Heat Transfer Analysis of the MHD Stagnation Point Flow of a Non-Newtonian Tangent Hyperbolic Hybrid Nanofluid past a Non-Isothermal Flat Plate with Thermal Radiation Effect

Heat transfer phenomena are used in a variety of industries, including chemical devices, shipbuilding, power plants, electronic devices, and medicinal plants. Propylene glycol, engine oil, water, and ethylene glycol are common single-phase heat transfer liquids used in a variety of industries, including chemical process industries and thermal power plants. Therefore, the authors are interested in investigating the magnetohydrodynamic ﬂ ow of a water-based hybrid nano ﬂ uid containing ferrous and graphene oxide nanoparticles past a ﬂ at plate. The stagnation points, as well as the e ﬀ ects of magnetic ﬁ eld and thermal radiation are taken into account in this analysis. The non-Newtonian tangent hyperbolic ﬂ ow, which is laminar and incompressible, is also considered to investigate the non-Newtonian behavior of the hybrid nano ﬂ uid ﬂ ow. The proposed model has been solved analytically with the help of HAM. The convergence of HAM is shown with the help of ﬁ gure. The hydrothermal characteristics of hybrid nano ﬂ uid ﬂ ow past a nonisothermal ﬂ at plate at a stagnation point are a ﬀ ected by the necessary parameters. The results show that the boosting volume fractions of the ferrous and graphene oxide nanoparticles have signi ﬁ cantly reduced the velocity ﬁ eld, while the thermal ﬁ eld has increased with the augmenting volume fractions of the ferrous and graphene oxide nanoparticles. The increasing power-law index has augmented the viscosity of the non-Newtonian hybrid nano ﬂ uid ﬂ ow due to which the velocity ﬁ eld escalated. However, this impact is opposite for the thermal ﬁ eld. Due to the direct relation between the Weissenberg number and relaxation time, the greater Weissenberg number has reduced the velocity pro ﬁ le, while increased the thermal ﬁ eld.


