Heat transfer analysis of the mixed convective flow of magnetohydrodynamic hybrid nanofluid past a stretching sheet with velocity and thermal slip conditions

The present study is related to the analytical investigation of the magnetohydrodynamic flow of Ag − MgO/ water hybrid nanoliquid with slip conditions via an extending surface. The thermal radiation and Joule heating effects are incorporated within the existing hybrid nanofluid model. The system of higher-order partial differential equations is converted to the nonlinear system of ordinary differential equations by interpreting the similarity transformations. With the implementation of a strong analytical method called HAM, the solution of resulting higher-order ordinary differential equations is obtained. The results of the skin friction coefficient, Nusselt number, velocity profile, and temperature profile of the hybrid nanofluid for varying different flow parameters are attained in the form of graphs and tables. Some important outcomes showed that the Nusselt number and skin friction are increased with the enhancement in Eckert number, stretching parameter, heat generation parameter and radiation parameter for both slip and no-slip conditions. The thermal profile of the hybrid nanofluid is higher for suction effect but lower for Eckert number, stretching parameter, magnetic field, heat generation and radiation parameter. For both slip and no-slip conditions, the hybrid nanofluid velocity shows an upward trend for both the stretching and mixed convection parameters.


Introduction
In recent decades, scientists and researchers have focused their attention on the study of a new class of fluid known as a hybrid nanofluid. Fluids with a higher thermal conductivity than nanofluids and base fluid are known as hybrid nanofluids. Hybrid nanofluids are formed by diffusing two sorts of nanoparticles in the carrier liquid. Mechanically to improve the transmission of heat in a variety of engineering and manufacturing processes the hybrid nanoparticles energy transition has been devolved. The hybrid nanofluid has strong thermal reliability as compared to the classical and base liquid. The numerous applications of hybrid nanofluid in industries and engineering mechanisms are heat pumps, heat capacitors microelectronics, digital cooling, car radiators, nuclear power stations, coolant in writing and machine tools, chemical producers, nano-drag shipments, automobile power creation, cancer treatment, and so forth. Several studies on hybrid nanofluid being conducted by many researchers and scientists as a result of their significant applications. Shoaib et al. [1] considered the spinning flow of nanofluid flow with radiation effects inside the stretching surface and noticed a variation of rotation parameter for velocity profile. Hanaya et al. [2] explained the micropolar hybrid nanofluid flow toward the curved stretching surface containing SWCNT and MWCNT nanoparticles and applied the bvp4c-shooting scheme for the examination of their model results. Liaquat et al. [3] numerically computed the MHD hybrid nanofluid flow toward the shrinking surface for stability analysis and dual solutions and their problem consists of C u − Al 2 O 3 − water the base fluid. They found that for increasing values of suction and radiation parameters temperature is enhanced (decrease) for both solutions. Shoaib et al. [4] scrutinized the phenomena of viscous dissipation in 3D MHD hybrid nanofluid flow via rotating disk. In this research work, it is noted that magnetic field reduced both the radial and azimuthal skin frictions coefficients. Ahmad et al. [5] pointed out the study of Go + Silver(Ag) in Maxwell hybrid nanoliquid for the improvement of thermal performance. Their concluding remarks illustrated that the heat transition is improved through the addition of silver volume fraction with graphene oxide. Alhussain et al. [6] inspected the influence of variable viscosity in a blood-based two-dimensional Casson hybrid nanofluid through the stretching sheet by introducing a magnetic field perpendicular in a flow field. From this analysis, the authors have demonstrated that with the increment of nanomaterials concentration in the base fluid, the thermal expansion rate is increased but the specific heat capacity decreased. Sreedevi et al. [7] perceived a numerical study for gyrotactic microorganism's past over a swirling cylinder and examined that microorganism density is raised for Weissenberg number. In addition, Roy et al. [8] inspected the second-grade nanoliquid flow across an extending surface along with heat propagation. They reported that the heat flux is condensed but wall shear stress is improved through the variation of velocity ratio parameter.
