Analysis of the electrically conducting magnetohydrodynamic hybrid nanofluid flow past a convectively heated stretching surface with suction/injection and non-linear thermal radiation

Fluid flow through a porous media has many industrial applications such as water flowing through rocks and soil and purification of gas and oil mixed in rocks. Also, heat transfer enhancement has been introduced in various thermal and mechanical systems by improving the thermal conductance of base fluids. In this article, the flow of an electrically conducting water-based hybrid nanofluid comprising GO and Fe3O4 nanoparticles over an extending sheet using a porous medium has been investigated. The space-dependent heat source, Joule heating, Brownian motion, thermophoresis, thermal radiation, chemical reaction, and activation energy impacts are taken into account. For the solution of the modeled equations, the homotopy analysis method is considered. The homotopic convergence is shown with the help of a figure. This analysis is contrasted with previous outcomes and has found a great agreement. The impacts of embedded factors on different flow characteristics, skin friction coefficient, and Nusselt and Sherwood numbers are displayed using figures and tables. The outcomes of the present analysis show that the increasing magnetic and suction factors have reduced the fluid motion while amplifying the thermal profiles. Additionally, the suction factor has a reducing impact on both temperature and concentration profiles. The thermal profiles have increased with the increasing thermal Biot number, Eckert number, thermophoresis, and Brownian motion factors. The Nusselt numbers have increased with the increasing thermal Biot number and stretching factor but reduced with the increasing thermal radiation and temperature difference factors.


