Pseudo-Algorithm

To accomplish the transformation from BVP to IVP, the representations *f* by *Y*<sup>1</sup> and *θ* by *Y*<sup>4</sup> have been put into action:

$$Y\_3' = (A\_1 \times A\_1)(Y\_2 - Y\_1)Y\_3 - \epsilon^2, \qquad \begin{aligned} Y\_1' &= Y\_2 & \quad Y\_1(0) = 0 \\ Y\_2(0) &= 1 \\ Y\_3(0) &= 1 \\ Y\_4' &= Y\_5 & \quad Y\_4(0) = 1 \end{aligned} \tag{17}$$

$$Y\_5' = \begin{pmatrix} A\_5 \\ \hline (A\_6 + (4/3)\lambda) \end{pmatrix} \ast P\_r \ast (2 \ast Y\_4 Y\_2 - Y\_1) Y\_{5\prime} \qquad \qquad Y\_5(0) = I\_2 \end{pmatrix}$$

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(0) = *I*<sup>1</sup> and *Y*5(0) = *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*<sup>1</sup> and *I*<sup>2</sup> until it is unable to satisfy the tolerance *ζ*, which is *max*{|*Y*3(*ηmax*)|, |*Y*5(*ηmax*)|} < *ζ* [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 *Cf* is used to calculate the shear stress at the stretched sheet which can be defined as:

$$\mathbf{c}\_{f} = -\frac{-2\mu\_{nf}}{\rho\_{f}\mathbf{u}\_{\overline{w}}^{2}}(\boldsymbol{u}\_{\boldsymbol{y}})\_{\boldsymbol{y}=\boldsymbol{0}}.\tag{18}$$

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

$$\frac{1}{2}Re\_x^{\frac{1}{2}}c\_f = -(1-\varrho)^{-2.5}f''(0), \text{ (for } Al\_2O\_3 - H\_2O)$$

$$\frac{1}{2}Re\_x^{\frac{1}{2}}c\_f = -(123\varrho^2 + 7.3\varrho + 1)f''(0), \text{ (for } \gamma - Al\_2O\_3 - H\_2O)$$

$$\frac{1}{2}Re\_x^{\frac{1}{2}}c\_f = -(306\varrho^2 - 0.19\varrho + 1)f''(0), \text{ (for } \gamma - Al\_2O\_3 - C\_2H\_6O\_2)$$

where the local Reynolds number is denoted by *Rex* <sup>=</sup> *xuw*(*x*) *vf* , which completely base on the stretching velocity *<sup>U</sup>*∞(*x*) and local skin friction coefficient *Re* <sup>1</sup> 2 *<sup>x</sup> c <sup>f</sup>* . The Nusselt number *Nux* can be defined as

$$Nu\_{\mathfrak{X}} = \frac{\mathfrak{x}\mathfrak{q}\_{\mathfrak{w}}}{\tilde{k}\_f (T\_{\mathfrak{w}} - T\_{\infty})},\tag{20}$$

where, the local surface heat flux is *q*˜ = *kn f*(*Ty*)*y*=0. Based on Equation (12), we obtain the following Nusselt number,

$$\begin{aligned} \operatorname{Re}\_x^{-\frac{1}{2}} \operatorname{Nu}\_x &= \left( \frac{k\_s + 2k\_f - 2\varrho(k\_f - k\_s)}{k\_s + 2k\_f + \varrho(k\_f - k\_s)} \right) (-\theta'(0)) \left( \operatorname{for } Al\_2O\_3 - H\_2O \right) \\\ \operatorname{Re}\_x^{-\frac{1}{2}} \operatorname{Nu}\_x &= (4.97\varrho^2 + 2.72\varrho + 1)(-\theta'(0)) \left( \operatorname{for } \gamma - Al\_2O\_3 - H\_2O \right) \\\ \frac{k\_{nf}}{k\_f} &= (28.905\varrho^2 + 2.87\varrho + 1)(-\theta'(0)) \left( \operatorname{for } \gamma - Al\_2O\_3 - \mathcal{C}\_2H\_6O\_2 \right) . \end{aligned} \tag{21}$$

#### **6. Discussion**

The graphical representation of the numerical findings is shown for the nanoparticles of Al2O3 and *γ*-Al2O3 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 , the dimensionless slip parameter *α*`, the solid volume fraction

*<sup>ϕ</sup>*, Prandtl number *Pr*, reduced skin friction coefficient *Re* <sup>1</sup> 2 *<sup>x</sup> c <sup>f</sup>* , reduced Nusselt number *Re*<sup>−</sup> <sup>1</sup> <sup>2</sup> *<sup>x</sup> Nux*, shear stress *f* (0), velocity profile *f* (*η*) and temperature profile *θ*(*η*) have been analyzed with regard to the slip boundary conditions on a stretching sheet conjunction with stagnation-point flow phenomena.

Table 4 presents the results of a comparison between the values of −*f* (0) that were reported for Al2O3-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.

**Table 4.** Results comparison with earlier findings for Al2O3 nanofluids.


The influence of the nanoparticle volume fraction on the velocity profile is described via the Figures 2–4 for Al2O3-water, Al2O3-ethylene glycol and *γ*-Al2O3-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 *γ*-Al2O3 nanofluids is greater than that of the Al2O3 nanofluids [16,45]. This is because the thickness of the momentum boundary layer in *γ*-Al2O3 nanofluids is much greater than in Al2O3 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 Figures 2–4 show that the Al2O3-water mixture has a lower velocity but the *γ*-Al2O3-Water mixture has a greater velocity.