The magnetohydrodynamic is in the domain of physical science that is concerned with the magnetic and electrically conducting fluids properties including saltwater, liquid metals, electrolytes, and plasma. The magnetohydrodynamic is engaged in a lot of engineering and industrial applications such as generators and electronic filters, heat exchangers, power pumps, cooling of the reactors, power generation, plasma, MHD sensor, magnetic drugs targeting, crystal intensification, and so forth. Many researchers and scientists are interested to analyze the behavior due to the influence of the magnetic field, notably in the context of boundary layer flow problems. Tlili et al. [9] scrutinized the MHD flow of hybrid ferrofluid with asymmetrical heat transfer rise and fall effect. They studied the impact of Fe 3 O 4 (magnetic oxide) and CoFe 2 O 4 (cobalt iron oxide) ferrous particles embedded in an H 2 O −EG (ethylene) (50-50%) mixture in their hybrid fluid model. Kempannagrari et al. [10] contemplated a problem to explain the MHD non-Newtonian fluid flow above the exponentially stretching surface along with joule heating effects. From this study, it is concluded that the temperature of the fluid is escalated with the escalation of the heat source parameter and Eckert number. Kumar et al. [11] considered the influence of Cattaneo-Christove heat flux on the problem of MHD electrically conducting fluid past a wedge and a cone. They made a comparison of momentum boundary layer thickness over the wedge and cone and found that the momentum boundary layer thicknesses are higher for flow on the wedge as compared to the cone. Ramudu et al. [12] presented a model of MHD Casson nanofluid flow under the stretching surface to check the effect of Brownian and thermophoresis behavior and obtained that the Lewis flow parameter decreased the concentration of the nanofluid. Kumar et al. [13] explored the characteristics of first and second-order velocity slip conditions on the MHD flow of micropolar nanofluid through the stretching surface and their result shows that second-order velocity slip conditions declined the temperature of the nanofluid but enhanced the velocity of the nanofluid. Ramdevi et al. [14] reported the solution of three-dimensional Carreau fluid flow on the bidirectional stretching sheet along with modified Fourier law and chemical reaction impact. Kumar et al. [15] analyzed the modeling of MHD flow of micropolar fluid toward the slandering stretching surface in the attendance of viscous dissipation. From this investigation, it is observed that the couple stress coefficient is higher for the microrotation parameter. Idowu and Falodun [16] presented the heat-mass transfer behavior in the magnetohydrodynamic non-Newtonian Casson nanofluid flow on the plate. For acquiring the solution of the problem, their converted ODEs were solved with the implementation of the spectral HAM. Islam et al. [17] discussed the MHD nano liquid flow through graphene oxide (Go) and copper (Cu) hybrid nanofluid. Their outcomes indicated that the Reynolds number declined the fluid temperature profile. Jain and Bhargava [18] scrutinized the MHD free convective non-Newtonian fluid flow inside a wavy enclosure with the use of the Meshfree method with the implementation of the magnetic field in the horizontal direction in which they noted that transfer of heat is much higher when Rayleigh number is high. El-Dabe et al. [19] used a numerical scheme for the study of MHD non-Newtonian power-law nanofluid flow in a non-Darcy porous media toward the inclined channel with Ohmic dissipation. In this evaluation, they found that nanoparticles concentration is the increasing function of the Eckert number. Kotha et al. [20] discussed the presence of the gyrotactic microorganism in two-dimensional MHD water-based nanofluid flow through the vertical plate and explained that the bioconvection Peclet number boosted the local Sherwood number impact. Waqas et al. [21] proposed the mathematical modeling of electrically conducting Oldroyd-B nanoliquid flow surrounded by a permeable medium across the rotating frame. Aziz and Shams [22] made the estimation of heat transmission in MHD non-Newtonian nanoliquid flow under the stretching sheet. They showed that the porosity parameter shows decrement in entropy generation, but the inverse trend of Brinkman number is observed. Gupta et al. [23] introduced the two-dimensional Williamson nanoliquid model with the variable thickness and also depicted the effects of different emerging parameters over fluid velocity, temperature concentration.
