Insight into stagnant point flow of Eyring-Powell hybrid nanofluid comprising on enlarged version of Yamada-Ota and Xue model

Abstract

This paper shows the stagnant point flow of Powell-Eyring hybrid nanofluids due to a stretching sheet. The Xue and Yamada-Ota based on hybrid nanoliquid have been subjected to a comparison study that has been scrutinized. The suspension of two nanoparticles $\mathrm{AA}7072$ and $\mathrm{AA}7075$ with base fluid Ethylene glycol ( $\mathrm{EG}$ ) is studied through hybrid nanoliquid. With the implementation of the noteworthy suitable alteration, the system of equations in terms of ODEs is established. The bvp4c technique is then applied to obtain the numerical solution of reduced ODEs. The significance of physical quantities over thermal, velocity, force friction, and heat transport are elaborated in tabular form and also in graphical form. During this analysis, the Xue model produced less heat transport as compared to the Yamada-Ota hybrid nanofluid. The result shows that the Xue model produces less heat gradient equated to the Yamada-Ota model. The velocity profile is enhanced and the thermal profile has decayed with the larger value of the fluid parameter.

Introduction

A nanofluid is a kind of colloidal suspension of nanoparticles that may be produced by dispersing a tiny proportion of nanoparticles into a base fluid with low thermal conductivity. Most nanoparticles consist of metals, metal oxides, or carbon-based compounds. Nanomaterials have drawn significant demand from a wide variety of fields as a direct result of the rapid growth of nanomaterials and the discoveries made in the scientific community. Nanofluids are primarily used in the areas of fiber technology and thermal engineering. Using nanoliquids in such designs to improve thermal properties and enhance the rate of heat characteristics is fundamental to achieving the goal of improved cooling. The remarkable thermophysical properties of nanofluids have inspired numerous scholars to investigate nanofluids in engineering and industrial features. These features include food processing, solar thermal system, underground water, oil drilling, and amongst others. The research done in the literature^1,2 has resulted in a comprehensive knowledge of the colloidal suspension of nanoparticles to produce nanofluids to improve thermal conductivity. Due to a recent upgrade, a special class of nanofluids known as hybrid nanoliquid has been included in obtaining nanofluids. The prime benefit of studying a hybrid nanoliquid is that it can be controlled by defining an appropriate mixing of two or more nanoparticles with the base fluids. This is the case because hybrid nanofluids include more than two nanoparticles. In addition to their outstanding ability of thermal conductivity, hybrid nanoliquids have the potential to deliver significant advantages since nanoparticles are effectively disseminated throughout the fluid. Xu ³ used an extended model of hybrid nanofluids to research the flow of hybrid nanofluids, including a wide variety of nanoparticles. Researchers from Fallah et al. ⁴ looked at the hybrid nano liquid flow with a revolving disc. Tassaddiq et al. ⁵ equated the nanoliquid and the hybrid nanoliquid while also doing a mass and heat transfer study. Farooq et al. ⁶ investigated the second law of thermodynamics through hybrid nanofluid flow. The impact of non-linear thermal radiative through hybrid nanoliquid flow was conducted by Usman et al. ⁷ Chu et al. ⁸ conducted a thermal analysis of a hybrid nanofluid using a numerical method. Wang et al. ⁹ explained the Carreau hybrid nanofluid with different nanoparticles immersed in a porous medium. Revathi et al. ¹⁰ pondered the significance of the tri-hybrid nanofluid with CMC water in the presence of chemically reactive and convective flow. Animasaun et al. ¹¹ examined the time-dependent flow of Darcian Forchheimer flow of tri-hybrid nanofluid over a horizontal surface.

The developing field of engineering known as hybrid nanoliquid has caught the attention of many researchers looking for behavior to increase the efficiency of coolant procedures used in the engineering industry. These researchers were finding behaviors to improve the effectiveness of coolant techniques. A nanoliquid is a fluid created when powerful particles with diameters smaller than 100 nm are propagated in fluids. Nanoliquids may be used in a variety of applications. One of the noticeable properties of a nanoliquid is its poor thermal properties, which are the characteristics that might hinder the role of heat transfer. Generally, heat transfer fluids, like water motor oil and (CH₂OH)₂, have a restricted rate of heat transfer abilities and cannot satisfy the cooling needs of today’s society due to this weakness. Other heat transfer fluids, such as kerosene, are more effective at transferring heat. Nanoparticles in suspension have the potential to enhance the movement of fluid with the heat transport of the ordinary liquid. The researchers are conducting extensive investigations to shed light on various facets related to nanofluids. Nanoliquid foam was investigated by Gul et al. ¹² in a revolving channel using CCH heat flux with a hybrid nanofluid. Ramzan and Shaheen ¹³ looked at the effects of the C–C heat flux on the nanoliquid flow while being influenced by the H–H reaction. Nanoliquid comprised of CNTs and Darcy-Forchheimer was first shown by Hayat et al. ¹⁴ They created their essential fluid out of water. The C–C influence on Tangent hyperbolic liquid flow merged with second-order slip was started by Ramzan et al. ¹⁵ They solved the issue by using a methodology known as the Runge–Kutta Fehlberg method. Abbasi et al. ¹⁶ discussed the impact of the slip mechanism on hybrid nanoparticles using a wave curve surface.