Introduction
Because of various applications in healthcare and engineering, such as microelectronics, solar collectors, process industries, cancer therapy, heat exchangers, and power production, the mechanisms of heat exchange incorporating nanomaterials have piqued the interest of researchers. Regular liquids such as glycol mixtures, engine oil, and water had moderately poor thermal properties and inadequate capacity to attain higher thermal efficiency. The use of nanoparticles to develop the thermal conductivity of various cooling fluids is a contemporary method. Nowadays, temperature distribution plays an essential function in a variety of scientific and technical disciplines. Heat transfer phenomena have a wide range of applications in sectors, shipbuilding, electronic devices, power plants, medicinal, and chemical devices. To design heat exchangers and discover the optimal geometry, radiators, condensers, evaporators, and boilers, heat transfer analysis and the related cooling process become invaluable. Propylene glycol, engine oil, water, and ethylene glycol are common single-phase heat transfer liquids used in a variety of chemical process industries and thermal power plants. Due to its low thermal conductivity, the single-phase traditional liquids are acknowledged to have poor heat transmission ability. This improvement in working fluid heat transport is critical for achieving energy and cost reductions. In order to boost-up the thermal conductivity of the base fluids, many researchers have worked to resolve these issues and improve the thermal conductivity of the base fluids. Solid materials have higher thermal conductivity than those of liquids. As a result, dispersing microscopic solid particles into a base liquid is a novel technique to increase the thermal conductivity of the base fluids. Khan et al. [1] investigated the Casson nanofluid flow past a rotating disk. Shah et al. [2] addressed the applications of radius, heat flux, and mass flux of the water-based copper nanoparticles. Gul T et al. [3] investigated the flow of carbon nanotube nanofluid past a rotating cone and disk. Sowmya et al. [4] addressed the effects of convective condition and internal heat generation in a nanofluid flow past a porous fin. Ashraf et al. [5] analyzed the magnetohydrodynamic peristaltic flow of the blood-based magnetite nanoparticles. Dawar et al. [6] studied the unsteady flow of carbon nanotube nanofluid with the magnetic field impact. Rasool and Wakif [7] examined the electromagnetohydrodynamic second-grade nanofluid flow over a Riga plate. Alghamdi et al. [8] presented the magnetohydrodynamic flow of sodium alginate-based nanofluid bounded by slender surface with heat source impact. Rout et al. [9] analyzed the water-and kerosene-based nanofluid flow with viscous dissipation. Alshomrani and Gul [10] investigated the dissipative flow of water-based Al 2 O 3 and Cu nanoparticles past a stretching cylinder with convective condition. Further related analyses can be studied in [11][12][13][14].
Hybrid nanofluid is a class of nanofluids, developed by integrating a certain class of nanoparticles inside a functional fluid which has recently been used. Two different nanomaterials are suspended in a conventional fluid to create hybrid nanofluids. Hybrid nanofluids are widely used in a diversity of disciplines of engineering as well as refrigeration, space planes, biomedicals, machining coolant, motor cooling, heat pipe reduction in medicine, and highperformance boats. Jana et al. [15] investigated the conductive nanomaterials like copper and gold nanoparticles and their hybrids. Khashi'ie et al. [16] analyzed heat transfer of a magnetohydrodynamic flow of a water-based hybrid nanofluid comprehending Cu and Al 2 O 3 nanoparticles. Their results show that the suction factor has a significant impact of heat transfer analysis. Additionally, they have computed the stability analysis as well. Nawaz and Nazir [17] studied the magnetohydrodynamic flow of an ethylene-based hybrid nanofluid flow containing MoS 2 and SiO 2 nanoparticles. They compared MoS 2 /ethylene-based and MoS 2 -SiO 2 /ethylene-based hybrid nanofluids. Their results showed that the thermal performance is greater for the MoS 2 -SiO 2 /ethylene-based as compared to MoS 2 /ethylene-based. Manjunatha et al. [18] presented the comparative analysis of the magnetohydrodynamic flows of Cu-H 2 O nanofluid and Cu-Al 2 O 3 /H 2 O hybrid nanofluid. They found that the nanoparticle volume fraction of the nanofluid and hybrid nanofluid has enhanced the velocity and thermal fields. Usman et al. [19] proposed the comparative analysis of the magnetohydrodynamic flow of a  [20] offered the comparative analysis of magnetohydrodynamic flows of SiO 2 / H 2 O nanofluid and MoS 2 -SiO 2 /H 2 O hybrid nanofluid considering different shapes of the nanoparticles. Their results showed that the nanofluid has slower flow as compared to hybrid nanofluid. Additionally, the lower temperature is observed for brick-shaped nanoparticles of the nanofluid, while the blade-shaped nanoparticle of the hybrid nanofluid has extreme temperature. Ghadikolaei et al. [21] offered the comparative investigation of magnetohydrodynamic flow of Cu/H 2 O and hybrid nanofluid containing TiO 2 -Cu/H 2 O at a stagnation point. They also considered three different shapes of the nanoparticles named as platelets, bricks, and cylinders. It is clear from of this research that using plateletshaped nanoparticles is more effective. Gul et al. [22] addressed the magnetohydrodynamic flow of hybrid nanofluid containing Cu and Fe 3 O 4 . Their results showed that the nanoparticle volume fractions of Cu and Fe 3 O 4 have significantly improved the thermal transmission and velocity field. Alghamdi et al. [23] addressed the comparative analysis of the magnetohydrodynamic flows of blood-based Cu nanofluid and blood-based Cu-CuO hybrid nanofluid. It has been introduced that the hybrid nanofluid flow has more effective thermal conductivity in a contracting channels. Acharya [24] probed the application of solar energy toward a hybrid nanofluid flow containing alumina and copper nanoparticles. In another article, Acharya and Mabood [25] addressed the water-based hybrid nanofluid flow containing ferrous and graphene oxide nanoparticles. Thumma et al. [26] investigated the Cu-CuO nanoparticles past a porous extending surface. Acharya et al. [27,28] analyzed the nanofluid and hybrid nanofluid flows under the impact of magnetic field.
According to the authors' knowledge, there is no study based on magnetohydrodynamic flow of water-based hybrid nanofluid containing ferrous and graphene oxide nanoparticles past a flat plate. The stagnation point along with the impacts of magnetic field and thermal radiation is taken in this consideration. The non-Newtonian tangent hyperbolic flow which is laminar and incompressible is also considered to investigate the non-Newtonian behavior of the hybrid nanofluid flow. The present analysis is composed of mathematical modeling which is shown in Section 2. HAM solution and convergence of HAM are presented in Sections 3 and 4, respectively. Section 5 is composed of results and discussion. In the last, the concluding remarks are shown in Section 6. Furthermore, the Hall current and thermal radiation effects are also considered. Following the above assumption, the leading equations are stated. Figure 1 shows the geometry of the hybrid nanofluid flow.

