta^{*}(=\delta/R_{0})<<1$ , and $\epsilon(=R_{0}/L_{0})=O(1)$ . As a consequence of the aforementioned process, the governing Eqns. (8)-(13) undergo modifications, which can be expressed as follows:

Continuity Equation:

(26) $\frac{\partial w_{1}^{*}}{\partial z_{1}^{*}}=0,$

Momentum Equation: $r_{1}$ -direction:

(27) $\frac{\partial p_{1}^{*}}{\partial r_{1}^{*}}=0,$

$Z_{1}$ -direction:

(28) $Re\frac{\rho_{hnf}}{\rho_{f}}\frac{\partial w_{1}^{*}}{\partial t_{1}^{*}}=-% \frac{\partial p_{1}^{*}}{\partial z_{1}^{*}}+\frac{1}{2r_{1}^{*}}\frac{% \partial}{\partial r_{1}^{*}}\bigg{[}\frac{\mu_{hnf}}{\mu_{f}}r_{1}^{*}\frac{% \partial w_{1}^{*}}{\partial r_{1}^{*}}\bigg{]}+\frac{(\rho\gamma)_{hnf}}{(% \rho\gamma)_{f}}\biggl{[}Gr{\theta}+Gc\phi-Rb{{\chi}_{1}}\biggr{]}\cos\eta% \delta_{ij}+U_{hs}q^{2}\Phi-\frac{\sigma_{hnf}}{\sigma_{f}}M^{2}w_{1}^{*}.$

Energy Equation:

(29) $\frac{(\rho C_{p})_{hnf}}{(\rho C_{p})_{f}}\frac{\partial{\theta}}{\partial t_% {1}^{*}}=\frac{1}{RePr}\frac{k_{hnf}}{k_{f}}\bigg{[}\frac{\partial^{2}{\theta}% }{\partial r_{1}^{*2}}+\frac{1}{r_{1}^{*}}\frac{\partial{\theta}}{\partial r_{% 1}^{*}}\bigg{]}+\frac{\sigma_{hnf}}{\sigma_{f}}\bigg{[}\frac{EcM^{2}}{Re}w_{1}% ^{*2}+\frac{(S_{z}+\tilde{Q})}{RePr}\bigg{]}+\frac{\mu_{hnf}}{\mu_{f}}\frac{Ec% }{Re}\bigg{[}\bigg{(}\frac{\partial w_{1}^{*}}{\partial r_{1}^{*}}\bigg{)}^{2}% \bigg{]}.$

Concentration Equation:

(30) $ReSc\frac{\partial\phi}{\partial t_{1}^{*}}=\frac{\partial^{2}\phi}{\partial r% _{1}^{*2}}+\frac{1}{r_{1}^{*}}\frac{\partial\phi}{\partial r_{1}^{*}}-Sc\xi\phi,$

Microorganism Equation:

(31) $ReSb\frac{\partial{{\chi}_{1}}}{\partial t_{1}^{*}}=\frac{\partial^{2}{{\chi}_% {1}}}{\partial r_{1}^{*2}}+\frac{1}{r_{1}^{*}}\frac{\partial{{\chi}_{1}}}{% \partial r_{1}^{*}}-Pe\sigma_{1}\biggl{(}\frac{\partial{{\chi}_{1}}}{\partial r% _{1}^{*}}\frac{\partial{\phi}}{\partial r_{1}^{*}}+(\sigma_{1}+{{\chi}_{1}})% \frac{\partial^{2}{\phi}}{\partial r_{1}^{*2}}\biggr{)},$

Electroosmotic Equation:

(32) $\frac{\partial^{2}\Phi}{\partial r_{1}^{*2}}+\frac{1}{r_{1}^{*}}\frac{\partial% \Phi}{\partial r_{1}^{*}}={q}^{2}\Phi.$

Here, Reynold’s viscosity model [39] has been utilised to illustrate the temperature-dependent viscosity. The model is expressed as follows:

(33) $\mu_{f}({\theta})=\mu_{0}e^{-\beta_{0}{\theta}}=\mu_{0}[1-\beta_{0}{\theta}]~{% }~{}\text{where}~{}~{}\beta_{0}<<1$

Upon substituting dimensionless variables in Eqn. (25), the resulting modified equation for the pressure gradient can be written as:

(34) $-\frac{\partial p_{1}^{*}}{\partial z_{1}^{*}}=B_{1}[1+e\cos(c_{1}t_{1}^{*})],$

where

$e=\frac{D_{1}}{D_{0}},B_{1}=\frac{D_{0}R_{0}^{2}}{\mu_{0}U_{0}},c_{1}=\frac{2% \pi R_{0}\nu_{p}}{U_{0}}.$

Upon applying non-dimensional values to the body acceleration Eqn. (19), the terms $F_{b}=\frac{\tilde{F}_{b}R_{0}^{2}}{\mu_{0}U_{0}}$ and $c_{2}=\frac{w_{b}R_{0}}{U_{0}}$ has taken the following form. The resulting form of the equation, by removing the bars, is as follows.

(35) $G(t_{1}^{*})=F_{b}\cos(c_{2}t_{1}^{*}+\psi),t_{1}^{*}>0$

The dimensionless form of the stenosis geometry is expressed as follows:

(36) $R_{1}^{*}(z_{1}^{*})=\begin{cases}1,&0<z_{1}^{*}\leq d\\ 1-\frac{6\delta^{*}}{5}(11(z_{1}^{*}-d)-47(z_{1}^{*}-d)^{2}+72(z_{1}^{*}-d)^{3% }-36(z_{1}^{*}-d)^{4}),&d<z_{1}^{*}\leq d+1\\ 1,&d+1<z_{1}^{*}\leq z_{1}\\ 1+r_{1}-\sqrt{\bigg{(}r_{1}^{2}-(z_{1}^{*}-z_{1})^{2}\bigg{)}}&z_{1}<z_{1}^{*}% \leq z_{2}\\ 2r_{0}\sec\eta+\bigg{(}z_{1}^{*}-z_{2}\bigg{)}\tan\eta&z_{2}<z_{1}^{*}\leq z_{% 4}+0.5\\ 2r_{0}\sec\eta+\bigg{(}z_{1}^{*}-z_{2}\bigg{)}\tan\eta-A^{*}&z_{4}+0.5<z_{1}^{% *}\leq z_{4}+2.5,\\ 2r_{0}\sec\eta+\bigg{(}z_{1}^{*}-z_{2}\bigg{)}\tan\eta,&z_{4}+2.5\leq z_{1}^{*% }\leq z_{max}\end{cases}$

Where $A^{*}=2\delta_{1}^{*}\bigg{\{}\cos\bigg{(}2\pi\bigg{(}\frac{z_{1}^{*}-d}{4}% \bigg{)}-0.25\bigg{)}-0.07\cos\bigg{(}128\pi\bigg{(}z_{1}^{*}-d-0.5\bigg{)}% \bigg{)}\bigg{\}}\tan(\eta)$ .

