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

Mehmood, Yasir; Alsinai, Ammar; Shafqat, Ramsha; Bilal, Muhammad; Khan, Allah Yar; Qahtan, Read S. A.

doi:https://doi.org/10.1155/2023/3763147

Journal of Mathematics

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Abstract Introduction Results and Discussion Data Availability Conflicts of Interest Authors’ Contributions References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 3763147 | https://doi.org/10.1155/2023/3763147

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

Yasir Mehmood,¹Ammar Alsinai,²Ramsha Shafqat,¹Muhammad Bilal,³Allah Yar Khan,¹and Read S. A. Qahtan⁴

Academic Editor: Qingkai Zhao

Received08 Feb 2023

Revised29 Mar 2023

Accepted02 Nov 2023

Published16 Nov 2023

Abstract

This study examines the flow and heat transfer of a finite thin layer of a hybrid nanofluid across an unstable stretching surface with varying thermal conductivity and viscous dissipation effects. A hybrid nanofluid model is considered to comprise two different types of nanoparticles, and , with kerosene oil used as a base fluid. To study the phenomenon of thermal conduction, a modified 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 differential equations is obtained by considering the similarity transformations over the obtained rheological system of partial differential equations which is then tackled by a well-known numerical approach, i.e., the bvp4c MATLAB technique. The rheological impacts of the power-law index, solid volume fraction, film thickness, Eckert number, and modified Prandtl number on temperature and velocity fields are graphically discussed and illustrated. In the presence of nanoparticles, the temperature of the working fluid is enhanced and the power-law index has an inverse relation with the velocity of the hybrid nanofluid.

1. Introduction

Thin film is a layer of material ranging from a fraction of nanometer to several micrometer in thickness. The thin film flow analysis is very valuable due to its inclusive applications in the engineering and technology field. The thin film flow problems have a broader area of interest and catch many fields, starting from flow inside the human lungs to industrial coating problems. It is much important to explore the thin film flow uses in structural mechanics, theology, and fluid mechanics. The liquid film has well-known uses such as the expulsion of polymer and metal, constant forming, foodstuff processing, drawing elastic sheets and devices, fluidization, and exchanges. For inspection of these uses and applications, the researchers are motivated to further investigate and make further development in the field of thin-film flow problems. Different approaches are adopted with modified geometries. In industrial applications, the stretching surface is a major topic. Gul et al. [1] worked on the flow of third-grade magnetohydrodynamic fluid with an impact on viscosity (temperature dependent) on a vertical belt. Later on, Bachicha [2] explored a method for producing cathodes from thin-filmed coatings which are produced by placing catholyte slime on a moving substrate. Aqil et al. [3] found that when the pure molybdenum thin film (by the DC magnetron sputtering technique) is deposited on the blank substrate, the deposition condition only changed for the deposition time, not for all other samples. Afterwards, Shah et al. [4] examined the effect of thermal radiations on a permeable medium and heat generation through a stretching surface during the Williamson liquid-film fluid flow. They considered the fluid flow in two dimensions of liquid films. Ullah et al. [5] worked on the flow (with skid conditions) of a generalized Maxwell fluid that is flowing on a nonisothermal cylindrical surface.