One of the models of non-Newtonian fluid complexity is the Powell-Eyring model. This model’s ability to be dubbed the viscosity relaxation theory makes it a candidate for this category. The Powell-Eyring fluid has favorable positions due to other non-Newtonian fluids in terms of their properties. ¹⁷ Instead of being based on an empirical correlation, the Eyring–Powell fluid is built using concepts from the theory of liquid kinetics. According to the kinetic theory of fluids, the bonds between liquid molecules may be of any strength or weakness. According to the Powell–Eyring paradigm, the link of a vital molecule coincides with the non-Newtonian rapport, whereas the bond of fewer molecules correlates with the Newtonian connection. Therefore, the fluid of Powell-Eyring can increase or decrease the shear rate. This indicates that it may overcome the challenges posed and the power law liquid. ¹⁶ Non-Newtonian fluid with various of the researchers due to different geometries were carried out by Refs.^18–21 Nazeer et al.^22,23 examined the Eyring–Powell fluid in a pored channel using numerical and analytical methods. Patel and Timol ²⁴ made the models of Powell–Eyring more significant and effective with their improvements. The nature of it was pretty complicated. The MHD flow of Eyring–Powell liquid was evaluated numerically by Sher Akbar et al. ²⁵ This strategy was used to analyze the magnetic flux across an extended plate. The MHD flow of Eyring–Powell fluid over a cylinder using Newtonian heating was elaborated by Hayat et al. ²⁶ Riaz et al. ²⁷ researched the impact of Eyring–Powell fluid in a rectangular channel. Recently, Ahmed et al. ²⁸ probed the MHD flow of Eyring Powell nanofluid due to three different geometries.

In real-life applications involving boundary layer flow developed by shrink or stretch surfaces in various industry procedures such as polymer production, glass production, modern extrusion procedures, and many others. These problems can be applied in a variety of contexts. At first, Crane ²⁹ thought of using a sheet that could be stretched to watch the fluid flow, and he explained the solution to the issue in its bunged form. Later, a variety of study has been produced due to the sheets contracting and expanding. Munawar et al. ³⁰ discovered the exact solution to the flow issue due to slip effects between two stretched sheets. The time-dependent flow of hybrid nanofluid transmission of heat phenomena across a stretch/shrink surface was described by Waini et al. ³¹ Manjunatha et al. ³² researched the boundary layer flow of a hybrid nanofluid past a stretchy surface. The authors, Devi and Devi, ³³ computed the 3-D flow of the combined effect of Newtonian heating with Lorentz force on a hybrid nanofluid due to stretchy sheets. Miklavčič and Wang ³⁴ described the time-independent flow behavior using a shrunk sheet as a visual aid. Fang et al. ³⁵ discovered the unsteady flow of viscous fluid flow that involves the participation of a shrinking surface. Fang ³⁶ and Fang and Zhang ³⁷ investigated the power-law fluid on the boundary layer flow due to a stretching sheet. Rohni et al. ³⁸ researched the transmission of heat with flow mechanisms using a shrinking sheet. Their research was based on the idea that suction effects occur in nanofluids. Stretching with different phenomena with effects and different geometries has been described in Refs.^39–42

