# Finite Difference Computation ofAu-Cu/Magneto-Bio-Hybrid Nanofluid Flow in an Inclined Uneven Stenosis Artery

Authors: H. Thameem Basha; Karthikeyan Rajagopal; N. Ameer Ahammad; S. Sathish; Sreedhara Rao Gunakala
Journal: Complexity
 (2022)

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## Abstract

The present study addresses the fluid transport behaviour of the flow of gold (Au)-copper (Cu)/biomagnetic blood hybrid nanofluid in an inclined irregular stenosis artery as a consequence of varying viscosity and Lorentz force. The nonlinear flow equations are transformed into dimensionless form by using nonsimilar variables. The finite-difference technique (FTCS) is involved in computing the nonlinear transport dimensionless equations. The significant parameters like angle parameter, the Hartmann number, changing viscosity, constant heat source, the Reynolds number, and nanoparticle volume fraction on the flow field are exhibited through figures. Present results disclose that the Lorentz force strongly lessens the hybrid nanofluid velocity. Elevating the Grashof number values enhances the rate of blood flow. Growing values of the angle parameter cause to reduce the resistance impedance on the wall. Hybrid nanoparticles have a superior wall shear stress than copper nanoparticles. The heat transfer rate is amplifying at the axial direction with the growing values of nanoparticles concentration. The applied Lorentz force significantly reduces the hybrid and unitary nanofluid flow rate in the axial direction. The hybrid nanoparticles expose a supreme rate of heat transfer than the copper nanoparticles in a blood base fluid. Compared to hybrid and copper nanofluid, the blood base fluid has a lower temperature.