Model Formulation
The relevance boundary conditions are defined as The radiative heat flux q r is defined as By using the Taylor series expansion, T 4 can be written as For the simulation of hybrid nanofluid flow, the thermophysical properties are defined as where p1 and p2 represent Fe 3 O 4 and GO nanoparticles, respectively, and ϕ 1 and ϕ 2 are the nanoparticle volume fractions of Fe 3 O 4 and GO, respectively. The numerical values of the thermophysical properties are defined in Table 1.   [29].

Journal of Nanomaterials
The similarity transformations are defined as Using the above similarity transformations, the leading equations are transformed as with boundary conditions The dimensionless parameters are defined as Physical quantities of importance like skin friction C f x and Nusselt number Nu x are defined as  Journal of Nanomaterials Using the similarity transformations defined in equation (6), the above quantities are reduced to

HAM Solution
To attain the analytical solution of the proposed model along with the relevant boundary conditions, HAM method which was introduced by Liao [30] is applied. The initial guesses and linear operators are defined as   Journal of Nanomaterials

HAM Convergence
Homotopy analysis method guarantees the convergence analysis of the highly linear and nonlinear differential equations. The auxiliary parameter h insures the convergence area of the modeled problem. The convergence areas for velocity and temperature profiles are −1:0 ≤ h ϒ ≤ 1:0 and − 2:5 ≤ h Θ ≤ 1:5, respectively, as shown in Figure 2.      Figure 5(a) shows the impact of ϕ 1 on ϒ ′ðζÞ when ϕ 2 = 0:05. The augmenting ϕ 1 declines ϒ ′ðζÞ. The increasing ϕ 1 declines the boundary layer thickness, which consequently reduces ϒ ′ðζÞ. Figure 5(b) shows the streamline patterns for ϕ 1 when ϕ 2 = 0:05. Figure 6(a) shows the impact of ϕ 2 on ϒ ′ ðζÞ when ϕ 1 = 0:05. A similar impact as of Fe 3 O 4 nanoparticle is found here. Figure 6(b) shows the streamline patterns for ϕ 2 when ϕ 1 = 0:05. Figure 7(a) signifies the consequence of M on ϒ ′ ðζÞ. The escalating magnetic parameter boosts up the veloc-ity field. As the dynamic growth upsurges, the boundary layer of the velocity profile gets thinner, showing that the magnetic parameter augments the flow mobility near the heated plate. The present model is computed along with stagnation point flow, thus the augmenting impact of the magnetic parameter has been reported here. Figure 7(b) shows the streamline patterns for M when ϕ 1 = ϕ 2 = 0:05. Figure 8(a) signposts the effect of We on ϒ ′ ðζÞ. The augmenting We reduces ϒ ′ ðζÞ. The maximum value of the parameter We increases ϒ ′ ðζÞ, because We is directly related to the relaxation time Γ. The relaxation time of the examined non-Newtonian hybrid nanofluid has increased. As a result of this physical property, the water-based flow encounters extra barrier in developing easily across the flow boundary, lowering the hybrid nanofluid velocity. Figure 8(b) shows the streamline patterns for We when  9 Journal of Nanomaterials ϕ 1 = ϕ 2 = 0:05. Figure 9(a) displays the effect of n on ϒ ′ðζÞ. The escalating n shows augmenting conduct against ϒ ′ðζÞ. The numerical value of the power-law index parameter is specified for two different fluids, namely, pseudoplastic ðn < 1Þ and dilatant ðn > 1Þ. Physically, the escalating n interconnects an important augmentation in the viscosity of the non-Newtonian hybrid nanofluid flow. That is why the velocity boundary layer thickness is declined; as a result, ϒ ′ ðζÞ is augmented. Figure 9(b) shows the streamline patterns for n when ϕ 1 = ϕ 2 = 0:05. Figure 10 shows the effect of ϕ 1 on ΘðζÞ when ϕ 2 = 