Thermal Radiation Energy Performance on Stagnation-Point Flow in the Presence of Base Fluids Ethylene Glycol and Water over Stretching Sheet with Slip Boundary Condition

Abstract

Nanoparticles are useful in improving the efficiency of convective heat transfer. The current study addresses this gap by making use of an analogy between Al${}_2$O${}_3$ and $\gamma$-Al${}_2$O${}_3$ nanoparticles in various base fluids across a stretched sheet conjunction with f. Base fluids include ethylene glycol and water. We address, for the first time, the stagnation-point flow of a boundary layer of $\gamma$-Al${}_2$O${}_3$ nanofluid over a stretched sheet with slip boundary condition. Al${}_2$O${}_3$ nanofluids employ Brinkman viscosity and Maxwell’s thermal conductivity models with thermal radiations, whereas $\gamma$-Al${}_2$O${}_3$ nanofluids use viscosity and thermal conductivity models generated from experimental data. For the boundary layer, the motion equation was solved numerically using the fourth-order Runge–Kutta method and the shooting approach. Plots of the velocity profile, temperature profile, skin friction coefficient and reduced Nusselt number are shown. Simultaneous exposure of the identical nanoparticles to water and ethylene glycol, it is projected, would result in markedly different behaviors with respect to the temperature profile. Therefore, this kind of research instills confidence in us to conduct an analysis of the various nanoparticle decompositions and profile structures with regard to various base fluids.

1. Introduction

Water, ethylene glycol and mineral oils are traditional heat transfer fluids with low thermal conductivities which may restrict their effectiveness in many industrial domains such as chemical processing, generation of power, air conditioning systems, microelectronics and transportation. Solid particle suspensions and fluids have a strong ability to enhance heat transfer. One may classify particles in a number of ways including metallic, non-metallic and polymeric. However, there are issues that arise when industries use macro-sized suspensions such as heat transfer erosion and flow channel blockage owing to poor suspension stability and a gradual but steady decrease in pressure. Therefore, researchers and engineers have been hammering away to conquer this fundamental barrier by dispersing particles as small as millimeters or micrometers in liquids since Maxwell (1873). Large particles quickly settle in fluids which is an issue. Extended surface technology has reached its limits in thermal management system designs; thus, innovations that might increase interest in which nanofluids are nanotechnological heat transfer fluids. Nanofluids are suited for engineering applications and have various benefits over convectional suspensions which include improved stability, high thermal conductivity and negligible pressure loss. Thus, nanofluid technology will emerge as a promising and interesting field of study in the twenty-first century [1,2,3,4,5,6,7,8,9].

Flow and heat transfer caused by the stretching sheet is a prominent process in many industrial applications, such as metallic sheet cooling, crystal growth in cooling baths, plastics and rubber sheets manufacturing, paper and glass fiber production, and the eviction of polymer and metals. Although enhancement of thermal and electrical conductivity can be achieved by using metallic particles was first proposed by Maxwell (1873), Choi learned from his work with micrometer-sized particle and fiber suspensions in the 1980s in which traditional particles in micro-channel flow passages are unusable [9,10,11]. However, advancements in nanotechnology can process and manufacture materials with average crystal-lite sizes below 50 nm. Thermal conductivity is key to developing energy-efficient heat transfer fluids. Because of the growing level of competitiveness on a worldwide scale, a number of different sectors are in desperate need of innovative heat transfer fluids that have much greater thermal conductivities than those that are already on the market [12]. In recent years, several studies have investigated the nanofluids boundary layer flow across a stretched surface using a wide range of metal and oxide nanoparticles [13,14,15,16,17]. Using a reworked version of Buongiorno’s model with verified thermo-physical correlations which examine the impact of Darcy–Forchheimer and Lorentz forces on radiative alumina-water nanofluid flows across a slippery curved geometry subject to numerous convective restrictions [18]. Later on, Saima et al. [19] used a finite volume method, a numerical approach to study micropolar nanofluids flow through a lid-driven cavity. The development of innovative hybrid $2D$-$3D$ graphene oxide diamond micro composite polyimide films to alleviate electrical and thermal conduction. It is believed to be a useful option for the thermal dissipation of the electronic components of electric machines as a result of its high and outstanding thermal conductivity [20]. A numerical investigation for two-dimensional Sutterby fluid flow which is bounded at a stagnation point with an inclined magnetic field and thermal radiation was conducted by Sabir et al. [21]. Meanwhile, another computational approach was performed on stagnation point pseudo-plastic nano-liquid flow towards a flexible Riga sheet by Azad et al. [22]. The stagnation-point flow of an incompressible non-Newtonian fluid over a non-isothermal stretching sheet is investigated by Rashidi et al. [23]. Baag et al. [24] investigated numerical methods into the flow of MHD micro-polar fluids toward a stagnation point on a vertical surface with a heat source and a chemical reaction.