The inner wall is represent as:

(37)

R_{2}^{*}(z_{1}^{*})=\begin{cases}0,&0<z_{1}^{*}\leq z_{3},\\ \sqrt{\tilde{r}_{1}^{2}-\bigg{(}(z_{1}^{*}-z_{3})-\tilde{r}_{1}\bigg{)}^{2}},&% z_{3}<z_{1}^{*}\leq z_{4},\\ (z_{1}^{*}-z_{2})\tan\eta,&z_{4}<z_{1}^{*}\leq z_{max}.\end{cases}

The lateral junction curvature $r_{1}$ and the flow divider radius $\tilde{r}_{1}$ in the dimensionless form (after ignoring bars) are given as:

(38) $r_{1}=\frac{1-2r_{0}\sec\eta}{\cos\eta-1},$

(39) $\tilde{r}_{1}=\frac{(z_{3}-z_{2})\sin\eta}{(1-\sin\eta)},$

where $z_{2}$ , $z_{3}$ and $z_{4}$ in the dimensionless form are specified as:

(40) $z_{2}=z_{1}+r_{1}\sin\eta,$

(41) $z_{3}=z_{2}+q_{1},$

and

(42) $z_{4}=z_{3}+\tilde{r}_{1}(1-\sin\eta).$

2.3 Radial Coordinate Transformation

In order to obtain a rectangular domain, it is necessary to apply the transformation $\bigg{(}x_{1}^{*}=\frac{r_{1}^{*}-R_{2}(z_{1}^{*})}{R(z_{1}^{*})}\bigg{)}$ to the geometry under consideration, where $R(z_{1}^{*})=R_{1}(z_{1}^{*})-R_{2}(z_{1}^{*})$ . Upon implementation of the aforementioned transformation, the equations denoted by Eqn. (28) through Eqn. (32) undergo a modification as follows:

(43) $\displaystyle Re\frac{\rho_{hnf}}{\rho_{f}}\frac{\partial w_{1}^{*}}{\partial t% _{1}^{*}}=B_{1}[1+ecos(c_{1}t_{1}^{*})]+\frac{1}{2}\bigg{(}\frac{1-\beta_{0}{% \theta}}{(1-\phi_{1})^{2.5}(1-\phi_{2})^{2.5}}\bigg{)}\bigg{[}\frac{1}{R^{2}}% \frac{\partial^{2}w_{1}^{*}}{\partial x_{1}^{*2}}+\bigg{(}\frac{1}{x_{1}^{*}R+% R_{2}}\bigg{)}\bigg{(}\frac{1}{R}\frac{\partial w_{1}^{*}}{\partial x_{1}^{*}}% \bigg{)}\bigg{]}-\frac{\beta_{0}}{2R^{2}((1-\phi_{1})^{2.5}(1-\phi_{2})^{2.5})% }\frac{\partial w_{1}^{*}}{\partial x_{1}^{*}}\frac{\partial{\theta}}{\partial x% _{1}^{*}}+\frac{(\rho\gamma)_{hnf}}{(\rho\gamma)_{f}}\biggl{[}Gr{\theta}+Gc{% \phi}-Rb{{\chi}_{1}}\biggr{]}\cos(\eta\delta_{ij})+U_{hs}q^{2}\Phi-\frac{% \sigma_{hnf}}{\sigma_{f}}M^{2}w_{1}^{*}+G(t_{1}^{*}),$

(44) $\displaystyle\frac{(\rho C_{p})_{hnf}}{(\rho C_{p})_{f}}\frac{\partial{\theta}% }{\partial t_{1}^{*}}=\frac{1}{RePr}\frac{k_{hnf}}{k_{f}}\bigg{[}\frac{1}{R^{2% }}\frac{\partial^{2}{\theta}}{\partial x_{1}^{*2}}+\bigg{(}\frac{1}{x_{1}^{*}R% +R_{2}}\bigg{)}\bigg{(}\frac{1}{R}\frac{\partial{\theta}}{\partial x_{1}^{*}}% \bigg{)}\bigg{]}+\frac{\sigma_{hnf}}{\sigma_{f}}\bigg{[}\frac{EcM^{2}}{Re}w_{1% }^{*}{{}^{2}}+\frac{S_{z}+\tilde{Q}}{RePr}\bigg{]}+\frac{1}{R^{2}}\bigg{(}% \frac{1-\beta_{0}{\theta}}{(1-\phi_{1})^{2.5}(1-\phi_{2})^{2.5}}\bigg{)}\frac{% Ec}{Re}\bigg{[}\bigg{(}\frac{\partial w_{1}^{*}}{\partial x_{1}^{*}}\bigg{)}^{% 2}\bigg{]},$

(45) $ReSc\frac{\partial{\phi}}{\partial t_{1}^{*}}=\frac{1}{R^{2}}\frac{\partial^{2% }{\phi}}{\partial x_{1}^{*2}}+\bigg{(}\frac{1}{x_{1}^{*}R+R_{2}}\bigg{)}\frac{% 1}{R}\frac{\partial{\phi}}{\partial x_{1}^{*}}-Sc\xi\phi_{1},$

(46) $ReSb\frac{\partial{{\chi}_{1}}}{\partial t_{1}^{*}}=\frac{1}{R^{2}}\frac{% \partial^{2}{{\chi}_{1}}}{\partial x_{1}^{*2}}+\bigg{(}\frac{1}{x_{1}^{*}R+R_{% 2}}\bigg{)}\frac{1}{R}\frac{\partial{{\chi}_{1}}}{\partial x_{1}^{*}}-\frac{Pe% \sigma_{1}}{R^{2}}\biggl{(}\frac{\partial{{\chi}_{1}}}{\partial x_{1}^{*}}% \frac{\partial{\phi}}{\partial x_{1}^{*}}+(\sigma_{1}+{{\chi}_{1}})\frac{% \partial^{2}{\phi}}{\partial x_{1}^{*2}}\biggr{)},$

(47) $\frac{1}{R^{2}}\frac{\partial^{2}\Phi}{\partial x_{1}^{*2}}+\bigg{(}\frac{1}{x% _{1}^{*}R+R_{2}}\bigg{)}\frac{1}{R}\frac{\partial\Phi}{\partial x_{1}^{*}}={q}% ^{2}\Phi.$

The boundary conditions specified in Eqns. (14) and (17) have been reduced in the following manner:

(48) $w_{1}^{*}=0,~{}~{}{\theta}=1,~{}~{}{\phi}_{1}=1,~{}~{}{\chi}_{1}=1,~{}~{}\text% {at}~{}~{}x_{1}^{*}=1~{}~{}~{}~{}~{}~{}\text{for all}~{}~{}~{}~{}z_{1}^{*},$

