Nonlinear Convective SiO 2 and TiO 2 Hybrid Nanofluid Flow over an Inclined Stretched Surface

The hybrid nano ﬂ uid is extensively used in manufacturing commercial applications due to its high exceptional capacity to increase the heat transfer rate. As a result, in the existence of nonlinear convection, the hybrid nano ﬂ uid is considered to ﬂ ow on an inclined plane. The nonlinear convection has many applications in real life and is more relevant to the natural ﬂ ow avoiding the ﬂ ow restrictions. The focus has been executed on the thermal and mass Grashof numbers to analyse the ﬂ uid motion in the presence of these parameters for nonlinear nature. Moreover, the hybrid nano ﬂ uid ﬂ ow analysis has been done to investigate the heat transfer analysis. The modelled equations are solved through an analytical approach. The heat and mass transfer rates and drag force are calculated under the in ﬂ uence of various physical parameters. The new parameter of the Grashof numbers improves the ﬂ uid motion for its larger values, and consequently, the ﬂ uid rapidly falls down from the inclined plane. The obtained outputs show that hybrid nano ﬂ uids are more e ﬀ ective in heat transfer analysis as compared to other conventional ﬂ uids.


Introduction
In the past few years, the utilization of thin-film liquid flow has fascinated researchers in the different area like engineering, technology, and industry; consequently, the study of thin-film liquid flow analysis regarding their application in many fields is necessary such as shipment through the flow in human lung and a lubricating process in the industry. These and a few more are considered to be the biggest subclass of thin-film liquid flow-related issues. The study of thin-film liquid flow in many active applications is a nice combination of structural mechanics, fluid mechanics, and theology. One of its main applications is the cable fiber undercoat. In addition, polymer and metal extrusion, food straightening, permanent formulation, elastic sheet drawing, and device fluidization, exchange, and chemical treatment apparatus are some of the well-known uses. Observing these applications, it became the principal subject for the investi-gator to study and analyse the behaviour of thin-film liquid flow at stretching surfaces. The flow behaviour for the first time regarding thin-film was investigated through Newtonian liquids and then expanded to non-Newtonian liquids. The non-Newtonian thin-film flow of nanoliquids has been analysed by Sandeep et al. [1]. Wang [2] has examined the transient thin-film liquid flow at the stretching surface. The motion of thin-film finite fluid at the time-dependent stretching surface was analysed by Usha et al. [3]. Liu et al. [4] have studied the movement of thin-film liquid regarding heat transfer behaviours at a stretching surface. Aziz et al. [5] examined thermal generation within thin-film liquid flow at a stretching sheet. The thin-film liquid motion with heat radiation regarding the behaviour of heat transmission has been analysed by Tawade et al. [6]. They solved their proposed model using Runge-Kutta-Fehlberg and Newton method. The analysis of heat transfer through thin-film liquid flow at a stretching surface has been done by Anderssona et al. [7]. Many researchers have worked independently on power-law liquid motion using thin-film time-dependent stretching sheets [8][9][10][11]. Megahe [12] has studied the flow of thin film of Casson liquid with heat transfer on a transient stretching sheet. In the proposed model, they also considered the slip velocity impact with heat flux and viscous dissipation. In the recent past, Tahir et al. [13] and Khan et al. [14] analysed the motion of thin-film nanoliquid through innovative approach. The growing interest in energy reserves is one of the most complicated issues for current researchers to fulfill the augmenting requirements for energy in modern scientific practices. Researchers are attempting to introduce new channel of energy that are convenient and reasonable for cooling and thermal applications. The most readily approachable means of renewable energy in the universe is solar energy. Solar energy is a good source of clean and renewable energy, so it will not cause environmental adulteration which is generated by conventional energy such as coal, oil during the process of use. The negative impacts of pollution on the earth cause global warming and lung and cancer disease. Solar energy is a good alternative to surmount this problem and reduce its detrimental impacts. Over the past 30 years, many developed and developing industrialized countries have focused on advancing solar technology. When there is oil, energy crisis, solar energy becomes more attractive. The oil crisis in general is a crisis of higher prices as OPEC countries raise fuel prices, so options include solar thermal, solar photovoltaic, wind energy, geothermal, marine, and wave energy. So scientists and researchers paid attention to all these things, and an interesting reality is that the energy that approaches the earth from 20 days of sunlight is equal to the energy stored in all the earth's storages of fossil fuels such as petroleum, coal, and natural gas. In this way, they play a prominent role in meeting the needs of the people in the world. Flat plates (solar collectors) use heat transfer liquids to transform solar energy into thermal energy. Rising energy needs around the world, including nonrenewable energy sources like fossil fuels, have minimal production of such resources, resulting in huge, detrimental effects on the environment, like global warming, climate change, and air pollution. To mitigate such losses, scientific approach has focused on improving the productivity of renewable energy processes, like solar energy. Solar energy is the cheapest and cleanest option of renewable energy, which converts solar energy into environmentally friendly electrical and thermal energy. For the conversion of solar energy into heat energy, a heat-changing liquid can be used in flat plate-type solar collectors [15]. Solar collectors receive solar rays through absorbing plates and converting such rays into a useful form of the energy (primarily water, water composition, and EG). Nevertheless, the major shortcomings are the low thermal properties of these conventional liquids, as they offer low thermal efficiency in the