In light of this, a comparison study was carried out on the flow with velocity and thermal boundary layers (BL) of Al${}_2$O${}_3$ and $\gamma$-Al${}_2$O${}_3$ nanofluids along various base fluids across a stretching sheet. We have been successful in developing a conjuncture between stagnation-point flow and nanofluids flow over the stretching sheet with slip boundary conditions, by the numerical technique. Models of viscosity and thermal conductivity are developed from data through experiments. For Al${}_2$O${}_3$, Maxwell’s thermal conductivity with radiation and Brinkman viscosity models are used in nanofluids flow. Therefore, this is a comparative new study and its contribution to the existing body of research will be significant.

1.1. Theoretical to Experimental Perspective on Nanofluids

The thermal properties of nanoparticles are explained through nanofluid theory, which supports physics and chemistry-based predictive models. The thermal conductivity of the nanofluids has not been explained for numerous reasons. First, nanofluids behave differently from solid-to-fluid suspensions or typical solid-to-solid composites. Reducing nanoparticle size increases nanofluid thermal conductivity. Second, nanofluids and traditional solid-to-liquid suspensions differ in thermal conductivity, concentration of the particles, and size. Thirdly, nanofluids are a new, highly multidisciplinary field that spans engineering, material science, physics, chemistry and colloidal science. Nanofluids hence need expertise in each field, which shows the difficulty to build a nanofluid theory [17,25,26,27]. Therefore, predictions are poorer when the nanoparticles are suspended in a liquid because these interactions include electromagnetic or particle-to-lattice heat transfer in addition to the lattice vibrational heat transfer predicted by the liquid models. Thus, a theory for nanofluid thermal conductivity can be developed by taking into account two crucial components, static and dynamic mechanisms. Near field radiation in nanofluids seems like a promising theory [26,28].

The most recent advances in fabrication technology have opened up exciting new possibilities for actively processing materials at nano-scale sizes. Materials that are either nanostructured or nanophase are composed of nanometer-sized components. Because of this, particles of a size less than 100 nm have characteristics that are distinct from those of traditional solids. The remarkable qualities that are associated with nanophase materials are a direct result of the relatively high surface area/volume ratio that these materials possess. This ratio is made possible by the presence of a significant number of constituent atoms that are located at the grain boundaries. Nanophase materials have superior thermal, mechanical, optical, magnetic, and electrical capabilities compared to traditional materials that have coarse grain patterns. These qualities include magnetism and electrical conductivity. As a consequence of this, the exploration of nanophase materials in research and development has attracted a significant amount of attention from both material scientists and engineers [29]. Several kinds of nanoparticles that are employed in nanofluids can be constructed out of a wide variety of materials, including oxide ceramics (Al${}_2$O${}_3$, CuO), carbide ceramic, metals (Cu, Ag, Au), semiconductors, composite materials and alloyed nanoparticles are some examples of advanced materials. Whereas, in the development of nanofluids a wide variety of liquids, including water, ethylene glycol and oil, have been employed as base fluids [29,30].

1.1.1. Volume Fraction and the Particle Size

Many experiments on nanofluid thermal conductivity have been described in recent years. Table 1 summarizes published experimental studies on nanofluids at room temperature. So, nanofluids with thermal conductivity higher than their base fluids, even at low nanoparticle concentrations, grow well with nanoparticle volume fraction. Below are several nanofluid thermal conductivity investigations.

Eastman et al. [31] first reported nanofluids’ increased effective thermal conductivity. Al${}_2$O${}_3$ and CuO nanoparticles dispersed in water increased thermal conductivity by $29\%$ to $60\%$ for $5\%$ nanoparticle volume fraction which later findings found a moderate increase in thermal conductivity for Al${}_2$O${}_3$ and CuO nanoparticles in water and ethylene glycol. Li et al. [32] recently studied the thermal conductivity of CuO and Al${}_2$O${}_3$ nanoparticles boosted waters thermal conductivity by $52\%$ and $22\%$ at $6\%$ fractional part of volume at a temperature of 34 ${}^{°}$C. Choi et al. [33] investigated multi-walled carbon nanotube-containing oil suspensions’ thermal conductivity. Thermal conductivity doubled with $1\%$ volumetric loading. Even at low volume fractions, nanotube addition increases conductivity non-linearly. Strong thermal field interactions between fibers might be to cause. TiO${}_2$ nanoparticles are used to evaluate the thermal conductivity in deionized water by Murshed et al. [34]. Their findings demonstrated, for the very first time, that there was no anomalous improvement in the thermal conductivity of nanofluids containing a very low volume proportion of particles. This conclusion is in direct opposition to Patel et al.’s [35] unusual finding which shows the incorrect Patel’s hypothesis. According to a comparison of the studies that have been conducted, the increases in thermal conductivities of various types of nanofluids are distinct from one another. The size and composition of the nanoparticles, in addition to the base fluids, both have an effect on the thermal conductivity of nanofluids. Particle size is essential to optimize results and build a relation to volume fraction which causes nano-scale mechanism in the suspensions. According to theoretical evidence, with a reduction in particle size, the effective thermal conductivity of nanofluids improved [36,37].

Table 1. An overview of various nanofluids and thermal conductivity.

