Computational Assessment of ZrO₂-Al₂O₃/EG and ZrO₂-Al₂O₃-Cu/EG Nanomaterial on Blasius-Rayleigh-Stokes Flow Influenced by an Aligned Magnetic Field

Abstract

In this work, the flow of a modified nanofluid is analysed as it passes over a moving surface to investigate the influence of nonlinear radiative heat transfer and the effects of magnetic fields that are aligned. In addition, ethylene glycol is used as the solvent while zirconium oxide and alumina are combined to generate a hybrid nanomaterial. Ternary nanomaterials consist of zirconium oxide, alumina, and copper dissolved in the ethylene glycol. For this mathematical model, Navier–Stokes equations were used to represent the assumed flow. The Navier–Stokes equations were approximated using the boundary layer method under the flow assumptions, yielding the PDE’s. Similarity transformations are used to translate this system into ODE’s. The bvp4c method is used to explain a dimensionless system. The impacts of the relevant physical parameters are elucidated quantitatively and visually. A greater temperature ratio parameter is observed to increase the temperature profile. In addition to this, when the magnetic field parameter is increased, the momentum layer becomes thicker.

1. Introduction

Numerous geophysical, astrophysical, and industrial applications include the magnetohydrodynamic (MHD) fluxes of an electrically conducting liquid. Engineers use MHD principles to design pumps, thermal safety, space vehicle propulsion, control, heat exchangers, reentry, and new power-generation systems. Using a magnetic field to separate molten metal from non-metallic impurities is another essential feature of MHD. In all of these MHD applications, it is vital to examine the MHD impacts. Over a vertical plate, Seth et al. [1,2] described a free convective unstable MHD flow that was rapidly moving and had a temperature gradient. Nevertheless, flow through porous medium is advantageous for reducing heat from nuclear fuel, disassembling debris, burying radioactive wastes underground, storing food items, manufacturing paper, and prospecting for oil, among other applications. Recently, Harshad et al. [3] demonstrated that variable convection MHD micropolar liquid pour is subject to Brownian thermophoresis movement and nonlinear thermal emission special effects caused by a nonlinear prolonged page in a porous medium under conditions of heat dissipation due to viscous dissipation, with convection boundary line conditions. Harshad [4] demonstrated that micropolar fluid transports heat and mass between two vertical walls, which has an effect on MHD flow due to heat radiation. Using an inclined asymmetric channel, Bhattacharyya et al. [5] capture peristaltic transport of chemically reactive pair stress liquids. Ebiwareme et al. [6] discussed the unsteady flow and heat transfer over a sheet under the influence of ohmic heating are compared using the Adomian decomposition approach and the differential transformation method. They found that the velocity and temperature distributions obtained by these methods when compared with benchmarked numerical solutions from the Keller box scheme showed excellent agreement. MHD flow across a stretched sheet by considering joule heating were studied by Rafique et al. [7], who looked at the effects of different nanoparticle shapes and quadratic velocities.

Nano-fluids, also known as nano-particle suspensions in a conventional fluid, are used in the process of simultaneously increasing heat transfer rates in order to achieve this goal. In the past few decades [8], nanoparticles have received a great deal of interest from the scientific community due to the wide range of possible applications in fields as diverse as mechanical technologies, biomedicine, electronics, biomaterials, and food processing. The concept of nano-fluids was created in [9] as a means of overcoming this obstacle. Increased heating efficiency and faster heat transmission may be achieved using a nanofluid consisting of nanoparticles made up of (1–100) of these components (atoms). Nanoparticles are very minute particles found in the base liquid that have the potential to amplify the phenomena [10]. Nano-fluids’ special qualities for altering the heat properties of different liquids allow them to overcome heat transfer disappointment. The nanoparticles in nano-fluids are so small that they can be transferred across a heat exchanger without affecting the fluid’s temperature. Some nano-fluids may be useful because of the special features they exhibit [11]. Nano-fluids have improved heat transfer properties over conventional fluids. The heat transfer coefficient of the nano-fluid with particles at 45 nm is larger than that of the nano-fluid with 150 nm particles. MHD flow including microorganisms with radiation and activation energy was studied by Majeed et al. [12], who performed a thermal study of the phenomenon. The rheological behaviour of a magnetite nanofluid was explored by Talebizadehsardari et al. [13], who conducted an experiment to examine this phenomenon. In addition, they discovered that the reduction in viscosity with increasing shear rates portrayed the non-Newtonian shear thinning behaviour. There has been a lot of study on nano-fluids, and many different types of findings have been documented [14,15,16].

In general, nanomaterials and hybrid nanomaterials are assumed to maintain a stable set of physical properties. More and more data suggest that nanomaterials and hybrid nanomaterials undergo changes in their physical features as a function of temperature. Since the viscosity of a fluid is assumed to be constant, heat transmitted in fluids owing to internal friction should have a constant temperature influence. Convective flow and heat transmission through a porous sheet was investigated by Layek and Mukhopadhyay [17], who looked at how the impacts of fluid, heat, and changing viscosity affected the processes. Vajravelu et al. [18] discussed the impact of viscosity variation on the transport of heat and the movement of different nanoparticles. Moreover, they discovered that the characteristics of heat transfer with different nanoparticles show numerous fascinating behaviors that demand additional research on the consequences of the “nano-solid-particles.” Using a curved stretching surface and varying viscosities, Nadeem et al. [19] studied the impact of carbon nanotubes on magneto nanofluid flow. They discovered that the nanofluid’s temperature distribution increased and flow resistance increased as the viscosity parameter increased. Additionally, it raises skin friction close to the fluid’s border. Numerical modelling of radiative slip flow with varying fluid parameters was studied by Khan et al. [20]. The process of non-depersonalization is modelled mathematically, and the von Kármán method is used to generate entropy. Khan et al. [21] conducted research on the constant squeezing hybrid nanofluid flow, which consisted of carbon nanotubes, ferrous oxide, and water, and looked at its influence of suction and injection. In spite of the fact that hybrid nanomaterial fluid over an exponentially stretched sheet would have several advantages over conventional fluids, a review of the literature on the features of variable fluids and hybrid nanofluids [22,23,24,25,26] indicates that this attempt has not been performed. Recently, Jadhav, et al. [27] discussed the relativistic theory to Compton effect for a spectroscopic detector.