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## Body

## 1. Introduction

In the physiology system, the heart is the primary organ that plays a vital role in circulating the oxygenated blood to other organs via arteries. The active and proper functioning of the heart is essential for a healthy life cycle. Improper blood transportation in the circulatory system and cardiac-related issues have been the cause of most physical illnesses and death in recent times. Based on the World Health Organization (WHO) report, 30% of deaths in 2008 were related to cardiac disease [1–3]. Most cardiac diseases are caused by cellular waste products, deposits of cholesterol, fibrin, calcium, and buildup of fatty substances in the lumen of the arteries or the formation of plaques. Such plaque formation in the arteries contributes to obstructing blood circulation flow, and this cardiac disease is perceived as stenosis. Besides, the accumulation of fatty substances on the blood vessels walls leads to heart attacks, blood clots, impeding the blood supply, myocardial infarction, and cerebral strokes. Changdar and De [4] scrutinized the impact of inclination on three different nanoparticles (silver, copper, and gold) cases in blood nanofluid flow in a multiple stenosis artery and observed that the wall exhibits low shear stress in the absence of angle parameter. Zaman et al. [5] reported the transport behaviour of silver and aluminium oxide hybrid blood nanofluid flow in a vertical stenotic artery and showed that the unity nanofluid has a higher temperature than the hybrid nanofluid. Tripathi et al. [6] utilized the explicit forward time step approach to scrutinize the flow of gold (Ag)-silver (Au)/blood hybrid nanofluid in irregular stenosis with the variable viscosity and noted that the variable viscosity parameter promotes the hybrid nanofluid axial velocity. Das et al. [7] explained the impacts of Hall current and inclination on hybrid blood nanofluid flow in a mild stenosis artery and reported that the growing values of the Grashof number elevate the blood velocity. Some studies about the blood fluid flow can be found in references [8–17].The physiological fluids that are affected by the external magnetic field and magnetization are dubbed biomagnetic fluids. In recent times, researchers give much attention to the examination of biomagnetic fluid because it has significant applications in biomedical and bioengineering, including targeted drug delivery, cell separation, magnetic wound treatment, reduction of bleeding during surgeries, medical devices (magnetic tracers and blood pumps), and magnetic hyperthermia. It is noticed that the blood is one of the relevant examples of a biomagnetic fluid because it have the cell membrane, haemoglobin compound interface, and intercellular protein. Besides, the unadulterated blood is experienced less impact with the magnetic field. Therefore, in such cases, notable magnetic fields strength is essential to influence its flow. It is observed that artificially suspending nanoparticles with magnetic behaviour can greatly promote the magnetization of the blood. Further, blood containing such magnetic nanoparticles refuses the diamagnetic material or discarding paramagnetic and leads to behaving as a ferromagnetic fluid; as a result, fluid velocity rises gradually. Misra and Shit [18] addressed the influence of magnetic dipole on the flow of viscoelastic biomagnetic fluid past an extending surface and observed that the ferromagnetic interaction parameter diminishes the biomagnetic axial velocity. Murtaza et al. [19] performed a numerical study with the aim to express fluid transport behaviour of maxwell biomagnetic fluid flow over an extending surface and pointed out the rate of heat transfer is reduced with the rising values of magnetic field. Maiti et al. [20] employed the Caputo-Fabrizio (CF) derivative model to examine the biomagnetic blood fluid flow in a porous vessel and found that the thermal radiation and the Schmidt number decline the blood concentration. The flow of biomagnetic fluid through a normal duct in the presence of magnetic dipole and Lorentz force was scrutinized by Mousavi et al. [21] and noticed that the Lorentz force near the constricted zone lessens the wall shear stress. Some notable studies on blood flow with the impact of the magnetic field are exposed through Refs. [22–29].Numerous researchers and engineers have paid significant attention to nanotechnology because it is used in several practical situations, for instance, biochemical engineering and medical industries. Several nanoparticles like silver, copper, gold, and ferrite particles are utilized in proteins, delivery of drugs, nucleic acids, vaccines, and genes [30–33]. Because these nanoparticles exhibit high biocompatibility, magnetic, chemical, unique mechanical, and thermal effects. It is observed that gold (atomic number = 79) nanoparticles have much popularity in biomedical applications for RNA quantification (via optical biosensors) and treating malignant tumours owing to unique quenching efficiencies, targeting ligands, significant surface modifiability, and imaging probes when comparing with other nanoparticles [34–38]. Besides, the gold nanoparticles exhibit a nontoxic behaviour in biological media. Gold nanoparticles have a unique optical behavior, which is more relevant for several therapeutic applications (photothermal and radiotherapy for eradicating cancer cells) and diagnostic approaches (cell imaging, computed tomography, and optical imaging). It is important to mention that the gold nanoparticles can be remarkably functionalized with DNA, proteins, antibodies, polyelectrolyte, and ligands. Koriko et al. [39] examined the heat transfer behaviour of gold-blood Carreau nanofluid flow with the influence of partial slip and that the elevating gold nanoparticle concentration tends to diminish the rate of heat transfer. Kumar and Srinivas [40] employed the Maxwell Garnett and Brinkman models (thermophysical property model) to analyze the flow of gold blood nanofluid in a channel and found that the gold nanoparticles have a less temperature in the blood than the aluminium oxide. Khan et al. [41] conducted a numerical study to determine the impact of nonlinear radiation on Casson gold–blood nanofluid flow over an extending spinning disk and noticed that the blood temperature rises by elevating nanoparticle volume fraction. Bhatti [42] explored the Jeffrey-gold intrauterine nanofluid flow through an asymmetric channel by means of linear thermal radiation and showed that thermal radiation maximizes Jeffrey blood temperature. Most science and engineering problems are usually in the form of nonlinear boundary value problems (BVPs). The solution of such nonlinear problems plays a significant role in understanding science and engineering systems. It is noticed that most of the nonlinear BVPs are partial differential equations (PDEs). Numerous numerical and analytical approaches are available in the literature to solve such PDEs. In that, the explicit finite-difference FTCS method is one of the notable approaches, and it is described in Hoffmann's book [43]. It is essential to mention that many researchers extensively used FTCS in fluid mechanical and biomechanical problems [44, 45]. Further, it is revealed that this scheme is stable, rapidly convergent, and easy to program.The motivation of this current analysis is that for the applications in targeted nano-drug delivery systems. Besides, this numerical simulation's primary purpose is to effectively carry out the decision-making process during arterial disease treatment. In recent times, targeting the nano-drugs at the stenosis region is the trending and influential approach compared to the conventional treatment method. The nano-drug delivery improves the stenosis throat's clotting formation; further, this computational simulation can also predict the effects of post-treatment processes. It is witnessed from the above literature that several studies explore the blood flow in different types of stenosis arteries with numerous physical aspects. However, no study has focused on the gold and copper/blood nanofluid flow in an uneven inclined stenosis artery in the presence of a magnetic field, viscous dissipation, and heat generation. Further, the gold and copper [46, 47] nanoparticles have predominant applications in drug targeting, wound treatment, cancer diagnosis, cardiovascular treatment, and chemotherapy. With these enthused, the current framework focuses on biomagnetic gold-copper blood hybrid nanofluid flow in an irregular inclined stenosis artery utilizing varying viscosity and Lorentz force. By means of nonsimilar variables, the dimensional flow equations are reduced in dimensionless form. The finite-difference approach is executed to compute the reduced flow equations. The physical parameters that arise from the regime equations are projected through graphs.