Researchers	Nanoparticle Size/Base-Fluid	Measurement	Thermal Conductivity (k) ↑
Eastman et al. [31]	Al${}_2$O${}_3$ (33)/water CuO (36)/water	transient hot-wire	$29\%$ for 5 vol % $60\%$ for 5 vol %
Murshed et al. [34]	TiO${}_2$/deionised-water	transient hot-wire	$30\%$ for 5 vol %
Li and Peterson [32]	Al${}_2$O${}_3$ (36)/water	SS method	$52\%$ for 6 vol %
Xuan and Li [37]	Cu (10)/water	transient hot-wire	$70\%$ for 3 vol %
Hwang et al. [38]	CuO (35.4)/water/EG	transient hot-wire	$9\%$ for 1 vol %
Liu et al. [39]	CuO (29)/EG	transient hot-wire	23% for 5 vol%
Kri. et al. [40]	Al${}_2$O${}_3$(20)/water	Unspecified	16% for 1 vol %
Wen and Ding [41]	TiO${}_2$ (34)/water	transient hot-wire	6% for 0.66 vol %

Table 1. An overview of various nanofluids and thermal conductivity.

Researchers	Nanoparticle Size/Base-Fluid	Measurement	Thermal Conductivity (k) ↑
Eastman et al. [31]	Al${}_2$O${}_3$ (33)/water CuO (36)/water	transient hot-wire	$29\%$ for 5 vol % $60\%$ for 5 vol %
Murshed et al. [34]	TiO${}_2$/deionised-water	transient hot-wire	$30\%$ for 5 vol %
Li and Peterson [32]	Al${}_2$O${}_3$ (36)/water	SS method	$52\%$ for 6 vol %
Xuan and Li [37]	Cu (10)/water	transient hot-wire	$70\%$ for 3 vol %
Hwang et al. [38]	CuO (35.4)/water/EG	transient hot-wire	$9\%$ for 1 vol %
Liu et al. [39]	CuO (29)/EG	transient hot-wire	23% for 5 vol%
Kri. et al. [40]	Al${}_2$O${}_3$(20)/water	Unspecified	16% for 1 vol %
Wen and Ding [41]	TiO${}_2$ (34)/water	transient hot-wire	6% for 0.66 vol %

1.1.2. The Influence of Temperature on Nanofluid

The temperature may affect nanofluid thermal conductivity. Despite the fact that nanofluids can be used at different temperatures, few studies have examined the temperature effect on their thermal conductivity.

Table 2. The Influence of Temperature on Nanofluids.

Researchers	Nanoparticle Size/Base-Fluid	Measurement	Thermal Conductivity (k) ↑
Das et al. [42]	Al${}_2$O${}_3$ (38.4)/water	Temperature Oscillations	For 4 vol %: $16\%$ at $36{}^{°}$C and 25% at $51{}^{°}$C
Li and Peterson [32]	Al${}_2$O${}_3$ (36)/water	SS method	For 2 vol %: $7\%$ at $27.5{}^{°}$C and $23\%$ at $36{}^{°}$C
Chon and Kihm [43]	Al${}_2$O${}_3$ (47)/water Al${}_2$O${}_3$ (150)/water	transient hot-wire	$6\%$ at $31{}^{°}$C and $11\%$ at $51{}^{°}$C $3\%$ at $51{}^{°}$C and $8.5\%$ at $51{}^{°}$C
Murshed et al. [44]	Al${}_2$O${}_3$ (150)/DIW	transient hot-wire	For 1 vol %: $11.4\%$ at $60{}^{°}$C

summarizes published studies on nanofluids in which thermal conductivity dependency on temperature. The nanofluids which comprise Al${}_2$O${}_3$ and CuO nanoparticles in water are used to measure the thermal conductivity which is investigated by Das et al. [42]. In their investigation, they employed a temperature oscillation technique to measure thermal diffusivity. They found that the thermal conductivity enhancement of these nanofluids increased by a factor of two to four over the temperature range of ${21}^{°}$ to ${51}^{°}$ Celsius. The different sizes of nanoparticles of Al${}_2$O${}_3$ dispersing in water to investigate the thermal conductivity of nanofluids are carried out by Chon and Kihm [43]. The nanoparticles ranged from 47 to 150 nanometers in size. They found that there was a slight increase in thermal conductivity in relation to temperature. When the temperature of the fluid was raised from $31{}^{°}$C to $51{}^{°}$C, there was a 6–11% increase in the thermal conductivity of nanofluids. It is interesting to note a little reduction in the thermal conductivity as the temperature increases, which is the opposite tendency seen in nanofluids containing spherical nanoparticles.

