Numerical Investigation of the Finite Thin Film Flow for Hybrid Nanofluid with Kerosene Oil as Base Fluid over a Stretching Surface along with the Viscous Dissipation and Variable Thermal Conductivity Effects

Tis study examines the fow and heat transfer of a fnite thin layer of a hybrid nanofuid across an unstable stretching surface with varying thermal conductivity and viscous dissipation efects. A hybrid nanofuid model is considered to comprise two diferent types of nanoparticles, Go and Ag, with kerosene oil used as a base fuid. To study the phenomenon of thermal conduction, a modifed version of Fourier’s law model is adopted because in the power-law model, the thermal conductivity depends on the velocity gradient. A system of nonlinear ordinary diferential equations is obtained by considering the similarity transformations over the obtained rheological system of partial diferential equations which is then tackled by a well-known numerical approach, i.e., the bvp4c MATLAB technique. Te rheological impacts of the power-law index, solid volume fraction, flm thickness, Eckert number, and modifed Prandtl number on temperature and velocity felds are graphically discussed and illustrated. In the presence of nanoparticles, the temperature of the working fuid is enhanced and the power-law index has an inverse relation with the velocity of the hybrid nanofuid.


Introduction
Tin flm is a layer of material ranging from a fraction of nanometer (10 − 9 ) to several micrometer (10 − 6 ) in thickness.Te thin flm fow analysis is very valuable due to its inclusive applications in the engineering and technology feld.Te thin flm fow problems have a broader area of interest and catch many felds, starting from fow inside the human lungs to industrial coating problems.It is much important to explore the thin flm fow uses in structural mechanics, theology, and fuid mechanics.Te liquid flm has well-known uses such as the expulsion of polymer and metal, constant forming, foodstuf processing, drawing elastic sheets and devices, fuidization, and exchanges.For inspection of these uses and applications, the researchers are motivated to further investigate and make further development in the feld of thin-flm fow problems.Diferent approaches are adopted with modifed geometries.In industrial applications, the stretching surface is a major topic.Gul et al. [1] worked on the fow of third-grade magnetohydrodynamic (MHD) fuid with an impact on viscosity (temperature dependent) on a vertical belt.Later on, Bachicha [2] explored a method for producing cathodes from thin-flmed coatings which are produced by placing catholyte slime on a moving substrate.Aqil et al. [3] found that when the pure molybdenum (Mo) thin flm (by the DC magnetron sputtering technique) is deposited on the blank Si substrate, the deposition condition only changed for the deposition time, not for all other samples.Afterwards, Shah et al. [4] examined the efect of thermal radiations on a permeable medium and heat generation through a stretching surface during the Williamson liquid-flm fuid fow.Tey considered the fuid fow in two dimensions of liquid flms.Ullah et al. [5] worked on the fow (with skid conditions) of a generalized Maxwell fuid that is fowing on a nonisothermal cylindrical surface.
Heat generation plays an important role in the fow of a thin flm.When any type of energy inside the body is transferred into heat energy, then the temperature of the body increases which causes heat generation.Te body is made up of atoms and molecules, and when any type of energy is applied to the body, the atoms and molecules start vibrating, which makes friction; as a result, heat is generated.In industries and engineering, the transfer of heat over the moving continuous stretching sheet is investigated frequently due to its many applications.To set the standard of any product in the industry, heat generation plays an important role.Heat generation is majorly afected by temperature difusion and the particle toppling rate in a chemical reaction.Chankha and Issa [6] worked on