## 2. Mathematical Formulation

This model considers time-dependent two-dimensional biomagnetic gold-copper hybrid nanofluid flow in an inclined irregular stenosis artery, and the schematic model is exhibited in Figure1. The nanoparticle volume fraction model is utilized to scrutinize the biomagnetic blood flow. It is noted that the gold and copper nanoparticles are suspended in the biomagnetic blood fluid, and the thermophysical properties of the base fluid (blood) and nanomaterial (gold and copper) are provided in Table 1. The strength of Lorentz force is executed in the transverse to the blood flow direction. For this modelling, the two-dimensional cylindrical coordinate r,θ,z system is employed, and r and z are represented as the artery radial and axial coordinates (Tripathi et al. [6]).(1)Rz=R0−2δcosz−d2−l042π−7100cosz−d−l0232π,d<z<d+l0R0,otherwise,where δ is the stenosis depth, d is the stenosis distance from origin, z is the axial co-ordinate, and l0 is the stenosis length.Figure 1
Physical configuration of the problem.Table 1
Values of the various physical parameters (blood, gold and copper) (Das et al. [7], Koriko [39], and Kumar and Srinivas [40]).
Physical propertiesBloodGold (Au)Copper (Cu)ρkg/m31063193208933CpJ/kgK3594129385γ×10−5K−10.181.41.67kW/mK0.492314401σS/m6.67×10−14.10×10759.6×106In this model, the blood nanofluid flowing is unsteady and bidirectionally; the velocity and temperature can be expressed as(2)Velocity:V=u¯r,z,t,0,w¯r,z,t,Temperature:T=Tr,z,t.,where u¯ and w¯ have denoted the of components radial and axial velocities. With these frameworks, the governing flow equations are written as follows (Zaman et al. [5], Tripathi et al. [6, 45], Das et al. [7], and Rathore and Srikanth [16]):(3)∂u¯∂r¯+u¯r¯+∂w¯∂z¯=0.(4)ρhnf∂u¯∂t¯+u¯∂u¯∂r¯+w¯∂u¯∂z¯=−∂p¯∂r¯+1r¯∂∂r¯μhnfT∂u¯∂r¯−2μhnfTu¯r¯+∂∂z¯μhnfT∂u¯∂z¯+∂w¯∂r¯−gγρhnfT−T1cosα.(5)ρhnf∂w¯∂t¯+u¯∂w¯∂r¯+w¯∂w¯∂z¯=−∂p¯∂z¯+1r¯∂∂r¯rμhnfT∂u¯∂z¯+∂w¯∂r¯+∂∂z¯2μhnfT∂w¯∂z¯+gγρhnfT−T1sinα−σhnfB02w.(6)ρCphnf∂T∂t¯+u¯∂T∂r¯+w¯∂T∂z¯=khnf∂2T∂r¯2+1r¯∂T∂r¯+∂2T∂z¯2+Q0.In the foregoing equation,α is the angle parameter, Q0 is the constant heat source parameter, B0 is the uniform magnetic field, ρhnf is the density, t¯ is the time, p¯ is the pressure, r¯ is the radial coordinate, khnf is the thermal conductive, μhnf is the dynamic viscosity, T is the temperature, γhnf is the thermal expansion, g is the gravitational acceleration, σhnf is the electrical conductivity, Cphnf is the specific heat capacity, and subscript hnf is the hybrid nanofluid.The relevant limiting conditions are expressed as follows:(7)wR,t=0,∂w0,t∂r=0,wr,0=0,TR,t=1,∂T0,t∂r=0,Tr,0=0..The governing nonlinear flow equations are transformed by suitable nonsimilar variables, which are shown as follows (Zaman et al. [5] and Tripathi et al. [6, 45]):(8)r=r¯R0,z=z¯l0,p¯=R02PU0l0μf,u=l0u¯δ∗U0,θ=T−T1Tw−T1,w=w¯U0,R=R¯R0,t=U0t¯R0,d=d¯l0.Here,U0 is the reference velocity, Gr=gρfR02γfTw−T1/U0μf is the Grashof number, Tw is the wall temperature, Ma=σf/μfB0R0 is the Hartmann number, ε=R0/l0 is the vessel aspect ratio, Pr=Cpfμf/kf is the Prandtl number, δ=δ∗/R0 is the stenosis height parameter, and Re=U0ρfR0/μf is the Reynolds number.By employing the above variables in equations (3)–(6), the transformed equations are as