(49) $\frac{\partial w_{1}^{*}}{\partial x_{1}^{*}}=0,~{}~{}\frac{\partial{\theta}}{% \partial x_{1}^{*}}=0,~{}~{}\frac{\partial{\phi}}{\partial x_{1}^{*}}=0,~{}~{}% \frac{\partial{\chi}_{1}}{\partial x_{1}^{*}}=0~{}~{}\text{at}~{}~{}x_{1}^{*}=% 0~{}~{}~{}~{}~{}~{}\text{for}~{}~{}~{}~{}0\leq z_{1}^{*}\leq z_{3},$

(50) $w_{1}^{*}=0,~{}~{}{\theta}=1,~{}~{}{\phi}=1,~{}~{}{\chi}_{1}=1,~{}~{}\text{at}% ~{}~{}x_{1}^{*}=0~{}~{}~{}~{}~{}~{}\text{for}~{}~{}~{}~{}z_{3}\leq z_{1}^{*}% \leq z_{max}.$

The wall shear stress at the outer wall of the bifurcated artery is given as below:

(51) $\tau_{w}=-\frac{1}{R}\bigg{(}\frac{\partial w_{1}^{*}}{\partial x_{1}^{*}}% \bigg{)}_{x_{1}^{*}=1},$

The flow rate for the parent artery and daughter artery is defined as follows:

(52) $Q_{d}^{f}=2\pi R\int_{0}^{1}w_{1}^{*}(x_{1}^{*}R+R_{2})dx_{1}^{*}.$

(53) $Q_{p}^{f}=\pi R\int_{0}^{1}w_{1}^{*}(x_{1}^{*}R+R_{2})dx_{1}^{*}.$

The resistance impedance for the the parent artery and daughter artery is given by:

(54) $\lambda_{p}=\left|\frac{z_{3}\bigg{(}-\frac{\partial p_{1}^{*}}{\partial z_{1}% ^{*}}\bigg{)}}{Q_{p}^{f}}\right|,~{}~{}~{}~{}~{}\text{for}~{}~{}~{}~{}~{}z_{1}% ^{*}<z_{3},$

(55) $\lambda_{d}=\left|\frac{(z_{max}-z_{3})\bigg{(}-\frac{\partial p_{1}^{*}}{% \partial z_{1}^{*}}\bigg{)}}{Q_{d}^{f}}\right|,~{}~{}~{}~{}~{}\text{for}~{}~{}% ~{}~{}~{}z_{1}^{*}\geq z_{3}.$

The Nusselt number at the outer wall of the bifurcated artery is computed as follows:

(56) $Nu_{x}=-\frac{1}{R}\bigg{(}\frac{\partial{\theta}}{\partial x_{1}^{*}}\bigg{)}% _{x_{1}^{*}=1}.$

The Sherwood number is given as:

(57) $Sh_{x}=-\frac{1}{R}\bigg{(}\frac{\partial{\phi}}{\partial x_{1}^{*}}\bigg{)}_{% x_{1}^{*}=1}.$

Similarly, the motile density number is given as:

(58) $\Lambda_{x}=-\frac{1}{R}\bigg{(}\frac{\partial\chi_{1}}{\partial x_{1}^{*}}% \bigg{)}_{x_{1}^{*}=1}.$

3. Solution Process

It is an established fact that there are several numerical techniques to compute the partial differential equations, but the finite difference scheme is the easiest and efficient technique for finding the solution these equations. In order to solve the PDEs, we adopted the Crank-Nicolson scheme and took step size of $\Delta x$ in the radial direction with a time step of $\Delta t=0.001$ to achieve the convergence of the numerical scheme. It is also observed that further change in $\Delta x$ and $\Delta t$ doesn’t bring any substantial changes in the results.

Discretization

The governing equations are discretized as:

(59) $\displaystyle\bigg{[}(1-\phi_{2})\bigg{[}(1-\phi_{1})+\phi_{1}\frac{\rho_{s1}}% {\rho_{f}}\bigg{]}+\phi_{2}\frac{\rho_{s2}}{\rho_{f}}\bigg{]}Re\bigg{[}\frac{w% _{i}^{k+1}-w_{i}^{k}}{dt}\bigg{]}=B_{1}[1+ecos(c_{1}t^{k})]\frac{1}{2}\bigg{\{% }\frac{1-\beta_{0}{\theta}}{(1-\phi_{1})^{2.5}(1-\phi_{2})^{2.5}}\bigg{\}}% \bigg{[}\frac{1}{R^{2}}\bigg{(}\frac{w_{i+1}^{k+1}-2w_{i}^{k+1}+w_{i-1}^{k+1}}% {dx^{2}}+\frac{w_{i+1}^{k}-2w_{i}^{k}+w_{i-1}^{k}}{dx^{2}}\bigg{)}+\frac{1}{R}% \frac{1}{Rx_{i}+R_{2}}\bigg{(}\frac{w_{i+1}^{k+1}-w_{i-1}^{k+1}}{2dx}+\frac{w_% {i+1}^{k}-w_{i-1}^{k}}{2dx}\bigg{)}\bigg{]}-\frac{1}{2R^{2}}\bigg{\{}\frac{% \beta_{0}}{(1-\phi_{1})^{2.5}(1-\phi_{2})^{2.5}}\bigg{\}}\bigg{[}\bigg{(}\frac% {w_{i+1}^{k+1}-w_{i-1}^{k+1}}{2dx}+\frac{w_{i+1}^{k}-w_{i-1}^{k}}{2dx}\bigg{)}% \bigg{(}\frac{{\theta}_{i+1}^{k}-{\theta}_{i-1}^{k}}{2dx}\bigg{)}\bigg{]}+U_{% hs}q^{2}\Phi+F_{b}cos(c_{2}t_{k}+\psi)+\bigg{[}(1-\phi_{2})\bigg{\{}(1-\phi_{1% })+\phi_{1}\frac{(\rho\gamma)_{s1}}{(\rho\gamma)_{f}}\bigg{\}}+\phi_{2}\frac{(% \rho\gamma)_{s_{2}}}{(\rho\gamma)_{f}}\bigg{]}\bigg{(}Gr{\theta}_{i}^{k}+Gc{% \phi}_{i}^{k}-Rb{{\chi}_{1}}_{i}^{k}\bigg{)}\cos(\eta\delta_{ij})-\frac{1}{2}% \frac{\sigma_{nf}}{\sigma_{f}}M^{2}(w_{i}^{k}+w_{i}^{k+1})$