2. Mathematical Formulation

Assume a two-dimensional laminar boundary layer incompressible flow with steady characteristics in which Al${}_2$O${}_3$ and $\gamma$-Al${}_2$O${}_3$ nanofluids move across a stretched sheet of various base fluids, such as water and C${}_2$H${}_6$O${}_2$, see from Figure 1. The flow that is related to nanofluids is produced as a result of the sheet being stretched along the x-axis by two forces that are identical in magnitude but act in opposite directions. The stretching velocity $u_w\left(x\right)$ is used to suppose that the flow is confined to $y>0$ and external velocity to boundary layer flow is $U_{\infty }$, such that $U_{\infty }=\tilde{a}x$, where $\tilde{a}$ is constant. It is observed that the ratio of the stretching surface velocity to the inviscid flow at the stagnation point determines the shape of a boundary layer produced in a stagnation-point flow of an incompressible viscous fluid towards a stretching surface. Therefore, the stagnation-point flow is represented by the term, ${\tilde{U}}_{\infty }{\displaystyle \frac{d{\tilde{U}}_{\infty }}{dx}}$ in the momentum equation of the fluid flow where ${\tilde{U}}_{\infty }$ is the velocity distribution for the free stream which is far away from the surface. The temperature profile at the surface which is being stretched is $T_{\omega }=T_{\infty }+\tilde{b}{x}^2$, with $\tilde{b}$ is constant and $T_{\infty }$ is the ambient temperature. In addition, it is assumed that the nanoparticles and the base fluids are in a state of thermal equilibrium and there is a slip between them [21,23,45].

Table 3. Thermophysical characteristics of Nanofluids.

	Density (kg/m${}^3$)	${\mathit{C}}_{\mathit{p}}$ (J/Kg K)	k (W/mK)	${\mathit{P}}_{\mathit{r}}$
H${}_2$O (Pure Water)	998.3	4182	0.60	6.96
C${}_2$H${}_6$O${}_2$ (Ethylene glycol)	1116.6	2382	0.249	204
Al${}_2$O${}_3$ (Alumina)	3970	765	40	-

summarizes the considered thermo-physical features of nanofluids.

Under these conditions, we can write down the steady boundary equation that controls the convective flow and heat transfer of nanofluids as:

$$\begin{array}{c}\hfill {\tilde{u}}_x+{\tilde{v}}_y=0,\end{array}$$

(1)

$$\begin{array}{c}\hfill \tilde{u}{\tilde{u}}_x+\tilde{v}{\tilde{u}}_y=\frac{{\mu }_{nf}}{{\rho }_{nf}}\frac{{\partial }^2\tilde{u}}{\partial y^2}+{\tilde{U}}_{\infty }{\displaystyle \frac{d{\tilde{U}}_{\infty }}{dx}}\end{array}$$

(2)

$$\begin{array}{c}\hfill \tilde{u}{\tilde{T}}_x+\tilde{v}{\tilde{T}}_y=\frac{k_{nf}}{{\left(\rho C_p\right)}_{nf}}\frac{{\partial }^2\tilde{T}}{\partial y^2}-\frac{1}{\tilde{\rho }{\tilde{C}}_p}\frac{\partial {\tilde{q}}_r}{\partial y},\end{array}$$

(3)

where $\tilde{u}$ and $\tilde{v}$ are velocity components along x and y directions, respectively. Moreover, $\tilde{\rho }$ is the density of the fluid and ${\tilde{C}}_p$ represents specific heat at constant pressure. For the reason that the intensity of radiant emission increases with increasing absolute temperature a very crucial factor in the heat transfer process. Therefore, the term $\frac{1}{\tilde{\rho }{\tilde{C}}_p}\frac{\partial {\tilde{q}}_r}{\partial y}$ represents the thermal radiation in which ${\tilde{q}}_r$ is the radiative heat flux. Such heat flux is defined by Rosseland approximation,

$$\begin{array}{c}\hfill {\tilde{q}}_r=-\left(\frac{4\tilde{\sigma }}{3\tilde{k}}\right)\frac{\partial T^4}{\partial y},\end{array}$$

(4)

where $\tilde{k}$ is the Rosseland coefficient of mean absorption and $\tilde{\sigma }$ is the Stefan Boltzmann constant. For the construction of the linear function of temperature $T^4$, apply the Taylor series about the free stream temperature $T_{\infty }$ and neglect the higher power, we have the following equation,

$$\begin{array}{c}\hfill T^4\approx (4T-3T_{\infty })T_{\infty }^3.\end{array}$$

(5)

Therefore, the radiative heat flux can be expressed as;

$$\begin{array}{c}\hfill {\tilde{q}}_r=-\left(\frac{16\tilde{\sigma }}{3\tilde{k}}\right)T_{\infty }^3\frac{\partial T}{\partial y}.\end{array}$$

(6)

Boundary conditions are:

$$\begin{array}{c}\hfill \tilde{u}=u_w+\breve{a}\frac{\partial u}{\partial y},\tilde{v}=0,\tilde{T}=T_{\omega }(T_{\omega }=T_{\infty }+\tilde{b}{x}^2)aty=0,\end{array}$$

(7)

$$\begin{array}{c}\hfill \tilde{u}\to 0,\tilde{T}\to T_{\infty }asy\to \infty \end{array}$$

(8)

where, u is the tangential velocity of the free fluid which is exterior normal to the stretching sheet and $\breve{a}={\displaystyle \frac{\sqrt{\breve{K}}}{\breve{\alpha }}}$ in which $\breve{K}$ is the permeability, $u_w\left(x\right)$ stretching velocity and $\breve{\alpha }$ is a dimensionless parameter which depends only on the properties of the fluid and permeable material.