follows:(9)δ∂u∂r+ur+∂w∂z=0,Reρhnfρfδε2∂u∂t+δεu∂u∂r+εw∂u∂z=−∂p∂r+δR0l02μ01r∂∂rμhnfθ∂u∂r+ε2R0μ0∂∂zμhnfθδεl0∂u∂z+1R0∂w∂r−2δε2μ0μhnfθur2−ργhnfργfεcosαGrθ,Reρhnfρf∂w∂t+δεu∂w∂r+εw∂w∂z=−∂p∂z+1r∂∂rrμhnfθμ0δε2∂u∂z+∂w∂r+ε2∂∂z2μhnfθμ0∂w∂z+ργhnfργfGrθsinα−σhnfσfMa2w,RePrkfkhnfρCphnfρCpf∂θ∂t+δεu∂θ∂r+εw∂θ∂z=∂2θ∂r2+1r∂θ∂r+ε2∂2θ∂z2+kfkhnfβ.Usually, in a biological system, the viscosity of blood is not constant in all cases since it varies due to several factors such as hematocrit ratio, vessel width, temperature, and axial or radial coordinates. Such a viscosity variation causes several reality cases, including decreasing blood thickness, rising of blood circulation, lowering coagulation factors, and maximizing blood flow. To capture these behaviours, in this model, the blood nanofluid viscosity is considered dependent on fluid temperature, which is given as (Zaman et al. [5])(10)μfθ=μ0e−η0θ,wheree−η0θ=1−η0θ,η0≪1,where η0 is the viscosity constant.The density, thermal conductivity, electrical conductivity, thermal expansion, dynamic viscosity, and specific heat capacity of the nanofluid and hybrid nanofluid are (Tripathi et al. [6] and Das et al. [7])(11)ρnf=ϕρs+1−ϕρf,σnf=σf3σs/σf−1ϕσs/σf+2−ϕσs/σf−1+1,ρCpnf=ϕρCps+1−ϕρCpf,μnf=μf1−ϕ2.5,knf=kfks+2kf−2ϕkf−ksks+2kf+ϕkf−ks,ργnf=ϕργs+1−ϕργf.,ρhnf=ϕ1ρs1+1−ϕ1ρf1−ϕ2+ϕ2ρs2,μhnf=μfθ1−ϕ12.51−ϕ22.5,σhnfσbf=σs2+2σbf−2ϕ2σbf−σs2σs2+2σbf+ϕ2σbf−σs2,khnfkbf=ks2+2kbf−2ϕ2kbf−ks2ks2+2kbf+ϕ2kbf−ks2,ρCphnf=ϕ1ρCps1+1−ϕ1ρCpf1−ϕ2+ϕ2ρCps2,ργhnf=ϕ1ργs1+1−ϕ1ργf1−ϕ2+ϕ2ργs2,whereσbfσf=σs1+2σf−2ϕ1σf−σs1σs1+2σf+ϕ1σf−σs1,kbfkf=ks1+2kf−2ϕ1kf−ks1ks1+2kf+ϕ1kf−ks1.,where μf, σf, Cpf, ρf,γf, andkf are the viscosity, electrical conductivity, specific heat capacity, density, thermal expansion, and thermal conductivity of the base fluid, ϕ1,ϕ2 is the nanoparticle volume fraction, subscript s1,bf,ands2 are the first solid particle, base fluid, and second solid particle.It is observed that compared with the artery radius, the stenosis maximum height is small, and the further length of the stenotic region and artery radius is of comparable magnitude. Therefore, the dimensionless flow equations are minimized with the following hypothesisδ≪1 and ε=O1. By employing these hypotheses, the reduced equations are(12)∂w∂z=0,∂p∂r=0,Reρhnfρf∂w∂t=−∂p∂z+1r∂∂rrμhnfθμ0∂w∂r+ργhnfργfGrθsinα−σhnfσfMa2w,ReρCphnfρCpfkfkhnfPr∂θ∂t=∂2θ∂r2+1r∂θ∂r+kfkhnfβ,Rz=1−2δ∗cosz−d2−142π−cosz−d−1232π7100,d<z¯<d+11,otherwise.According to Burton [48], the pulsatile pressure gradient is denoted as(13)−∂p∂z=A0+A1tcos2πwp,t>0,where A0 represents the mean pressure gradient and A1 represents the amplitude of the pulsatile component that controls systolic and diastolic pressures.By employing (8) in (13), the simplified equation is(14)−∂p∂z=B11+ecosc1t,where e=A1/A0,B1=A0a2/μ0U0 and c1=2πawp/U0.Incorporating the (14) in the blood hybrid nanofluid axial velocity, one can get:(15)Reρhnfρf∂w∂t=B11+ecosc1t+1r∂∂rrμhnfθμ0∂w∂r+ργhnfργfGrθsinα−σhnfσfMa2w.The set of equations are transformed in the radial coordinatex=r/Rz form since the governing flow equations are incorporated with limiting conditions:(16)Reρhnfρf∂w∂t=B11+ecosc1t+μhnfθμ01R2∂2w∂x2+1x∂w∂x+ργhnfργfGrθsinα−σhnfσfMa2w,ρCphnfρCpfkfkhnfRePr∂θ∂t=1R2∂2θ∂x2+1x∂θ∂x+kfkhnfβ.The radial coordinate form of boundary conditions are(17)x=0:∂w0,t∂x=0,∂θ0,t∂x=0,x=1:w1,t=0,θ1,t=1,t=0:wx,0=0,θx,0=0.Wall shear stressτs, the Nusselt number Nu∗, volumetric flow rate QF, and resistance impedance λ of the present model are written as(18)τs=−1Rz∂w∂xx=1,Nu∗=−1Rz∂θ∂xx=1,QF=Rz22π∫01wx dx,λ=L∂p/∂zQF=LB11+ecos2πtRz22π∫01wx dx..