Heat generation plays an important role in the flow of a thin film. When any type of energy inside the body is transferred into heat energy, then the temperature of the body increases which causes heat generation. The 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 affected by temperature diffusion 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 flat surface. Afterwards, the impact of heat generation, as well as radiation due to the linearly stretching sheet through an incompressible micropolar fluid, is highlighted by Reddy [7]. Mehmood et al. [8] worked on the numerical investigation of MWCNT and SWCNT fluid flow along with the activation energy effects over quartic autocatalytic endothermic and exothermic chemical reactions. Pavithra and Gireesha [9] numerically examined a boundary flow and heat transfer for the dusty fluid 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 Yeşiller et al. [10]. Ganga et al. [11] illustrated the heat generation on the boundary layer magnetohydrodynamic flow of nanofluids. The heat generation effects over the stretching surface in the flow of non-Newtonian nanofluid are explored by Awais et al. [12]. Aziz et al. [13] studied the impact of heat generation on magnetonanofluid. Later on, the influence of heat generation over MHD Oldroyd B fluid 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 flow. Nanofluid is a fluid having nanometer-size particles, called nanoparticles. The nanoparticles are naturally made of metals, fibers, tubes, wires, rods, sheets, droplets, oxides, or carbon nanotubes. Nanofluids are mostly used to improve the thermophysical properties such as viscosity, thermal conductivity, thermal diffusivity, and heat transfer (depending on thermal conductivity) coefficients to differentiate base fluids (water or oil). They are very valuable for the cooling of a microsystem. It has many applications in many fields. 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 nanofluids. A review (theoretical and experimental) of nanofluid for the magnification of thermal conductivity is studied by Kleinstreuer and Feng [17]. Fluids in which two or more nanometer-size particles of different materials are mixed into the base fluid to get the desired physical properties are called hybrid nanofluids. These are being used to further enhance the heat transfer rate, pressure drop properties, better thermal network, and collaborative effect of nanomaterials. In the case of nanofluids, we have to face some major issues. One of them considered the stability of nanofluids, and it has become a big challenge to attain the required characteristics. The properties of hybrid nanofluids are investigated by researchers to find out their applications, advantages, and disadvantages. Madhesh and Kalaiselvan [18] experimentally used the tubular heat exchanger to study the heat transfer through a hybrid nanofluid. In their work, they explored the effects of the thermal characteristics and also analyzed the hybrid nanofluids as coolants. Later on, Hayat and Nadeem [19] explained the use of as a hybrid nanofluid 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 nanofluid. Similarly, Minea and Moldoveanu [21] worked on the development and benefits of hybrid nanofluids. The steady, laminar, incompressible two-dimensional flow over the static channel using hybrid nanofluid is examined by Dinarvand et al. [22]. A mass-based approach has been presented for the hybrid nanofluid model. Bumataria et al. [23] analyzed the rate of heat transfer through a pipe and discussed it with applications for hybrid nanofluids. Some recent work related to nanofluid and hybrid nanofluid can be cited in [24–26].

When any shear force is applied to the fluid, 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 field 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 identified viscous dissipation by using nondimensional Eckert numbers. In gravitational fields, such as large planets, the remarkable effects of viscous dissipation may also exist. Massoudi and Christie [27] worked on the thermodynamically frictionless, incompressible third-grade fluid flow to investigate the reaction of viscosity (variable) and viscous dissipation. Later on, Mehmood et al. [28] numerically examined the boundary layer nanofluid flow due to mass flux conditions and viscous dissipation in a bent stretching surface. Pandey and Kumar [29] explored the MHD nanofluid flow, with injection and viscous dissipation over the porous medium and slip flow. Afterwards, Ahmed et al. [30] studied a nanofluid flow with the additional impact of viscous dissipation. The KKL model is implemented for the thermal conductivity of nanofluids and the solid nanoparticles of copper oxide. Mishra and Kumar [31] explored the effects of heat absorption on nanofluid (Ag) in the existence of suction/injection through a vertical Riga plate. Using the homotopy perturbation technique, the fluid flow due to viscous dissipation and variable temperature-dependent conductivity along with magnetohydrodynamic over a flat plate is studied by Saxena [32]. Masood et al. [33] worked on the MHD flow of a nanofluid in the biomedical field to describe the viscous dissipation impact on velocity and temperature properties.

Pseudoplastic fluids are a type of non-Newtonian fluid. These are shear-thinning fluids because viscosity decreases in these fluids. Viscosity decreases when shear forces are increased. Because of their many applications in industry, these fluids have much importance. Guedda and Kersner [34] studied the exact and analytical results of non-Newtonian pseudoplastic fluids. Afterwards, Ma et al. [35] examined the turbulent and laminar pseudoplastic fluid flow containing CMC-water through a square duct by using microphones and ultrasonic Doppler velocity. The fluctuating pressure and duct velocity for different flow rates were measured. Das et al. [36] worked on entropy analysis with heat convection by using a porous channel of pseudoplastic MHD nanofluid flow. Ayoubi Ayoubloo at el. [37] used a porous layer on a cylindrical cavity (vertical, partially filled) to investigate the heat transfer and natural convection flow behavior of the pseudoplastic fluid.

The main objective of the present study is to discuss the numerical investigation of finite thin-film flow for hybrid nanofluid with kerosene oil as base fluid 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. The heat transition rate of GO-K is 15.03% greater than that of regular fluid devoid of GO. Theoretical simulations supported by documentation may be more beneficial 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 modified Fourier’s law is adopted to discuss the phenomenon of pseudoplastic fluid. Instead of using a simple fluid or nanofluid specifically in this article, we have considered hybrid nanofluids and the influence of internal viscous dissipation is also taken into account. As far as we are aware, no such study has been carried out. The bvp4c MATLAB function is employed to solve the governing problem numerically. The effects on temperature and velocity field for the viscous dissipation, power-law index, thickness parameter, unsteadiness parameter, and Prandtl number are studied and graphically shown.