In the research carried out by Abbas et al., ⁴³ an investigation of the Xue and Yamada-Ota model characteristics of a hybrid nanofluid was carried out while it passed a stretching surface. They investigated the flow problem that occurred behind a stagnation point. We have extended their work and built a mathematical model for the flow and heat transfer of hybrid nanofluid (AA7072-AA7075/EG) across a stretching surface with a stagnation point as a result of being motivated by the works that have been shown above, particularly the study that was conducted by Abbas et al. ⁴³ The current article has the originality to examine the impact of the Eyring-Powell hybrid nanoliquid toward stagnant flow using the Xue and Yamada-Ota model. This research, to establish a hybrid nanofluid by the suspension of two nanoparticles AA7072 and AA7075 with ethylene glycol (EG) base fluid. In addition, this research investigates an additional consequence that arises from the interaction of two models of hybrid nanofluid (Yamada-Ota and Xue) with the occurrence mentioned earlier. The calculation is carried out using the bvp4c procedure to accomplish this goal. The goal of this study was to find answers to the following related research questions:

• I. How does the velocity of the dynamics of chemically reactive EG conveying AA7072 and AA7075 compare to the case of EG conveying AA7072 and AA7075 at different levels of stagnant velocity, volume fraction, and fluid velocity? Also, how does this compare to the case of EG conveying Xue and Yamada-Ota model.

• II. In the presence of varying amounts of thermal radiation, volume fractions of the overall hybrid nanofluid, and heat sources, to what extent does the addition of AA7072 nanoparticles to the already present chemically reactive EG conveying AA7072 and AA7075 nanoparticles affect the temperature distribution across the domain? This question will be addressed in the following way:

An examination is made of a graphic depiction of the Xue and Yamada-Ota models for the properties of the fluid, comprising the velocity profile and temperature distribution related to the limitations of the different variables. The numeric values of Skin friction and Nusselt number connected to the variables are assessed and visually identified.

Model formulation

Assume that an incompressible, unsteady boundary layer flows toward the stagnant flow past a stretching sheet with the convective condition. For the hybrid nanofluid, Non-Newtonian Powell–Eyring mathematical model is considered. Two nanoparticles namely $\mathrm{AA}7072$ and $\mathrm{AA}7075$ with base fluid Ethylene glycol ( $\mathrm{EG}$ ) are studied to show the comparison result of Yamada and Ota ⁴⁴ and Xue model. ⁴⁵ The coordinates are taken with the horizontal and the vertical surface along the stretching velocity $U_w=\frac{\mathrm{ax}}{1-\delta t}$ and free stream velocity is $U_e=\frac{\mathrm{bx}}{1-\delta t}$ . The convection of heat from an evenly heated fluid spread across the plate warms its outer lateral surface, and the temperature does not change as one moves further away from the plate. Radiative heat transmission is considered using Rosseland approximations. The geometry of the problem is seen in Figure 1.

Figure 1.

Geometrical representation of the flow problem.

Based on the aforementioned presumptions, the expression in terms of partial differential equations is used to create the mathematical model. The Eyring-Powell hybrid nanofluid model can be presented in terms of a system of differential equations.^44–46

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

(1)

$$\begin{array}{c}\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=U_e\frac{\partial U_e}{\partial x}+({\nu }_{\mathrm{hnf}}+\frac{1}{a_{1\varrho {\rho }_{\mathrm{hnf}}}})\hfill \\ \frac{{\partial }^{2}u}{\partial y^2}-\frac{1}{2\varrho a_1^2{\rho }_{\mathrm{hnf}}}{\left(\frac{\partial u}{\partial y}\right)}^2\frac{{\partial }^{2}u}{\partial y^2}\hfill \end{array}$$

(2)

$$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{k_{\mathrm{hnf}}{\partial }^{2}T}{(\rho C_p{)}_{\mathrm{hnf}}\partial y^2}-\frac{1\partial q_r}{(\rho C_p{)}_{\mathrm{hnf}}\partial y\prime }$$

(3)

$q_r$ defines as

$$q_r=\frac{-4\sigma }{3k}\frac{\partial T^4}{\partial y}=\frac{-16\sigma T_{\infty }^3}{3k}\frac{\partial T}{\partial y}$$

(4)

The boundary constraints are applied as follows:

$$\begin{array}{c}u=U_w(x,t),v=0,-k_{\mathrm{hnf}}\frac{\partial T}{\partial y}=h_f(T_f-T)\mathrm{at}y=0,\hfill \\ u\to U_e(x,t),T\to T_{\infty }\mathrm{as}y\to \infty .\hfill \end{array}$$

(5)

Here, $T$ shows the fluid temperature, $a_1\wedge \varrho$ shows the parameter of fluid, ${\upsilon }_{\mathrm{hnf}}$ represent the kinematic viscosity, ${\rho }_{\mathrm{hnf}}$ represent the density, $k_{\mathrm{hnf}}$ represent the thermal conductivity, and ${\left(\rho C_p\right)}_{\mathrm{hnf}}$ represent the heat capacity.