Introduction
The selection of the coolant of different devices/equipment at the industrial level and for various engineering applications is one of the most challenging tasks. Recently, thermal flow enhancement has been introduced in various thermal and mechanical systems by improving the thermal conductance of base fluids. Different fluids which are considered as pure base fluids are engine/kerosene oil, water, ethylene glycol, etc. The suspension of non-size particles in a pure fluid has been familiarized by Choi and Eastman (1995) first to expand the thermal flow characteristics of the base fluid. Acharya et al. (2022) discussed the fluid flow and thermal phenomenon for a time-based MHD nanoliquid on a spinning surface and explored that the thermal properties of the nanofluid were enhanced by 84.61% in comparison to the normal fluid. Shah et al. (2020) mixed gold particles in blood to enhance its thermal properties by incorporating radiation and rotational effects in the flow system. Khan et al. (2021) introduced radiative effects in the revolving motion of an MHD nanofluid on a spinning cylinder and noticed a difference in temperature growth at the wall as well as on the surface of the cylinder. Bhatti et al. (2022) explored a bioconvective MHD Williamson nanofluid flowing amid two spinning circular surfaces placed in a penetrable medium. Hussain et al. (2022) solved numerically the radiative EMHD Williamson fluid flow on a stretching surface. Rasheed et al. (2022) discussed the Brownian three-dimensional motion of a thin-film nanofluid past a stretched and rotary sheet and found that the Nusselt number augmented with growth in magnetic and Brownian factors and the concentration of nanoparticles. Akbar et al. (2022) scrutinized the exact solution of an unsteady thermal conductive pressure on peristaltic transport with temperature nanofluid viscosity. Carbon nanotubes (CNTs) are grasped as nanoparticles in an irregular channel. Akram et al. (2022) investigated the electroosmotic flow of peristaltic transport of a nanofluid over a curved microchannel.
It has further been noticed experimentally that the thermal conductance of a base fluid can be additionally enhanced by suspending two unlike natures of nanoparticles in it and is characterized as a hybrid nanofluid. Chu et al. (2022) examined the impact of different nanoparticle shapes for unsteady hybrid nanofluid flow amid two plates of infinite length and determined that velocity weakened whereas the temperature of the fluid upsurged with augmentation in the number of hybrid nanoparticles. Zhang et al. (2022) examined the effects of magnetic field on hybrid nanoparticle flow over an elastic sheet and noticed that motion of the fluid propagated faster with progression in tantalum nanoparticles and the Darcy number while the temperature of the fluid declined with an upsurge in nickel and tantalum nanoparticle concentration. Guedri et al. (2022) discussed the trihybrid radiative nanofluid motion over a non-linear extended sheet using the impression of the Darcy-Forchheimer model and noticed that the heat of the fluid amplified with growth in Brownian, temperature ratio, and thermophoretic parameters. Salahuddin et al. (2022) examined flow and thermal behavior for a highly magnetized wavy heated cylinder on which hybrid nanofluids flow. Alrabaiah et al. (2022) estimated the bioconvective hybrid nanoparticles' flow in the cavity of a cone/ disk using the influences of dissipation and microorganisms. Lone et al. (2022) explored MHD micro-rotational hybrid nanoparticles' flow past a flat plate using thermally radiated effects and mixed convection. Khan et al. (2022a) examined hybrid dissipative nanofluid flow on a heated revolving needle using microorganisms and Hall current effects. Maraj et al. (2017) studied the closed-form solution of mixed convective MHD carbon nanotube nanofluid flow in a rotating channel. In this study, one can see that temperature is the enhancing function against the improving estimations of the volume fraction parameter. Habib and Akbar (2021) proposed the incorporation of novel nanofluids in clinical isolates to battle Staphylococcus aureus. Akram et al. (2021) reported water-based hybrid (Ag-Au) nanofluids electroosmotically pumped through an inclined asymmetric microfluidic channel in a porous setting. With the help of the Debye-Hückel and lubrication linearization principles, the governing equations of the current model are linearized.
The study of electrically conducted fluids associated with magnetic effects such as salty water and plasma is called magnetohydrodynamics (MHD). Such fluids are crucial in many engineering and industrial applications, for instance, design of nuclear reactors, MHD generators, and flow meters. Asjad et al. (2022) explored the impact of activated energy and magnetic effects on a Williamson fluid using bioconvective effects on an exponentially stretched surface. Bejawada et al. (2022) inspected radiated MHD fluid motion on a non-linear inclined sheet using a permeable Forchheimer surface and concluded that the motion of liquid degenerated while temperature expanded with progression in the magnetic factor. Kodi and Mopuri (2022)  number supported the fluid motion and thermal characteristics, whereas growth in the magnetic factor declined the velocity and upsurged the Nusselt number. Wahid et al. (2022) examined an MHD nanofluid at the stagnant point of a shrinking surface with viscously dissipative Joule heating effects and noticed that 25% growth in melting effects augmented skin friction by 5%, whereas the flow phenomenon can be sustained as laminar by taking alumina nanoparticles as 2% instead of 1%. Xuan (2022) reviewed non-linear electro-kinetic fluid flow taking the effects of induced charge to Joule heating. Abbas et al. (2022) debated on the influence of the Darcy-Forchheimer model on dissipative MHD fluid flow using Joule heating effects on a porous sheet and explored that with progression in the thermal diffusion factor, the temperature, concentration, and velocity characteristics augmented. Khan et al. (2022b) examined the production of irreversibility for hydromagnetic fluid flow using the Darcy-Forchheimer model and Joule heating effects and explained the thermal flow phenomenon both for prescribed thermal flux and surface temperatures. Waqas et al. (2022) deliberated on MHD nanoliquid flow on a radiated stretched surface using Joule heating as well as dissipative effects and noticed that the density of microbes degenerated with growth in the Peclet number. Saleem et al. (2022) discussed the bio-mathematical model for the flow of blood through an artery using Joule heating. Kumar et al. (2022) discussed numerically the chemically reactive MHD fluid slip flow with Joule heating on an exponentially extended sheet and explored that fluid motion weakened while the Nusselt number, as well as skin friction, amplified with an upsurge in magnetic effects. Xie et al. (2023a) designed experimental and numerical evaluations of a novel bearing's fluid-structure interaction lubricating abilities. Xie et al. (2023b) demonstrated the fluid-structure-acoustic coupling dynamics of a new water-lubricated bearing being studied theoretically and experimentally.
Thermal radiation is another main factor that plays a significant role in thermal flow analysis. Rehman et al. (2022) thermally inspected the radiated MHD Jeffery fluid flow with a comparative analysis upon plan/cylindrical surfaces and deduced that fluid flow on cylindrical surfaces has better thermal flow properties than on plain surfaces. For instance, the Nusselt number has greater values in case of cylindrical surfaces. Shaw et al. (2022) inspected MHD cross-liquid motion with effects of linear as well as non-linear heat radiations using an arbitrary Prandtl number and highlighted the thermal flow characteristics for Prandtl numbers within the interval 10 −4 ≤ Pr ≤ 10 4 in case of the linear as well as non-linear thermal radiation factor. Bilal et al. (2022) explored the impact of heat radiation on liquid motion over a linear stretched surface and explained that with elevation in porosity and radiation factors, there is a growth in skin friction. Adnan (2022) discussed numerically the effects of radiative and convective thermal conduction on nanoliquid flow on a non-linear extended sheet and determined that fluid motion weakened with growth in the radiation factor, which, on the other hand, augmented both the Nusselt number and skin friction. Recently, thermal flow enhancement has been introduced in various thermal and mechanical systems by improving the thermal conductance of base fluids. Different fluids which are considered as base fluids are pure engine/kerosene oil, water, ethylene glycol, etc. This work investigates the flow of an electrically conducting hybrid nanofluid over an extending surface using a porous medium. The space-dependent heat source, Joule heating, Brownian motion, thermophoresis, thermal radiation, and chemically reactive activation energy impacts are taken into consideration. Section 2 comprises the main body of the study. Section 3 shows the homotopic solution of the model. The homotopic convergence is shown in Section 4. Section 5 shows the results and discussion part of the work. The final remarks are listed in Section 6.