3. Thermophysical Characteristics of Al₂O₃ and $\mathbf{\gamma }$-Al₂O₃ Nanofluids

The heat capacitance ${\left(\rho C_p\right)}_{nf}$ and the effective dynamic density ${\rho }_{nf}$ of the nanofluids have been provided by the following expressions

$$\left.\begin{array}{c}\hfill {\rho }_{nf}=(1-\phi ){\rho }_f+\phi {\rho }_s,\\ \\ \hfill {\left(\rho C_p\right)}_{nf}=(1-\phi ){\left(\rho C_p\right)}_f+\phi {\left(\rho C_p\right)}_s,\end{array}\right\}$$

(9)

where the solid volume fraction of nanofluids is denoted by $\phi$ whereas the nanofluid’s dynamic viscosity is characterized by

$$\left.\begin{array}{c}\hfill \frac{{\mu }_{nf}}{{\mu }_f}={(1-\phi )}^{-2.5}(forA{l}_2{O}_3-water)\\ \hfill \frac{{\mu }_{nf}}{{\mu }_f}=123{\phi }^2+7.3\phi +1(for\gamma -A{l}_2{O}_3-water)\\ \hfill \frac{{\mu }_{nf}}{{\mu }_f}=306{\phi }^2-0.19\phi +1(for\gamma -A{l}_2{O}_3-C_2{H}_6{O}_2).\end{array}\right\}$$

(10)

An expression for the nanofluid’s effective thermal conductivity is

$$\left.\begin{array}{c}\hfill \frac{k_{nf}}{k_f}=\frac{k_s+2k_f-2\phi (k_f-k_s)}{k_s+2k_f+\phi (k_f-k_s)}(forA{l}_2{O}_3-water)\\ \hfill \frac{k_{nf}}{k_f}=4.97{\phi }^2+2.72\phi +1(for\gamma -A{l}_2{O}_3-water)\\ \hfill \frac{k_{nf}}{k_f}=28.905{\phi }^2+2.87\phi +1(for\gamma -A{l}_2{O}_3-C_2{H}_6{O}_2)\end{array}\right\}$$

(11)

4. Non-Dimensionalization through Similarity Transformation

By making use of similarities, transformation is achieved by

$$\eta =\sqrt{\frac{a}{v_f}}y,\tilde{u}=ax{f}^{\prime }\left(\eta \right),\tilde{v}=-{\left(a{v}_f\right)}^{\frac{1}{2}}f\left(\eta \right),\theta =\frac{\tilde{T}-T_{\infty }}{T_{\omega }-T_{\infty }}$$

(12)

4.1. Momentum Equations

The Equation (2) regulate the boundary layer into non-dimensional ordinary differential equations which can be written in the following way,

$$\left.\begin{array}{c}\hfill f^{‴}-{(1-\phi )}^{-2.5}\left(1-\phi \left(\frac{{\rho }_s}{{\rho }_f}\right)\right)(f^{\prime }-f{f}^{″})+{\epsilon }^2=0(forA{l}_2{O}_3-H_{2}O)\\ \\ \hfill f^{‴}-\left(\frac{1-\phi \left(\frac{{\rho }_s}{{\rho }_f}\right)}{306{\phi }^2-0.19\phi +1}\right)(f^{\prime }-f{f}^{″})+{\epsilon }^2=0(for\gamma -A{l}_2{O}_3-water)\\ \\ \hfill f^{‴}-\left(\frac{1-\phi \left(\frac{{\rho }_s}{{\rho }_f}\right)}{123{\phi }^2+7.3\phi +1}\right)(f^{\prime }-f{f}^{″})+{\epsilon }^2=0(for\gamma -A{l}_2{O}_3-C_2{H}_6{O}_2).\end{array}\right\}$$

(13)

4.2. Energy Equations

The temperature Equation (3) regulate the boundary layer into non-dimensional ordinary differential equations which can be explained as,

$$\left.\begin{array}{c}\hfill {\theta }^{″}\left(\frac{{\tilde{k}}_{nf}}{{\tilde{k}}_f}+\frac{4}{3}R\right)-p_r\left({(1-\phi )}^{2.5}+\phi \left(\frac{{\left(\rho C_p\right)}_s}{{\left(\rho C_p\right)}_f}\right)\right)(f{\theta }^{\prime }-2\theta f^{\prime })=0forA{l}_2{O}_3-H_{2}O,\\ \hfill {\theta }^{″}\left(\frac{{\tilde{k}}_{nf}}{{\tilde{k}}_f}+\frac{4}{3}R\right)-p_r\left({(1-\phi )}^{2.5}+\phi \left(\frac{{\left(\rho C_p\right)}_s}{{\left(\rho C_p\right)}_f}\right)\right)(f{\theta }^{\prime }-2\theta f^{\prime })=0for\gamma -A{l}_2{O}_3-water,\\ \hfill {\theta }^{″}\left(\frac{{\tilde{k}}_{nf}}{{\tilde{k}}_f}+\frac{4}{3}R\right)-p_r\left({(1-\phi )}^{2.5}+\phi \left(\frac{{\left(\rho C_p\right)}_s}{{\left(\rho C_p\right)}_f}\right)\right)(f{\theta }^{\prime }-2\theta f^{\prime })=0for\gamma -A{l}_2{O}_3-C_2{H}_6{O}_2.\end{array}\right\}$$