Similarity transformation

It is necessary to carry out the appropriate transformation that follows.

$$\begin{array}{c}\zeta =y\sqrt{\frac{a}{v_f(1-\delta t)}},u=\frac{\mathrm{ax}}{1-\delta t}F\prime \left(\zeta \right),v=-\sqrt{\frac{a{v}_f}{1-\delta t}}F\left(\zeta \right),\hfill \\ \vartheta \left(\zeta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},h\left(t\right)=\frac{d}{\sqrt{1-\delta t}}\hfill \end{array}$$

(6)

The noteworthy nanofluid models have been carried out through Yamada-Ota ⁴⁴ and Xue ⁴⁵ models. These models were considered in an extension for hybrid nanofluid flow by Abbas et al.^43,47 and considered the solid nanoparticles with ordinary fluid.

In addition, the thermophysical characteristics of the Yamada-Ota ⁴⁴ and Xue ⁴⁵ hybrid nanofluid are demarcated as:

$$\begin{array}{c}\frac{k_{\mathrm{hnf}}}{k_f}=\hfill \\ \frac{1+(1-\frac{k_{\mathrm{bf}}}{\mathrm{ks}2}){\varphi }_2\frac{L}{R}{\varphi }_2^{0.2}+\frac{k_{\mathrm{bf}}}{\mathrm{ks}2}{\varphi }_2\frac{L}{R}{\varphi }_2^{0.2}+2{\varphi }_2\ln \left(\frac{\mathrm{ks}2+k_{\mathrm{bf}}}{2\mathrm{ks}2}\right)}{1+2{\varphi }_2\left(\frac{k_{\mathrm{bf}}}{\mathrm{ks}2-k_{\mathrm{bf}}}\right)\ln \left(\frac{\mathrm{ks}2+k_{s2}}{2\mathrm{ks}2}\right)-{\varphi }_2}\hfill \end{array}$$

(7)

$$\begin{array}{c}\frac{k_{\mathrm{bf}}}{k_f}=\hfill \\ \frac{1+(1-\frac{k_f}{\mathrm{ks}1}){\varphi }_1\frac{L}{R}{\varphi }_1^{0.2}+\frac{k_{\mathrm{bf}}}{\mathrm{ks}2}{\varphi }_1\frac{L}{R}{\varphi }_1^{0.2}+2{\varphi }_1\ln \left(\frac{\mathrm{ks}1+k_{\mathrm{bf}}}{2\mathrm{ks}1}\right)}{1+2{\varphi }_1\left(\frac{k_f}{\mathrm{ks}1-k_f}\right)\ln \left(\frac{\mathrm{ks}1+k_f}{2k_f}\right)-{\varphi }_1}\hfill \end{array}$$

(8)

The Xue model ⁴² can be designed in the form of nanofluid and hybrid nanofluid

$$\begin{array}{c}\multicolumn{1}{c}{\frac{k_{\mathrm{hnf}}}{k_f}=\frac{1-{\varphi }_2+2{\varphi }_2\left(\frac{k_{s2}}{\mathrm{ks}2-k_{\mathrm{bf}}}\right)\ln \left(\frac{\mathrm{ks}2+k_{\mathrm{bf}}}{2k_{\mathrm{bf}}}\right)}{1-{\varphi }_2+2{\varphi }_2\left(\frac{k_{\mathrm{bf}}}{\mathrm{ks}2-k_{\mathrm{bf}}}\right)\ln \left(\frac{\mathrm{ks}2+k_{\mathrm{bf}}}{2k_{\mathrm{bf}}}\right)}}\end{array}$$

(9)

$$\begin{array}{c}\frac{k_{\mathrm{bf}}}{k_f}\hfill \\ =\frac{1+(1-\frac{k_f}{\mathrm{ks}1}){\varphi }_1\frac{L}{R}{\varphi }_1^{0.2}+\frac{k_{\mathrm{bf}}}{\mathrm{ks}2}{\varphi }_1\frac{L}{R}{\varphi }_1^{0.2}+2{\varphi }_1\ln \left(\frac{\mathrm{ks}1+k_{\mathrm{bf}}}{2\mathrm{ks}1}\right)}{1+2{\varphi }_1\left(\frac{k_f}{\mathrm{ks}1-k_f}\right)\ln \left(\frac{\mathrm{ks}1+k_f}{2k_f}\right)-{\varphi }_1}\hfill \end{array}$$

(10)