In the course of the last four years, a brand-new research field has emerged, which focuses on the production of three particle nanofluids. These nanofluids mix a variety of nanoparticles in order to produce an ideal nanofluid. These are often referred to as ternary nanofluids in common parlance. Because of the synergistic effects that they have, ternary nanofluids, which are very similar to hybrid nanofluids, may have substantially greater thermophysical characteristics than molecular and hybrid nanofluids. In their research, Gupta and Rana [28] conducted an investigation on the flow characteristics of a tri-hybrid nanofluid at a 3D magnetic stagnation point, considering various slip effects Sahoo [29] investigated the thermo-hydraulic characteristics of a try hybrid that was composed of nanoparticles with a variety of morphologies. The researchers Shahzad et al. [30] investigated the Hall and electromagnetic forces that were operating on ternary nanofluid as it flowed over a spinning disc. Alharbi and colleagues [31] investigated the flow of ternary-hybrid nanofluid towards an extensible cylinder while taking into account the effects of induction.

After some time had passed, Larki et al. [32] studied the impact of a series of detachable vortex generators on the dynamics of phase change materials inside an energy storage system. The study conducted by Sohail et al. [33] focused on examining the phenomenon of triple-mass diffusion species and energy transfer inside Carreau-Yasuda material. The researchers placed specific emphasis on understanding the influence of activation energy and heat source on these two variables. In their study, Khan et al. [34] conducted a computational analysis to examine the comparative heat transport in nanofluids consisting of Cu-H₂O and Cu-Al₂O₃/H₂O. In their study, Puneeth et al. [35] examined the influence of gyrotactic microorganism movement on the heat and mass transport properties of Casson nanofluid. Guedri et al. [36] investigated the thermal efficiency of hybrid (Al₂O₃-CuO/H₂O) and tri-hybrid (Al₂O₃-CuOCu/H₂O) nanofluids in converging/diverging channels, taking into account the presence of viscous dissipation function. The field of study that deals with the development and implementation of algorithms for solving mathematical problems using numerical methods. The transfer characteristics of Casson nanofluid refer to the properties and behaviour of this particular kind of nanofluid in terms of heat transfer, mass transfer, or any other relevant transfer phenomena. Ullah [37] conducted a study on the flow of a hybrid nanofluid, combining radiative and Darcy-Forchheimer effects, across an inclined stretched surface. The study considered the influence of nonlinear convection and homogeneous heterogeneous reactions.

Thermal radiation serves as a mechanism for the transfer of thermal energy among molecules inside a liquid medium. The field of engineering encompasses a wide range of industrial processes that need high temperatures. These processes include the production of paper plates, the cooling of metallic components, the fabrication of electronic chips, and the operation of petroleum pumps. These examples underscore the significant impact of radiation on magnetohydrodynamic movement. A significant phenomenon in the field of heat transfer is the emission of thermal energy as a result of radiation, which is seen in industrial processes that need elevated temperatures. The field of engineering and other industrial processes that involve high temperatures, such as the production of paper plates, cooling of metal components, fabrication of electronic chips, and operation of petroleum pumps, demonstrate the significant influence of radiation. The comprehension and incorporation of this phenomena play a crucial role in the design of thermal systems. The investigation and assessment of the radiative heat transmission of conventional and hybrid nanofluids across a stretched sheet have been the focus of several scholarly inquiries [38,39]. Recently, a study regarding micropolar pulsatile blood flow through an artery was carried out by Shit and Roy [40]. In this paper, impact of magnetic field and body acceleration was taken into account together with heat-transfer characteristics. It was found that wall shear stress reduced with the effect of Hartmann number while heat transfer increased. The problem of Darcy Forchheimer flow with radiative and convective thermal aspects on micropolar nanofluids in rotating frame of reference was studied by Alzahrani et al. [41]. Raju et al. [42] numerically explored the time dependent nonlinear convective Casson fluid flow over a rotating cone saturated in a porous medium.

In this particular study, we want to investigate the effects of flow and heat transfer over a moving surface that has nonlinear radiation and a magnetic field that is aligned. It should be emphasized that we came up with the concept of a modified ternary hybrid nanomaterial. This is the starting point from which the examples of hybrid nanomaterial and ternary hybrid nanomaterial may be recovered as special instances. Furthermore, zirconium oxide and alumina are combined to create a hybrid nanomaterial, with ethylene glycol serving as the solvent. Ternary nanomaterials consist of zirconium oxide, alumina, and copper dissolved in ethylene glycol. With the help of appropriate self-similarity transformations, the system of flow and temperature transfer equations are transformed into an ordinary differential system, and its solution is found with the help of the bvp4c with MATLAB solver. Graphs and tables are used to illustrate the computation results.

