Flow and irreversible mechanism of pure and hybridized non-Newtonian nanofluids through elastic surfaces with melting effects

Nomenclature

a , b > 0

Constant ( s − 1 )

x , y

Cartesian coordinates

V → = ( u , v , 0 )

Velocity field

u , u w , u slip

Velocities ( m s − 1 )

u ∞

Far-field velocity ( m s − 1 )

T , T w , T ∞

Fluid, wall, and ambient temperatures ( K )

S →

Cauchy stress tensor

∇ C , ∇ T

Gradient of concentration and temperature

D B

Brownian diffusion ( m 2 s − 1 )

D T

Thermophoresis diffusion ( m 2 s − 1 )

( ρ c p ) f

Fluid heat capacitance

( ρ c p ) p

Nanoparticle’s heat capacitance

p

Pressure ( N m − 2 )

c p

Specific heat ( J kg − 1 K − 1 )

k f

Thermal conductivity ( W m − 1 K − 1 )

D

Slip coefficient

δ

Heat potential

C s

Solid surface heat capacity ( J kg − 1 K − 1 )

T m

Melting temperature ( K )

f

Dimensionless velocity

Θ

Dimensionless temperature

ψ

Dimensionless concentration

N 1

Slip parameter

τ w

Stress tensor component at the wall

q w

Wall heat flux ( W m − 2 )

q m

Wall mass flux ( k g m − 2 s − 1 )

Re

Reynold number

Pr

Prandtl number

Ω , ϵ

Material parameters

Rd

Radiation parameter

N b

Brownian diffusion parameter

N t

Thermophoresis diffusion parameter

σ

Electrical conductivity ( S m − 1 )

σ *

Stephan–Boltzmann constant ( W m − 2 K − 4 )

φ hnf = φ 1 + φ 2

Volume friction of φ 1 and φ 2

A 1

Rivlin–Ericksen tensor

q r

Radiative heat flux

k *

Mean absorption coefficient ( c m − 1 )

J

Current density ( A m − 2 )

B 0

Magnetic field ( W m 2 )

β

Material parameter (1 / P a )

γ

Second material parameter (1 / s )

C

Fluid concentration ( kg m − 3 )

C ∞ , C 0

Ambient and reference concentration ( kg m − 3 )

ρ f

Fluid density ( kg m − 3 )

ρ p

Nanoparticle’s density ( kg m − 3 )

μ f

Dynamic viscosity ( kg m − 1 s − 1 )

η

Similarity variable

E G

Entropy generation ( W m − 3 K − 1 )

R

Molar gas constant ( J K − 1 mol − 1 )

N g

Dimensionless entropy

Br

Brinkman number

C f

Skin friction

D 1

Diffusion parameter

Nu

Nusselt number

Sh

Sherwood number

Be

Bejan number

B

Ratio parameter

M

Magnetic number

Ec

Eckert number

Le

Lewis number

M t

Melting parameter

A 1 , A 2

Temperature and concentration ratio parameters

1 Introduction

A stagnating point is a location in fluid mechanics where the prevailing fluid velocity is stationary. This point designates where the flow splits and travels on both sides of the surface as it moves towards it. When a fluid confronts an impermeable solid barrier, such as on an airplane wing or a vibrating cylinder submerged in fluid, stagnation-point flows take place. These flow patterns exhibit a fluid stagnation point, and the streamlines around it show a local resemblance to those around a saddle point. Blood flow at an arterial junction is another instance that is really intriguing. Stagnation-point flows, which describe the fluid flow near the stagnant area at the leading edge of a blunt-nosed physique, exist on all moving solid objects. The amount of pressure, energy transfer, and mass deposit rates in the stagnation area are all at their highest levels. In light of many potential applications of this subject in fields like the aviation sector, nuclear reactor cooling, and fan-cooled chilling of electronic equipment, the boundary layer dynamics around stagnation points are highly valued by researchers. Hiemenz [1], for the first time, investigated the two-dimensional fluid of a stagnant state flow of viscous liquids in 1911. Petschek et al. [2] carried out preliminary experimental studies on the stagnation point flow in 1968. They used dog ex vivo blood, which was drawn and injected into the flow cavity. Microscopically, the accumulation of blood elements was observed. The enhanced technique utilized by Reininger et al. [3] involved the leading passage of platelet-rich plasma (PRP) into the flow cavity. Homann [4] provided the precise solution of an axisymmetric flow subject to an incompressible viscous fluid model. Ishak et al. [5] refined the boundary layer flow at the stagnation point when a magnetic field is present. Van Gorder and Vajravelu [6] investigated the stagnant point flow of second-grade fluid via a flat stretched surface. Fang et al. [7] discussed the unsteady incompressible flow with mass transfer under the region of the stagnation point. Bachok et al. [8] provided a computational solution to the stagnation point boundary layer flow through the stretching surface and the assumption of nanosized particles.