0:05. The increasing ϕ 1 augments ΘðζÞ. Figure 11 shows the effect of volume fraction ϕ 2 on ΘðζÞ when ϕ 1 = 0:05. The increasing ϕ 2 augments ΘðζÞ. Figure 12 shows the effect of power-law index n on ΘðζÞ when ϕ 1 = ϕ 2 = 0:05. The rising n declines ΘðζÞ. The increasing n thickens the temperature boundary layer which diminishes the temperature of the hybrid nanofluid flow. Thus, a declining impact is found here. Figure 13 exhibits the effect of We on ΘðζÞ when ϕ 1 = ϕ 2 = 0:05. The increasing We augments ΘðζÞ. The increasing We shows that the increased quantity of thermal energy provided to the nanofluidic system due to resistive nanofluid motion can explain this thermal behavior physically. Figure 14 displays the impact of M on ΘðζÞ when ϕ 1 = ϕ 2 = 0:05. The augmenting M escalates ΘðζÞ of the hybrid nanofluid flow. Physically, as the magnetic parameter increases, the movement of particles of hybrid nanofluid escalates. Thus, both the thermal boundary and temperature of the hybrid nanofluid augment. Figure 15 designates the effect of Rd on ΘðζÞ when ϕ 1 = ϕ 2 = 0:05. The increasing radiation parameter boosts up ΘðζÞ. Physically, the increasing radiation parameter augments the surface heat of the hybrid nanofluid flow which makes the hybrid nanofluid hotter. Thus, the escalating conduct is observed here. Figure 16 displays the effect of Eckert number Ec on ΘðζÞ when ϕ 1 = ϕ 2 = 0:05. The increasing Eckert number augments ΘðζÞ. The link between kinetic energy and enthalpy in a flow is described by the Eckert number. It denotes the effort expended in converting kinetic energy to internal energy in the face of viscous fluid forces. An increase in the Eckert number implies that the fluid has a high kinetic energy; consequently, the intermolecular collisions take place which enhances the particles vibration. So, the increased molecule collisions increase heat dissipation in the boundary layer region, causing ΘðζÞ to climb.

Conclusion
The magnetohydrodynamic flow of water-based hybrid nanofluid containing ferrous and graphene oxide nanoparticles past a flat plate has been studied in this article. The stagnation point along with the impacts of magnetic field and thermal radiation is taken in this consideration. The non-Newtonian tangent hyperbolic flow which is laminar and incompressible is also considered to investigate the non-Newtonian behavior of the hybrid nanofluid flow. The hydrothermal characteristics of the hybrid nanofluid flow past a nonisothermal flat plate at a stagnation point are affected by the necessary parameters. Key points of this analysis are as follows: (1) The increasing volume fractions of the ferrous and graphene oxide nanoparticles have significantly reduced the velocity field, while the thermal field has increased with the augmenting volume fractions of the ferrous and graphene oxide nanoparticles 10 Journal of Nanomaterials (2) The augmenting magnetic parameter has considerably enhanced the velocity and thermal fields (3) Due to the direct relation between the Weissenberg number and relaxation time, the greater Weissenberg number has reduced the velocity profile, while increased the thermal field (4) The increasing power-law index has augmented the viscosity of the non-Newtonian hybrid nanofluid flow due to which the velocity field escalated. However, this impact is opposite for the thermal field Re Temperature: T (K) Free-stream velocity: u e ðxÞ (ms −1 ) Velocity components: ðu, vÞ (ms −1 ) Weissenberg number: We Cartesian coordinates: ðx, yÞ (m).

Data Availability
All the supporting data are within the manuscript.

Conflicts of Interest
The authors declare that they have no conflict of interest.