## 3. Numerical Method

The FTCS (forward time central space) finite-difference technique is employed to solve the present mathematical model's nonlinear coupled dimensionless flow equations subject to the appropriate initial and boundary condition. A rectangular region of the flow field is chosen in the explicit approach. The region is divided into a grid of lines parallel to axes, and it is displayed in Figure2. The spatial domain is first discretized in this approach and after that the velocity and temperature values are obtained from each node xj. Besides, instant step ti is found over the time. The first derivative is discretized by forward differencing and the second derivative for central differencing. The N+1 steps are used to discretize the spatial variable with the Δx/N+1 step size. ti expresses the time change, and ti=Δti−1 finds its value. It is noticed that the Δt is a small difference in time. The blood velocity and the temperature are calculated in a different time step.(19)∂w∂t≅wji+1−wjiΔt;∂θ∂t≅θji+1−θjiΔt∂w∂x≅wj+1i−wj−1i2Δx;∂θ∂x≅θj+1i−θj−1i2Δx∂2w∂x2≅wj+1i−2wji+wj−1i2Δx;∂2θ∂x2≅θj+1i−2θji+θj−1i2Δx.Figure 2
Finite difference space grid.By employing the above expression, the discretized form of equations is written as follows:(20)wji+1=wji+ΔtRe1−ϕ21−ϕ1+ϕ1ρs1/ρf+ϕ2ρs2/ρfB11+ecosc1ti+1−η0θji1−ϕ12.51−ϕ22.51R2∂2w∂x2+1x∂w∂x+1−ϕ21−ϕ1+ϕ1ργs1ργf+ϕ2ργs2ργfGrθjisinα−σhnfσfMa2wji,θji+1=θji+khnf/kfΔtRePr1−ϕ21−ϕ1+ϕ1ρCps1/ρCpf+ϕ2ρCps2/ρCpf1R2∂2θ∂x2+1x∂θ∂x+kfkhnfβ.With the boundary conditions are(21)wj1=θj1=0,att=0,wj+1i=wji,θj+1i=θji,atx=0,wN+1i=0,θN+1i=1,atx=1..The stability of this scheme completely depends on time incrementΔt and step size Δx; thus, Δt=0.0001 and Δx=0.025 are fixed to tackle the stability condition. Several studies [5, 6, 43, 45, 46] proved that these values are suitable for stability and convergence of the FTCS approach. Further, Hoffmann's book [43] exposed that the above-used time and spatial values confirm this scheme's stability and convergence. Figure 3a and Figure 3b shows the local error and sum of time and space error on the dimensionless radius, respectively. From these figures, it is observed that the velocity and temperature of blood maintain 10−10 error at each spatial node. As similar, the sum of time and space error is 10−6. This is evidence that the current numerical approach provides efficient results. The FTCS approach has been employed in numerous previous simulations, including heat transfer enhancement for solar energy absorber in a permeable annular [49], the blood flow of viscoelastic fluid in tapered overlapping even stenosed artery [50], and the exploration of nano-Bingham-Papanastasiou fluid in a diseased curved artery [51]. The studies manifest above have extensively validated that the FTCS approach is efficient for blood flow computation in complex geometries.Figure 3
(a) Local error onw and θ. (b) Sum of time and space error on w and θ.
(a)(b)