The improvements in fluid heat transfer rate have attracted a lot of attention from scientists and engineers because of their extensive and practical uses in medical, industrial, and technical domains. The heat transmission through the hybrid nanofluid has a broad spectrum of uses in serval mechanical and engineering science, chemical analysis and biological sectors such as food processing insulating materials, polymers solar energy and reactor fluidizations, cooling of microchips, heat exchangers, heat pipes, solar water heating, extraction of geothermal power, glass manufacturing, hydraulic breaks and so on. Several researchers have analyzed the effect of heat transition on various flow conditions as a result of these varied uses. AshwinKumar et al. [24] presented the numerical solution of two-dimensional MHD hybrid nanofluid flow with heat transfer effect toward the different geometries by using the Runge-Kutta method. In their investigation, they take water as a base fluid and the nanoparticles are CuO and Al 2 O 3 . Samrat et al. [25] introduced the mathematical modeling of magnetohydrodynamic dusty-nano and dusty hybrid-nano liquid with the occurrence of heat transfer behavior. They found that higher heat transfer occurs in dusty hybrid-nano liquid as compared to the dusty nano liquid. Chalavadi et al. [26] considered the study of MHD Sakiadis flow of Casson/Carreau hybrid nanofluid flow over the moving needle in the presence of heat transfer in which they discussed that with the enhancement of nanoparticles volume fraction the temperature distribution of Carreau hybrid fluid model is augmented than the Casson fluid model. Ashwin Kumar et al. [27] adopted the regular perturbation technique for the investigation of the heat transfer effect over the unsteady MHD flow of Al 50 Cu50 / Ag − water nanofluid through the elongate surface. Mabood et al. [28] studied the MHD flow of an unsteady hybrid nanofluid past a slandering surface along with the influence of heat transfer impact. They noted that the skin friction coefficient of the hybrid nano-liquid is reduced when the magnetic field parameter is increased. Mabood et al. [29] described the impacts of heat transfer and nonlinear radiation on the three-dimensional unsteady MHD flow of Fe 3 O 4 /graphene-water hybrid nanofluid. They observed that the enlargement in heat transfer is more favorable in hybrid nanofluid as compared to the usual nanofluid. Tlili et al. [30] deliberated the theoretical and experimental study of three-dimensional MHD flow of AA7072−AA7075/methanol hybrid nanofluid on the uneven thickness surface with heat transfer effects. They noticed that the Nusselt number of hybrids nanofluid is raised for wall thickness parameter. Akbari et al. [31] explained the heat propagation in a non-Newtonian hydrodynamic nanoliquid flow containing the volume fraction 0.5%-1.5% of the solid nanoparticles Alumina. They figured out that betterment in heat transfer is occur because of the enhancement in the quantity of nanoparticulate. Xiong et al. [32] conducted the experimental analysis of the power-law in Cu − water nanoliquid model by using energy transmission and flow nature and used finite volume method for the description of the problem and found that the ratio between Nusselt number and skin friction becomes high up to 3.57% and 0.08% when the Grahsof and Reynolds number increases. Mahabaleshwar et al. [33] designed a model of MHD mixed convective-radiative flow in a permeable medium toward the accelerating surface. Their findings from this study show that the boundary layer thickness is reduced for all the involving functions because of the increment of the Richardson parameter. Mukthar et al. [34] initiated the influence of energy transition and thermal radiation in a flow of MHD Maxwell Copper-water Cu − H 2 O and Molybdenumdisuflide MoS 2 − H 2 O nanofluid past a stretched flat plate. They experimented that, Cu − H 2 O based nanoliquid has higher thermal efficiency as compared to the MoS 2 − H 2 O nanofluid. Hashim et al. [35] introduced the mass and energy transmission characteristics in 2D time-dependent Williamson nanoliquid flow via heated sheet and concluded that the thermophoresis parameter improved the fluid temperature. Hayat et al. [36] discoursed the MHD flow of unsteady Jeffrey nanofluid through the stretching cylinder. From this observation, we can analyze that fluid viscosity is intensifying with the increasing of Schmidt number Sc. Khan et al. [37] investigated the energy diffusion processes in Walter-B ferrofluid using gyrotactic microorganisms in free convective nanocomposites slip flow. In this paper, they observed that the bioconvection Peclet and Lewis number declined the motile density profile. In another investigation of heat transfer, Waini et al. [38] reported the hybrid MHD nanofluid flow over the permeable stretching wedge by considering the heat transition and magnetic effects and also explained the comparison of heat transfer between hybrid nanoliquid and regular nanofluid and studied that hybrid nano-fluid increases more heat transfer rate as compared to regular nanofluid.