2. Mathematical Formulation

Consider the erratic 2D flow and heat transfer of an incompressible viscous-based nanomaterial as it moves over a heated, moving, semi-infinite plate. The nanomaterial is flowing in a direction that is perpendicular to the plate’s direction of motion. The surface is pushing its way out of a moving slot in the $x$-axis direction. When measured in terms of time, the fluid is motionless at the point indicated by the notation $t=0$. A magnetic field is supplied at an acute angle $\omega$ to the flow, and the direction of the flow is predicted to be a function of the radial distance from the centre of the field, as shown by $B\left(x\right)=B_0/{\left(\mathit{cos}\left(\omega \right){\nu }_{f}t+\mathit{sin}\left(\omega \right)\left(\frac{{\nu }_{f}x}{u_w}\right)\right)}^{1/2}$ using $B_0\ne 0$ in the location along the plate indicated by $x$ as the position.

The boundary layer equations that are applicable to hybrid nanomaterials may be written as follows:

$$\frac{\partial u}{\partial x}+\frac{\partial u}{\partial x}=0,$$

(1)

$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\frac{{\partial }^{2}u}{\partial y^2}+\frac{\sigma B^2}{{\rho }_{hnf}}{\sin }^2\omega u,$$

(2)

$$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}={\alpha }_{nf}\frac{{\partial }^{2}T}{\partial y^2}+\frac{1}{{\left(\rho c_p\right)}_{hnf}}\frac{\partial q_r}{\partial y}+\frac{\mu }{{\left(\rho c_p\right)}_{hnf}}{\left(\frac{\partial u}{\partial y}\right)}^2+\frac{\sigma B^2}{{\left(\rho c_p\right)}_{hnf}}{\sin }^2\omega { u}^2,$$

(3)

The equations that regulate the boundary layer for ternary hybrid nanomaterials are presented as follows:

$$\frac{\partial u}{\partial x}+\frac{\partial u}{\partial x}=0,$$

(4)

$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\frac{{\mu }_{thnf}}{{\rho }_{thnf}}\frac{{\partial }^{2}u}{\partial y^2}+\frac{\sigma B^2}{{\rho }_{thnf}}{\sin }^2\omega u,$$

(5)

$$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}={\alpha }_{nf}\frac{{\partial }^{2}T}{\partial y^2}+\frac{1}{{\left(\rho c_p\right)}_{thnf}}\frac{\partial q_r}{\partial y}+\frac{\mu }{{\left(\rho c_p\right)}_{thnf}}{\left(\frac{\partial u}{\partial y}\right)}^2+\frac{\sigma B^2}{{\left(\rho c_p\right)}_{thnf}}{\sin }^2\omega { u}^2,$$

(6)

These are the relevant boundary conditions:

$$\begin{array}{l}u=0, v=0, T=T_w\mathrm{at}y=0,\\ u=u_w, T=T_{\infty }\mathrm{as}y\to \infty \end{array}$$

(7)

The thermophysical properties ${\mu }_{hnf}, k_{hnf}, {\left(\rho C_p\right)}_{hnf}$, and ${\rho }_{hnf}$ for the hybrid nanomaterial (zirconium oxide/alumina) are defined as follows:

$$\begin{array}{l}\frac{{\rho }_{hnf}}{{\rho }_f}=\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right){\rho }_f+{\varphi }_1{\rho }_{s1}\right]+{\varphi }_2{\rho }_{s2} ,\\ {\left(\rho C_p\right)}_{hnf}=\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right){\left(\rho C_p\right)}_f+{\varphi }_1{\left(\rho C_p\right)}_{s1}\right]+{\varphi }_2{\left(\rho C_p\right)}_{s2}.\\ \frac{{\mu }_{hnf}}{{\mu }_f}=\frac{1}{{\left(1-{\varphi }_1\right)}^{2.5}{ \left(1-{\varphi }_2\right)}^{2.5}}.\\ \frac{k_{hnf}}{k_{bf}}=\frac{k_2+2k_{bnf}-2{\varphi }_2(k_{bnf}-k_2)}{k_2+2k_{bnf}+{\varphi }_2(k_{bnf}-k_2)} , \frac{k_{bf}}{k_f}=\frac{k_1+2k_f-2{\varphi }_1(k_f-k_1)}{k_1+2k_f+{\varphi }_1(k_f-k_1)}. \\ \frac{{\sigma }_{hnf}}{{\sigma }_{nf}}=\frac{{\sigma }_2+2{\sigma }_{bf}-2{\varphi }_2({\sigma }_{bf}-{\sigma }_2)}{{\sigma }_2+2{\sigma }_{bf}-2{\varphi }_2({\sigma }_{bf}-{\sigma }_2)},\frac{{\sigma }_{bf}}{{\sigma }_f}=\frac{{\sigma }_1+2{\sigma }_f-2{\varphi }_1({\sigma }_f-{\sigma }_1)}{{\sigma }_1+2{\sigma }_f-{\varphi }_1({\sigma }_f-{\sigma }_1)}.\end{array}$$

The expression for ${\mu }_{hnf}\left(\beta \right), k_{hnf}, {\left(\rho C_p\right)}_{hnf}$, and ${\rho }_{hnf}$ of the ternary hybrid nanomaterial (zirconium oxide/alumina/cupper) is as follows:

$$\begin{array}{l}\frac{{\mu }_{thnf}}{{\mu }_f}=\frac{1}{{\left(1-{\varphi }_{1 }\right)}^{2.5}{\left(1-{\varphi }_{2 }\right)}^{2.5}{\left(1-{\varphi }_{3 }\right)}^{2.5}}.\\ \frac{{\rho }_{thnf}}{{\rho }_f} =\left(1-{\varphi }_1\right)\left\{\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_3\right)+\frac{{\varphi }_3{\left(\rho Cp\right)}_{s3}}{{\rho }_f}\right]+\frac{{\varphi }_2{\left(\rho Cp\right)}_{s2}}{{\rho }_f}\right\}+\frac{{\varphi }_1{\left(\rho Cp\right)}_{s1}}{{\rho }_f}.\\ \frac{{\rho }_{thnf}}{{\left(\rho C_p\right)}_f}=\frac{{\varphi }_1{\left(\rho Cp\right)}_{s1}}{{\rho }_f}+\left(1-{\varphi }_1\right)\left\{\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_3\right)+\frac{{\varphi }_3{\left(\rho Cp\right)}_{s3}}{{\rho }_f}\right]+\frac{{\varphi }_2{\left(\rho Cp\right)}_{s2}}{{\rho }_f}\right\}.\\ \frac{k_{thnf}}{k_{hnf}}=\frac{k_3+2k_{hnf}-2{\varphi }_3\left(k_{hnf}-k_3\right)}{k_3+2k_{hnf}+{\varphi }_3\left(k_{hnf}-k_3\right)},\frac{k_{hnf}}{k_{bf}}=\frac{k_2+2k_{bnf}-2{\varphi }_2\left(k_{bnf}-k_2\right)}{k_2+2k_{bnf}+{\varphi }_2\left(k_{bnf}-k_2\right)},\frac{k_{bf}}{k_f}=\frac{k_1+2k_f-2{\varphi }_1\left(k_f-k_1\right)}{k_1+2k_f+{\varphi }_1\left(k_f-k_1\right)}.\\ \frac{{\sigma }_{thnf}}{{\sigma }_{hnf}}=\frac{{\sigma }_3+2{\sigma }_{hnf}-2{\varphi }_3({\sigma }_{hnf}-{\sigma }_3)}{{\sigma }_3+2{\sigma }_{hnf}-2{\varphi }_3({\sigma }_{hnf}-{\sigma }_3)},\frac{{\sigma }_{hnf}}{{\sigma }_{nf}}=\frac{{\sigma }_2+2{\sigma }_{bf}-2{\varphi }_2({\sigma }_{bf}-{\sigma }_2)}{{\sigma }_2+2{\sigma }_{bf}-2{\varphi }_2({\sigma }_{bf}-{\sigma }_2)},\frac{{\sigma }_{bf}}{{\sigma }_f}=\frac{{\sigma }_1+2{\sigma }_f-2{\varphi }_1({\sigma }_f-{\sigma }_1)}{{\sigma }_1+2{\sigma }_f-{\varphi }_1({\sigma }_f-{\sigma }_1)}.\end{array}$$

In Equation (5), the equation for the radiative heat flow is provided by the following:

$$q_r=-\frac{4{\sigma }^{*}}{3k^{*}}\frac{\partial T^4}{\partial z}=-\frac{16{\sigma }^{*}}{3k^{*}}{T}^3\frac{\partial T}{\partial z}=\frac{16{\sigma }^{*}}{3{\left(\rho c_p\right)}_{thnf}{k}^{*}}\left[T^3\frac{{\partial }^{2}T }{\partial z^2}+3T^2{\left(\frac{\partial T}{\partial z}\right)}^2\right],$$

(8)

Now introduce the following similarity variables:

$$\begin{array}{l}\psi \left(x,y,t\right)=u_w\sqrt{\cos \left(\omega \right){\nu }_{f}t+\sin \left(\omega \right)\left(\frac{{\nu }_{f}x}{u_w}\right)} f(\eta ), T=T_{\infty }\left(1+\left({\theta }_w-1\right)\theta \left(\eta \right)\right),\\ \eta =y/\sqrt{\cos \left(\omega \right){\nu }_{f}t+\sin \left(\omega \right)\left(\frac{{\nu }_{f}x}{u_w}\right)}.\end{array}$$

(9)

These ordinary differential equations are derived from the governing Equations (1)–(6) using Equations (8) and (9).

For the hybrid nanomaterials case

$$\frac{{\mu }_{hnf}}{{\mu }_f}{f}^{‴}+\frac{{\rho }_{hnf}}{{\rho }_f}\left(\frac{1}{2}\cos \left(\omega \right)\eta f^{″}+\frac{1}{ 2}\sin \left(\omega \right)f{f}^{″}\right)+\frac{{\sigma }_{hnf}}{{\sigma }_f}Qsi{n}^2\left(\omega \right)f\prime =0,$$

(10)

$$\begin{array}{r}\frac{k_{hnf}}{k_f}\left({\theta }^{″}+R\left[\begin{array}{c}{\left(1+\left({\theta }_w-1\right)\theta \right)}^3{\theta }^{″}+\\ 3\left({\theta }_w-1\right){\theta }^{\prime 2}{\left(1+\left({\theta }_w-1\right)\theta \right)}^2\end{array}\right]\right)+\frac{Pr}{2}\left(1-{\varphi }_2\right)\left[\begin{array}{c}\left(1-{\varphi }_1\right)+\\ {\varphi }_1\left(\frac{{\left(\rho Cp\right)}_{s1}}{{\left(\rho Cp\right)}_f}\right)\end{array}\right]+{\varphi }_2\left(\frac{{\left(\rho Cp\right)}_{s2}}{({\rho Cp)}_f}\right)\\ \left(\eta \cos \left(\omega \right)+f\sin \left(\omega \right)\right){\theta }^{\prime }+EcQsi{n}^2\left(\omega \right)f^{\prime 2}+Ec{f}^{″2}=0,\end{array}$$

(11)