transformation process. Converting conventional working liquids into nanoliquids is one of the initiatives, which has received more attention over the past few years in enhancing the thermal efficiency of this technology. A stable synthesis of solids components between 1 nm and 100 nm is referred to nanoliquids [16]. Nanoliquid is also widely utilized in solar energy depots [17,18], heat exchangers [19], and freezing methods [20]. Mebarek-Oudina [21] studied nanofluid using various basic fluids. An analytical study of MHD nanoliquid motion, for heat transfer analysis, was done by Saeed and Gul [22]. Rehman et al. [23] analysed the motion of nanofluids by implementing the induced changeable magnetic field using the liquid film flow model. The motion of Darcy-Forchheimer nanoliquid through a mathematical model at the curved surfaces was analysed by Sajjad et al. [24]. Several recent analyses have been done in different energy and thermal environments using analytical and numerical methods, for handling heat exchanges and nanoliquid flow behaviour, by Sheikholeslami [25], Zhang et al. [26], and Gul et al. [27]. The magnetic properties of an electrically conducting liquid are referred to as MHD. In the natural and industrial spheres, we can notice that the behaviour of liquid motion is influenced by magnetic fields. The MHD phenomenon occurs when the velocity and magnetic field are combined. Instances of these kinds of fluids are liquid metals, electrolytes, plasma, etc. The concept of magnetohydrodynamic has been invented by Hannes Alfven [28]. On this great success, he was given the Nobel Prize in Physics in 1970. MHD has many applications in engineering and technologies such as MHD generators, plasma, and nuclear reactors. The use of solar energy collectors also plays a significant role in medicine such as cancer therapy and MRI. This study is about the movements of ionized atoms or components and their liaison with the electric fields and neighbouring liquids. This safeguards against particle (atoms or molecules) and fluid transport phenomena such as electrorotation, dielectrophoresis, electro-osmosis, and electrokinesis. This has wide range of applications in many areas including gas pumps, drag reduction, increasing drying rate, and plasma actuators. At the outset, electrohydrodynamic fluid motion has been examined by Woodson and Melcher [29]. The impact of substitutive current and thermal transport on the dielectric viscous peristaltic fluid motion has been examined by Sayed et al. [30]. Khan et al. [31] investigated the irreversible behaviour of the electromagnetic hydrodynamic convective flow of viscous fluid. Rashid et al. [32] scrutinized the micro polar electromagnetohydrodynamic radiative fluid flow with convection state at a stretchable permeable sheet. In the recent past, scientists have focused on the fluids flowing at the permeable space and particles of various shapes inside the porous region. Their use can be understood in the multiple fields like nuclear engineering, environmental sciences, solar thermal engineering, bioinformatics, and construction engineering. Several processes that require the movement of fluids in a porous region include the utilization of geothermal energy, the flow of blood to lungs or veins, below-ground power lines, porous heating pipes, and chemically catalytic combiners. To understand the movements of fluid in porous space, Darcy's law is used frequently. Darcy's notion of high velocity and turbulent impacts in the porous space is wrong. Mebarek-Oudina et al. [21,[33][34][35] investigated the MHD hybrid nanofluid flow through various configuration and geometries in the presence of different nanomaterials. Journal of Nanomaterials Forchheimer [36] updated the momentum expression through the inclusion of second-order polynomial to adjust the impact of inertia on relative permeability. To analyse the impacts of inertia at relative permeability, a term of secondorder polynomial in the momentum equation has been introduced by Forchheimer [36]. Muskat [37] pointed out that component as a Forchheimer component. Many researchers have investigated the fluid motion via porous media by utilizing the Darcy-Forchheimer idea in various geometries. Few of them are presented here. Saif et al. [24] explained the motion of nanoliquid in the porous gap and concluded that the fluctuation in the fluid flow forms the surface of the stretchable curve. The behaviour of nanofluid motion regarding Darcy-Forchheimer effects produced through a stretching sheet, as explored by Rasool et al. [38]. A Darcy-Forchheimer flow of liquid at a spinning disc has been explored by Sadiq et al. [39]. A non-Darcy liquid flow within a transparent gap is explained by Sheikholeslami et al. [40]. The impacts of Darcy-Forchheimer and EMHD on the movement of viscous liquid in the presence of Joule heating and heat flux at a stretching surface were scrutinized by Hayat et al. [41,42]. In addition, they studied the process of entropy generation with the aid of the second law of thermodynamics. Kumar et al. [43] calculated the numerical outcomes of CNT nanofluids movement by the numerical scheme in divergent and convergent channels under the effects of thermal radiation. Akgül et al. [44][45][46][47][48][49] presented different novel technique for investigating fractional differential equations including the Atangana-Baleanu fractional derivative. The relevant and latest literature can be seen as [50][51][52].
The main objective of the ongoing research is to analyse the impact of electro-hydrodynamic Darcy-Forchheimer liquid movement and its use in augmenting the capacity of solar collectors through inclined plates. The energy equation has been developed under Joule heating, heat radiation, and viscous dissipation. The proposed model of the fluid flow has been formulated through PDEs and subsequently solved analytically in Mathematica using HAM technique. The analysis of different significant emerging parameters is elucidated in terms of temperature, velocity, and concentration. The numerical findings of the proposed model have been validated through findings in the literature. It reveals that the outcomes of the model are real and applicable in many fields of engineering and science.
The newness of the proposed model is as follows.
(I) The inclined plane with nonlinear mixed convection is used for the first time