The flow and heat-mass transfer mechanism of an incompressible non-Newtonian and viscous fluid flow over a stretchable surface has attracted the attention of researchers. This phenomenon exhibits wide-range applications in manufacturing and scientific processes, such as the fabrication of paper and fiberglass, wire drawing, polymer, and high temperatures such as nuclear power plants, gas turbines, thermal energy, solar power gadgets, and electrical power generation. Several investigators have focused on the investigation of magnetohydrodynamic (MHD) flow across a stretched sheet because of its significant use in manufacturing and industrial processes. Metallurgy, MHD production, solar and stellar structures, and radio transmission via the stratosphere are a few of these potential uses. Crane [9] first explored the extensible sheet problem for the fluid flow using the closed-form solution. Munawar et al. [10] established the exact solution to the flow problem with slip effects between two extending plates. Waini et al. [11] studied the time-dependent flow behavior and heat transfer phenomena through stretched/shrunk sheets containing hybrid nanofluid. In order to investigate the effects of variable viscosity on the boundary layer flow past a stretched surface, Manjunatha et al. [12] investigated the effects of hybrid nanofluids. Devi and Devi [13] investigated the effects of Newtonian heating and the Lorentz force on the 3D flow characteristics of a hybrid nanofluid interacting with a stretched sheet. Miklavčič and Wang [14] discussed the time-independent flow behavior using a shrunk sheet. They concluded that the mass suction works to create the flow across a contracting sheet. Tie-Gang et al. [15] established the viscous time-dependent concept of fluid flow using a shrinking sheet. Fang [16] and Fang and Zhang [17] investigated the power-law importance of the boundary layer flow process. Rohni et al. [18] carried out the heat transfer and flow mechanism using a shrinking sheet under the assumption of suction effects.

The poor thermal conductivity of the functional fluid in the border layer flow and energy transfer problems restrict the improvement in the transfer of heat. However, from the perspective of energy conservation, the continued miniaturization of electronic systems necessitates further advancements in heat transmission. The nanofluid is the consequence of the mixing of the working fluid and nanoparticles (NPs) with distinctive chemical and physical characteristics. It is anticipated that the NPs will increase the thermal conductivity of nanofluid and, hence, significantly improve its ability to transmit heat. Choi and Eastman [19] originated the concept of nanofluid, which was noted for its outstanding heat transmission and cooling capabilities. Choi introduced nanotechnology as a mixture of nanomaterials ( < 100 nm in size) and discovered that the regular fluid and nanofluid treatment had significant chemical and physical characteristics. As reported in previous studies [20, 21,22], the fundamental composition of nanomaterials has the potential to influence the variation in temperature. Following the single-phase machines of the nanofluid model suggested by Tiwari and Das [23], the effect of small particle fractional volume on the hydrodynamic migration within two perpendicular permeable surfaces advancing in reverse directions was explored. They experienced a decrease in the nanoliquid temperature distribution after increasing the particle fractional volume. Buongiorno [24] addressed the use of hybridization platforms, controlling temperature, and thermal exchangers in household freezers and clarified this concept by studying the thermophoretic and Brownian movement of nanoparticles. Several frameworks for studying nanofluids have been proposed by researchers using the concept given by Buongiorno [24], and Tiwari and Das's concepts have received the utmost attention. Rafiq et al. [25] examined the effects of Hall and particle slip on the peristaltic velocity of the nanofluid using the Buongiorno nanofluid framework for biomedical purposes and discovered that when thermophoresis and Brownian motion increase, the temperature consequently increases. Mekheimer et al. [26] investigated the properties of the blood-gold transport as a third-grade nanofluid in the catheter using the Buongiorno nanofluid model and discovered that the body temperature of the nanofluid raises via a greater thermophoresis force. Ahmed and Nadeem [27] solved a blood flow model using NPs ( Cu , A l 2 O 3 , Ti O 2 ) as antimicrobial agents in sick arteries using the Cauchy–Euler technique. NPs are extremely fragile because they quickly interact with other substances, causing the size of the particles to change.