(60) $\displaystyle\bigg{[}(1-\phi_{2})\bigg{\{}(1-\phi_{1})+\phi_{1}\frac{(\rho C_{% p})_{s_{1}}}{(\rho C_{p})_{f}}\bigg{\}}+\phi_{2}\frac{(\rho C_{p})_{s_{2}}}{(% \rho C_{p})_{f}}\bigg{]}\bigg{[}\frac{{\theta}_{i}^{k+1}-{\theta}_{i}^{k}}{dt}% \bigg{]}=\frac{1}{RePr}\frac{k_{hnf}}{k_{f}}\bigg{[}\frac{1}{2R^{2}}\bigg{(}% \frac{{\theta}_{i+1}^{k+1}-2{\theta}_{i}^{k+1}+{\theta}_{i-1}^{k+1}}{dx^{2}}+% \frac{{\theta}_{i+1}^{k}-2{\theta}_{i}^{k}+{\theta}_{i-1}^{k}}{dx^{2}}\bigg{)}% +\frac{1}{2(Rx_{i}+R_{2})}\bigg{(}\frac{{\theta}_{i+1}^{k+1}-{\theta}_{i-1}^{k% +1}}{2dx}+\frac{{\theta}_{i+1}^{k}-{\theta}_{i-1}^{k}}{2dx}\bigg{)}\bigg{]}+% \frac{1}{R^{2}}\bigg{\{}\frac{1-\frac{\beta_{0}}{2}({\theta}_{i}^{k+1}+{\theta% }_{i}^{k})}{(1-\phi_{1})^{2.5}(1-\phi_{2})^{2.5}}\bigg{\}}\frac{Ec}{Re}\bigg{[% }\frac{1}{2}\bigg{(}\frac{w_{i+1}^{k+1}-w_{i-1}^{k+1}}{2dx}+\frac{w_{i+1}^{k}-% w_{i-1}^{k}}{2dx}\bigg{)}\bigg{]}^{2}+\frac{\sigma_{hnf}}{\sigma_{f}}\bigg{[}% \bigg{(}\frac{1}{2}\frac{EcM^{2}}{Re}\bigg{(}w_{i}^{k+1}+w_{i}^{k}\bigg{)}% \bigg{)}^{2}+\frac{S_{z}+\tilde{Q}}{RePr}\bigg{]}.$

(61) $\displaystyle ReSc\bigg{[}\frac{{\phi}_{i}^{k+1}-{\phi}_{i}^{k}}{dt}\bigg{]}=% \bigg{[}\frac{1}{2R^{2}}\bigg{(}\frac{{\phi}_{i+1}^{k+1}-2{\phi}_{i}^{k+1}+{% \phi}_{i-1}^{k+1}}{dx^{2}}+\frac{{\phi}_{i+1}^{k}-2{\phi}_{i}^{k}+{\phi}_{i-1}% ^{k}}{dx^{2}}\bigg{)}+\frac{1}{2(Rx_{i}+R_{2})}\bigg{(}\frac{{\phi}_{i+1}^{k+1% }-{\phi}_{i-1}^{k+1}}{2dx}+\frac{{\phi}_{i+1}^{k}-{\phi}_{i-1}^{k}}{2dx}\bigg{% )}\bigg{]}-\frac{Sc\xi}{2}\bigg{[}{\phi}_{i}^{k+1}+{\phi}_{i}^{k}\bigg{]}$

(62) $\displaystyle ReSb\bigg{[}\frac{{{\chi}_{1}}_{i}^{k+1}-{{\chi}_{1}}_{i}^{k}}{% dt}\bigg{]}=\bigg{[}\frac{1}{2R^{2}}\bigg{(}\frac{{{\chi}_{1}}_{i+1}^{k+1}-2{{% \chi}_{1}}_{i}^{k+1}+{{\chi}_{1}}_{i-1}^{k+1}}{dx^{2}}+\frac{{{\chi}_{1}}_{i+1% }^{k}-2{{\chi}_{1}}_{i}^{k}+{{\chi}_{1}}_{i-1}^{k}}{dx^{2}}\bigg{)}+\frac{1}{2% (Rx_{i}+R_{2})}\bigg{(}\frac{{{\chi}_{1}}_{i+1}^{k+1}-{{\chi}_{1}}_{i-1}^{k+1}% }{2dx}+\frac{{\phi}_{i+1}^{k}-{\phi}_{i-1}^{k}}{2dx}\bigg{)}\bigg{]}-\frac{Pe% \sigma_{1}}{2}\bigg{[}\bigg{\{}\frac{{{\chi}_{1}}_{i+1}^{k+1}+{{\chi}_{1}}_{i-% 1}^{k+1}}{2dx}+\frac{{{\chi}_{1}}_{i+1}^{k}+{{\chi}_{1}}_{i-1}^{k}}{2dx}\bigg{% \}}\bigg{(}\frac{{\phi}_{i+1}^{k}-{\phi}_{i-1}^{k}}{2dx}\bigg{)}+\bigg{\{}% \sigma_{1}+\frac{{{\chi}_{1}}_{i}^{k+1}+{{\chi}_{1}}_{i}^{k}}{2}\bigg{\}}\bigg% {(}\frac{{\phi}_{i+1}^{k}-2{\phi}_{i}^{k}-{\phi}_{i-1}^{k}}{dx^{2}}\bigg{)}% \bigg{]}$

(63) $\frac{\Phi_{i+1}-2\Phi_{i}+\Phi_{i-1}}{dx^{2}}+\frac{1}{(Rx_{i}+R_{2})}\bigg{% \{}\frac{\Phi_{i+1}-\Phi_{i-1}}{2dx}\bigg{\}}=q^{2}\Phi_{i}$

The discretized governing Eqns. (59) and (63) are then converted to a tri-diagonal system of equations, which is subsequently solved through the utilization of the Tri-Diagonal Matrix Algorithm (TDMA).

4. Results and Graphical Analysis

A MATLAB-based computer code was developed to gain insight into the mathematical and physical aspects of the current problem being considered. The code was designed to generate graphical representations of velocity, temperature, concentration, microorganisms, flow rate, impedance, Nusselt, and Sherwood profiles. This study examines the hemodynamic characteristics and blood rheology in pathological conditions such as stenosis on the arterial walls of bifurcated arteries. The thermophysical characteristics of nanoparticles and the parameters of nanofluids are shown in Tables 1,2, respectively. Table 3 (Ref. [6, 10, 32, 34, 40, 41, 42, 43, 44, 45]) shows the possible values explored for the different flow parameters. Table 4 shows the numerical values of Nusselt, Sherwood, wall shear stress, and motile density numbers.

Table 1.Thermophysical parameters of nanofluid and hybrid nanofluid.