Formulation of the problem
We consider the two-dimensional flow of an MHD hybrid nanofluid containing GO and Fe 3 O 4 nanoparticles past a stretching sheet using porous media. A magnetic field of strength B 0 with an acute angle γ is employed for the hybrid nanofluid flow. The stretching velocity of the sheet is depicted by u w (x) ax. The surface of the sheet is kept with a constant temperature and concentration, T w (such that T w < T f ) and C w , respectively, while the corresponding ambient temperature and concentration are T ∞ and C ∞ , respectively. Here, T f is the reference temperature. A geometrical representation of the flow problem is shown in Figure 1. Additionally, the space-dependent heat source, Joule heating, Brownian motion, thermal radiation, thermophoresis, chemical reaction, and activation energy impacts are taken into consideration. The leading equations are (Reddy et al., 2020;Dawar et al., 2022a) subject to the following boundary conditions (Dawar et al., 2022b): where q r is given as The thermophysical properties are defined as For the aboveproposed model, the similarity variables are defined as Using Equation 10, the leading equations are reduced as In the abovementioned equations, the embedded factors are defined as Pr In the abovementioned equations, Rd is the thermal radiation factor, Nb is the Brownian motion parameter, Pr is the Prandtl number, Kr is the chemical factor, Nt is the thermophoretic factor, γ is the porosity factor, Sc is the Schmidt number, Bi T is the thermal Biot number, Ec is the Eckert number, Q e is the heat source factor, M is the magnetic parameter, S is the suction/ injection factor, E is the activation energy factor, and δ is the temperature difference parameter. Furthermore, A 1 , A 2 , A 3 , A 4 , and A 5 are defined as The quantities of interest such as C fx , Nu x , and Sh x are defined as Here, Eq. 18 is reduced as Here, Re x ] bf is the local Reynolds number.

Solution by HAM
The initial guesses are described as The linear operators are defined as with the following properties: where Z 1 − Z 7 are the arbitrary constants. The chart shown in Figure 2 explains the procedure of the homotopy analysis method (Liao, 1999;Liao, 2010).