Figures 5–7 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 *γ*-Al2O3 and Al2O3 nanofluids rise as well. Therefore, nanofluids based on water have a steeper temperature profile compared to those on ethylene glycol (shallower profile) [16,45–47]. This is owing to the fact that the thermal diffusivity of water is significantly greater than that of ethylene glycol whereas, the Prandtl number *Pr* of water is substantially lower than that of ethylene glycol [34,40,45]. Comparing nanoparticles, Al2O3-water has a higher temperature profile than *γ*-Al2O3-water, while *γ*-Al2O3-ethylene glycol is higher. It is possible to draw the conclusion that the Al2O3 nanoparticles and the *γ*-Al2O3 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].

**Figure 2.** Velocity profile as a function of nanoparticle volume fraction *ϕ* with fix parameters  = 0.02 *a*˘ = 0.01 and *Pr* = 6.96.

**Figure 3.** Velocity profile as a function of nanoparticle volume fraction *ϕ* with fix parameters  = 0.02 *a*˘ = 0.01 and *Pr* = 204.

**Figure 4.** Velocity profile as a function of nanoparticle volume fraction *ϕ* with fix parameters  = 0.02 *a*˘ = 0.01 and *Pr* = 6.96.

**Figure 5.** Temperature profile as a function of nanoparticle volume fraction *ϕ* with fix parameters  = 0.02 *a*˘ = 0.01, *Pr* = 6.96 and *R* = 1.

**Figure 6.** Temperature profile as a function of nanoparticle volume fraction *ϕ* with fix parameters  = 0.02 *a*˘ = 0.01, *Pr* = 204 and *R* = 1.

**Figure 7.** Temperature profile as a function of nanoparticle volume fraction *ϕ* with fix parameters  = 0.02 *a*˘ = 0.01, *Pr* = 6.96 and *R* = 1.

The impacts of different physical parameters in the flow model on the velocity of the nanofluids within the boundary layer are depicted in Figures 8–10, while Figures 11–13 show the results for temperature profile. The influence of the slip parameter *a*˘ against the similarity variable *η* 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 8.** The influence of the slip parameter *a*˘ against *η* on the velocity profile with *ϕ* = 3%,  = 0.01 and *R* = 1.

**Figure 9.** The influence of the slip parameter *a*˘ against *η* on the velocity profile with *ϕ* = 3%,  = 0.01 and *R* = 1.

**Figure 10.** The influence of the slip parameter *a*˘ against *η* on the velocity profile with *ϕ* = 3%,  = 0.01 and *R* = 1.

**Figure 11.** The slip parameters *a*˘ against *η* on the temperature profile with *ϕ* = 2%,  = 0.01 and *R* = 1.

**Figure 12.** The slip parameters *a*˘ against *η* on the temperature profile with *ϕ* = 2%,  = 0.01 and *R* = 1.

**Figure 13.** The slip parameters *a*˘ against *η* on the temperature profile with *ϕ* = 2%,  = 0.01 and *R* = 1.

Figures 14–16 illustrate the effect that the stagnation parameter  has on the velocity profile in which the value of  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  will result in a decrease in the stretch velocity.

**Figure 14.** The effect of the stagnation parameter  on the velocity profile with fix parameters *ϕ* = 3%, *Pr* = 6.96 and *a*˘ = 0.01.

**Figure 15.** The effect of the stagnation parameter  on the velocity profile with fix parameter *ϕ* = 3%, *Pr* = 204 and *a*˘ = 0.01.

**Figure 16.** The effect of the stagnation parameter  on the velocity profile with fix parameters *ϕ* = 3%, *Pr* = 6.96 and *a*˘ = 0.01.

Figures 17–19 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.

Figures 20 and 21 illustrate the value fluctuation <sup>1</sup> <sup>2</sup>*Re* <sup>1</sup> 2 *<sup>x</sup> c <sup>f</sup>* which is the local (reduced) skin friction coefficient, as well as *Re*<sup>−</sup> <sup>1</sup> <sup>2</sup> *<sup>x</sup> Nux*, located along the *y*-axis and the nanoparticle volume fraction *ϕ* along the *x*-axis. It has been shown that an increase in *ϕ* 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 *γ*-Al2O3 than it is for Al2O3. For different base fluids, the amount of skin friction coefficient by Al2O3 with base fluid as water is greater than that produced by Al2O3 ethylene glycol as base fluid. However, *γ*-Al2O3 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 *γ*-Al2O3 nanoparticles is much greater.

**Figure 17.** The radiative heat flux *R* against *η* with fix parameters *ϕ* = 2%, *a*˘ = 0.02 and *Pr* = 204.

**Figure 18.** The radiative heat flux *R* against *η* with fix parameters *ϕ* = 2%, *a*˘ = 0.02 and *Pr* = 6.96.

**Figure 19.** The radiative heat flux *R* against *η* with fix parameters *ϕ* = 2%, *a*˘ = 0.02 and *Pr* = 6.96.

**Figure 20.** Frictional efficiency of skin as a function of *ϕ*.

19

**Figure 21.** Impact of *ϕ* on Nusselt number.