Properties	Mathematical expression for nanofluid and hybrid nanofluid
Viscosity	$\mu_{nf}=\dfrac{\mu_{f}}{(1-\phi_{1})^{2.5}}$
	$\mu_{hnf}=\dfrac{\mu_{f}}{(1-\phi_{1})^{2.5}(1-\phi_{2})^{2.5}}$
Density	$\rho_{nf}=(1-\phi_{1})\rho_{f}+\phi_{1}\rho_{s_{1}}$
	$\rho_{hnf}=[(1-\phi_{2})\{(1-\phi_{1})\rho_{f}+\phi_{1}\rho_{s_{1}}\}]+\phi_{2% }\rho_{s_{2}}$
Heat Capacity	$(\rho C_{p})_{nf}=(1-\phi_{1})(\rho C_{p})_{f}+\phi_{1}(\rho C_{p})_{s_{1}}$
	$(\rho C_{p})_{hnf}=[(1-\phi_{2})\{(1-\phi_{1})(\rho C_{p})_{f}+\phi_{1}(\rho C% _{p})_{s_{1}}\}+\phi_{2}(\rho C_{p})_{s_{2}}]$
Thermal Conductivity	$\dfrac{k_{nf}}{k_{f}}=\dfrac{k_{s_{1}}+(m-1)k_{f}-(m-1)\phi_{1}(k_{f}-k_{s_{1}% })}{k_{s_{1}}+(m-1)k_{f}+\phi_{1}(k_{f}-k_{s_{1}})}$
	$\dfrac{k_{hnf}}{k_{nf}}=\dfrac{k_{s_{2}}+(m-1)k_{nf}-(m-1)\phi_{2}(k_{nf}-k_{s% _{2}})}{k_{s_{2}}+(m-1)k_{nf}+\phi_{2}(k_{nf}-k_{s_{2}})}$
Electrical Conductivity	$\dfrac{\sigma_{nf}}{\sigma_{f}}=\dfrac{\sigma_{s_{1}}+(m-1)\sigma_{f}-(m-1)% \phi_{1}(\sigma_{f}-\sigma_{s_{1}})}{\sigma_{s_{1}}+(m-1)\sigma_{f}+\phi_{1}(% \sigma_{f}-\sigma_{s_{1}})}$
	$\dfrac{\sigma_{hnf}}{\sigma_{nf}}=\dfrac{\sigma_{s_{2}}+(m-1)\sigma_{nf}-(m-1)% \phi_{2}(\sigma_{nf}-\sigma_{s_{2}})}{\sigma_{s_{2}}+(m-1)\sigma_{nf}+\phi_{2}% (\sigma_{nf}-\sigma_{s_{2}})}$
Thermal Expansion Coefficient	$\dfrac{(\rho\gamma)_{nf}}{(\rho\gamma)_{f}}=[(1-\phi_{1})+\phi_{1}\frac{(\rho% \gamma)_{s1}}{(\rho\gamma)_{f}}]$
	$\dfrac{(\rho\gamma)_{hnf}}{(\rho\gamma)_{nf}}=(1-\phi_{2})[(1-\phi_{1})+\phi_{% 1}\frac{(\rho\gamma)_{s1}}{(\rho\gamma)_{f}}]+\phi_{2}\frac{(\rho\gamma)_{s2}}% {(\rho\gamma)_{f}}$

Table 2.Thermophysical properties of nanoparticles.

Thermophysical Properties	Blood	Al ${}_{2}$ O ${}_{3}$	Cu
Density [ $\rho(kg/m^{3})$ ]	1060	3970	8933
Thermal Expansion Coefficient [ $\gamma\times 10^{-5}(K^{-1})$ ]	0.18	0.85	5
Electrical Conductivity [ $\sigma(S/m)$ ]	0.667	3.5 $\times$ $10^{7}$	10 $\times$ 10 ${}^{-10}$
Thermal Conductivity [K (W $/$ mK)]	0.492	40	314
Heat Capacitance [ $C_{p}(J/kgK)$ ]	3770	3970	8933

Table 3.Default values of emerging parameters with their sources.

Parameters	Values	References
Magnetic field ( $M^{2}$ )	0–5	[6, 40]
Grashof number (Gr)	0–5	[41, 42, 43]
Rayleigh number (Rb)	0–6	[32, 34]
Prandtl number (Pr)	14–25	[44]
Heat source parameter $(\tilde{Q})$	0–1	[10, 45]

Table 4.The numerical values of Nusselt, Sherwood, wall shear stress, and motile density numbers.

Pr	$\tilde{Q}$	$Nu_{x}$	$\xi$	Sc	$Sh_{x}$	M	Gr	$\tau_{w}$	$P e$	Sb	$\Lambda_{x}$
19	0.2	–2.7343	1.5	1.5	–0.9260	0	0.5	6.7637	0	2.5	–0.8547
23	0.2	–3.0589	2.5	1.5	–1.2907	1	0.5	6.5131	1.5	2.5	–1.5250
25	0.2	–3.2107	3	1.5	–1.4607	2	0.5	5.9094	3	2.5	–2.2929
23	0.5	–3.0156	3	1	–1.0015	2	0	4.6583	1.5	2	–1.2187
23	1	–2.9432	3	0.5	–0.5238	2	1	7.0397	1.5	1	–0.5804

4.1 Validation

The validation of our work is consummated with the published work of Tripathi et al. [46]. The present study employs a finite difference methodology, specifically the Crank-Nicolson method, to compute the governing equations.

In contrast, the previously published work of [46] utilised the FTCS scheme. The present study compares two research works by analysing the impact of different parameters, namely Solutal Grashof number $(Gc=0)$ , Rayleigh number $(Rb=0)$ , magnetic field parameter $(M=0)$ , Debye-Huckel parameter $(q=0)$ , heat source $(Q=0)$ , Eckert number $(Ec=0)$ , and Joule heating parameter $(S_{z}=0)$ . Additionally, the inner wall $(R_{2}(z)=0)$ has been considered. The velocity and temperature profile for pure blood (without the presence of nanoparticles) are illustrated in Figs. 2,3, respectively. The graphical representations illustrate a high degree of concurrence between our study and prior research [46] concerning velocity and temperature profiles.

Fig. 2.

Velocity profile for pure blood.

Fig. 3.

Temperature profile for for pure blood.

