**5. Results and Discussion**

The impacts of numerous pertinent parameters on the temperature, velocity, heat transfer and drag force are discussed and presented in tabular form and as well as graphically (see Figures 1–21). Table 5 shows the assessment of −*F* <sup>00</sup>(0) with current outcomes through the outcomes reported by Shafie [49] and Chamkha [50].

**Figure 1.** Impact of *φ* on *F* 0 (*η*).

**Figure 2.** Impact of *φ* on *θ*(*η*).

The outcomes depict a superb conformity. The significant parameters for computational purposes are considered as *φ* = 0.02, *M* = 0.1, *ε* = 10, *ξ* = 0.5, *θ<sup>w</sup>* = 0.1, *Ec* = 0.5, *A*<sup>0</sup> = *B*<sup>0</sup> = 0.1 and *R<sup>d</sup>* = 02, with the variations shown in Figures 1–21.

Figures 1 and 2 describe the influence of volume fraction *φ* on the velocity *F* 0 (*η*) and fluid temperature *θ*(*η*). Figures 1 and 2 confirm that the *F* 0 (*η*) and *θ*(*η*) accelerate gradually for larger values of *φ*. Physically, the nanofluid density under consideration decreases due to the larger amount of *φ*, which consequently augments the velocity and temperature. Thus, the inter-molecular forces between the particles of nanofluids become weaker, and as a result, the fluid velocity accelerates. It is also clear from Figure 2 that the temperature is higher in the case of water and lower in case of ethylene glycol. The justification for this result is that water has a smaller Prandtl number than ethylene glycol, and as a result, the water thermal diffusivity is much superior to that of ethylene glycol. In addition, C2H6O<sup>2</sup> nanoliquids can be utilized for the purpose of cooling.

**Figure 3.** Impact of *M* on *F* 0 (*η*).

**Figure 4.** Impact of *M* on *θ*(*η*).

**Figure 5.** Impact of *λ* > 0 on *F* 0 (*η*).

**Figure 6.** Impact of *λ* < 0 on *F* 0 (*η*).

**Table 5.** Comparison of the values of −*F* <sup>00</sup>(0), when *M* = *φ* = *λ* = 0.


The influence of *M* on *F* 0 (*η*) and *θ*(*η*) is portrayed in Figures 3 and 4. Figure 3 suggests that the velocity declines due to *M* in both H2O\C2H6O<sup>2</sup> based nanofluids.

Physically, the existence of magnetic function engenders a type of resistive force (or Lorentz force) in the flow region, which holds the nanofluid motion. In contrast, the temperature profile (Figure 4) rises as a result of *M*. The physics behind this are that an enhancement in magnetic function causes an upsurge in electro-magnetic force, which controls the motion of fluid and consequently increases the temperature as well

as the thickness. Figures 5–8 show the impact of *λ* on *F* 0 (*η*) and *θ*(*η*) for assisting and opposing flows.

**Figure 7.** Impact of *λ* > 0 on *θ*(*η*).

**Figure 8.** Impact of *λ* < 0 on *θ*(*η*).

It is clear from Figure 5 that the velocity increases with *λ* in the assisting flow, while the velocity as shown in Figure 6 declines in the opposing flow. Physically, a greater amount of *λ* generates a substantial buoyancy force that ultimately generates greater kinetic energy. The reverse is true for the opposing flow. Figure 7 shows that the temperature diminishes due to *λ* for assisting flow in both γAl2O<sup>3</sup> − H2O and γAl2O<sup>3</sup> − C2H6O<sup>2</sup> nanofluids, whereas the temperature increases in the opposing flow, as depicted in Figure 8. Physically, the fluid attains the heat from the sheet, and later on, heat energy is transmuted into different forms of energy, like kinetic energy. As expected, the temperature is lower for γAl2O<sup>3</sup> − C2H6O<sup>2</sup> than γAl2O<sup>3</sup> − H2O due to the greater Prandtl number. The nature of the temperature profiles is observed in Figures 9–11 for changed values of *R<sup>d</sup>* , *Ec* and *ξ*.