For the ternary hybrid nanomaterials case

$$\frac{{\mu }_{thnf}}{{\mu }_f}{f}^{‴}+\frac{{\rho }_{thnf}}{{\rho }_f}\left(\frac{1}{2}\cos \left(\omega \right)\eta f^{″}+\frac{1}{ 2}\sin \left(\omega \right)f{f}^{″}\right)+\frac{{\sigma }_{thnf}}{{\sigma }_f}Qsi{n}^2\left(\omega \right)f\prime =0,$$

(12)

$$\begin{array}{r}\frac{k_{thnf}}{k_f}\left({\theta }^{″}+R\left[\begin{array}{c}{\left(1+\left({\theta }_w-1\right)\theta \right)}^3{\theta }^{″}\\ +3\left({\theta }_w-1\right){\theta }^{\prime 2}\\ {\left(1+\left({\theta }_w-1\right)\theta \right)}^2\end{array}\right]\right)\frac{Pr}{2}\left[\frac{{\varphi }_1{\left(\rho Cp\right)}_{s1}}{{\rho }_f}+\left(1-{\varphi }_1\right)\left\{\begin{array}{c}\left(1-{\varphi }_2\right)\left[\begin{array}{c}\left(1-{\varphi }_3\right)\\ +\frac{{\varphi }_3{\left(\rho Cp\right)}_{s3}}{{\rho }_f}\end{array}\right]\\ +\frac{{\varphi }_2{\left(\rho Cp\right)}_{s2}}{{\rho }_f}\end{array}\right\}\right]\\ \left(\eta \cos \left(\omega \right)+f\sin \left(\omega \right)\right){\theta }^{\prime }+EcQsi{n}^2\left(\omega \right)f^{\prime 2}+Ec{f}^{″2}=0,\end{array}$$

(13)

Depending on the boundary conditions

$$\begin{array}{l}f\left(\eta \right)=0, f^{\prime }\left(\eta \right)=0, \theta \left(\eta \right)=1, \mathrm{at} \eta =0,\\ f^{\prime }\left(\eta \right)=1, \theta \left(\eta \right)=0, \mathrm{as} \eta \to \infty .\end{array}$$

(14)

where ${\theta }_w=\frac{T_w}{T_{\mathrm{\infty }}}$—temperature ratio parameter, $Pr=\frac{{\alpha }_{nf}}{{\nu }_{nf}}$—Prandtl number, $R=\frac{16{\sigma }^{\mathrm{*}}{T}_{\mathrm{\infty }}^3}{3k_{nf}{k}^{\mathrm{*}}}$—radiation parameter, $Ec=\frac{{u_w}^2}{(T_w-T_{\mathrm{\infty }})c_{pf}}$—Eckert number, and $Q=\frac{\sigma B_0^2}{{\rho }_{nf}}$—magnetic parameter.

The local skin friction coefficient ($C_{fx}$) and the local Nusselt number ($N{u}_x$) are the quantities of importance in this study from a purely physical standpoint. They are defined as follows:

$$C_{fx}=\frac{{\tau }_w}{{\rho }_f{u}_{\infty }^2} \mathrm{and} N{u}_x=\frac{x{q}_w}{\left(T_w-T_{\infty }\right)},$$

where${\tau }_w$ and $q_w$ are given by

$${\tau }_{wx}={\mu }_{thnf}{\left(\frac{\partial u}{\partial z}\right)}_{y=0} \mathrm{and} q_w=-k_{thnf}{\left(\frac{\partial T}{\partial z}\right)+q_r}_{y=0},$$

The reduced local skin friction ($\sqrt{R{e}_x}{C}_{fx}$) coefficient and the local Nusselt number $\left(\frac{N{u}_x}{\sqrt{R{e}_x}}\right)$ can be written as follows:

$$\sqrt{R{e}_x}{C}_{fx}=\frac{{\mu }_{thnf}}{{\mu }_f}\frac{f^{″}\left(0\right)}{\cos \left(\omega \right)t+\sin \left(\omega \right)} \mathrm{and} \frac{N{u}_x}{\sqrt{R{e}_x}}=\frac{k_{thnf}}{k_f}\frac{\left(-\left[1+R{\theta }_w^3\right]{\theta }^{\prime }\left(0\right)\right)}{{\left(\cos \left(\omega \right)\tau +\sin \left(\omega \right)\right)}^{1/2 }}.$$

3. Numerical Method

The calculation of the effect of a large number of physical elements on the behaviour of velocity, in addition to the distribution of temperature, which are both dimensionless, is accomplished via the use of the numerical approach that was just described. Hence, the resultant equations are ODEs of a higher degree. We start by transforming these ODEs of higher order into their simpler first order form.

$$\mathrm{Let} f=y_1, f^{\prime }=y_2, f^{″}=y_3, f^{‴}=y_4, \theta =y_5, {\theta }^{\prime }=y_6, {\theta }^{″}=y_7.$$

For the hybrid nanomaterials case

$$y_4=-\frac{{\mu }_f}{{\mu }_{hnf}}[\frac{{\rho }_{hnf}}{{\rho }_f}\left(\frac{1}{2}\cos \left(\omega \right)\eta y_3+\frac{1}{ 2}\sin \left(\omega \right)y_1{y}_3\right)+\frac{{\sigma }_{hnf}}{{\sigma }_f}Qsi{n}^2\left(\omega \right)y_2],$$

