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

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.


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. e 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][2][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][9][10][11][12][13][14][15][16][17].
e 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. erefore, 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. e 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][23][24][25][26][27][28][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][31][32][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][35][36][37][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).
e 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. e 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. e nanodrug 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 2 Complexity 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. e finite-difference approach is executed to compute the reduced flow equations. e physical parameters that arise from the regime equations are projected through graphs.

Mathematical Formulation
is 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 Figure 1.
e 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. e 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]).
where δ is the stenosis depth, d is the stenosis distance from origin, z is the axial co-ordinate, and l 0 is the stenosis length. In this model, the blood nanofluid flowing is unsteady and bidirectionally; the velocity and temperature can be expressed as Velocity: V � [u(r, z, t), 0, w(r, 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]):

Complexity
In the foregoing equation, α is the angle parameter, Q 0 is the constant heat source parameter, B 0 is the uniform magnetic field, ρ hnf is the density, t is the time, p is the pressure, r is the radial coordinate, k hnf is the thermal conductive, μ hnf is the dynamic viscosity, T is the temperature, c hnf is the thermal expansion, g is the gravitational acceleration, σ hnf is the electrical conductivity, (C p ) hnf is the specific heat capacity, and subscript hnf is the hybrid nanofluid.
e relevant limiting conditions are expressed as follows: e 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]):   [39], and Kumar and Srinivas [40]).

Complexity
Here, U 0 is the reference velocity, By employing the above variables in equations (3)-(6), the transformed equations are as follows: 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]) where η 0 is the viscosity constant. e 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]) where , and k f are the viscosity, electrical conductivity, specific heat capacity, density, thermal expansion, and thermal conductivity of the base fluid, (ϕ 1 , ϕ 2 ) is the nanoparticle volume fraction, subscript s 1 , bf, and s 2 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. erefore, the dimensionless flow equations are minimized with the following hypothesis δ ≪ 1 and ε � O(1). By employing these hypotheses, the reduced equations are According to Burton [48], the pulsatile pressure gradient is denoted as where A 0 represents the mean pressure gradient and A 1 represents the amplitude of the pulsatile component that controls systolic and diastolic pressures. By employing (8) in (13), the simplified equation is Incorporating the (14) in the blood hybrid nanofluid axial velocity, one can get: e set of equations are transformed in the radial coordinate (x � r/R(z)) form since the governing flow equations are incorporated with limiting conditions: e radial coordinate form of boundary conditions are Wall shear stress (τ s ), the Nusselt number (Nu * ), volumetric flow rate (Q F ), and resistance impedance (λ) of the present model are written as

Numerical Method
e 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. e region is divided into a grid of lines parallel to axes, and it is displayed in Figure 2. e spatial domain is first discretized in this approach and after that the velocity and temperature values are obtained from each node x j . Besides, instant step t i is found over the time.
e first derivative is discretized by forward differencing and the second derivative for central differencing. e N + 1 steps are used to discretize the spatial variable with the Δx/N + 1 step size. t i expresses the time change, and t i � Δt(i − 1) finds its value. It is noticed that the Δt is a small difference in time. e blood velocity and the temperature are calculated in a different time step.
By employing the above expression, the discretized form of equations is written as follows: With the boundary conditions are e stability of this scheme completely depends on time increment (Δt) 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 3 (a) and Figure 3 (b) 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 . is is evidence that the current numerical approach provides efficient results. e 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]. e studies manifest above have extensively validated that the FTCS approach is efficient for blood flow computation in complex geometries.