Using the similarity transmission explained in equation (6), equations (2) and (3) have rendered into the ODEs

$$\begin{array}{l}\left(\frac{1}{{\Lambda }_1{\Lambda }_2}+\frac{\alpha }{{\Lambda }_1}\right)F^{‴}+F^{″}F-{F\prime }^2-\frac{\alpha \beta }{{\Lambda }_1}{F}^{″}2F^{‴}\\ +{\epsilon }^2-\lambda (F{F}^{\prime }+0.5\zeta F^{″}-1)+\lambda \epsilon =0\end{array}$$

(11)

$$\vartheta \prime \prime ({\Lambda }_3+\frac{4}{3}\mathrm{Rd})+\mathrm{Pr}\frac{{\Lambda }_3}{{\Lambda }_4}(F\vartheta \prime -F\prime \vartheta -\lambda (\vartheta +0.5\vartheta \prime \zeta ))=0.$$

(12)

The boundary conditions are described as:

$$\begin{array}{c}F\left(0\right)=0,F\prime \left(0\right)=1,{\Lambda }_4\vartheta \prime \left(0\right)=-\mathrm{Bi}(1-\vartheta \left(0\right)),\hfill \\ \multicolumn{1}{c}{F\prime \left(\infty \right)\to \epsilon ,\vartheta \left(\infty \right)\to 0.}\end{array}\}$$

(13)

In which:

$$\mathrm{Dynamicviscosity}={\Lambda }_1=\frac{{\mu }_{\mathrm{hnf}}}{{\mu }_f}=\frac{1}{{(1-{\varphi }_1-{\varphi }_2)}^{2.5}},$$

$$\mathrm{Density}={\Lambda }_2=\frac{{\rho }_{\mathrm{hnf}}}{{\rho }_f}=(1-{\varphi }_2)[(1-{\varphi }_1)+\frac{{\varphi }_1{\rho }_{1s}}{{\rho }_f}]+\frac{{\varphi }_2{\rho }_{2s}}{{\rho }_f},$$

$$\begin{array}{c}\mathrm{Thermalconductivity}={\Lambda }_3=\frac{k_{\mathrm{hnf}}}{k_f}=\frac{k_{2s}+2k_f-2{\varphi }_2(k_f-k_{2s})}{k_{2s}+2k_f+{\varphi }_2(k_f-k_{2s})}\hfill \\ \times \left(k_{\mathrm{nf}}\right)k_{\mathrm{nf}}=\frac{k_{1s}+2k_f-2{\varphi }_1(k_f-k_{1s})}{k_{1s}+2k_f+{\varphi }_1(k_f-k_{1s})},\hfill \end{array}$$

$$\begin{array}{c}\mathrm{Heatcapacity}={\Lambda }_4=\frac{{\left(\rho c_p\right)}_{\mathrm{hnf}}}{{\left(\rho c_p\right)}_f}\hfill \\ =(1-{\varphi }_2)[(1-{\varphi }_1)+\frac{{\varphi }_1{\left(\rho c_p\right)}_{1s}}{{\left(\rho c_p\right)}_f}]+\frac{{\varphi }_2{\left(\rho c_p\right)}_{2s}}{{\left(\rho c_p\right)}_f}\hfill \end{array}$$

In the above expression, the formulation symbols ${\varphi }_1$ and ${\varphi }_2$ exhibits the solid nanoparticles volume fraction of the (AA7072) and (AA7075) nanoparticles, respectively. However, the subscripts $f,\mathrm{hnf},1s$ , and $2s$ show the ordinary fluid, hybrid nanoliquid, and the two different solid nanoparticles. Table 1 displays the hybrid nanoparticles’ physical significance characteristics, along with the essential base fluid.

Table 1.

Thermophysical significance of the hybrid nanofluid.

Physical properties	$C_p(J/\mathrm{kgK})$	$\rho (\mathrm{kg}/m^3)$	$k(W/\mathrm{mK})$
$\mathrm{EG}$	2415	1114	0.252
$\mathrm{AA}7072$	893	2720	222
$\mathrm{AA}7075$	960	2810	173

However, the transformation equations non-dimensional physical quantities can defined as the unsteady variable $(\lambda =\frac{a}{\delta })$ , the fluid parameters are $(\alpha =\frac{1}{{\mu }_f\varrho a_1})$ and $(\beta =\frac{U_w^3}{2{\varrho }^2{v}_f})$ , stretching variable $(\epsilon =\frac{b}{\delta }),$ the Radiation parameter $\mathrm{Rd}=\frac{4\sigma *T_{\infty }^3}{k*k_f}$ , the Prandtl number $\mathrm{Pr}{\mu }_f$ , and the Biot number $\mathrm{Bi}=h_f\sqrt{\frac{{\upsilon }_f}{a}}\times \frac{1}{k_f}$ .