(15)

$$y_7=-\frac{1}{\left(1+R{\left(1+\left({\theta }_w-1\right)y_5\right)}^3\right)}\left[\begin{array}{c}3R\left({\theta }_w-1\right)y_6^2{\left(1+\left({\theta }_w-1\right)y_5\right)}^2\\ +\frac{k_f}{k_{hnf}}\left\{\begin{array}{c}\frac{Pr}{2}\left(1-{\varnothing }_2\right)\left[\begin{array}{c}\left(1-{\varnothing }_1\right)+\\ {\varphi }_1\left(\frac{{\left(\rho Cp\right)}_{s1}}{{\left(\rho Cp\right)}_f}\right)\end{array}\right]+{\varphi }_2\left(\frac{{\left(\rho Cp\right)}_{s2}}{({\rho Cp)}_f}\right)\\ \left(\eta \cos \left(\omega \right)+y_1\sin \left(\omega \right)\right)y_6+\\ EcQsi{n}^2\left(\omega \right)y_2^2+Ec{y}_3^2\end{array}\right\}\end{array}\right].$$

(16)

For the ternary hybrid nanomaterials case

$$y_4=-\frac{{\mu }_f}{{\mu }_{thnf}}\left[\frac{{\rho }_{thnf}}{{\rho }_f}\left(\frac{1}{2}\cos \left(\omega \right)\eta y_3+\frac{1}{ 2}\sin \left(\omega \right)y_1{y}_3 \right)+\frac{{\sigma }_{thnf}}{{\sigma }_f}Qsi{n}^2\left(\omega \right)y_2\right],$$

(17)

$$y_7=-\frac{1}{\left(1+R{\left(1+\left({\theta }_w-1\right)y_5\right)}^3\right)}\left[\begin{array}{c}3R\left({\theta }_w-1\right)y_6^2{\left(1+\left({\theta }_w-1\right)y_5\right)}^2+\\ \frac{k_f}{k_{thnf}}\left\{\begin{array}{c}\frac{Pr}{2}\left[\begin{array}{c}{\varnothing }_1\frac{{\left(\rho C_p\right)}_{s1}}{{\left(\rho C_p\right)}_f}+\left(1-{\varnothing }_1\right)\\ \left\{\begin{array}{c}{(1-\varnothing }_2)\\ \left[{(1-\varnothing }_3)+{\varnothing }_3\frac{{\left(\rho Cp\right)}_{s3}}{({\rho Cp)}_f}\right]\\ +{\varnothing }_2\frac{{\left(\rho Cp\right)}_{s2}}{({\rho Cp)}_f}\end{array}\right\}\end{array}\right]\\ \left(\eta \cos \left(\omega \right)+y_1\sin \left(\omega \right)\right)y_6\\ +EcQsi{n}^2\left(\omega \right)y_2^2+Ec{y}_3^2\end{array}\right\}\end{array}\right] .$$

(18)

subject to the following boundary conditions:

$$\begin{array}{l}{y}_1\left(\eta \right)=0, y_2\left(\eta \right)=0, y_5\left(\eta \right)=1 \mathrm{at} \eta =0,\\ y_2\left(\eta \right)=1, y_5\left(\eta \right)=0, \mathrm{as} \eta \to \infty ,\end{array}$$

(19)

4. Results and Discussion

Here, we exhibit the results of varying velocity and heat distributions for a number of parameters, which graphically convey the underlying physical relevance of the data. These distributions are exhibited as a result of the modulating velocity. In addition, graphs have been used to record illustrative findings for the $N{u}_{x}R{e}_x^{-1/2}$ and $C_{fx}R{e}_x^{-1/2}$ for a range of parameter values applicable to the $Zr{O}_2$-$A{l}_2{O}_3$-$EG$ as well as the $Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ case. These results are exemplary in nature. Thermophysical characteristics of $Zr{O}_2$-$A{l}_2{O}_3$-$EG$ as well as $Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ are outlined in

Table 1. Thermal and physicochemical characteristics of the base fluid and the nanoparticles.

Physical Properties	Base Fluid $\left(\mathit{E}\mathit{G}\right)$	Hybrid Nanoparticles		Ternary Hybrid Nanoparticles
		Alumina $\mathit{A}{\mathit{l}}_2{\mathit{O}}_3$	Zirconium Oxide$\left(\mathit{Z}\mathit{r}{\mathit{O}}_2\right)$	Copper $\left(\mathit{C}\mathit{u}\right)$	Alumina $\left(\mathit{A}{\mathit{l}}_2{\mathit{O}}_3\right)$	Zirconium Oxide$(\mathit{Z}\mathit{r}{\mathit{O}}_2$)
$\mathit{\rho }(\mathbf{k}\mathbf{g}/{\mathbf{m}}^{\mathbf{3}})$	989	3970	5680	8933	3970	5680
${\mathit{\rho }\mathit{c}}_{\mathit{p}}(\mathbf{j}/\mathbf{k}\mathbf{g}·\mathbf{K})$	4175	765	502	385	765	502
$\mathit{k}(\mathbf{w}/\mathbf{m}·\mathbf{K})$	0.6376	40	1.7	401	40	1.7
$\sigma {\left(\frac{\mathsf{\Omega }}{\mathrm{m}}\right)}^{-1}$	$2.6\times {10}^{-4}$	$3.96\times {10}^6$	$0.3\times {10}^2$	$5.96\times {10}^6$	$3.96\times {10}^6$	$0.3\times {10}^2$

. There is a tabular representation of the computational values of $N{u}_{x}R{e}_x^{-1/2}$ and $C_{fx}R{e}_x^{-1/2}$, when the conditions are ${\varphi }_1={\varphi }_2={\varphi }_3=10\%, \delta =0.4, Pr=6, \omega =\frac{\pi }{4}, R=0.5$, and ${\theta }_w=1.3$.

Table 2. Comparison of present results.