Hybridized nanofluid is one of the novel categories of nanofluids and has recently piqued the interest of researchers. Hybrid nanofluids are made in two ways: two or more types of nanomaterials are entangled with the base fluid, and composite NPs are dispersed in the base fluid. The implementation of these nanofluids can improve the heat transfer rates. Due to their lower production costs, researchers and industrialists are taking an interest in this subject. Metal NPs’ intrinsic qualities, such as zinc oxide ( ZnO ) and iron ( F e 2 O 3 ), are primarily defined by their size, content, crystallization, and shape. The physical, chemical, mechanical, functional, morphological, and spectral characteristics of materials can be altered by shrinking them to the nanoscale. These reformed properties enable NPs to interact with cell biologic molecules in a novel way, facilitating the physical transport of NPs into inner cellular structures [28]. Nanostructured materials contain a higher percentage of atoms on their surface, and they have a high surface reactivity. Thus, nanostructures have recently gained prominence in basic and applied sciences, as well as nanotechnology in biological sciences. Zinc is identified naturally as assisting molecules that aid approximately 300 enzymes associated with diverse body functions. After iron, zinc is the most important and abundant significant component of the human body, accounting for 30 mm (2–4 g) of the total body zinc content. Most zinc is found in the eye, kidneys, liver, muscles, bones, prostate gland, and brain [29]. Zinc deficiency has been linked to vision problems such as hazy cataracts and poor night vision. A lack of zinc puts patients at risk for alopecia (hair loss from the eyelids and brows), increased susceptibility to infections, and mental lethargy. In humans, zinc deficiency promotes cell damage and the development of malignancies, which can lead to cancer. Thus, zinc-accelerated cancer chemoprevention is useful in the prevention and treatment of many malignancies. Zinc insufficiency is typically caused by insufficient zinc absorption or ingestion, increased zinc losses in the body, or increased zinc plea [30]. Loss of appetite, growth retardation, and impaired immune function are symptoms of zinc deficiency [31,32]. Taste changes, weight loss, reduced cognition, delayed appetite, and prolonged wound healing are all possible causes of insufficiency of zinc [33,34,35]. Superparamagnetic iron oxide NPs, on the other hand, provide appealing potential for improving site-specific drug administration due to their transportation to selected locations by an external magnetic field [36,37]. They have also been proposed for improving the contrast of magnetic resonance imaging in cancer detection [38,39], which may be combined with localized heat treatments in a so-called theragnostic method [40]. Despite this potential advantage, only a small number of NPs containing iron centers have received clinical approval [41,42,43]. The US Food and Drug Administration licensed Ferumoxides as the first NP-based iron oxide diagnostic agents for the detection of liver lesions in 1996. Several investigations on non-Newtonian nanofluid flow may be found in previous studies [44,45,46].

Clinical and medical investigators have been attracted by the flow of blood in a blood vessel since it is important in cell tissue engineering, medication targeting, and multiple therapies such as hyperthermia and cancer. Numerous non-Newtonian rheological fluids, such as Carreau fluid, tangential hyperbolic fluid, Eyring–Powell fluid, and viscoelastic fluid, exhibit blood flow characteristics. The Powell–Eyring fundamental model is widely used to illustrate blood viscoelasticity. Blood is the most important non-Newtonian viscoelastic multi-component liquid in nature, consisting of plasma, platelets, white blood cells, red blood cells, and so on. Powell and Eyring developed a new fluid model known as the Eyring–Powell fluid model in 1944 [47]. These viscoelastic fluids play an important role in physiology and industry. The Eyring–Powell nanofluid model, among other models, can be used to obtain reliable findings for viscous nanofluid at low and high shear rates. Human blood is the greatest example of an Eyring–Powell nanofluid with low shear rates (<100 s − 1 ) . The investigation of NPs subject to the Eyring–Powell fluid was avoided by many researchers due to the complicated mathematical description of this model in a stretched surface flow. Asha and Sunitha [48] conducted additional research on the mixed convection peristaltic flow of the Eyring–Powell nanofluid with a magnetic field in a non-uniform channel. Noreen and Qasim [49] investigated the peristaltic flow of the MHD Eyring–Powell fluid in a channel.