2. Mathematical Model

Presume a parallel stretching surface with a small gap through which a pseudoplastic hybrid nanofluid is flowing in a thin finite layer under shear-thinning conditions. Figure 1 depicts a schematic physical model and coordinate system. The fluid is made to move as a rapid response to the sheet that is stretching along the -axis. The sheet at is riveted with the -axis which has the temperature distribution and moves with velocity . A thin film reclines on a stretching surface with thickness . Hybrid nanofluid is considered which is incompressible, and thermal equilibrium holds for both base fluid and hybrid nanoparticles. It is also presumed that the viscous dissipation phenomenon is also contained in the stretching sheet. The sheet is stretched, giving the liquid film a flat surface and preventing the production of waves. The heat flow and viscous shear stress are eliminated by this adiabatic-free surface. Furthermore, the flow is considered laminar, incompressible, and irrotational. Thermophysical properties of the base fluid 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]:

The coefficients from equation (2), and and from equation (3) represent the property of the pseudoplastic fluid. Following are the boundary conditions [39]:where and represents the components of velocity along the and axes, respectively, shows time, is the shear stress, is the nanofluid effective viscosity, is the coefficient of modified consistency viscosity, shows the nanofluid effective density, and shows the power-law index. In the case of Newtonian fluid , for pseudoplastic fluid, , while describes a dilatant fluid. shows temperature, and are the specific heat and effective heat capacities, respectively, and is the nanofluid film thickness. For the fluid motion, a kinematics constraint is imposed by . The power-law viscosity effects are considered on the temperature field by assuming that temperature and velocity fields are similar and thermal conductivity is dependent on velocity, for example, and shows the effective thermal diffusivity. The effect of velocity dissipation is composed by the term . When the wall surface is stretched at , this causes a flow such that the sheet moves with velocity along -direction aswhere both and are constants having a positive value with dimensions. The temperature of wall surface differ with slot distance and time aswhere shows modified kinematic viscosity and shows modified viscosity for base fluid, is the origin temperature, is the base fluid density, and shows reference temperature that may be considered as the constant. The values of and from equations (5) and (6), respectively, help us to develop a transformation that converts governing PDEs into ODEs. Moreover, is considered as the hybrid nanofluid fluid viscosity and dilute suspension is contained by and is defined by Brinkman aswhere stands for the solid volume fraction of nanoliquid. To properly define the stream function, such as and , some important transformation variables are used as follows:

The nondimensional parameters which are used for the conversion purpose are as follows:where is the nanoliquid film thickness, shows the local Reynolds number, shows Prandtl number, represents the unsteadiness parameter, and shows the Eckert number. The rest of the physical properties of nanofluid are defined aswhere and represent the solid density and solid heat capacity, respectively, stands for the base fluid heat capacity, and show modified thermal conductivity and thermal conductivity of base fluid, while is the thermal conductivity of solid particles. All abovementioned nondimensional parameters and dimensionless transformations are used for individualized growing of PDEs. The value of velocity components and are given as

After simplification, we convert them into the following ODEs:

The boundary conditions are

3. Numerical Solution

The higher-order ordinary differential equations are transformed into the system of order ODEs by using , and as , and with six reformed boundary conditions. The parameters 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 five. This reformed system is numerically solved by using the MATLAB built-in function . 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. The obtained results are quite evident that these are aligned with the previous work in true spirit.

4. Results and Discussion

MATLAB is used to obtain the solution using . The toleration is maintained at . The behavior of velocity and temperature profiles has been inspected by varying the numeric values of distinguished parameters. The kerosene oil as base fluid is taken with as a hybrid nanoparticle. The overall working of the model is shown in Figure 2, in which the whole model is divided into three main steps. The rheological values of are given for different parameters mentioned in Table 3.

Figures 3 and 4 illustrate the impact of , which ultimately affects (velocity) and (temperature) of hybrid nanofluid, while all the other physical parameters are fixed, i.e., , and . The value of (velocity) is turned down due to an expansion in the power-law index . So, upon higher values of , the velocity of the hybrid nanofluid becomes lower. As we know that for , we have a Newtonian fluid, shear thickening fluid for , and for , we have shear-thinning fluid. While enhancing the power-law index, the shear thickening impact is produced that happens due to higher viscosity and ultimately reduces the working fluid’s velocity. For the larger , 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 (solid volume fraction) and (velocity) and (temperature), respectively. For hybrid nanofluids, solid volume fraction contains two different types of nanoparticles; first, is treated as a variable, i.e., while is considered fix. Here, it is important to note that denotes the volume fraction of graphene oxide and of silver. As the numeric values of lead to a rise, the velocity profile moves up. The difference in velocity between the wall and free surface is striking due to the exploits of . A similar impact on the velocity profile is eminent for of the second type of nanoparticles. Again, the velocity rises for the higher values of .