(14)

4.3. Boundary Conditions

The related boundary conditions are as follows:

For Momentum equation

$$f\left(0\right)=0,f^{\prime }\left(0\right)=1\&f^{\prime }(\infty )=0.$$

(15)

For Energy equation

$$\theta \left(0\right)=1\&\theta (\infty )=0.$$

(16)

4.4. Solution Methodology

In this part, an overview of the solution to the non-linear ODE with regard to the boundary conditions is offered. A well-known method known as the shooting approach is used in order to show the solution for these non-linear ODEs. MATLAB’s built-in solver called bvp4c is used to do an analysis of the numerical results, with the domain set at zero to ${\eta }_{max}$.

Pseudo-Algorithm

To accomplish the transformation from BVP to IVP, the representations f by $Y_1$ and $\theta$ by $Y_4$ have been put into action:

$$\left.\begin{array}{c}\hfill Y_1^{\prime }=Y_2,Y_1\left(0\right)=0\\ \hfill Y_2^{\prime }=Y_3,Y_2\left(0\right)=1\\ \hfill Y_3^{\prime }=(A_1\times A_1)(Y_2-Y_1)Y_3-{ϵ}^2,Y_3\left(0\right)=I_1\\ \hfill Y_4^{\prime }=Y_5,Y_4\left(0\right)=1\\ \hfill Y_5^{\prime }=\left({\displaystyle \frac{A_5}{(A_6+(4/3)\lambda )}}\right)\ast P_r\ast (2\ast Y_4{Y}_2-Y_1)Y_5,Y_5\left(0\right)=I_2\end{array}\right\}$$

(17)

The preceding IVP is supplied by making use of a well-known shooting strategy in conjunction with the Runge–Kutta scheme. The starting conditions that are not present are represented by the notation $Y_3\left(0\right)=I_1$ and $Y_5\left(0\right)=I_2$, respectively. Newton’s technique, in its usual form, may be used to fill in missing beginning circumstances and still obtain accurate results. The numerical strategy of the shooting is applied to the missing values of $I_1$ and $I_2$ until it is unable to satisfy the tolerance $\zeta$, which is $max\left\{\right|Y_3\left({\eta }_{max}\right)|,|Y_5\left({\eta }_{max}\right)\left|\right\}<\zeta$ [20,21].

5. Important Physical Characteristics

The skin friction coefficient (shear stress rate) and the Nusselt number (rate of heat transfer) are two physical characteristics of importance in engineering problems. The skin friction coefficient $C_f$ is used to calculate the shear stress at the stretched sheet which can be defined as:

$$\begin{array}{c}\hfill c_f=-\frac{-2{\mu }_{nf}}{{\rho }_f{u}_w^2}{\left(u_y\right)}_{y=0}.\end{array}$$

(18)

By utilizing Equation (18) into Equation (13), we have the following expressions:

$$\left.\begin{array}{c}\hfill \frac{1}{2}R{e}_x^{\frac{1}{2}}{c}_f=-{(1-\phi )}^{-2.5}{f}^{″}\left(0\right),(forA{l}_2{O}_3-H_{2}O)\\ \hfill \frac{1}{2}R{e}_x^{\frac{1}{2}}{c}_f=-(123{\phi }^2+7.3\phi +1)f^{″}\left(0\right),(for\gamma -A{l}_2{O}_3-H_{2}O)\\ \hfill \frac{1}{2}R{e}_x^{\frac{1}{2}}{c}_f=-(306{\phi }^2-0.19\phi +1)f^{″}\left(0\right),(for\gamma -A{l}_2{O}_3-C_2{H}_6{O}_2)\end{array}\right\}$$

(19)

where the local Reynolds number is denoted by $R{e}_x={\displaystyle \frac{x{u}_w\left(x\right)}{v_f}}$, which completely base on the stretching velocity $U_{\infty }\left(x\right)$ and local skin friction coefficient $R{e}_x^{\frac{1}{2}}{c}_f$. The Nusselt number $N{u}_x$ can be defined as

$$\begin{array}{c}\hfill N{u}_x=\frac{x{\tilde{q}}_w}{{\tilde{k}}_f(T_{\omega }-T_{\infty })},\end{array}$$

(20)

where, the local surface heat flux is $\tilde{q}=k_{nf}{\left(T_y\right)}_{y=0}$. Based on Equation (12), we obtain the following Nusselt number,

$$\left.\begin{array}{c}\hfill R{e}_x^{-\frac{1}{2}}N{u}_x=\left(\frac{k_s+2k_f-2\phi (k_f-k_s)}{k_s+2k_f+\phi (k_f-k_s)}\right)(-{\theta }^{\prime }\left(0\right))(forA{l}_2{O}_3-H_{2}O)\\ \hfill R{e}_x^{-\frac{1}{2}}N{u}_x=(4.97{\phi }^2+2.72\phi +1)(-{\theta }^{\prime }\left(0\right))(for\gamma -A{l}_2{O}_3-H_{2}O)\\ \hfill \frac{k_{nf}}{k_f}=(28.905{\phi }^2+2.87\phi +1)(-{\theta }^{\prime }\left(0\right))(for\gamma -A{l}_2{O}_3-C_2{H}_6{O}_2).\end{array}\right\}$$