Entropy production and minimization (EGM) is regarded as a viable strategy for improving the functionality of energy devices. The thermodynamic features of any mechanical structure are demarcated utilizing the thermodynamic fundamental laws, which assert that the evolution of entropy continuously increases in thermodynamic systems. The production of entropy is connected to the conservation nature of thermodynamics, e.g., if irreversibility increases, the value of the whole system's work shrinks. The well-known emergence of tiny molecules in normally administered materials can have a profound effect on the overall foundation of entropy. As a result, using nanotechnology in controlled heat domains depresses the heat of the structure, lowering the prospective involvement of energy transfer to the overall rate of entropy. Bejan [50] initially examined the entropy in convective heat transmission. Afterward, multiple models for lowering entropy generation for various geometries have been suggested by Hayat et al. [51]. The effects of thermodynamics and Bejan on Powell–Eyring liquid were addressed by Nisar et al. [52]. This qualitative study found that the amount of entropy rate decreases as fluid variables increase. Through a divergent channel, Basha and Sivaraj [53] examined the coefficient of entropy for the peristaltic radiant Powell–Eyring nanoliquid. They found that the total entropy rate decreases as the temperature divergence parameter increases. Using electrostatic pressure and heat radiation, Guedri et al. [54] reported the peristaltic transport of a modified hybridized nanoliquid within an entangled channel. They found that the irreversibility parameter diminishes close to the walls.

The stagnation point flow of the Eyring–Powell fluid over an extensible sheet in the presence of Eyring–Powell viscous dissipation under an inclined Lorentz force and the contribution of entropy generation with single- and two-phase nanofluid models effects have not been studied so far. We took into account the Brownian motion and thermophoresis in the context of Buongiorno [24] and Tiwari and Das [23] hybrid nanofluid volume friction concept in order to fulfill the discrepancy in the existing literature. To the best of our knowledge, no research has been done on the impact of entropy production on the stagnation point flow of the Eyring–Powell fluid model through a stretched sheet with heat and mass transfer within the framework of two distinct nanofluid models. The Eyring–Powell model, resembling blood rheology, is considered as blood in this communication. Two different NPs (zinc ZnO and iron oxide F e 2 O 3 NPs) with homogenous sizes and different thermo–chemi–physical attributes are dipped in an Eyring–Powell fluid, and hence a mixture of hybrid blood nanofluid is established. The single-phase nanofluid accounts for the physical features of NPs by incorporating the density, viscosity, heat capacitance attributes, and electrical conductivity of two different NPs with precise volumetric friction in a working liquid. Meanwhile, the two-phase model accounts for the slip mechanism of NPs, including Brownian and thermophoretic phenomena. With the deployment of the stress tensor of the Eyring–Powell fluid, Lorentz force as body forces, and Buongiorno theory in the conservation equation, the governing equations are established. Realistic boundary conditions, such as the melting effect and wall slips, are considered to attain real-world applications. These equations are resolved computationally using the Runge–Kutta–Fehlberg technique together with the shooting procedure. In-depth analyses are performed on the effects of various physical quantities. The findings of this study can be interpreted with significant implications for biomedical science, the drug sector, biological product polymer amalgamated materials, environment-responsive demos, microbial lubrication extraction, biosensors, and bioengineering.