Joule heating effects occur when an electric current passes through a conductor and produces heat. The use of the Joule heating effect is in multiple devices and industrial processes.
The food processing, scanning electron microscope, infrared-thermal images, electric radiative space heater, incandescent light bulb, laboratory water bath, electric tabletop hotplate, clothes iron, soldering iron, fan heater, hairdryer and cartridge heater are some important applications of the Joule heating effects. The researchers and engineers discussed the Joule heating effect in their research area in light of the above applications of the Joule heating effect. Alaidrous et al. [39] explored the 3D magnetic flow of a hyperbolic tangent nanofluid in porous media toward the extending surface and obtained that fluid temperature is increased for joule heating parameter. Hafeez et al. [40] studied the non-Newtonian Oldroyed-B nanofluid flow along with the characteristics of Cattaneo-Christove heat flux theory. In this work, they disclosed that the concentration of the Oldroyed-B nanofluid is raised for the thermophoresis parameter. Khan et al. [41] carried out the numerical study of hydromagnetic three-dimensional Casson nanofluid flow via rotating cylinder along with entropy and Joule heating effects and found that magnetic field parameter upsurges the nanofluid velocity profile. Kumar  From the above-mentioned literature, it is noted that no attention is paid on the study of MHD flow of Ag − MgO/water hybrid nanoliquid and heat transition toward the stretching and shrinking surface in the existing of radiation and Joule heating effects. The heat source/ sink and slip conditions are also included in the present investigation. To an acquired solution of coupled higher-order non-linear ordinary equations, the homotopy analysis method is hired. The flow and heat transmission of a hybrid nanofluid across an exponentially stretched sheet with mixed convection and Joule heating effects are substantial and advantageous in various engineering, practical and industrial applications. Microelectronics, manufacturing, naval structures, nuclear system cooling, biomedical and drug reduction are only a few of the realworld (practical) and industrial applications of hybrid nanofluid challenges. The skin friction and Nusselt number have been estimated for distinct parameters. Moreover, the behavior of hybrid nanofluid velocity and temperature are calculated in graphical form for various parameters.
Some important applications related to the present physical model are discussed in detail. In the area of PVC (Poly Vinyl Chloride), pipe production, wiredrawing, casting of metals, hot rolling, etc. The concepts of the stretching sheet involve the cooling of continuous strips or filaments by drawing them through a quiescent fluid. In industrial applications, the problem of the heat transfer flow related to the stretching surface has great importance, for instance, in the metallurgical processes, like tinning and annealing of the copper wires, drawing of continuous filaments and manufacturing of the rubber and plastic sheets through the quiescent fluids, fiber spinning and continuous cooling, glass blowing and crystal growing, in addition to extensive applications in various engineering developments like as manufacturing of the paper and foods, stretching of the wire drawing, plastic films, and polymer extrusion, fiber production, and the continuous casting, and many other processes.

Problem formulation
The steady MHD flow of hybrid nanofluid flow of Ag(silver) and MgO(Magnesium oxide) nanoparticles and water as a base fluid is considered via exponentially stretching sheet. Thermal radiation and Joule heating effects are taken into account. The influence of heat source/ sink is used to discuss the generation and absorption of heat phenomena. The slip conditions are taken in the description of the existing model. For the computation of heat transport behavior, the viscous dissipation impact is employed. Along y-axis, a magnetic field B in which a is positive constant. Fig 1 displays the hybrid nanofluid flow mechanism. The temperature at the surface is T w and ambient temperature is T 1 . In view of the above assumption, the leading equations are [47][48][49][50][51]. The relevant boundary conditions are The electrical conductivity and heat capacity of the hybrid nanofluid are expressed by σ hnf and (ρC p ) hnf , respectively, the acceleration due to gravity is g, T is the temperature, Q 0 is the heat sink/source, λ is the stretching parameter, A 1 and B 1 are the slip constants.
The thermophysical properties are defined as The calculated values of the thermophysical features of the base fluid and nanoparticles are presented in Table 1.