the impact of heat generation and thermogenesis in hydromagnetics with mass and heat transfer through a permeable fat surface.Afterwards, the impact of heat generation, as well as radiation due to the linearly stretching sheet through an incompressible micropolar fuid, is highlighted by Reddy [7].Mehmood et al. [8] worked on the numerical investigation of MWCNT and SWCNT fuid fow along with the activation energy efects over quartic autocatalytic endothermic and exothermic chemical reactions.Pavithra and Gireesha [9] numerically examined a boundary fow and heat transfer for the dusty fuid in the existence of internal heat generation and viscous dissipation over an explosive/ accelerating stretching surface.Later on, a comprehensive review of heat generation in various types of wastes was taken by Yes ¸iller et al. [10].Ganga et al. [11] illustrated the heat generation on the boundary layer magnetohydrodynamic fow of nanofuids.Te heat generation efects over the stretching surface in the fow of non-Newtonian nanofuid are explored by Awais et al. [12].Aziz et al. [13] studied the impact of heat generation on magnetonanofuid.Later on, the infuence of heat generation over MHD Oldroyd B fuid with radiation on an inclined stretching sheet is examined by Mabood et al. [14].Khan et al. [15] explored the heat generation impact on the heated surface during the chemically reactive fow.Nanofuid is a fuid having nanometer-size (10 − 9 ) particles, called nanoparticles.Te nanoparticles are naturally made of metals, fbers, tubes, wires, rods, sheets, droplets, oxides, or carbon nanotubes.Nanofuids are mostly used to improve the thermophysical properties such as viscosity, thermal conductivity, thermal difusivity, and heat transfer (depending on thermal conductivity) coefcients to diferentiate base fuids (water or oil).Tey are very valuable for the cooling of a microsystem.It has many applications in many felds.Fluids such as water, oil, and ethylene glycol in these processes have been considered as cooling liquids but have a limited heat transfer rate.Wong and De Leon [16] focused on the broader scope of heat and the applications that demand nanofuids.A review (theoretical and experimental) of nanofuid for the magnifcation of thermal conductivity is studied by Kleinstreuer and Feng [17].Fluids in which two or more nanometer-size particles of diferent materials are mixed into the base fuid to get the desired physical properties are called hybrid nanofuids.Tese are being used to further enhance the heat transfer rate, pressure drop properties, better thermal network, and collaborative efect of nanomaterials.In the case of nanofuids, we have to face some major issues.One of them considered the stability of nanofuids, and it has become a big challenge to attain the required characteristics.Te properties of hybrid nanofuids are investigated by researchers to fnd out their applications, advantages, and disadvantages.Madhesh and Kalaiselvan [18] experimentally used the tubular heat exchanger to study the heat transfer through a hybrid nanofuid.In their work, they explored the efects of the thermal characteristics and also analyzed the hybrid nanofuids as coolants.Later on, Hayat and Nadeem [19] explained the use of Ag + CuO as a hybrid nanofuid to improve the heat transfer rate.Afterwards, Sundar et al. [20] took a review for the friction factor, heat transfer, and thermal properties of hybrid nanofuid.Similarly, Minea and Moldoveanu [21] worked on the development and benefts of hybrid nanofuids.Te steady, laminar, incompressible two-dimensional fow over the static channel using hybrid nanofuid (TiO 2 + CuO/water) is examined by Dinarvand et al. [22].A massbased approach has been presented for the hybrid nanofuid model.Bumataria et al. [23] analyzed the rate of heat transfer through a pipe and discussed it with applications for hybrid nanofuids.Some recent work related to nanofuid and hybrid nanofuid can be cited in [24][25][26].