2 Problem declaration and model development

On an elongated sheet, a time-independent, incompressible flow with a stagnation region (Power–Eyring fluid) is considered. We estimate the thermal–chemical parameters of hybrid nanofluids using the renowned Buongiorno’s and Tiwari–Das models, which enables us to manipulate the appealing aspects of Brownian motion and thermophoretic diffusion and NP volumetric friction in conjunction. An inclined transverse magnetic field B 0 located at an angle ω is applied to the plate, with the influence of the induced magnetic field being ignored due to the lower magnetic Reynolds number. Figure 1(a) and (b) shows the physical framework execution. With a velocity of u = u ∞ = bx (where b is the positive constant), the surface of the sheet is stretching. Thermal and concentration transport takes place because of the utilization of the magnetic effect B 0 along the y-axis. The sheet temperature is maintained at T w while the wall temperature far away from the surface is kept at T ∞ . As illustrated in Figure 1(a) and (b), the system of Cartesian coordinates is selected so that the x-axis runs down the vertically expanding surface at y = 0 and the liquid filled area is considered at y > 0 . The velocity near the boundary is considered to be u = u w + u slip . The mainstream flow is believed to be in the direction of the y-axis, designated along the length of the moving plate, while the y -axis is perpendicular to it, and the x coordinate of the velocity components u varies in the y directions. It is assumed that the shape and size of the nanomaterials in the hybrid nanofluid are homogenous, and the impact of these particles clumping together is ignored. In this work, zinc ( ZnO ) and iron oxide ( F e 2 O 3 ) NPs are taken into account. The base fluid and the NPs are combined in a stable manner while creating hybrid nanofluids. The temperature of the blood is higher than that of the vein walls T > T w , while T w > T ∞ . The vein wall internal fats can be melt with the utilization of higher melting heat. Since the proper representation of blood flow in veins, especially in the aorta, can be scrutinized with the inclusion of stretching surfaces and stagnation points, they act as a plaque at the artery's internal interface. The main equations for continuity, the momentum, mass concentration, and heat in laminar boundary layer flow in a blood-based hybrid nanofluid can be stated as follows under these presumptions and using the model proposed by Tiwari–Das and Buongiorno model.

Figure 1

(a) and (b) Flow geometry and distribution of the flow.

Starting from the conservation laws and thermodynamics principle [55,56,57],

where S → is the Cauchy stress tensor, p is the fluid pressure, ρ hnf is the nanofluid viscosity, and I is the unit tensor. k hnf , ( ρ c p ) hnf , and ( ρ c p ) p are the thermal conductivity and heat capacitances of hybrid nanofluid and NPs, respectively. D B and D T are the coefficients of Brownian and thermophoretic diffusion, ∇ C and ∇ T are the concentration and temperature differentials, and T ∞ , T , and C are the ambient and blood temperature and concentration, respectively. Furthermore, A 1 denotes the Rivlin–Ericksen tensor given by [58]

where V → = ( u , v , 0 ) is the velocity vector. The velocity components u , v signify the flow in the x , directions. denotes the matrix transpose, μ f is the dynamic viscosity, β and γ are E-Powell fluid properties with dimensions 1 / P a and 1 / s , respectively.

Using the second-order approximations of the sin h − 1 function and ignoring the upper term, we obtain

J denotes the density of an electrical current in motion under the action of a magnetic field B [59].

Furthermore, assuming an electrical field E = 0 , the physical relations shown below apply. Thus,

The conceptual framework of radiant heat flow can be stated using the fast-acting Rosseland estimation as follows [60]:

In the previously written equation, k * and σ * represent the absorption and Stefan–Boltzmann quantities. The preceding equation can also be expressed as follows:

The nanofluid originated by dissolving iron F e 2 O 3 and ZnO NPs in uncontaminated blood (base fluid) with a molar volumetric friction of φ hnf . Thus, at a contact volume, friction of φ 1 , magnesium ZnO and F e 2 O 3 NPs were used. To create a hybrid nanofluid, φ 1 and φ 2 are disseminated in the fluid. The volumetric concentration coefficient of hybrid nanoliquid is demarcated as φ hnf = φ 1 + φ 2 . The P–Eyring fluid model constitutive equations originated from the molecular liquid model. The P–Eyring fluid model exhibits viscous fluid at high and low deformation rates. The power-law fluid model, on the other hand, describes an unlimited realistic viscosity at a low shear rate, restricting its applicability. The y-momentum equation is now easily simplified to ∂ p ∂ y ≅ 0 , using the traditional boundary layer approach. That is, particularly for viscoelastic fluids, the applied pressure remains constant across the boundary layer, allowing it to be calculated from the inviscid flow outside the boundary layer. Now, using the Bernoulli equation (without taking into account the hydrostatic factor) to substitute − 1 ρ f ∂ p ∂ y = u ∞ d u ∞ d x , one can ignore viscoelastic effects beyond the boundary layer [61]. The resulting governing equations and realistic boundary conditions are as follows[62,63]:

The transportation problem pertinent boundary prerequisites are [64,65]

The physical interest parameters are as follows:

3 Entropy modeling

In the current situation, entropy development entails four processes: viscous extinction, bulk diffusion, transfer of heat, and Joule dissipation. For an Eyring–Powell nanofluid, the entropy equation is [66,67,68]

The final shape of entropy development is characterized as

The dimensionless form of the entropy equation after incorporating Eq. (13) is

The Bejan number is established as follows:

A 1 = ( T ∞ − T 0 ) T ∞ , A 2 = ( C ∞ − C 0 ) C ∞ , D 1 = R D B ( C ∞ − C 0 ) k f , Br = μ f a 2 x 2 ( T ∞ − T 0 ) k f .

Table 2 displays the thermo–chemical–physical characteristics of fluids and NPs.

4 Details of the solution scheme

The conditions at the boundary equations (Eqs (24) and (25)) in conjunction with the dimensionless standard differential equations (Eqs (21)–(23)) are precisely nonlinear. Calculating the correct responses to the aforementioned boundary value problem is quite challenging. By using any suitable computational scheme, we must calculate the approximations of the acceptable solutions to the problem at hand. Here, in the current investigation, we have used the Runge–Kutta–Fehlberg technique together with the shooting procedure to find computational solutions of dimensionless differential equations (Eqs (21)–(23)) together with boundary conditions. This computational system resolves an initial value problem. In this quantitative approach, the non-dimensional differential equations (Eqs (21)–(23)) are first converted into first-order differential equations, and five initial conditions are then required. However, the missing (unknown) requirements are four initial conditions, each of Y 1 ( 0 ) , Y 2 ( 0 ) , Y 3 ( 0 ) , Y 4 ( 0 ) , and Y 6 ( 0 ) . With the help of a quantitative shooting strategy, these deficient conditions are used to develop initial estimations that are known. The selection of the appropriate meshes for η ∞ is a crucial component of this numerical technique. By using the framework of Newton–Raphson for illuminating unidentified conditions, we have made some early assumptions. The following requirements Y 2 ( ∞ ) , Y 4 ( ∞ ) , Y 6 ( ∞ ) are satisfied by these initial hypotheses. In the current investigation, we used a relative precision threshold of 1 0 − 5 and a step size of ∆ η = 0.01, keeping η ∞ = 10 .

Announcing new variables in order to reduce the system into first order:

5 Parametric outcomes and physical explanation

The assessment of blood flow and heat-mass transfer is considered with the help of different parametric effects. The following physical data, such as Pr ( = 21 ) , c p ( = 14.65 J k g − 1 K − 1 ) , R e x < 500 (for laminar flow), ν f = 4 × 1 0 − 4 m 2 s − 1 , σ f = 0.6 S s − 1 , k f ( = 2.2 × 1 0 − 3 J m − 1 s − 1 K − 1 ) , μ f ( = 3.2 × 10 − 3 k g − 1 m − 1 s − 1 ) , for human blood at T = 36.5 ° C are accounted as suggested in previous studies [75,76,77,78]. The velocity and blood temperature are accounted for pure and nanoparticle-loaded blood, while the physical quantities are measured in the case of slip scenario.