The similarity transformations are defined as [48, 49, 51]: After applying similarity transformations, Eqs (2) and (3) are reduced as: Pr The transformed conditions are In the above equations, η is the similarity variable, the radiation parameter is denoted by Þ is used for Eckert number, the magnetic field parame- is for mixed convection parameter. The dimensionless velocity and temperature are signified by f and θ. The dimensionless velocity slip parameter is ffi ffi ffi ffi ffi ffi p . Two cases arises on S, for mass suction S > 0, (ν 0 < 0), and for mass injection S < 0, (ν 0 > 0). Mathematically, the significant physical quantities are stated as: where By using the similarity transformations defined in (6), the dimensionless form are:

Solution of the problem
The analytical solution of the higher-order ordinary differential equations is gained through the homotopy analysis method via Mathematica 10. The initial guesses and linear operators are defined as: with where C i (i = 1 − 5) are the arbitrary constants.

m th -order deformation problem
Now the m th deformation of the problem is explained as: The R f m mðZÞ and R y m m Z ð Þ are defined below: The general solution is achieved through the use of particular solutions:

Results and discussion
For the description of the current study, the most powerful technique called the homotopy analysis technique is employed. The influence of emerging parameters on the hybrid nanofluid velocity and temperature are computed and debated in detail.

PLOS ONE
Heat transfer analysis of the mixed convective flow of magnetohydrodynamic hybrid nanofluid distinct parameters. Table 2, demonstrates the influence of Eckert number Ec, stretching parameter λ, suction parameter S, mixed convection parameter λ 1 , heat generation Q, magnetic field M and radiation parameter Rd on the skin fraction coefficient of the hybrid nanofluid for both slip A > 0 and no-slip A = 0 conditions. From Table 2, it is clear that the skin fraction ffi ffi ffi ffi ffi ffi ffi Re x p C fx of the hybrid nanofluid is enhanced with the increasing of Ec, λ, λ 1 , Q and Rd   conditions due to varied values of Eckert number Ec, stretching parameter λ, suction parameter S, mixed convection parameter λ 1 , heat generation parameter Q, magnetic field parameter M and radiation parameter Rd are discussed in Table 3. Table 3 shows that the results of Nusselt number 1 ffi ffi ffi ffi ffi     Table 3.

PLOS ONE
Heat transfer analysis of the mixed convective flow of magnetohydrodynamic hybrid nanofluid   Fig 7. Higher values of magnetic field produced more and more frictional forces that have to decrease the fluid velocity. Physically, the heat transfer rate is enhanced due to the production of Lorentz forced by larger values of magnetic field parameter. Further, the heat transfer rate is increased due to the magnetic field parameter because there will be a large amount of energy is store in the fluid. Moreover, it is examined that the thermal and momentum boundary layer of the hybrid nanofluid becomes thinner with the increase of magnetic field parameter. conditions via discrete estimation of ϕ 2 are designed in Figs 13 and 14. When the nanoparticles concentration of MgO(Magnesium oxide) ϕ 2 is large, then the hybrid nanofluid velocity is small in these figures for both no-slip and slip conditions. Physically, the weakening characteristic of the momentum boundary layer is caused by increasing values of nanoparticles volume fractions ϕ 1 and ϕ 2 , therefore the velocity of the hybrid nanofluid is decreased.