HAM convergence
We are assured of the convergence of the series solution by the homotopy analysis approach. Our series solutions' convergence area is controlled and adjusted by the significant auxiliary parameter -. Because of this, we have displayed the -−curves in Figure 3. The acceptable value for the velocity profile is −0.4 ≤f ≤ 0.1, the temperature profile is −0.41 ≤θ ≤ 0.12, and the concentration profile is −0.65 ≤φ ≤ 0.2.

Results and discussion
This section presents the discussion on the impacts of different embedded factors on the flow profiles of an electrically conducting hybrid nanofluid over an extending surface using a porous medium. The space-dependent heat source, Joule heating, Brownian motion, thermophoresis, thermal radiation, chemical reaction, and activation energy impacts are taken into consideration. Figure 4A shows the influence of the magnetic parameter M on the velocity profile (f′(η)). It is observed that the increasing M reduces f′(η) significantly. Physically, when we increase M, an opposing force in the direction of fluid is created. This force is actually the Lorentz force, which resists the fluid particles' motion. This force slows down the motion of the flow particles and, thus, f′(η) reduces. So, the velocity profile reduces with the increasing M. The effect of the suction/injection factor S on the velocity profile f′(η) is depicted in Figure 4B. When S > 0, f′(η) decreases. The heated fluid is physically forced away from the surface by the increased blowing factor, which causes the viscosity to drop and the fluid to accelerate. The momentum boundary layer is, however, thinned by the wall suction S > 0, which imposes a drag force near the surface. The impact of λ on the velocity profile f′(η) is shown in Figure 4C. f′(η) shows increasing conduct via λ. This is due to the fact that increasing λ decreases the viscous influence on the flow. As a result, rising

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frontiersin.org 05 λ reduces the thickness of the momentum boundary layer, and hence, f′(η) increases. The effect of the porosity factor γ on f′(η) is shown in Figure 4D. It has been seen that f′(η) reduces with the upsurge in γ. This causes the motion of the fluid to reduce, which consequently reduces the thickness of the velocity boundary layer. So, f′(η) declines with the inclement in γ. The consequence of M over the temperature profile θ(η) is shown in Figure 5A. According to this figure, the impact of the magnetic parameter results in the rise of θ(η) as M increases. It is also important to note that the application of M has a positive impact on the thickness of thermal boundary layers because the thickness increases in the presence of M. Figure 5B depicts how the injection/suction factor S affects the temperature profile θ(η). It is understandable that when S increases, the thickness of the thermal boundary layer decreases. In addition, the rate of deformation from the wall towards the fluid accelerates with the increasing S. The effect of the Brownian motion factor Nb on θ(η) is displayed in Figure 5C. Increasing Nb improves θ(η). Additionally, Nb has an increasing effect on the thermal boundary layer. According to the definition, when Nb grows, θ(η) rises because the fluid particles have more kinetic energy. The influence of Nt on θ(η) is shown in Figure 5D. From Figure 5D, we can see that θ(η) increases due to the rising Nt. Figure 5E displays the variation in the temperature profile θ(η) due to Ec. It is seen that θ(η) rises as Ec increases. To understand the thermal performance of fluid flow, Ec is important. By raising Ec, the rising intermolecular interaction will increase the kinetic energy, which will increase θ(η) and allow Ec to be utilized as a hot agent. Figure 5F and Figure 5G show how the radiation factor Rd and the temperature ratio factor θ w affect the temperature profile θ(η). By increasing θ w and Rd, the fluid temperature rises significantly.
Physically, the fluid particles are supported and activated by the rise in θ w and Rd as a result of obtaining thermal energy. The temperature of the boundary layer rises as a result of this. Increasing the thermal diffusion and thermal distribution, in turn, causes the boundary layer thickness to grow and its temperature to rise. Figure 5H shows how the temperature profile θ(η) is influenced by the space-dependent heat source parameter Q e . θ(η) increases when we raise Q e . Physically, whenQ e > 0, the thermal boundary layer produces energy which causes augmentation in θ(η). Figure 5I illustrates the influence of the thermal Biot number Bi T on the temperature profile θ(η). When Bi T increases, θ(η) also increases. Physically, an enhancement in Bi T results in more supported convection, which causes the increasing conduct in θ(η). Therefore, higher values of Bi T increase the thermal boundary layer thickness, which, in turn, results in higher θ(η). The influence of Nb on the concentration profile φ(η) is shown in Figure 6A. φ(η) significantly decreases with higher values of Nb. Physically, the increasing Nb produces the random movement of the nanoparticles in fluid; as a result, φ(η) reduces. Figure 6B portrays the impact of the thermophoresis factor Nt on φ(η). It is observed that the increasing Nt increases φ(η). Physically, a rise in Nt is followed by an upsurge in the thermal energy, which promotes the liquid's temperature. As a result, the kinetic energy increases, and more collisions happen, which is enough to make the distribution of the concentration of nanoparticles large under the influence of Nt. The effect of activation energy E on the concentration profile φ(η) is seen in Figure 6C. φ(η) increases when E increases. Increasing values of E diverge from the modified Arrhenius function, which increases the rate of generative chemical reactions. The concentration is increased as a result. The impact of the Schmidt number Sc on the concentration profile φ(η) is exhibited in Figure 6D. φ(η) decreases as Sc increases. The concentration boundary layer gets thinner because higher values of