## 4. Results and Discussion

This section affords the physical aspects of emerging parameters like variable viscosityη0=0,0.1,0.3,0.5, constant heat source β=0,0.3,0.6,0.9, angle parameter α=0,π/6,π/4,π/2, the Hartmann number Ma=1,2,3,4, the Grashof number Gr=0,0.1,0.3,0.5, the Reynolds number Re=2,3,4,5, and nanoparticle volume fraction ϕ1+ϕ2=0.01,0.02,0.03,0.05 on the biomagnetic blood hybrid nanofluid velocity w, temperature θ, resistance impedance λ, wall shear stress τs, the Nusselt number Nu∗, and volumetric flow rate QF. The parametric values [4, 5, 6, 46] such as B1=1.41,d=0.5,δ∗=0.1,e=0.5,Pr=14,c1=1,β=0.1,Gr=0.5,Re=2,Ma=0.5,α=π/2,η0=0.2,ϕ1=0.025, and ϕ2=0.025 are considered for computation. In this model, the significance of gold-copper hybrid nanofluid and copper nanofluid characteristics is analyzed via graphs. The system of equations is reduced with the help of nonsimilar variables, and FTCS obtains the solution. For obtaining scheme validation, the present result is compared with earlier results in Table 2. It is witnessed that the present results are in valid agreement. It is essential to note that the hybrid nanofluid turns into a mono nanofluid in the absence of gold nanoparticles ϕ1=0. To show the variation in figures, the solid line is used for the hybrid nanofluid case, and the dashed line is used for the nanofluid case. The behaviour of the base fluid, nanofluid, and hybrid nanofluid on temperature is compared in Table 3. This table shows that the blood base fluid generates a lower temperature than the nanofluid and hybrid nanofluid cases.Table 2
Validation of biomagnetic blood axial velocity with Zaman et al. [5] and Tripathi et al. [6].
RZaman et al. [5]Tripathi et al. [6]Present result00.58590.58810.589000.10.58290.58450.582890.20.57250.57250.571700.30.55400.55180.553390.40.52610.52150.525930.50.48640.48020.487270.60.43230.42570.438430.70.36050.35490.370830.80.26710.26370.278910.90.14830.14730.157151.0000Table 3
Comparison of base fluid, nanofluid, and hybrid nanofluid on temperature.
RTemperatureBlood basefluidCu/blood nanofluidAu-Cu/blood hybrid nanofluid00.0606070.0920540.0939890.10.0670320.0998590.1018550.20.0873740.1240160.1261760.30.1246220.1665250.1689020.40.1829080.2299360.2325050.50.2662310.3162890.3189370.60.3767970.4258980.4284280.70.5133560.5562860.5584560.80.6701140.7016530.7032270.90.8367820.8531460.8539591.0111Variation of the Reynolds numberRe on biomagnetic blood hybrid nanofluid axial velocity and temperature is visualised for hybrid nanofluid and Cu nanofluid cases in Figures 4 and 5, respectively. In this study, due to the laminar flow case, meagre Reynolds numbers are assumed, so the viscosity is dominant in the regime. It is clear from these figures that the biomagnetic blood axial velocity significantly reduces while growing values of Re. Further, blood temperature experiences a similar nature. The lower value of ReRe=2 exhibits less impact on the blood velocity and temperature; however, its magnitudes variation is higher by growing ReRe=4,6,8. Due to stenosis prohibition and nanoparticles, blood flow reduction occurs in the channel, even though the Reynolds number promotes the inertial force. Due to this reason, biomagnetic blood velocity and temperature are behaving with the impact of Re.Figure 4
Behaviour ofRe on w.Figure 5
Behaviour ofRe on θ.Figures6 and 7 demonstrate the characteristics of nanoparticles volume fraction ϕ1+ϕ2 on blood velocity and temperature in hybrid nanofluid and nanofluid cases. It is seen from Figure 6 that the augmentation of gold-copper ϕ1+ϕ2 nanoparticle concentration and copper ϕ2 nanoparticle concentration from 0.01 to 0.05 tends to promote the blood velocity in the artery. This characteristic of hybrid nanofluid is predominant to clinicians because it may help to promote the blood flow in the capillary tubes and stenosis during surgery. Figure 7 is drawn to discuss the behaviour of nanoparticles volume fraction ϕ1+ϕ2 on blood temperature. An increment in nanoparticles volume fraction remarkably elevates the blood temperature. Further, coupling the gold nanoparticle ϕ1 with copper nanoparticles ϕ2 highly contributes to blood temperature growth than copper nanoparticles suspension in the blood. It is revealed from this figure that the thermal diffusion of blood rises with nanoparticles in the stenotic vessel, which leads to elevating the transport process. These results show that the hybrid nanoparticles and copper nanoparticles help to accelerate the blood flow in the regular and irregular stenosis artery region.Figure 6
Behaviour ofϕ1+ϕ2 and ϕ2 on w.Figure 7
Behaviour ofϕ1+ϕ2 and ϕ2 on θ.Figure8 is plotted to exhibit the influence of the Hartmann number Ma on blood velocity. In the case of Ma=0, the biomagnetic blood nanofluid behave as a nonmagnetic blood nanofluid. It is found that the fluid momentum has a higher magnitude difference for Ma values 1–5. The gold and copper nanoparticles are highly dragged with the strength of Lorentz force in the artery. This creates a resistive behaviour in the blood flow. Due to this reason, blood velocity declines. Further, it is noticed that the present outcome accords with the results of Das et al. [7]. The greater diffusion of nanoparticles is the prime objective in medical applications, so that this characteristic of Ma is remarkable in synthesising bio-nanomaterials. Also, due to the characteristic of the magnetic field, it is employed to control the blood flow in the artery. It can be used to treat several cardiovascular diseases as an effective tool. In particular, this may be useful in wound treatments and pharmacological, for instance, healing skin contusions and burns.Figure 8