4.2 Velocity Profile

The composition of blood is multifaceted, encompassing haemoglobin, plasma, white blood cells, and diverse ionic constituents. The human circulatory system consists of millions of red blood cells and other ionic components, which render it capable to exhibit biomagnetic properties. The primary objective of red blood cells (RBCs) is to transport oxygen to different tissues and organs within the human body. A comparative study was conducted to explore the impact of magnetic field parameters on the stenosed artery, motivated by its blood magnetic property. Fig. 4a depicts the velocity profile variation for different magnetic field parameter values in both parent and daughter arteries. The maximum velocity of the fluid is achieved in the absence of the magnetic field that is $(M=0)$ , which reduces as the strength of the magnetic field enhances from $M=0$ to $M=\sqrt{5}$ . This has occurred due to the application of a transversal magnetic field in the direction of the flow, which produces the resistance Lorentz force and slows the fluid motion. The profile of the velocity for a variable electro-kinetic parameter, or Debye Huckel parameter $q$ , is shown in Fig. 4b. The enhancement in the velocity profile can be explained by the electro-kinetic acceleration term present in the momentum equation, i.e., $U_{hs}q^{2}\Phi$ . The Debey Huckel parameter is expressed as the quotient of the tube radius denoted by $R_{0}$ and the Debye length represented by $q_{e}$ . It has been observed that the Debey Huckel parameter exhibits an inverse relationship with the thickness of the electric double layer (EDL). Hence, raising the value of the Debey Huckel parameter reduces the EDL width, increasing the electro-osmotic forces that counteract fluid drag and boost fluid velocity. Bioconvection phenomena can be attributed to the upward swimming of motile microorganisms in a hybrid nanofluid solution. Typically, the density of microorganisms exhibits a slightly higher value than that of the hybrid nanofluid solution, resulting in the migration of microorganisms towards the upper surface of the solution. The observed phenomenon involves generating an uneven and unstable solution, resulting in the downward displacement of microorganisms and the subsequent initiation of a bioconvection process. The association between the bioconvection Rayleigh number and the momentum equation substantially impacted the velocity profile, as seen in Fig. 4c. The increase in the Rayleigh number strengthens the microorganism’s convection, which works against the buoyancy force acting on the fluid particles. This results in a decrement in the velocity profile. The variation in velocity profile by varying nanoparticle concentration is depicted in Fig. 4d. Nanoparticles play a pivotal role in facilitating the targeted delivery of therapeutic agents to specific tissues through the circulatory system, enabling the treatment of a broad spectrum of medical conditions. Upon the introduction of copper nanoparticles into the bloodstream, the blood flow velocity increases, resulting in an elevated velocity profile compared to pure blood. The figure suggests that the velocity profile reaches its maximum when introducing copper nanoparticles into the circulatory system. In contrast, the minimum velocity profile is observed in the presence of aluminium oxide nanoparticles. Thus, assisting surgeons in regulating blood flow near the diseased segment during surgical procedures. Fig. 4e illustrates the velocity profile of parent and daughter arteries by considering the different arterial cross sections. In particular, the value of $B_{1}=1.41$ corresponds to the coronary artery, whereas the value of $B_{1}=6.6$ is indicative of the femoral artery. At the stenotic position, the velocity profile displays an augmentation in its profile for increasing arterial cross section. The combined effect of the Grashof number and the solutal Grashof number is depicted in Fig. 4f. Grashof number represents the ratio of buoyant force to viscous force. Clearly, it can be deduced that the fluid velocity is enhanced with an increment of Gr from 0 to 1. This has happened due to enhancement in the buoyant forces as compared to viscous forces, which accelerates the fluid velocity. A similar trend is observed for the solutal Grashof number Gc. This observed trend is corroborated in the work of Sharma et al. [47].

Fig. 4.

Velocity profile.

4.3 Temperature, Concentration and Microorganisms Profile

The variation in temperature for varying Prandtl number is depicted in Fig. 5a. In both the parent and daughter arteries, the temperature distribution decreases as the value of Pr enhances from 19 to 23. The prandtl number represents the ratio of momentum and thermal diffusivity. The prandtl value for pure blood is 21, which is higher as compared to water and other base fluids. The smaller Prandtl number has higher thermal conductivity, which signifies the heat transmitted faster from the arterial wall as compared to higher-Pr fluids. Fig. 5b signifies the argumentation in the temperature profile for an increasing heat source parameter.

Fig. 5.

Temperature, Concentration and Microorganisms profile.

The enhancement occurs due to additional heat produced by the heat source that raises the temperature profile. The result of Fig. 5b may serve as a promising application in the drug delivery system where the metallic nanoparticles can be used as carriers to treat cancerous cells. The tumour cells present downstream of the stenotic region can be treated by enhancing the temperature. Fig. 5c,d depict the concentration profile for Schmidt and chemical reaction parameters, respectively. The concentration profile illustrates how a growing Schmidt number leads to a decreasing concentration. Sc denotes the ratio of the kinematic viscosity to the molecular diffusion coefficient. Since diffusivity is inversely proportional to Sc, a lower Sc number leads to higher diffusivity. The more highly diffusive species have a more noticeable impact of slowing down the concentration distribution. As the parameter for the chemical reaction increases, the concentration profile begins to fall. This has happened because the consumption of additional species will lead to the suppressed concentration profile. The impact of bioconvective Pe on the microorganism’s distribution is seen in Fig. 5e. The Peclet number is the most prominent component that highly influences the density of microorganisms in the blood. Pe was defined as the ratio of the maximal swimming speed of a cell to the diffusion rate of microorganisms. The process by which a substance moves from an area of higher concentration to a lower concentration is commonly known as diffusion. From the figure, there is a direct correlation between the rise in the Pe value from 0 to 3, which results in a reduction in the microbes’ overall dispersion. It has been discovered that an increase in the bioconvective Peclet number results in an increase in the speed of motile microorganisms, which decreases the density of microorganisms.

The combined impact of the bioconvective Lewis number and the microbial concentration differences parameter on the dispersion of microorganisms is shown in Fig. 5f. The figure shows that the density decreases for an upsurge in magnitude of parameter $\sigma_{1}$ . This results from a more significant density differential being formed between the gyrotactic microorganisms and the base fluid, which causes the gyrotactic bacteria to flow back and suppresses the concentration profile. The mounting value of Sb reduces the motile density of the fluid, and this has happened due to a decrease in the microorganism’s diffusivity process. This shows that the microorganism’s density reduces in both sections of the bifurcated artery (parent and daughter arteries) as the magnitude of Sb and $\sigma_{1}$ enhances.

4.4 Flow Rate

\&

Impedance Profile

In stenotic conditions, the hemodynamic factors play a crucial role in assessing the risk of atherosclerosis progression induced by flow disorders. Thus, it is essential to study these factors to reduce its risk and address it at the correct time for better treatment. The volumetric flow rate is defined as the amount of fluid that passes through the arteries in a given amount of time. In contrast, fluid resistance, known as Impedance, is determined by the ratio of pressure drop to flow rate. Fig. 6a shows the flow rate profile for varying magnetic field parameters. The flow rate profile depicts the declining nature as the magnetic field parameter enhances from $M=0$ to $M=\sqrt{5}$ . The flow rate is maximum in the absence of a magnetic field ( $M=0$ ), which clearly shows that the flow rate can be regulated by the magnetic field. The fluid experiences the resistive Lorentz force that retard the fluid motion, and from the figure; it can also be interpreted that the fluid experiences the disturbance at the bifurcated point. Fig. 6b illustrates the flow rate change when the bifurcation angle is changed from $\pi/12$ to $\pi/6$ . According to the inferences drawn from the figure, the flow rate profile remains unchanged in the parent artery but undergoes substantial changes following the bifurcation. The daughter artery’s flow rate profile increases with enhancement in the bifurcation angle. The flow rate as a function of the Grashof number is seen in Fig. 6c. This results from an increase in the temperature differential between the two regions, which has caused the buoyant force to predominate compared to the viscous forces. As a consequence, there has been an increase in the flow rate profile. The decrease in flow rate that occurs when the value of the Schmidt number grows from 0.5 to 1.5 is seen in Fig. 6d. The reduction in molecular diffusion is shown physically by an increase in the value of Sc, whereas a reversal of behaviour was seen concerning the velocity profile, which led to a drop in the flow rate profile. Fig. 6e illustrates the decrement in the velocity profile as the bifurcated angle increases from $\pi/12$ to $\pi/6$ . It can be observed that for change in the bifurcation angle, the impedance profile depicts no change in the parent artery but shows the changes in the daughter artery as the bifurcated angle caused the fluid to change its behaviour and led to the decreasing profile for an increasing bifurcation angle. Fig. 6f shows the growing nature of the impedance profile by enhancing the Rayleigh number. The impedance profile is minimum when $Rb=0$ and increases for a positive value of Rb. It has been noticed that the convection produced by microorganisms lowers the buoyant force that hinders the mobility of the fluid and raises the impedance distribution.