(21)

6. Discussion

The graphical representation of the numerical findings is shown for the nanoparticles of Al${}_2$O${}_3$ and $\gamma$-Al${}_2$O${}_3$ combined with water and ethylene glycol as the base fluids. An investigation into the effects of a number of disparate parameters, including the stagnation point parameter $ϵ$, the dimensionless slip parameter $\stackrel{‘}{\alpha }$, the solid volume fraction $\phi$, Prandtl number $P_r$, reduced skin friction coefficient $R{e}_x^{\frac{1}{2}}{c}_f$, reduced Nusselt number $R{e}_x^{-\frac{1}{2}}N{u}_x$, shear stress $f^{″}\left(0\right)$, velocity profile $f^{\prime }\left(\eta \right)$ and temperature profile $\theta \left(\eta \right)$ have been analyzed with regard to the slip boundary conditions on a stretching sheet conjunction with stagnation-point flow phenomena.

Table 4. Results comparison with earlier findings for Al${}_2$O${}_3$ nanofluids.

$\mathit{\phi }$	Hamid et al. [16]	Vishnu et al. [45]	Present Work
0.05	1.00538	1.00537	1.00530
0.10	0.99877	0.99877	0.98866
0.15	0.98185	0.98184	0.97132
0.2	0.95992	0.95591	0.94581

presents the results of a comparison between the values of $-f^{″}\left(0\right)$ that were reported for Al${}_2$O${}_3$-water by Hamad et al. [16] and Vishnu et al. [45]. We are confident in continuing to utilize the existing code since the results indicate great agreement. Through the use of the figures, the authors examine the impact that the nanoparticle volume friction has on the velocity profile, temperature profile, skin friction coefficient and the reduced Nusselt number.

The influence of the nanoparticle volume fraction on the velocity profile is described via the Figure 2, Figure 3 and Figure 4 for Al${}_2$O${}_3$-water, Al${}_2$O${}_3$-ethylene glycol and $\gamma$-Al${}_2$O${}_3$-water nanofluids. It has been determined, based on the findings presented here, that higher values of nanoparticle volume fraction result in an increase in the velocity of oxide nanofluids. It may be said that the velocity of the $\gamma$-Al${}_2$O${}_3$ nanofluids is greater than that of the Al${}_2$O${}_3$ nanofluids [16,45]. This is because the thickness of the momentum boundary layer in $\gamma$-Al${}_2$O${}_3$ nanofluids is much greater than in Al${}_2$O${}_3$ nanofluids. When comparing nanofluids according to different base fluids, those based on ethylene glycol have a faster velocity than those based on water. Nanofluids based on water have a smaller momentum boundary barrier thickness compared to nanofluids based on ethylene glycol. The results which are depicted from Figure 2, Figure 3 and Figure 4 show that the Al${}_2$O${}_3$-water mixture has a lower velocity but the $\gamma$-Al${}_2$O${}_3$-Water mixture has a greater velocity.

Figure 5, Figure 6 and Figure 7 show the impact of the nanoparticle volume fraction on the temperature profile of the nanofluids. It has been observed that when the values of the nanoparticle volume fraction rise, the temperatures of both $\gamma$-Al${}_2$O${}_3$ and Al${}_2$O${}_3$ nanofluids rise as well. Therefore, nanofluids based on water have a steeper temperature profile compared to those on ethylene glycol (shallower profile) [16,45,46,47]. This is owing to the fact that the thermal diffusivity of water is significantly greater than that of ethylene glycol whereas, the Prandtl number $P_r$ of water is substantially lower than that of ethylene glycol [34,40,45]. Comparing nanoparticles, Al${}_2$O${}_3$-water has a higher temperature profile than $\gamma$-Al${}_2$O${}_3$-water, while $\gamma$-Al${}_2$O${}_3$-ethylene glycol is higher. It is possible to draw the conclusion that the Al${}_2$O${}_3$ nanoparticles and the $\gamma$-Al${}_2$O${}_3$ nanoparticles have opposing impacts on the temperature profile when used in conjunction with various base fluids such as water and ethylene glycol. To achieve cooling effects, it is possible to make use of nanofluids that include ethylene glycol functioning as the basis fluid [45].

The impacts of different physical parameters in the flow model on the velocity of the nanofluids within the boundary layer are depicted in Figure 8, Figure 9 and Figure 10, while Figure 11, Figure 12 and Figure 13 show the results for temperature profile. The influence of the slip parameter $\breve{a}$ against the similarity variable $\eta$ on the velocity and temperature profiles show that an increase in the value of a parameter known as the slip velocity results in a reduction in the velocity of the nanofluids also an increase in the value of the same parameter results in a rise in the temperature [45,47].