When any shear force is applied to the fuid, then this force passes through the adjacent layers and hence the work done because of this force is transferred into heat; this process is called viscous dissipation.High rotational speed devices which have a strong impact on the gravitational feld also have an essential part of viscous dissipation for their natural convection.Viscous dissipation also has an important part in geographical and nuclear physics.Many researchers have identifed viscous dissipation by using nondimensional Eckert numbers.In gravitational felds, such as large planets, the remarkable efects of viscous dissipation may also exist.Massoudi and Christie [27] worked on the thermodynamically frictionless, incompressible third-grade fuid fow to investigate the reaction of viscosity (variable) and viscous dissipation.Later on, Mehmood et al. [28] numerically examined the boundary layer nanofuid fow due to mass fux conditions and viscous dissipation in a bent stretching surface.Pandey and Kumar [29] explored the MHD nanofuid fow, with injection and viscous dissipation over the porous medium and slip fow.Afterwards, Ahmed et al. [30] studied a nanofuid fow with the additional impact of viscous 2 Journal of Mathematics dissipation.Te KKL model is implemented for the thermal conductivity of nanofuids and the solid nanoparticles of copper oxide.Mishra and Kumar [31] explored the efects of heat absorption on nanofuid (Ag) in the existence of suction/injection through a vertical Riga plate.Using the homotopy perturbation technique, the fuid fow due to viscous dissipation and variable temperature-dependent conductivity along with magnetohydrodynamic over a fat plate is studied by Saxena [32].Masood et al. [33] worked on the MHD fow of a nanofuid in the biomedical feld to describe the viscous dissipation impact on velocity and temperature properties.Pseudoplastic fuids are a type of non-Newtonian fuid.Tese are shear-thinning fuids because viscosity decreases in these fuids.Viscosity decreases when shear forces are increased.Because of their many applications in industry, these fuids have much importance.Guedda and Kersner [34] studied the exact and analytical results of non-Newtonian pseudoplastic (Cu, Al 2 O 3 , and TiO 2 ) fuids.Afterwards, Ma et al. [35] examined the turbulent and laminar pseudoplastic fuid fow containing CMC-water through a square duct by using microphones and ultrasonic Doppler velocity.Te fuctuating pressure and duct velocity for diferent fow rates were measured.Das et al. [36] worked on entropy analysis with heat convection by using a porous channel of pseudoplastic MHD nanofuid fow.Ayoubi Ayoubloo at el. [37] used a porous layer on a cylindrical cavity (vertical, partially flled) to investigate the heat transfer and natural convection fow behavior of the pseudoplastic fuid.
Te main objective of the present study is to discuss the numerical investigation of fnite thin-flm fow for hybrid nanofuid with kerosene oil as base fuid over a stretching surface.In it, we have considered graphene oxide and silver as the nanoparticles.Kerosene oil (K) with a graphene oxide (GO) basis may be able to aid in parabolic trough surface accumulator (PTSC) performance monitoring.Te heat transition rate of GO-K is 15.03% greater than that of regular fuid devoid of GO.Teoretical simulations supported by documentation may be more benefcial for enhancing solar thermal energy plans.Applications for a PTSC include heating homes, distillation, cooling systems, manufacturing heat, power plants, and irrigation water pumps.A modifed Fourier's law is adopted to discuss the phenomenon of pseudoplastic fuid.Instead of using a simple fuid or nanofuid specifcally in this article, we have considered hybrid nanofuids and the infuence of internal viscous dissipation is also taken into account.As far as we are aware, no such study has been carried out.Te bvp4c MATLAB function is employed to solve the governing problem numerically.Te efects on temperature and velocity feld for the viscous dissipation, power-law index, thickness parameter, unsteadiness parameter, and Prandtl number are studied and graphically shown.