$\mathbf{Pr}$	Ghadikolaei et al. [33]	Hosseinzadeh et al. [34]	Reddy et al. [3]	Present Results
0.7	0.4538	0.4541	0.4539	0.45415
2.0	0.9113	0.9114	0.9113	0.91133
7.0	1.8954	1.8954	1.8954	1.89545

presents the findings obtained by analysing the Prandtl number in relation to other research that has been previously published. The conclusion that can be inferred from this is that the present results are in great harmony with previous studies, and this conclusion can be formed as a result of the fact that this is the case.

Table 3. Comparison of the values of $R{e}^{-1/2}Nu$ for different physical parameter values for both the $Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ and $Zr{O}_2$-$A{l}_2{O}_3$/$EG$ case.

$\mathbf{Pr}$	$\mathit{E}\mathit{c}$	$\mathit{Q}$	${\mathit{\varphi }}_1$	${\mathit{\varphi }}_2$	${\mathit{\varphi }}_3$	$\mathbf{Nusselt} \mathbf{Number} \left(\frac{\mathit{N}{\mathit{u}}_{\mathit{x}}}{\sqrt{\mathit{R}{\mathit{e}}_{\mathit{x}}}}\right)$
						$\mathit{Z}\mathit{r}{\mathit{O}}_2$-$\mathit{A}{\mathit{l}}_2{\mathit{O}}_3$-$\mathit{C}\mathit{u}$ / $\mathit{E}\mathit{G}$	$\mathit{Z}\mathit{r}{\mathit{O}}_2$-$\mathit{A}{\mathit{l}}_2{\mathit{O}}_3$ / $\mathit{E}\mathit{G}$
0.1						0.166076	0.147526
0.3						0.176270	0.151266
0.5						0.201610	0.162602
	0.1					0.262315	0.171266
	0.3					0.238968	0.166031
	0.5					0.218061	0.161655
		0.1				0.184255	0.179411
		0.2				0.187977	0.180097
		0.3				0.192973	0.183820
			0.1			2.452295	2.457524
			0.2			2.359885	2.349007
			0.3			2.258100	2.241073
				0.1		2.632480	2.527524
				0.2		2.429885	2.415200
				0.3		2.351073	2.338100
					0.1	2.691420	2.537524
					0.2	2.643150	2.197524
					0.3	2.452295	2.457524

shows the comparison result of hybrid and ternary hybrid nanofluid. From this table, one can observe that the performance of ternary nanofluid is much better than that of hybrid nanofluid.

Figure 1 displays the effect of the magnetic parameter ($Q$) on the velocity ($f\prime \left(\eta \right)$) profile for both the $Zr{O}_2$-$A{l}_2{O}_3$-$EG$ and $Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ cases. In both ternary and hybrid conditions, increasing the magnetics ($Q$) parameter may be accomplished by cutting down on the velocity profile ($f\prime \left(\eta \right)$). This transformation is also caused by a reduction in the inter-relevance thickness of the corresponding layer for highly valued magnetic parameters ($Q$). Physically, the magnitude of the force that is acting in opposition to us, as measured by the magnetic parameter ($Q$), grows in proportion to the escalating strength of the magnetic field strength. This has a direct impact on the velocity profile, which causes it to decrease.

Figure 2 illustrates the influence of $\omega$ against the thermal profile ($\theta \left(\eta \right)$) for situations involving $Zr{O}_2$-$A{l}_2{O}_3$-$EG$ as well as $Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$. The temperature profile reduces by a considerable amount whenever there is an increase in the value of $\omega$. This phenomenon occurs in both the ternary and hybrid examples. Furthermore, the thickness of the inter-relevant layer reduced in both the cases. For the $Zr{O}_2$-$A{l}_2{O}_3$-$EG$ as well as the $Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ examples, the impact of ${\varphi }_1$, ${\varphi }_2$, and ${\varphi }_2$ on the $f\prime \left(\eta \right)$ profile is shown in Figure 3, Figure 4 and Figure 5. In both the $Zr{O}_2$-$A{l}_2{O}_3$-$EG$ and $Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ cases, it was discovered that the inclusion of nanoparticle volume friction into the nanomaterial caused a drop in the velocity curve. This was the consequence of the interaction between the two. This outcome may be attributed to collisions that occur between nanoparticles that are extensively dispersed.

Figure 6 displays the influence of $R$ on the relevant $\theta \left(\eta \right)$ profiles for the $Zr{O}_2$-$A{l}_2{O}_3$-$EG$ as well as $Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ cases. A higher $R$ value indicates a more powerful heat transmission to the liquid, which ultimately results in a thicker thermal layer. This is true for both the $Zr{O}_2$-$A{l}_2{O}_3$-$EG$ and $Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ cases. A higher $R$ allows for better heat dispersion, which in turn makes the influence of atmospheric conditions more apparent. When $R$ values are high, a greater amount of heat is transferred to the fluids in the system.

In Figure 7, we compare the properties of $\theta \left(\eta \right)$ for the $Zr{O}_2$-$A{l}_2{O}_3$-$EG$ and $Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ conditions as a function of increasing values of ${\theta }_w$. For both $Zr{O}_2$-$A{l}_2{O}_3$-$EG$ and $Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ conditions, the $\theta \left(\eta \right)$ and layer thickness increase exponentially with increasing ${\theta }_w$ as seen in the image. In addition, raising the fluid’s temperature improves the whole fluid situation for high values of ${\theta }_w.$ More importantly, the fluid temperature enchantment is most significant in the $Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$Eg$ instance, followed by the $Zr{O}_2$-$A{l}_2{O}_3$/$Eg$ case. Physically, when you mix all of these components in the “$Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ nanomaterial”, you are building a composite material that benefits from the high thermal conductivity of copper while still preserving the stability and other desired features of the ceramics and the base fluid. In other words, you are making a material that can take advantage of copper’s high thermal conductivity. This combination results in improved heat transmission, which makes it possible for heat to be transported effectively via the nanomaterial. As a consequence, the nanomaterial is a more effective medium for the transfer of heat when compared to the “$Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ nanomaterial fluid”.