The drag friction $C_f$ and the local Nusselt number $N{u}_x$ which are presented by:

$$C_f=\left(\frac{{\tau }_w}{{\rho }_f{u}_w^2}\right),\mathrm{Nu}=\frac{x}{k_f(T_w-T_{\infty })}(q_w-{\left(q_r\right)|}_{y=0}),$$

(14)

The $q_w$ and ${\tau }_w$ shear stress is

$$q_w=-k_{\mathrm{hnf}}[1+\frac{16\sigma T_{\infty }^3}{3k*k_f}]\left(\frac{\partial T}{\partial y}\right){\vee }_{y=0},{\tau }_w=$$

(15)

Given equations (6) and (15), we get

$$\begin{array}{c}{{\Re }_x}^{1/2}{C}_f[(\frac{1}{{\Lambda }_1}+\alpha )F^{″}\left(0\right)-\frac{\alpha \beta }{3}{\left(F^{″}\left(0\right)\right)}^3],\hfill \\ {\Re }^{-1/2}\mathrm{Nu}=-({\Lambda }_3+\frac{4}{3}\mathrm{Rd}){\vartheta }^{\prime }\left(0\right).\hfill \end{array}$$

(16)

Where ${\Re }_x\frac{u_{w}x}{{\upsilon }_f}$ called the local Reynolds number.

Numerical solution

Using the Bcp4c approach, a set of nonlinear ordinary differential equations (11) and (12) along with the boundary conditions (13) are utilized numerically. In the beginning, the set of highly ordinary differential equations (11) and (12) and (13) are changed into first-order differential equations by following the approach that is outlined

$$F″=T_3,F″\prime =T_3\prime ,\theta =T_4,\theta \prime =T_5,\theta ″=T_5\prime$$

$$T_3\prime =\frac{-T_3{T}_1+T_2^2-{\epsilon }^2+\lambda (T_1{T}_2+0.5\zeta T_3-1)-\lambda \epsilon }{(\frac{1}{{\Lambda }_1{\Lambda }_2}+\frac{\alpha }{{\Lambda }_1})+\frac{\alpha \beta }{{\Lambda }_1}{T}_3^2}$$

(17)

$$\begin{array}{c}\multicolumn{1}{c}{T_5\prime =\frac{-\mathrm{Pr}\frac{{\Lambda }_3}{{\Lambda }_4}(T||1T_5-T_4{T}_2+\lambda (T_4+0.5\zeta T_4))}{({\Lambda }_3+\frac{4}{3}\mathrm{Rd})}}\end{array}$$

(18)

Boundary conditions is

$$\begin{array}{c}{T}_1\left(0\right)=0,T_2\left(0\right)=1,{\Lambda }_4{T}_5\left(0\right)\hfill \\ =-\mathrm{Bi}(1-T_4\left(0\right)),T_2\left(\infty \right)\to \epsilon ,T_4\left(\infty \right)\to 0.\hfill \end{array}$$

(19)

Iterations are done with the numbers until an accuracy of ${10}^{-6}$ is reached.

Result and discussion

Using graphical illustrations, the role of the numerous mechanisms of the flow profiles containing velocity $F\prime \left(\zeta \right)$ and temperature $\vartheta \left(\zeta \right)$ are debated for the model of Xue and Yamada-Ota due to various cases. The following are the constant values that are associated with the parameters Pr = 40, $\lambda$  = 0.2, Rd = 0.2, $ϵ$  = 2.2, $\alpha$  = 0.5, $\beta$  = 0.2, and $\gamma$  = 0.2. Through the investigation of the Xue and Yamada-Ota model, the characteristics of pertinent quantities over velocity $F\prime \left(\zeta \right)$ and thermal $\vartheta \left(\zeta \right)$ fields are scrutinized by graphical representation. Moreover, a comparison of $F(0)$ with the outcomes that were accessible in the limiting circumstances was carried out using Table 2 and the results were found to be a fantastic accord. Tables 3 and 4 elaborates that the drag friction and heat transport are mathematically elaborated for the Xue and Yamada-Ota models. The value of Pr = 40 for the Ethylene glycol has been fixed for the entire computation. ⁴⁶

Table 2.