Results and Discussion
is section affords the physical aspects of emerging parameters like variable viscosity (η 0 � 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 e parametric values [4,5,6,46] such as B 1 � 1.41, d � 0.5, δ * � 0.1, e � 0.5, Pr � 14, c 1 � 1, β � 0.1, G r � 0.5, R e � 2, M a � 0.5, α � π/2, η 0 � 0.2, ϕ 1 � 0.025, and ϕ 2 � 0.025 are considered for computation. In this model, the significance of gold-copper hybrid nanofluid and copper nanofluid characteristics is analyzed via graphs. e 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 (ϕ 1 � 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. e behaviour of the base fluid, nanofluid, and hybrid nanofluid on temperature is compared in Table 3. is table shows that the blood base fluid generates a lower temperature than the nanofluid and hybrid nanofluid cases.
Variation of the Reynolds number (R e ) 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 R e . Further, blood temperature experiences a similar nature. e lower value of R e (R e � 2) exhibits less impact on the blood velocity and temperature; however, its magnitudes variation is higher by growing R e (R e � 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 R e . Figures 6 and 7 demonstrate the characteristics of nanoparticles volume fraction (ϕ 1 + ϕ 2 ) on blood velocity and temperature in hybrid nanofluid and nanofluid cases. It is seen from Figure 6 that the augmentation of gold-copper (ϕ 1 + ϕ 2 ) nanoparticle concentration and copper (ϕ 2 ) nanoparticle concentration from 0.01 to 0.05 tends to promote the blood velocity in the artery.
is 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 (ϕ 1 + ϕ 2 ) on blood temperature. An increment in nanoparticles 10 Complexity volume fraction remarkably elevates the blood temperature. Further, coupling the gold nanoparticle (ϕ 1 ) with copper nanoparticles (ϕ 2 ) 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. ese results show that the hybrid nanoparticles and copper nanoparticles help to accelerate the blood flow in the regular and irregular stenosis artery region. Figure 8 is plotted to exhibit the influence of the Hartmann number (M a ) on blood velocity. In the case of M a � 0, the biomagnetic blood nanofluid behave as a nonmagnetic blood nanofluid. It is found that the fluid momentum has a higher magnitude difference for M a values 1-5. e gold and copper nanoparticles are highly dragged with the strength of Lorentz force in the artery. is 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]. e greater diffusion of nanoparticles is the prime objective in medical applications, so that this characteristic of M a 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 9 explores the changes of the Grashof number (G r ) on blood hybrid nanofluid axial velocity. An increase in G r causes to lift the blood velocity for both the hybrid and unitary nanofluid cases. It is noted that G r is the ratio between buoyancy force and viscous force. e concentration differences of nanoparticles tend to grow the diffusion of nanoparticle species in the blood.
us, 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. e gold and copper nanoparticles are energized with the impact of β. Besides, the heat slightly triggers the buoyancy force. erefore, the blood nanofluid and hybrid nanofluid velocity and temperature increases. Figure 12 displays the influence of changing viscosity parameter (η 0 ) on blood velocity. It is observed that it has the same characteristic as G r on fluid velocity. Physically, the viscosity of the blood lessens owing to the rising values of η 0 . 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. erefore, 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. Complexity Figures 4 and 8). M a and R e follow the same behaviour expressed in axial blood velocity on the volumetric flow rate. e response in volumetric flow rate to G r and α parameters is provided in Figures 16 and 17. From these figures, it is clear that both the parameters have a similar trend. e amplification in G r and α tends to enhance Q F . Further, the absence of G r and α expresses a lower Q F ; however, magnifying these parameters manifests a greater Q F in the artery. e behaviours of the Grashof number (G r ), angle parameter (α), changing viscosity parameter (η 0 ), and the Hartmann number (M a ) on the wall shear stress (τ s ) 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 M a 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 G r , α, and η 0 on wall shear stress. ese figures show that higher values of G r , α, and η 0 result in increasing the wall shear stress for nanofluid and hybrid nanofluid cases. It is cleared from these figures that wall shear stress (τ s ) is highly sensitive to copper (ϕ 2 ) nanoparticles rather than goldcopper (ϕ 1 + ϕ 2 ) hybrid nanoparticles. to the flow rate, which is exhibited in (18). Magnifying the magnetic field (M a ) slowdown the resistance impedance in the artery, whereas the opposite trend is found for α. is 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. is result shows that controlling the blood movement in the artery thus help to stabilize the patients instantly. 14 Complexity Hartmann number (M a ) 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 M a . e influences of ϕ 1 , ϕ 2 , 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 ϕ 1 , ϕ 2 , and β . Further, it is clear that the hybrid nanofluid has a supreme heat transfer rate than the copper nanofluid case.

Conclusions
e 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. e mild stenosis approximation is considered to simplify governing flow equations. e nondimensionalized flow equations are solved by deploying a finite-difference approach. e 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. e 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) e hybrid nanoparticle and copper nanoparticle volume fraction promote the axial velocity and temperature of the blood (III) e 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) e 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) e blood axial velocity in the stenosis artery decreases by means of the growing Reynolds number Data Availability e data used in the study are provided in the respective text part, and no additional data are employed for these outcomes.

Conflicts of Interest
e authors declare that they have no conflicts of interest.