Behaviour ofMa on w.Figure9 explores the changes of the Grashof number Gr on blood hybrid nanofluid axial velocity. An increase in Gr causes to lift the blood velocity for both the hybrid and unitary nanofluid cases. It is noted that Gr is the ratio between buoyancy force and viscous force. The concentration differences of nanoparticles tend to grow the diffusion of nanoparticle species in the blood. Thus, the thermal buoyancy force increases in the channel, whereas viscous force decreases. As a result, the blood velocity rises. Figures 10 and 11 demonstrate the nature of constant heat source β on blood velocity and temperature. In the channel, the generation of energy takes place as a result of the increase of β. The gold and copper nanoparticles are energized with the impact of β. Besides, the heat slightly triggers the buoyancy force. Therefore, the blood nanofluid and hybrid nanofluid velocity and temperature increases. Figure 12 displays the influence of changing viscosity parameter η0 on blood velocity. It is observed that it has the same characteristic as Gr on fluid velocity. Physically, the viscosity of the blood lessens owing to the rising values of η0. As a consequence, blood velocity is accelerated. Numerous values of angle parameter α on biomagnetic blood nanofluid velocity is displayed in Figure 13. It is evident that the blood velocity augments with the increasing values of α. In this model, α is incorporated with the thermal buoyancy term in the momentum equation. Therefore, when α is absent, it eliminates the thermal buoyancy effect in the regime. Further, rising the α highly enhances the buoyancy effect, and it accelerates the blood velocity.Figure 9
Behaviour ofGr on w.Figure 10
Behaviour ofβ on w.Figure 11
Behaviour ofβ on θ.Figure 12
Behaviour ofη0 on w.Figure 13
Behaviour ofα on w.Figures14–17 depict the influences of the Grashof number Gr, angle parameter α, the Reynolds number Re, and the Hartmann number Ma on volumetric flow rate QF for hybrid and single nanofluid cases. It is cleared that the integration of variable blood velocity concerning the channel radial direction is known as volumetric flow rate. Besides, the volumetric flow rate reveals the nature of the axis velocity. The impacts of Ma and Re on volumetric flow rate are illustrated in Figures 14 and 15. It is confirmed that Ma and Re highly reduce the axial blood velocity (see Figures 4 and 8). Ma and Re follow the same behaviour expressed in axial blood velocity on the volumetric flow rate. The response in volumetric flow rate to Gr and α parameters is provided in Figures 16 and 17. From these figures, it is clear that both the parameters have a similar trend. The amplification in Gr and α tends to enhance QF. Further, the absence of Gr and α expresses a lower QF; however, magnifying these parameters manifests a greater QF in the artery.Figure 14
Behaviour ofMa on QF.Figure 15
Behaviour ofRe on QF.Figure 16
Behaviour ofGr on QF.Figure 17
Behaviour ofα on QF.The behaviours of the Grashof numberGr, angle parameter α, changing viscosity parameter η0, and the Hartmann number Ma on the wall shear stress τs are sketched in Figures 18–21. It is evident that endothelial proliferation and turbulent flow happen in the system when low shear stress occurs. Besides, laminar flow happens with high shear stress. Figure 18 shows the impact of Ma on wall shear stress. When this term is magnified, the wall shear stress is reduced. It is revealed that the magnetic field plays a major role to maintain a laminar flow in the system. Figures 19–21 are drawn to explore the influences of Gr, α, and η0 on wall shear stress. These figures show that higher values of Gr, α, and η0 result in increasing the wall shear stress for nanofluid and hybrid nanofluid cases. It is cleared from these figures that wall shear stress τs is highly sensitive to copper ϕ2 nanoparticles rather than gold-copper ϕ1+ϕ2 hybrid nanoparticles.Figure 18
Behaviour ofMa on τs.Figure 19
Behaviour ofGr on τs.Figure 20
Behaviour ofη0 on τs.Figure 21
Behaviour ofα on τs.Figures22 and 23 present the effects of the Hartmann number Ma and angle parameter α on resistance impedance. The resistance impedance is inversely related to the flow rate, which is exhibited in (18). Magnifying the magnetic field Ma slowdown the resistance impedance in the artery, whereas the opposite trend is found for α. This flow pattern exhibits that the growing magnetic field declines the impedance of the blood rheology. Further, the magnetic field behaves like a controlling parameter to tune the impedance effects. This result shows that controlling the blood movement in the artery thus help to stabilize the patients instantly. Figures 24 and 25 exhibit the changes in blood flow rate for several values of the Hartmann number Ma and angle parameter α. It is cleared that the flow rate in the axial coordinate is elevated with the increasing values of α, whereas a reverse trend is found for Ma. The influences of ϕ1,ϕ2, and β on the Nusselt number Nu∗ are depicted in Figures 26 and 27, respectively. It is ascertained that the Nusselt number is highly improved with the impacts of ϕ1,ϕ2, and β . Further, it is clear that the hybrid nanofluid has a supreme heat transfer rate than the copper nanofluid case.Figure 22
Behaviour ofMa on λ.Figure 23
Behaviour ofα on λ.Figure 24
Behaviour ofMa on QF at Z.Figure 25
Behaviour ofα on QF at Z.Figure 26
Behaviour ofϕ1+ϕ2 and ϕ2 on Nu∗ at Z .Figure 27
Behaviour ofβ on Nu∗ at Z.