The relation between (solid volume fraction) and (temperature) is perceived in Figures 7 and 8 by presuming two cases regarding nanoparticles and . In Figure 7, grips as a variable, i.e., , while that shows the volume fraction of silver and all the other parameters are taken constantly. It is noted that as the elevated, the temperature escalates. Due to the presence of hybrid fluid, the thermal conductivity enhances and the increment in thermal conductivity increases the temperature. The consequences are almost similar to the previous case when is considered a variable, while and all other parameters are considered constant. So, there is a direct relation between and , i.e., as the film solid volume portion rises so does its temperature.

Figure 9 describes the effects on (temperature) due to the presence of viscous dissipation (Eckert number) with hybrid nanofluid. The 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 , the temperature drops down and the temperature difference between the wall and free surface increases. So, the temperature profile of the thin film swiftly decreases from the wall surface to the free surface. Figure 10 explores the behavior of on the temperature profile . The graph shows that with the increasing values of , the temperature decreases. A higher implying lower thermal diffusivity is the main reason for the lowering in temperature distribution. Figure 11 portrays the effect of on (velocity) that shows an increasing pattern. A higher unsteadiness parameter will increase the height of the thin film thickness, and it will allow the fluid to move more quickly.

5. Concluding Remarks

Heat transfer of hybrid nanofluid comprising graphene oxide plus silver/kerosene oil in a finite thin film on a stretching surface with variable thermal conductivity is inspected. The conducting PDEs are reduced into a couple of ODEs (nonlinear) through similarity transformation. The MATLAB bvp4c technique is used to gain its numerical solutions. Some important results obtained are as follows:(i)The Nusselt number, temperature, and velocity fields are highly affected by the power-law index. Temperature and velocity fields are inversely proportional to the power-law index.(ii)Solid volume fractions of hybrid nanofluid impose great effects on the local Nusselt number, velocity, and temperature profiles.(iii)The velocity of hybrid nanofluid decreases while the temperature increases with the increase in the value of solid volume fraction.

This article can further be enhanced by adding more prominent and valuable impacts such as thermal radiations and Joule heating. A magnetic field across the thin film can also be employed. Entropy analysis can also be done.

Nomenclature

Symbols

:	Velocity component along x and y directions
:	The power-law index
:	Fluid temperature
:	Time
:	Specific heat
:	The effective thermal conductivity of nanofluid
:	Sheet velocity
:	Temperature at the origin
:	Reference temperature
:	Surface temperature of the nanofluid film
:	Unsteadiness parameter
:	Film thickness
:	Modified Prandtl number
:	Local Reynolds number
:	Eckert number
:	For the nanofluid
:	For the hybrid nanofluid
:	For the base fluid
:	For the solid (nanoparticles)
:	Solid volume fraction
:	Density
:	Effective density of nanofluid
:	Heat capacity of nanofluid
:	Viscosity coefficient of nanofluid
:	Viscosity coefficient of base fluid
:	Diffusivity of the nanofluid
:	Thermal diffusivity of nanofluid
:	Thermal conductivity of nanofluid
:	Shear stress
:	Stream function
:	Similarity variable
:	Dimensionless temperature
:	Dimensionless film thickness.

Data Availability

No new data were created in this study.

Conflicts of Interest

The authors declare that they have no conflicts of interests.

Authors’ Contributions

Ammar Alsinai conceptualized the study; Yasir Mehmood designed the methodology; Read S. A. Qahtan was responsible for software; Muhammad Bilal was responsible for formal analysis; Allah YarKhan, Read S. A. Qahtan, and Ammar Alsiani were responsible for the investigation of the study; Ammar Alsiani was responsible for the resources; Allah YarKhan, Ammar Alsinai, and Ramsha Shafqat wrote the original draft; Read S. A. Qahtan was responsible for reviewing and editing; Allah YarKhan and Ammar Alsiani supervised the study; Ammar Alsinai was responsible for project administration; Read S. A. Qahtan was responsible for funding acquisition.

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Copyright © 2023 Yasir Mehmood et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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