Comparison of $F\prime \prime \left(0\right)$ for different values of the unsteadiness parameter $\lambda$ when rest of the parameters are zero.

$\lambda$	$F\prime \prime \left(0\right)$
	Ali and Zaib 48	Current outcomes
0.8	1.260691	1.260691
1.2	1.377625	1.377625

Table 3.

Numerical values of ${\Re }_x^{1/2}{C}_{\mathrm{fx}}$ for the numerous values of physical parameters.

$\alpha$	$\beta$	$\epsilon$	$\lambda$	${\Re }^{1/2}{C}_f$
				Yamada	Xue
0.1	0.1	0.1	0.1	0.9013	0.9005
0.2				0.9616	0.9607
0.3				1.0919	1.0910
0.1	0.1	0.2	0.1	0.8433	0.8424
0.2				0.8999	0.8988
0.3				0.9538	0.9527
0.1	0.2	0.2	0.2	0.7977	0.7965
0.2				0.8511	0.8501
0.3				0.9022	0.9013
0.1	0.4	0.3		0.7176	0.7165
0.2				0.7655	0.7645
0.3				0.8114	0.8104
0.1	0.5	0.4		0.4567	0.4557
0.2				0.4881	0.4871
0.3				0.5182	0.5173
0.1	1.0	0.6		0.2679	0.2668
0.2				0.2872	0.2861
0.3				0.3056	0.3045
0.1	1.1	0.7		0.1297	0.1286
0.2				0.1396	0.1385
0.3				0.1491	0.1481

Table 4.

Numerical values of ${\Re }_x^{-1/2}\mathrm{Nu}$ for the numerous values of physical parameters.

$\mathrm{Bi}$	$\mathrm{Rd}$	$\alpha$	${\Re }^{-1/2}{N}_u$
			Yamada	Xue
0.1	0.3	0.1	0.0957	0.0946
0.2			0.1711	0.1704
0.3			0.2322	0.2313
0.1	0.4	0.1	0.1024	0.1012
0.2			0.1822	0.1812
0.3			0.2463	0.2454
0.1	0.5	0.2	0.1095	0.1084
0.2			0.1945	0.1934
0.3			0.2626	0.2615
0.1	0.8	0.4	0.1162	0.1152
0.2			0.2056	0.2046
0.3			0.2767	0.2755
0.1	0.9	0.5	0.1229	0.1219
0.2			0.2169	0.2158
0.3			0.2907	0.2899
0.1	1.0	0.6	0.4617	0.4606
0.2			0.4508	0.4499
0.3			0.4399	0.4387
0.1	1.1	0.7	0.1295	0.1284
0.2			0.2276	0.2265
0.3			0.3046	0.3035
0.1	1.2	0.8	0.1361	0.1354
0.2			0.2384	0.2373
0.3			0.3182	0.3174

Velocity effect

Figure 2(a) to (c) presents the significance of fluid variable $\alpha$ , stretching parameter $ϵ,$ and unsteady parameter $\lambda$ on the velocity curve $F\prime \left(\zeta \right)$ . The plot of the velocity distribution divulged increasing phenomena with the larger value of fluid quantity $\alpha$ . As $\alpha$ is opposite to the ordinary fluid viscosity, augmentation in the quantity of $\alpha$ reduces the viscosity of the base fluid as well as elevates the shear rate inside the boundary layer. Consequently, convectional velocity and hybrid intensify inside the boundary. The significance of stretching quantity $ϵ$ is sketched in Figure 2(b). When stretching quantity $ϵ$ augments having the boundary layer limit $0\to 1$ . Moreover, the velocity of fluid augments with $ϵ<1$ , when the velocity augments due to the higher magnitude of stretching parameter $ϵ$ and $ϵ>1$ , since no boundary pattern is noted as $ϵ=0$ . Figure 2(c) exhibited the fluctuation of an unsteady parameter $\alpha$ over the velocity curve $F\prime \left(\zeta \right)$ . The velocity curve exhibited depreciating behavior causing an enhancement of the unsteady parameter $\lambda$ . The unsteady parameter $\lambda$ relies upon stronger buoyancy, therefore the flow is higher and the velocity curve of the fluid is reduced.

Figure 2.

(a–c) $F\prime \left(\zeta \right)$ curve over $\alpha ,ϵ$ , and $\lambda$ .