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frontiersin.org 06 Sc result in a faster mass transfer rate. So, φ(η) shows a decreasing impact against Sc. Figure 6E shows how the chemical reaction factor Kr affects the concentration profile φ(η). It is obvious that φ(η) decreases as Kr increases. The effect of the temperature difference parameter δ on the concentration profile φ(η) is seen in Figure 6F. φ(η) shows diminishing behavior when the temperature difference values are increased. Physically, a greater δ causes a decrease in molecular diffusivity, which lowers φ(η). Table 1 shows the thermophysical  properties of the base fluid and nanoparticles. The comparison of the present results with published results are shown in Table 2. The values of C fx are shown in Table 3 for different stretching ratio parameters λ and magnetic field parameters M. We determined the negative values for C fx . When friction force is negative, it indicates that the sheet is causing the fluid to move more slowly. In terms of quality, the effects of λ and M on C fx caused by the flow are equivalent. In other words, C fx is the decreasing function of M and λ. The numerical values of the Nusselt number Nu x and Sherwood number Sh x are shown in Tables 4  and Table 5 for various values of Rd, θ w , Bi T , M, and λ. Nu x , in this case, increases for high values of Bi T and λ while decreasing for larger values of Rd, θ w , and M. Sh x is the increasing function of Bi T , and M is a decreasing function of Rd, θ w , and λ.

Conclusion
In this article, the authors have presented an electrically conducting hybrid nanofluid flow over an extending surface using a porous medium. The homotopic approach is tackled for the solution of the modeled equations. The spacedependent heat source, Joule heating, Brownian motion, thermophoresis, thermal radiation, and chemically reactive activated energy impacts are used. The following are the concluding points of this study: • The growing magnetic and suction factors reduced the velocity profiles, while they enlarged the thermal profiles by magnetic factor. Additionally, the suction factor has a reducing impression on the thermal profile. • Motion of the fluid reduced with the increasing porosity factor, while it increased with the increasing stretching factor. • The thermal profiles increased with the increasing thermal Biot number, Eckert number, thermophoresis, space-based heat source, Brownian motion, and non-linear thermal radiation factors. • The concentration profiles reduced with the increasing Brownian motion, chemical reaction, and temperature difference factors, while they were increased by the activation energy factor. • The magnetic and stretching factors augmented the surface drag coefficient. • The Nusselt numbers increased with the increasing thermal Biot number and stretching factor, while they reduced with the   increasing thermal radiation and temperature difference factors.
• The Sherwood numbers increased with the thermal Biot number and magnetic factor, while they reduced with the increasing thermal radiation, temperature difference, and stretching factors.

Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.