Mathematical Formulation
A 2D steady fluid motion at a stretchable inclined plate is studied, which makes θ angle along with vertical axis, as shown in Figure 1, as a solar collector schematic outlook. The space is presumed as a porous Darcy-Forchheimer.
In addition, the energy equation is updated through the inclusion of heat radiation, viscous dissipation, and Joule heating.
Þ, E stands for electric field, and it is presumed that E ! = 0. In addition, T w , T ∞ , C w , and C ∞ are wall's temperature, free-stream temperature, wall's concentration, and free-stream concentration, respectively. Using the above principles, the mathematical formulation of the proposed model is [21,22,25,53] ∂u ∂x Acceptable boundary conditions are where K represents the porous medium permeability and u and v denote the velocity components in the x and y direction.
Using the following pertinent transformation variables, The reduced form of Equations (1),(2),(3),(4),(5) in the 3 Journal of Nanomaterials light of Equation (7) is as follows: With interrelated boundary conditions, Pr The above physical quantities are Grashof number, Prandtl number, heat source/sink factor, and Schmidt number.
Furthermore, the additional most key physical number are skin friction coefficient (C f x ), Nusselt number (Nu x ), and Sherwood number written as where τ w is the shear stress and q w denotes heat flux near the surface. Utilizing Equation (7), Equation (18) x y g Ω Figure 1: Geometry of the problem. 4 Journal of Nanomaterials
The convergence of HAM algorithm via ℏ curves for the three important profiles like temperature, velocity, and concentration has been sketched in various graphs (2)(3)(4), correspondingly. Figure 2 is the h curve for the velocity profile, Figure 3 is the h curve for the temperature field, and Figure 4 is the h curve for the concentration profile. Figure 5 explains the behaviour of the proposed flow model through the various values of Grashof number (Gr). The sketch reveals that an increment in the value of Gr augments the liquid velocity. The reality furthers is, since (Ther-mal Grashof Number) integrates both hydrodynamic forces and the thermal buoyancy force, which occurs in the boundary layer due to variation in temperature. Therefore, increasing the thermal buoyancy effect of the liquid permits the specified fluid to cool the hot plate.
The impact of the Grashof number (Gr * ) which is caused by nonlinear convection enhances the movement of fluid relatively larger and more related to natural phenomena and shown in Figure 6.  The influence of Sc (Schmidt number) at the concentration profile is sketched in Figure 9, indicating that the enhancement of the value of Sc decrements the liquid concentration profile. The ratio of two properties such as momentum diffusion to mass diffusion is referred to as Sc. Therefore, for the larger value of Sc, the mass diffusion is due to momentum diffusion. While Sc increments, because of less mass diffusion and smaller D B , the ΦðηÞ profile diminishes.
The thermal Grashof number and nonlinear thermal Grashof enhance the fluid motion and consequently decline the skin friction as shown in Figures 10 and 11. 7 Journal of Nanomaterials Furthermore, the effect is comparatively more impressible using the hybrid nanofluid.
The mass Grashof number and nonlinear mass Grashof strengthen the fluid velocity and consequently decline the skin friction as shown in Figures 12 and 13. Furthermore, the effect is comparatively more impressible using the hybrid nanofluid.
The hybrid nanofluids have the tendency to improve the thermal efficiency as shown in Figure 14. The heat transfer rate is comparatively more effective using the hybrid nanofluids. Physically, the thermal combined thermal conductivity of these nanoparticles is more reliable for the enhancement of heat transfer.
The mass transfer declines with the increasing amount of Sc as shown in Figure 15. The Schmidt number also declines the concentration for its increasing values; that is why the mass transfer also declines.
The present work is compared with the published work [60] as shown in Table 1 by considering the common parameters. It is concluded that the obtained results closely agreed with the published work.

Conclusion
In the present section, we evaluated the nonlinear convective fluid flow over an inclined plane considering heat and mass transfer analysis. The nonlinear convection is mainly focused to study the flow field down the inclined plane.
The highlights of the current work are as follows:

Data Availability
All the relevant data exist in the manuscript.

Conflicts of Interest
The authors declare that no such interest exists.