Fig. 6.

Flow rate and Impedance profile.

4.5 WSS, Nusselt Number and Sherwood Number

Wall shear stress (WSS) is defined as the force per unit area on the fluid produced by the arterial wall along the tangential direction. The research concluded that WSS is a critical component in the biomedical industry for elucidating the pattern of atherosclerotic lesion development. This research has clinical potential for assessing WSS’s temporal and spatial distribution, which may aid in the early diagnosis of stenosis. Fig. 7a depicts the WSS distribution by illustrating the effect of varying the stenotic depth in the bifurcated artery. In both sections of the arteries (parent and daughter arteries), the WSS profile decreases as the stenotic depth increases. The findings of this research corroborate those of Zhang’s experimental work [48], which also found that arterial lesion development decreased WSS. Fig. 7b indicates the influence of the solutal Grashof number on the WSS profile. The study reveals that the WSS profile rises as the parameter Gc increases from 0 to 1. WSS profile shows the minimum profile when $Gc=0$ and increases as the Gc enhances, leading to the emergence of a buoyant force that improves the flow near the arterial wall and enhances the WSS profile. Table 4 illustrates the reduction in the wall shear stress ( $\tau_{w}$ ) profile with an escalation in $M$ , whereas the converse pattern is discernible for Gr. The parameter $\tau_{w}$ experiences an augmentation from 4.65 to 7.03 as the Gr value ascends from 0 to 1. Conversely, $\tau_{w}$ exhibits a decrement from 6.76 to 5.90 as the magnetic field parameter ( $M$ ) progresses from 0 to 2. The ratio of convective heat transmission to conductive heat transfer in the arterial walls is represented by the dimensionless Nusselt number. The decrease in the Nusselt number that occurs with an increasing Prandtl number is seen in Fig. 7c and Table 4. It can be inferred from the figure that the efficiency of transferring heat from the arterial wall to blood reduces as the Pr value enhances. This has happened due to the low thermal conductivity of higher-Pr fluids. Fig. 7c visually represents the relationship between the Nusselt and Prandtl numbers. The figure demonstrates that as the Prandtl number increases, there is a notable decrease in the Nusselt number. The observed pattern indicates that heat transfer efficiency from the arterial wall to the blood decreases as the Prandtl values increase.

Fig. 7.

WSS, Nusselt number and Sherwood profile. WSS, wall shear stress.

In the context of arterial heat transfer, it has been observed that fluids with higher Prandtl numbers tend to demonstrate diminished efficacy in the conduction of heat compared to fluids with lower Prandtl numbers. Consequently, the Nusselt number decreases with increasing Prandtl number, indicating a reduced ability to transfer heat from the arterial wall to the blood. Fig. 7d and Table 4 depict the relation between the heat source parameter and the Nusselt profile. The utilisation of a heat source has been found to have potential applications in therapeutic procedures. By selectively targeting the affected region, heat energy can be generated without causing harm to nearby tissues. This localised heating can serve multiple purposes, including the dilation of arteries to facilitate increased blood flow to the affected area. Therefore, it can be utilised as a potential intervention to mitigate stenosis. From the Fig. 7d, it can be inferred that there is an apparent upward trend in the Nusselt profile associated with an escalation in the heat source parameter. An increase in the heat source parameter from 0 to 1 induces a rise in the heat generation rate within the blood, thereby causing an escalation in the temperature, enhancing the Nusselt number profile. Similarly, the Table 4 show the enhancement in the Nusselt profile as the heat source parameter increases from 0.2 to 1 at the axial position $z_{1}^{*}=6.12$ .

Thermal treatment is one of the finest ways to expose blood tissue and cancerous cells to high temperatures in biomedical area; nevertheless, it must be performed under safety recommendations to prevent damage to healthy tissues. The effect of the chemical reaction parameter and Schmidt number on the Sherwood profile is demonstrated in Fig. 7e,f, respectively. The statistics suggest that nature is deteriorating, with a rise in $\xi$ and $S c$ . This pattern may be described by the fact that the molecular diffusivity lowers, indicating a lesser mass transfer across the wall, which leads to a diminishing nature in the Sherwood profile. Table 4 presents the numerical values of the Sherwood profile for $\xi$ and Sc at the axial position $z_{1}^{*}=6.12$ . The table illustrates the decrease in the Sherwood profile as the values of $S c$ and $\xi$ increase. Additionally, Table 4 depicts the variation in the motile density number $\Lambda_{x}$ with increasing values of Pe and Sb. Clearly, it can be inferred that the $\Lambda_{x}$ profile decreases with an increase in the parameters Pe and Sb, resulting from a reduction in the microorganism’s density.

4.6 Velocity Contour

This section presents visual representations of the velocity pattern as influenced by various parameters. This facilitates an enhanced visual and comprehensive depiction of the hemodynamic flow in close proximity to the constricted area along the walls of the bifurcated artery.

The velocity contour for varying Grashof numbers is depicted in Fig. 8a–c. The figure demonstrates a positive correlation between the elevation of the trapping bolus and the magnitude of Gr. The maximum velocity achieved is 0.35 for both scenarios when the Grashof number ( $G r$ ) is equal to 0 and 0.5. Additionally, a maximum velocity of 0.25 is explicitly observed for the case when $G r$ is equal to 0. The maximum fluid speed is observed to occur in the parent region for all three cases. This phenomenon is attributed to the reduction in the value of Gr in the daughter artery, which is caused by the inclination angle $\eta/2$ . The velocity contour demonstrates the relationship between the augmentation in velocity and the accompanying elevation in Gr, which can be ascribed to the amplification of the buoyant thermal force and its subsequent influence on the velocity profile. The effect of the magnetic field on the velocity contour is seen in Fig. 8. Fig. 8d–f show that the velocity field decreases when the magnetic field parameter increases. Without a magnetic field, the fluid can only reach a maximum speed of 0.5 , but when the magnetic field parameter is enhances, the fluid’s speed reduces. The resistive Lorentz force is responsible for the declination in the velocity, which can be predicted by the velocity contour.

Fig. 8.

Velocity contour for Grashof number and Magnetic field parameter.

The velocity contour for different values of Rb is depicted in Fig. 9a–c. The data suggest a negative correlation between velocity and Rb, indicating that an increase in Rb results in a decrease in velocity. In the context of Rb deficiency, it has been noted that the maximum velocity profile is achieved in both the parent and daughter arteries. According to our findings, maximum resistance resulting from the overlapping stenosis occurs at $Rb=0.3$ . In the region pertaining to the daughter, a decrease in the quantity of trapped bolus is observed as the value of $R b$ varies from 0.3 to 0.5. The presence of overlapping and irregular stenosis is observed to cause resistance and decrease fluid velocity in the vicinity of the affected segments. The observed decrease in velocity profile can be attributed to the strengthening of Rb, which resulted in the microorganism’s convection counteracting the buoyancy force exerted on the fluid particles.