## 5. Conclusions

The impacts of changing viscosity and Lorentz force on the biomagnetic hybrid nanofluid flow through an inclined irregular stenosis artery have been scrutinized in this study. To exhibit the characteristics of hybrid nanofluid, the volume fraction model is adopted. The mild stenosis approximation is considered to simplify governing flow equations. The nondimensionalized flow equations are solved by deploying a finite-difference approach. The present numerical solution is validated with Zaman et al. [5] and Tripathi et al. [6] for various axial velocity values, which is portrayed in Table 2. From this Table 2, it is evident that the adopted numerical method has a decent agreement. Variations caused by the influences of growing parameters on blood hybrid nanofluid velocity, temperature, and physical quantities (wall shear stress, resistance impedance, flow rate) are shown through graphs. The preeminent findings of the present analysis are itemized as follows:(I)
It is cleared that the hybrid nanoparticles have a better fluid flow and heat transfer than the unitary nanoparticles(II)
The hybrid nanoparticle and copper nanoparticle volume fraction promote the axial velocity and temperature of the blood(III)
The hybrid nanoparticles express a supreme flow rate than the copper nanoparticles(IV)
In the absence of the Grashof number and angle parameter, the wall shear stress and flow rate have similar trends for hybrid nanofluid and copper nanofluid cases(V)
The variable viscosity parameter exhibits the peak magnitude on wall shear stress by magnifying values(VI)
In the 2.5-to-3.5-time region, the Hartmann number and angle parameter manifest a greater resistance impedance on the blood hybrid nanofluid flow(VII)
Variable viscosity, heat source/sink, and angle parameter cause to enhance the blood hybrid nanofluid velocity(VIII)
The blood axial velocity in the stenosis artery decreases by means of the growing Reynolds number



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*Source: 2078372-2022-04-12.xml*