Mathematical Model
Presume a parallel stretching surface with a small gap through which a pseudoplastic hybrid nanofuid is fowing in a thin fnite layer under shear-thinning conditions.Figure 1 depicts a schematic physical model and coordinate system.Te fuid is made to move as a rapid response to the sheet that is stretching along the x-axis.Te sheet at y � 0 is riveted with the x-axis which has the temperature distribution T w (x, t) and moves with velocity u w (x, t).A thin flm reclines on a stretching surface with thickness h(x, t).Hybrid nanofuid is considered which is incompressible, and thermal equilibrium holds for both base fuid and hybrid nanoparticles.It is also presumed that the viscous dissipation phenomenon is also contained in the stretching sheet.Te sheet is stretched, giving the liquid flm a fat surface and preventing the production of waves.Te heat fow and viscous shear stress are eliminated by this adiabatic-free surface.Furthermore, the fow is considered laminar, incompressible, and irrotational.Termophysical properties of the base fuid and nanoparticles (graphene oxide and silver) are given in Table 1.According to all these premises, governing equations for momentum and energy can be written as follows [38,39]: (3)

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Te coefcients μ hnf /ρ hnf |zu/zy| n− 1 from equation ( 2), and α hnf |zu/zy| n− 1 and μ hnf /(ρC p ) hnf |zu/zy| n− 1 from equation ( 3) represent the property of the pseudoplastic fuid.Following are the boundary conditions [39]: where u and v represents the components of velocity along the x and y axes, respectively, t shows time, τ xy � μ hnf |zu/zy| n− 1 zu/zy is the shear stress, μ � μ hnf |zu/zy| n− 1 is the nanofuid efective viscosity, μ hnf is the coefcient of modifed consistency viscosity, ρ hnf shows the nanofuid efective density, and n shows the power-law index.In the case of Newtonian fuid n � 1, for pseudoplastic fuid, 0 < n < 1, while n > 1 describes a dilatant fuid.T shows temperature, C p and ρC p are the specifc heat and efective heat capacities, respectively, and h(x, t) is the nanofuid flm thickness.For the fuid motion, a kinematics constraint is imposed by v � uzh/zx + zh/zx.Te power-law viscosity efects are considered on the temperature feld by assuming that temperature and velocity felds are similar and thermal conductivity is dependent on velocity, for example, k � α hnf (ρC p ) hnf |zu/zy| n− 1 and α hnf |zu/zy| n− 1 shows the effective thermal difusivity.Te efect of velocity dissipation is composed by the term μ hnf /(ρC p ) hnf |zu/zy| n− 1 (zu/zy) 2 .
When the wall surface is stretched at y � 0, this causes a fow such that the sheet moves with velocity u w along x-direction as where both a and b are constants having a positive value with (sec) − 1 dimensions.Te temperature T w of wall surface difer with slot distance x and time t as where c f � μ f /ρ f shows modifed kinematic viscosity and μ f shows modifed viscosity for base fuid, T o is the origin temperature, ρ f is the base fuid density, and T ref shows reference temperature that may be considered as the constant.Te values of u w and T w from equations ( 5) and ( 6), respectively, help us to develop a transformation that converts governing PDEs into ODEs.Moreover, μ hnf is considered as the hybrid nanofuid fuid viscosity and dilute suspension is contained by μ f and is defned by Brinkman as where ϕ stands for the solid volume fraction of nanoliquid.
To properly defne the stream function, such as u � zψ/zy and v � −zψ/zx, some important transformation variables are used as follows: Te nondimensional parameters which are used for the conversion purpose are as follows: where β is the nanoliquid flm thickness, Re x shows the local Reynolds number, Pr shows Prandtl number, S represents the unsteadiness parameter, and Ec shows the Eckert number.Te rest of the physical properties of nanofuid are defned as where ρ s1 , ρ s2 and (ρC p ) s1 , (ρC p ) s2 represent the solid density and solid heat capacity, respectively, (ρC p ) f stands for the base fuid heat capacity, k nf and k f show modifed thermal conductivity and thermal conductivity of base fuid, while k s is the thermal conductivity of solid particles.All abovementioned nondimensional parameters and dimensionless transformations are used for individualized growing of PDEs.Te value of velocity components u and v are given as After simplifcation, we convert them into the following ODEs: Te boundary conditions are Journal of Mathematics

Numerical Solution
Te higher-order ordinary diferential equations are transformed into the system of 1 st order ODEs by using f, f ′ , f ″ , θ, and θ ′ as y 1 , y 2 , y 3 , y 4 , and y 5 with six reformed boundary conditions.Te parameters β and S are the two key features that represent the thickness and unsteadiness, respectively, and also have a dependence relation.Due to this relation, six boundary conditions are reduced into fve.Tis reformed system is numerically solved by using the MAT-LAB built-in function bvp4c.Let the convective notations which are used with their boundary conditions are Table 2 displays a role of comparison of the numerical results which are considered for the limiting case.Te obtained results are quite evident that these are aligned with the previous work in true spirit.

Results and Discussion
MATLAB is used to obtain the solution using bvp4c.Te toleration is maintained at 10 − 5 .Te behavior of velocity and temperature profles has been inspected by varying the numeric values of distinguished parameters.Te kerosene oil as base fuid is taken with GO + Ag as a hybrid nanoparticle.Te overall working of the model is shown in Figure 2, in which the whole model is divided into three main steps.Te rheological values of −θ ′ (0) are given for diferent parameters mentioned in Table 3.