Temperature effect

The impact of fluid variable $\alpha$ and radiation parameter $\mathrm{Rd}$ over temperature distribution $\vartheta \left(\zeta \right)$ is indicated in Figure 3(a) and (b). Figure 3(a) manifested the role of fluid parameters $\alpha$ on temperature distribution $\vartheta \left(\zeta \right)$ . The temperature distribution $\vartheta \left(\zeta \right)$ is sketched to reduce with a larger value of radiation parameter $\mathrm{Rd}$ . Since it steadily enhances, the fluid produces hotness. Fluid heats as it rises. Radiation is heat wave emission. Figure 4(a) and (b) confirms the role of unsteady variables $\lambda$ and Bi over the thermal distribution $\vartheta \left(\zeta \right)$ . Figure 4(a) reveals fluctuation in the thermal distribution for numerous magnitudes of unsteady parameter $\lambda$ . The ascending trend in temperature distribution is noted for the rising value of $\lambda$ and this rises the thickness of the thermal layer. The magnitude of Bi intensified which accelerated the temperature distribution $\vartheta \left(\zeta \right)$ . It is because the Bi has led to the heat gradient related to the hot fluid $h$ and thermal resistance is lessened to the h coefficient, that’s the reason improvements in Bi values lead to an increase in temperature distribution $\vartheta \left(\zeta \right)$ .

Figure 3.

(a and b) $\vartheta \left(\zeta \right)$ curve over $\alpha$ and $\mathrm{Rd}$ .

Figure 4.

(a and b) $\vartheta \left(\zeta \right)$ curve over $\lambda$ and $\mathrm{Bi}.$

Figure 5(a) and (b) reveals the ${\Re }_x^{1/2}{C}_{\mathrm{fx}}$ drag force and ${\Re }_x^{-1/2}\mathrm{Nu}$ Nusselt number against various values of $\alpha$ and $\mathrm{Rd}$ . From the figure, for risingmagnitude of $\lambda$ and $\mathrm{Bi}$ augmentation is noted in ${\Re }_x^{1/2}{C}_{\mathrm{fx}}$ and ${\Re }_x^{-1/2}\mathrm{Nu}$ . Computational values of the bar graph for ${\Re }_x^{1/2}{C}_{\mathrm{fx}}$ drag force and ${\Re }_x^{-1/2}\mathrm{Nu}$ in the case of Xue and Yamada-Ota model is elaborated in Figure 6(a) and (b).

Figure 5.

(a and b) ${\Re }_x^{1/2}{C}_{\mathrm{fx}},{\Re }_x^{-1/2}\mathrm{Nu}$ with a variation of ${\lambda }_1$ and $\mathrm{Bi}.$

Figure 6.

(a and b) Bar plot of ${\Re }_x^{1/2}{C}_{\mathrm{fx}},{\Re }_x^{-1/2}\mathrm{Nu}$ against $\lambda$ and $\mathrm{Rd}.$

Xue and Yamada-Ota models are plotted for stream line as sketched in Figure 7(a) and (b). While the mixture of Xue and Yamada-Ota model has been exhibited in Figure 8. Figure 9(a) and (b) is sketched to view the contour plot for Xue and Yamada-Ota models.

Figure 7.

Stream plot of $\left(a\right)$ Yamada-ota and (b) Xue.

Figure 8.

Mixture of Stream plot of Yamada-Ota and Xue model.

Figure 9.

(a and b) Contour plot of Yamada-Ota and Xue model.

Conclusion

In this exploration, the unsteady stagnant flow of Eyring-Powell hybrid nanofluid $(\mathrm{AA}7072-\mathrm{AA}7075/\mathrm{EG})$ over a stretched sheet is discussed. Thermal radiation and convective flow are also studied. A comparison is analyzed for Xue and Yamada-Ota models. The main points are given below:

• The temperature curve $\vartheta \left(\zeta \right)$ exhibited to be decaying for larger values of unsteady parameter $\lambda$ .

• ${\Re }_x^{1/2}{C}_{\mathrm{fx}}$ drag force and ${\Re }_x^{-1/2}\mathrm{Nu}$ Nusselt number enhanced due to larger values of $\lambda$ and $\mathrm{Bi}$ .

• The temperature distribution $\vartheta \left(\zeta \right)$ viewed reducing which boosts the values of fluid parameter $\alpha$ . While the opposite trend is noted in velocity profile $F\prime \left(\zeta \right)$ .

• Thermal radiation $\mathrm{Rd}$ has enhanced results and enhanced temperature distribution $\vartheta \left(\zeta \right)$ .

• The Xue model produces fewer values as compared to the Yamada-Ota model.