Fig. 9.

Blood flow patttern for Rayleigh number and nanoparticle volumetric concentration.

Fig. 9d–f displays the velocity contour related to manipulating volumetric nanoparticle concentration. The maximum velocity achievable by pure blood has been found to be 0.4. Additionally, the presence of trapped bolus can be observed in the vicinity of overlapping stenosis. Upon insertion of the copper nanoparticle into the bloodstream, there is an observed increase in the maximum velocity of the fluid, as illustrated in the Fig. 9e. Fig. 9f depicts the velocity contour of copper and aluminium oxide suspended in the base fluid (blood). Although the maximum velocity remains constant in both cases Fig. 9e and Fig. 9f, the velocity profile is reduced when aluminium oxide nanoparticles are doped in copper/blood solution. These findings provide novel insights for evaluating the precision of theoretical investigations on complex systems and comprehending the impact of blood properties on diverse nanoparticles. Consequently, the surgeon surgeon possesses the ability to regulate the blood flow during the surgical intervention.

5. Conclusions

The current study presents a mathematical model describing the hemodynamic flow through a bifurcated artery with overlapping and irregular stenosis at the parent and daughter arteries, respectively. The investigation has focused on implementing the Al ${}_{2}$ O ${}_{3}$ -Cu/Blood hybrid nanofluid in conjunction with electroosmotic force, a heat source, and microorganisms. Nanoparticles are colloidal suspensions composed of microscopic-shaped substances dispersed in a base fluid. The favourable heat transfer capabilities and stability of nanofluids make them an ideal choice for applications in the biomedical field. The pursuit of stable, non-aggregating, and bio-compatible medicines is a primary objective of researchers. The antibacterial and antiviral properties of copper nanoparticles are attributed to their large surface area-to-volume ratio. The chemical stability of aluminium oxide nanoparticles (AlNPs) in abrasive environments, their low cost, and ease of acquisition render them a viable option for application in biomedicine. The favourable characteristics exhibited by the nanoparticles have prompted us to investigate the potential of utilising the Al ${}_{2}$ O ${}_{3}$ -Cu/Blood hybrid nanofluid in the context of the bifurcated artery. The significant outcomes are outlined below:

• It is noticed that the velocity profile decreases in both parent and daughter artery with enhancement in magentic field parameter while reverse trend is observed for Debye-Huckel parameter.

• Temperature profile enhances with an upsurge in the heat source parameter.

• The mounting value of Sb reduces the motile density of the fluid, attributed to a reduction in microorganism diffusivity.

• Nusselt number profile decline with an enhancement in Pr while the opposite behaviour is observed for heat source parameter.

• Sherwood profile decreases with an enhancement in both chemical reaction parameter and Schmidt number due to lower molecular diffusivity.

The future scope of this study includes extending the investigation to incorporate solute dispersion in the Darcy-Brinkman-Forchheimer porous medium with compliant walls. Additionally, the temperature-dependent viscosity model employed in this study can be substituted with a hematocrit-dependent viscosity model. Further exploration can involve replacing the Newtonian fluid with a non-Newtonian fluid and incorporating the influence of nanoparticle size. Additionally, the wall conditions can be modified to include velocity-slip and thermal-slip conditions.

Nomenclature
$u_{1}^{*}$	Radial velocity (ms ${}^{-1}$ )	$f_{p}$	Heart pulse frequency
$w_{1}^{*}$	Axial direction (ms ${}^{-1}$ )	$G r$	Thermal Grashof Number
$z_{1}^{*}$	Axial direction (m)	$G c$	Solutal Grashof Number
$r_{1}^{*}$	Radial direction (m)	$E c$	Eckert Number
$U_{0}$	Reference velocity (ms ${}^{-1}$ )	$Q_{p}^{f},Q_{d}^{f}$	Volumetric flow rate in parent and daughter artery (m ${}^{3}$ /s)
$\tilde{T}^{*}$	Temperature (K)	$e$	Systolic to diastolic pressure ratio
$\tilde{C}^{*}$	Concentration (K)	$R b$	Rayleigh number
$g$	Acceleration due to gravity (ms ${}^{-2}$ )	$P r$	Prandtl Number
$t_{1}^{*}$	Time (s)	$k_{f}$	Thermal conductivity (W/(m $\cdot$ K))
$\tilde{T}^{*}_{1}$	Reference Temperature (K)	Greek letters
$\tilde{T}^{*}_{w}$	Temperature at wall (K)	$(C_{p})_{hnf}$	Specific heat at constant pressure (J Kg ${}^{-1}$ K ${}^{-1}$ )
$\tilde{C}^{*}_{1}$	Reference Concentration (mol m ${}^{-3}$ )	$\sigma$	Electrical conductivity (S/m)
$\tilde{C}^{*}_{w}$	Concentration at wall (mol m ${}^{-3}$ )	$\delta_{i}$	Stenosis depth (m)
$f_{b}$	Frequency of body acceleration	$\rho_{nf}$	Density of nano-fluid (Kg/m ${}^{3}$ )
$R_{0}$	Radius of normal artery (m)	$\tau_{w}$	Shear stress at the wall (Pa)
$\tilde{Q}$	Heat Source parameter	$\lambda$	Impedance ( $\Omega$ (Ohm))
B	Uniform Magnetic Field (T)	$\omega_{b}$	Circular frequency
$R e$	Reynold’s Number	$\mu_{f}$	Blood’s viscosity (Pa $\cdot$ s)
$F_{b}$	Body acceleration parameter	$\mu_{0}$	Reference viscosity (Pa $\cdot$ s)
$D_{0}$	Amplitude of pulsatile component	$\gamma$	Thermal expansion coefficient (K ${}^{-1}$ )
$D_{1}$	Amplitude of pressure gradient

Availability of Data and Materials

The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request.

Author Contributions

UK–drafting the manuscript, methodology, investigation, software handling, review and editing; BKS–conceptualization, drafting the manuscript, review and editing; BA–analysis, interpretation of data, review and editing; MMB–methodology, investigation, software handling, interpretation of data. All authors contributed to editorial changes in the manuscript. All authors read and approved the final manuscript. All authors have participated sufficiently in the work and agreed to be accountable for all aspects of the work.

Ethics Approval and Consent to Participate

Not applicable.

Acknowledgment

The research is supported by Researchers Supporting Project number (RSP2024R158), King Saud University, Riyadh, Saudi Arabia.

Funding

The research is supported by Researchers Supporting Project number (RSP2024R158), King Saud University, Riyadh, Saudi Arabia.

Conflict of Interest

The authors declare no conflict of interest.

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