Figure 14, Figure 15 and Figure 16 illustrate the effect that the stagnation parameter $ϵ$ has on the velocity profile in which the value of $ϵ$ rises, the velocity of the nanofluids also rises but the temperature and the nanoparticle volume fraction fall [3,13,46]. If we assume that the velocity of the stream remains the same then an increase in the value of $ϵ$ will result in a decrease in the stretch velocity.

Figure 17, Figure 18 and Figure 19 show the result for different values of radiative heat flux parameter R. When radiative heat flux increases, the temperature profile shows enhancement whereas the velocity profile shows a downfall. Even though the velocity profile shows a trend that is slightly decreasing as well as a trend that is slightly increasing as the values of thermal radiation increased [12,16,34]. These results explain that an increase in the value of the parameter R, thermal radiation, has an insignificant impact on the fluid velocity. This is the case despite the fact that the velocity profile shows a slight decreasing trend [17,29,47]. In point of fact, the fluid viscosity has a propensity to grow with increased resistance to distortion, which results in a reduction within the velocity profile. On the other hand, it has a propensity to drop as internal heat production and thermal radiation both increase.

Figure 20 and Figure 21 illustrate the value fluctuation $\frac{1}{2}R{e}_x^{\frac{1}{2}}{c}_f$ which is the local (reduced) skin friction coefficient, as well as $R{e}_x^{-\frac{1}{2}}N{u}_x$, located along the y-axis and the nanoparticle volume fraction $\phi$ along the x-axis. It has been shown that an increase in $\phi$ values, the skin friction coefficient and Nusselt number are found to increase as well [46,48,49]. Therefore, the skin friction coefficient is greater for nanofluid $\gamma$-Al${}_2$O${}_3$ than it is for Al${}_2$O${}_3$. For different base fluids, the amount of skin friction coefficient by Al${}_2$O${}_3$ with base fluid as water is greater than that produced by Al${}_2$O${}_3$ ethylene glycol as base fluid. However, $\gamma$-Al${}_2$O${}_3$ nanoparticles have been shown to exhibit the reverse tendency. Nanofluids that are based on ethylene glycol have a greater Nusselt number than nanofluids that are based on water. When compared to other nanoparticles, the Nusselt number for $\gamma$-Al${}_2$O${}_3$ nanoparticles is much greater.

7. Conclusions

The boundary layer flow of nanofluids of Al${}_2$O${}_3$ and $\gamma$-Al${}_2$O${}_3$ with various base fluids is carried out on a stretching sheet. We have been successful in developing a numerical solution for the steady BL flow and heat transfer at the stagnation point of nanofluids with a slip boundary condition. The velocity of the sheet’s stretching in its own plane $u_w$, is different from the velocity of the external flow $U_{\infty }$. The similarity ordinary differential equations that were produced are solved using the shooting method in conjunction with the $RK-4$ approach. For $\gamma$-Al${}_2$O${}_3$ nanofluids, we use models of viscosity and thermal conductivity that are built from actual experiments. For Al${}_2$O${}_3$, the viscosity model from Brinkman and the thermal conductivity model from Maxwell, are used along thermal radiation. The following are some particular findings that has be drawn from this investigation.

• Both the momentum and the thermal boundary layer, thickness increase with increasing nanoparticle volume fraction of Al${}_2$O${}_3$ and $\gamma$-Al${}_2$O${}_3$ nanoparticles. The velocity of nanofluids $\gamma$-Al${}_2$O${}_3$, is greater than that of Al${}_2$O${}_3$, in a comparison of nanoparticles [16,45,46,47]. Nanofluids made from ethylene glycol move more quickly than those made from water.
• However, in terms of the temperature distribution, $\gamma$-Al${}_2$O${}_3$-C${}_2$H${}_6$O${}_2$ (ethylene glycol) is greater than that of Al${}_2$O${}_3$-ethylene glycol, whereas that of Al${}_2$O${}_3$-water is more than that of $\gamma$-Al${}_2$O${}_3$-water. The temperature profile increased with increasing values of parameter R, indicating that the thermal boundary layer thickness increases with increasing thermal radiation. As a consequence, a higher radiation parameter causes more heat to be produced in the flow, resulting in an increase in the temperature profile of the fluid. Nanofluids that are based on ethylene glycol have a lower temperature profile than those based on water. There is potential for cooling applications of nanofluids based on ethylene glycol [16,45,47].
• When compared to Al${}_2$O${}_3$ nanofluids, $\gamma$-Al${}_2$O${}_3$ nanofluids have greater skin friction. The skin friction is greater for Al${}_2$O${}_3$-water than it is for Al${}_2$O${}_3$-ethylene glycol. However, $\gamma$-Al${}_2$O${}_3$ nanoparticles exhibit the opposite tendency. The Nusselt number of ethylene glycol-based nanofluids is greater than that of water-based nanofluids. The Nusselt number for $\gamma$-Al${}_2$O${}_3$ nanoparticles is greater than that of other nanoparticles.