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Te value of f ′ (η) (velocity) is turned down due to an expansion in the power-law index n.So, upon higher values of n, the velocity of the hybrid nanofuid becomes lower.As we know that for n � 1, we have a Newtonian fuid, shear thickening fuid for n > 1, and for n < 1, we have shearthinning fuid.While enhancing the power-law index, the shear thickening impact is produced that happens due to higher viscosity and ultimately reduces the working fuid's velocity.For the larger n, speed goes down, so ultimately, the kinetic energy of the molecules gets slower and resultantly shows a decrement in the temperature.So, they also have an inverse relation.
Figures 5 and 6 represent the relationship between ϕ (solid volume fraction) and f ′ (η) (velocity) and θ(η) (temperature), respectively.For hybrid nanofuids, solid volume fraction contains two diferent types of nanoparticles; frst, ϕ 1 is treated as a variable, i.e., ϕ 1 � (0%, 1.5%, 3%, 4.5%, ) while ϕ 2 is considered fx.Here, it is important to note that ϕ 1 denotes the volume fraction of graphene oxide and ϕ 2 of silver.As the numeric values of ϕ 1 lead to a rise, the velocity profle moves up.Te diference in velocity between the wall and free surface is striking due to the exploits of ϕ 1 .A similar impact on the velocity profle is eminent for ϕ 2 of the second type of nanoparticles.Again, the velocity rises for the higher values of ϕ 2 .8 Journal of Mathematics Te relation between ϕ (solid volume fraction) and θ(η) (temperature) is perceived in Figures 7 and 8 by presuming two cases regarding nanoparticles ϕ 1 and ϕ 2 .In Figure 7, ϕ 1 grips as a variable, i.e., ϕ 1 � (0.0%, 1.5%, 3%, 4.5%), while ϕ 2 that shows the volume fraction of silver and all the other parameters are taken constantly.It is noted that as the ϕ 1 elevated, the temperature escalates.Due to the presence of hybrid fuid, the thermal conductivity enhances and the increment in thermal conductivity increases the temperature.Te consequences are almost similar to the previous case when ϕ 2 is considered a variable, while ϕ 1 and all other parameters are considered constant.So, there is a direct relation between ϕ and θ(η), i.e., as the flm solid volume portion rises so does its temperature.
Figure 9 describes the efects on θ(η) (temperature) due to the presence of viscous dissipation (Eckert number) Ec with hybrid nanofuid.Te results demonstrate that the heat transfer process is dependent on the Eckert number.It is observed that when the value of the Eckert number grows      (Ec � 0, 2, 4, 6), the temperature drops down and the temperature diference between the wall and free surface increases.So, the temperature profle of the thin flm swiftly decreases from the wall surface to the free surface.Figure 10 explores the behavior of Pr on the temperature profle θ(η).Te graph shows that with the increasing values of Pr � 0.1, 0.5, 1, 5, the temperature decreases.A higher Pr implying lower thermal difusivity is the main reason for the lowering in temperature distribution.Figure 11 portrays the efect of S on f ′ (η) (velocity) that shows an increasing pattern.A higher unsteadiness parameter will increase the height of the thin flm thickness, and it will allow the fuid to move more quickly.

Concluding Remarks
Heat transfer of hybrid nanofuid comprising graphene oxide plus silver/kerosene oil in a fnite thin flm on a stretching surface with variable thermal conductivity is inspected.Te conducting PDEs are reduced into a couple of ODEs (nonlinear) through similarity transformation.Te MATLAB bvp4c technique is used to gain its numerical solutions.Some important results obtained are as follows: (i) Te Nusselt number, temperature, and velocity felds are highly afected by the power-law index.
Temperature and velocity felds are inversely proportional to the power-law index.(ii) Solid volume fractions of hybrid nanofuid impose great efects on the local Nusselt number, velocity, and temperature profles.(iii) Te velocity of hybrid nanofuid decreases while the temperature increases with the increase in the value of solid volume fraction.
Tis article can further be enhanced by adding more prominent and valuable impacts such as thermal radiations and Joule heating.A magnetic feld across the thin flm can also be employed.Entropy analysis can also be done.

Symbols u, v:
Velocity component along x and y directions n: Te power-law index T:

Table 1 :
Termophysical properties of base fuid and nanoparticles.

Table 2 :
Comparison of current numerical outcomes with the earlier work in the limiting case.

Table 3 :
Numerical results of the Nusselt number for diferent parameters.