Figure 2 highlights how the ratio parameter B affects the flow of a hybrid nanofluid f ′ ( η ) . The ratio parameter B = b a estimates the ratio of the free stream to the sheet velocity. As the strained motion increases close to the stagnating region, the external flow accelerates more quickly, leading to the boundary layer thickness to drop as b a increases, and hence the velocity increases. The velocity-restricted layer thickness displays identical extending and receding characteristics to the open stream velocity. For B = 1 , the liquid and the surface proceed at a comparable speed; hence, there is also an insufficient boundary layer. When B < 1 ( 0 − 0.9 ) , the mobility is stronger at the edge of the wall, and when B > 1 ( 1.1 − 1.9 ) , the momentum is higher outside of the wall. It is important to note that stagnation spots occur when two coronary arteries meet or the junction where veins are partially or completely clogged by cholesterol plaques on their interior walls, which restricts blood circulation inside the vessels. On the other hand, such obstructions experience the highest pressure, which causes angina or stroke. Figure 3 shows how the strong magnetic variable M affects the apparent horizontal velocity of fluid for the nanofluid solution. The width of the outer boundary layer and the velocity contour both shrink when the magnetic factor M expands. Technically, this occurs because the Lorentz force generates delaying forces, which serve as drag forces in the contrary orientation of the flow. This can be clarified by the reality that blood, a circulating electrically conductive fluid, will produce magnetic as well as electric fields if a magnetic field is applied to it. Because of the interplay between these fields, an external force known as the Lorentz force is created, which tends to resist fluid motion and slows down blood circulation in the human artery system. The effect of inclined magnetic field parameter ω on the flow stream of a hybrid nanofluid made of blood and zinc/iron oxide is shown in Figure 4. Physically, when the degree of orientation increases, the velocity field drops, which alternately causes an elevated magnetic and Lorentz force. The Lorentz force shows resistance against blood flow and finally detracts from the blood stream. Due to the delayed transmission of the hybrid nanofluid, hypomagnesemia sufferers rehabilitate slowly. To deliver a suitable dose of zinc-iron oxide particles to the damaged area, a magnetic field can be helpful to guide the drug delivery. The volatile nature of fluid flow in relation to the viscosity parameter is shown in Figure 5 for various values of ϵ versus the normalized variable η . Increasing the estimated values of ϵ obviously reduces the fluid viscosity and fluid drift. The reliability of our numerical method is guaranteed by a similar pattern, as was revealed by Aljabali et al. [79]. Additionally, a positive variation in ϵ decreases the velocity and slows the motion of the blood. Fluid viscosity changes the strain and stress within the blood. When φ np = 0 , the boundary layer flow of the blood is higher than that of φ np = 0.01 : ( F e 2 O 3 + ZnO ) for distinct mixture of NPs. Physically, these behaviors are greatly influenced by synthetic procedures, crystallization, form, size, and composition of iron oxide NPs [80]. Figure 6 depicts the influence of the slip variable N 1 on velocity profiles. It is interesting to note from the illustration that the velocity threshold along the sheet decreases when the slip velocity exists within the boundary layer. The distribution of velocity is less close to the sheet surface than it is farther away. Physically, a velocity slip exists when the slip obstruction/plaque ( N 1 ≠ 0 ) exists, which means that the flow velocity close to the sheet slows down relative to the speed of the stretching sheet. Additionally, decreasing the value of N 1 will result in a slower flow rate in the slip situation. In actuality, the magnetic force and slippage effects work as retarding forces that prevent the transit of fluid. Thus, in the presence of the magnetic flux and slip, which are severe obstructions to the blood flow. Finally, in both the cases φ np = 0.01 : ( F e 2 O 3 + ZnO ) and φ np = 0.0 , the thickness of the momentum boundary layer decreases. Additionally, magnetic parameters can be used to lower backflows whenever they occur in stretched sheets. Its participation also prevents the split.

Figure 2

Normalized profile of velocity f ′ ( η ) in the perspective of flow across a stretched sheet for classified values of B (ratio parameter).

Figure 3

Normalized profile of velocity f ′ ( η ) in the perspective of flow across a stretched sheet for classified values of M (magnetic parameter).

Figure 4

Normalized profile of velocity f ′ ( η ) in the perspective of flow across a stretched sheet for classified values of ω (inclination parameter).

Figure 5

Normalized profile of velocity f ′ ( η ) in the perspective of flow across a stretched sheet for classified values of ϵ (material variable of the Powell–Eyring fluid).