The presence of ceramics and a base fluid, in addition to copper nanoparticles with high thermal conductivity, synergistically improve the overall heat transmission properties. This explains why the “$Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ nanomaterial” enhances heat transmission at a higher rate in comparison to the “$Zr{O}_2$-$A{l}_2{O}_3$-/$EG$ nanomaterial fluid”.

Figure 8 depicts the behaviour of the $\theta \left(\eta \right)$ profile in relation to the $Pr$ number for both the $Zr{O}_2$-$A{l}_2{O}_3$-$EG$ and $Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ conditions. For both the $Zr{O}_2$-$A{l}_2{O}_3$-$EG$ and $Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ conditions, the liquid temperature drops considerably with increasing $Pr$. The Prandtl number $\left(Pr\right)$ is a measure of the thickness of a viscous material in relation to its thermal diffusivity. Figure 8 shows that for both the $Zr{O}_2$-$A{l}_2{O}_3$-$EG$ and $Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ examples, an increase in $Pr$ considerably decreases the $\theta \left(\eta \right)$ profile and the thermal boundary layer.

Figure 9, Figure 10 and Figure 11 highlight the effect of the ${\varphi }_1$, ${\varphi }_2$, and ${\varphi }_2$ parameters on the $\theta \left(\eta \right)$ profile for both the $Zr{O}_2$-$A{l}_2{O}_3$-$EG$ and $Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ conditions. The heat that is produced is the result of the nanoparticle releasing the energy that it has been storing. Therefore, the mixing of more nanoparticles may need more energy (see Refs. [43,44,45]), which will result in an increase in the temperature and the width of the boundary layer in both the $Zr{O}_2$-$A{l}_2{O}_3$-$EG$ and $Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ nanomaterial cases. In addition, the temperature of the fluid may be better controlled in scenarios involving the $Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ nanomaterials than in instances involving the $Zr{O}_2$-$A{l}_2{O}_3$/$EG$ nanomaterials.

As can be seen in Figure 12, the $Zr{O}_2$-$A{l}_2{O}_3$-$EG$ and $Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ $\theta \left(\eta \right)$ profiles improve as the Eckert number $Ec$ increases.

The energy profile enhances with the upshot values of the Eckert number $Ec$ for both $Zr{O}_2$-$A{l}_2{O}_3$-$EG$ and $Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ as shown in Figure 12. This is due to the fact that when the Eckert number changes, the wall stretching increases while the specific heat capacity of the fluid decreases (see Refs. [46,47,48]). Additionally, the thermal thickness of the layer also maximizes in both the $Zr{O}_2$-$A{l}_2{O}_3$-$EG$ and $Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ cases for higher values of the $Ec$ number.

The behaviour of the friction factor when subjected to changing values of ${\varphi }_1$ and $\omega$ parameters is seen in Figure 13. Higher estimations of both ${\varphi }_1$ and $\omega$ parameters result in an increase in the friction factor. In addition, the performance of fluid velocity is much improved in the $Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ instance in comparison to the $Zr{O}_2$-$A{l}_2{O}_3$/$EG$ case. The effect that ${\varphi }_1$ and $\omega$ parameters have on $N{u}_{x}R{e}_x^{-1/2}$ is seen in Figure 14. The $N{u}_{x}R{e}_x^{-1/2}$ value drops when the ${\varphi }_1$ parameter is increased, as this figure clearly demonstrates. On the other hand, when $\omega$ is given bigger values, the opposite pattern is seen. Figure 15 illustrates the variation in characteristics of the primary parameters $Pr$ and $R$ on the $N{u}_{x}R{e}_x^{-1/2}$ curve. The variation is due to the fact that the $N{u}_{x}R{e}_x^{-1/2}$ parameter decreases as the $Pr$ and $R$ parameters increase.

5. Final Remarks

This paper presents $Zr{O}_2$-$A{l}_2{O}_3$/$EG$ nanomaterials, the $Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ nanomaterial model, and the repercussions of slip while taking into account flow and radiative heat transfer across a stretched surface. Both numerical and graphical analyses are used to expound on the impacts of the physical elements. Some of the most important findings are as follows:

• The $Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ nanomaterial enhances heat transmission at a higher rate than the $Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ nanomaterial fluid.
• An improvement in the velocity profile may be achieved by raising the ${\varphi }_1,{\varphi }_2$, and ${\varphi }_3;$
• Increasing the Eckert number increases wall stretching while decreasing the specific heat capacity of the fluid (see Refs. [46,47,48]);
• When $Pr$ values are higher, the presence of a thermal layer cannot be supported;
• When the value of δ parameter was made larger, the fluid moved at a more leisurely pace;
• The higher the values of the ${\theta }_w$ parameter become as the temperature of the fluid gets higher;
• A higher $R$ allows for better heat dispersion, which in turn makes the influence of atmospheric conditions more apparent. In addition, when $R$ values are high, a greater amount of heat is transferred to the fluids in the system;
• The temperature profile goes up when ${\varphi }_1,{\varphi }_2$, and ${\varphi }_3$ parameters reach their maximum levels;
• Since more energy is required for mixing more nanoparticles (see Refs. [30,31,32]), the breadth of the thermal boundary layer increases for both $Zr{O}_2$-$A{l}_2{O}_3$-$EG$ and $Zr{O}_2$-$A{l}_2{O}_3$-$Cu$/$EG$ nanomaterials.