Temperature profile
The consequence of Eckert number Ec, stretching parameter λ, magnetic field parameter M, heat generation parameter Q, radiation parameter Rd, suction parameter S and nanoparticles concentration ϕ 1  over fluid temperature for the no-slip condition (B = 0). It is noted that the improvement in Ec boosted the hybrid nanofluid temperature. The performance of the Eckert number on the hybrid nanofluid temperature for slip condition (B > 0) is determined in Fig 16. It is distinguished that with the augmentation of Eckert number, the temperature of the hybrid nanofluid is amplified. The process of conversion of mechanical energy into thermal energy becomes faster due to the increase of Eckert number that leads to enlarged the fluid temperature. The Eckert number is a term that describes the relationship between the flow of kinetic energy and heat enthalpy variation. It means that with the increase of Eckert, the kinetic energy of the hybrid nanofluid is amplified. Furthermore, the temperature is commonly defined as the average kinetic energy. That's why the temperature of the hybrid nanofluid is enhanced with the enhancement of Eckert number. it is noticed that the temperature profile is boosted when the magnetic field parameter is upgraded. As it is detected from the above-mentioned velocity profile that fluid motion is decreased due to an increase of magnetic field parameter. Because of the slow motion of the fluid, the kinetic energy is transformed into heat energy, causing the temperature to rise. Moreover, it is examined that the thermal and momentum boundary layer of the hybrid nanofluid becomes thinner with the enhancement of the magnetic field. The inverse relation between the density of the hybrid nanofluid and magnetic parameter is observed. As a result, increasing the magnetic field parameter causes the fluid density to decrease, resulting in temperature increases. The influence of fluid temperature profile for different values of heat generation parameter Q for no-slip condition (B = 0) is inspected in Fig  21). It shows that the temperature profile increases as the value of heat generation Q augments. The role of Q on the temperature of the fluid for slip condition (B > 0) is studied in Fig 22. It is comprehended that the estimation in heat generation parameter Q leads to the escalation in temperature of the hybrid nanofluid. Fig 23 is used to detect the effect of radiation parameter Rd for no-slip condition (B = 0) on fluid temperature profile which explained that temperature profile is enhanced for larger values of Rd. The enrichment in boundary layer thickness is occur due to the rise of fluid temperature. An increment in radiation parameter provides an additional amount of heat to hybrid nanofluid and as a result the fluid temperature heightens. Also, the thermal diffusivity of the hybrid nanofluid is enhanced for higher values of radiation parameter, that's why the hybrid nanofluid temperature is increased. Fig 24 portrays the influence of radiation parameter for slip condition (B > 0) on the fluid temperature profile. The expansion in temperature is exhibited for intensifying values of radiation parameter. In Fig 25, for the no-slip condition (B = 0), the disparity of temperature profile for increasing values of S is discussed. The outcome from this figure shows that decrement in hybrid nanofluid temperature profile is perceived for varied values of S. Fig 26 presents the variation in temperature profile against S for the case of slip condition (B > 0). It revealed that for rising values of S, the fluid temperature profile is reduced which results to decaying conduct in the boundary layer. Physically, a considerable amount of fluid flows outside the walls of the sheet during the process of suction which slows down the motion and boundary layer thickness of the fluid, that's why the fluid temperature is diminished. Figs 27 and 28 are displayed to describe the influence of nanoparticles concentration ϕ 1 on the hybrid nanofluid temperature for the slip (B > 0) and no-slip (B = 0) conditions. For both no-slip and slip conditions, the hybrid nanofluid temperature is increased with the rise of nanoparticles concentration (ϕ 1 ) of Ag(silver). Figs 29 and 30 shows the disparity of hybrid nanofluid temperature profile for serval estimation of nanoparticles concentration (ϕ 2 ) of MgO(Magnesium oxide). From this analysis, it is observed that with the augmentation of nanoparticles concentration ϕ 2 , the temperature of hybrid nanofluid is heightened. By adding the nanoparticles to base fluids improves thermal conductivity and so improves the thermal efficiency of the hybrid nanofluid. That's why the hybrid nanofluid temperature is increased against growing values of nanoparticles volume fraction.

Conclusion
This article aims to investigate the MHD flow of hybrid nanofluid toward the stretching surface in the presence of Joule heating, thermal radiation and heat source/sink effects. The present model consists of momentum and temperature equations along with slip conditions. The fluid motion is generated due to the stretching of the surface. The homotopy analysis method (HAM) is employed for the discretization of the current study. The main finding of the present study is • The skin friction coefficient ffi ffi ffi ffi ffi ffi ffi Re x p C fx of the hybrid nanofluid is increased with the increase in Eckert number Ec, stretching parameter λ, heat generation parameter Q and radiation parameter Rd for both no-slip (A = 0) and slip (A > 0) conditions.
• The growth in suction and magnetic field parameters lead to a decline the skin friction coefficient ffi ffi ffi ffi ffi ffi ffi Re x p C fx of the hybrid nanofluid for both no-slip (A = 0) and slip (A > 0) conditions.
• For the case of slip (B > 0) and no-slip (B = 0) conditions, the improvement in Eckert number Ec, stretching parameter λ suction parameter S, mixed convection parameter λ 1 , heat generation Q and radiation parameter Rd have augmented the Nusselt number 1 ffi ffi ffi ffi ffi Re x p Nu x .
• The declining conduct in 1 ffi ffi ffi ffi ffi