Figure 9 confirms that temperature increases with *R<sup>d</sup>* for H2O\C2H6O<sup>2</sup> based γ − Al2O<sup>3</sup> nanofluids. The coefficient of absorption declines as radiation increases, and due to this, an enhancement occurs in the temperature distribution. Similar behavior is noticed for the Eckert number, owing to fractional heating as illustrated in Figure 10. Larger inference of *Ec* implies that the heat of thermal dissipation is stocked in the fluid, which ultimately increases the temperature. The convective parameter causes upsurges in the distribution of temperature (Figure 11) for H2O\C2H6O<sup>2</sup> based γ − Al2O<sup>3</sup> nanofluids.

**Figure 9.** Impact of *R<sup>d</sup>* on *θ*(*η*).

**Figure 10.** Impact of *Ec* on *θ*(*η*).

The sheet temperature gradient increases due to commanding convective heating. This permits the thermal influence to penetrate deeper in the sluggish fluid. Thus, the temperature increases. Figures 12 and 13 demonstrate the influence of heat sink/source on the *θ*(*η*) profile. It is clear from these profiles that the heat source increases the temperature, while the heat sink reduces the temperature, as expected.

Physically, the impact of the heat source (A<sup>0</sup> > 0, B<sup>0</sup> > 0) adds extra energy within the boundary layer, which ultimately increases the temperature, while the heat sink (A<sup>0</sup> < 0, B<sup>0</sup> < 0) absorbs the energy, which causes a reduction in the temperature.

Figures 14–16 illustrate the behavior of entropy generation for distinct parameters *φ*, Re*<sup>L</sup>* and *Br* for H2O\C2H6O<sup>2</sup> based γ − Al2O<sup>3</sup> nanofluids. Figure 14a,b show that the entropy increases due to *φ* in both nanofluids. It is interesting to note that ethylene-glycolbased nanofluid has greater impact on the entropy due to the huge Prandtl number and lower thermal diffusivity. Figure 15a,b suggest that the entropy enhances due to Re*<sup>L</sup>* in both nanofluids owing to friction nanofluid and heat transport within the boundary layer for γAl2O<sup>3</sup> − C2H6O<sup>2</sup> and as well as γAl2O<sup>3</sup> − H2O nanofluids. Similarly, the impact of γAl2O<sup>3</sup> − C2H6O<sup>2</sup> on the entropy is greater than γAl2O<sup>3</sup> − H2O. Figure 16a,b confirm that the entropy depicts the growing function of *Br* due to fluid friction for both nanofluids.

**Figure 11.** Impact of *ξ* on *θ*(*η*).

**Figure 12.** Impact of A<sup>0</sup> > 0, B<sup>0</sup> > 0 on *θ*(*η*).

**Figure 13.** Impact of A<sup>0</sup> < 0, B<sup>0</sup> < 0 on *θ*(*η*).

**Figure 14.** Impact of *φ* on EG (**a**) γ − Al2O<sup>3</sup> − H2O; (**b**) γ − Al2O<sup>3</sup> − C2H6O2.

**Figure 15.** Impact of Re*<sup>L</sup>* on EG (**a**) γ − Al2O<sup>3</sup> − H2O; (**b**) γ − Al2O<sup>3</sup> − C2H6O2.

**Figure 16.** Impact of *Br* on EG. (**a**) γ − Al2O<sup>3</sup> − H2O; (**b**) γ − Al2O<sup>3</sup> − C2H6O2.

**Figure 17.** Impact of *M* on the friction factor.

**Figure 18.** Impact of *φ* on the friction factor.

**Figure 19.** Impact of *R<sup>d</sup>* on the Nusselt number.

**Figure 20.** The streamline patterns for (**a**) γAl2O<sup>3</sup> − H2O and (**b**)γAl2O<sup>3</sup> − C2H6O2.

**Figure 21.** The isotherm patterns for (**a**) γAl2O<sup>3</sup> − H2O and (**b**)γAl2O<sup>3</sup> − C2H6O2.

The trend of significant parameters versus Re0.5 *<sup>x</sup> C<sup>F</sup>* and Re−0.5 *<sup>x</sup> Nu<sup>x</sup>* for γAl2O<sup>3</sup> − C2H6O<sup>2</sup> and γAl2O<sup>3</sup> − H2O is seen in Tables 6 and 7.


**Table 6.** The numerical values of <sup>−</sup>Re0.5 *<sup>x</sup> C<sup>F</sup>* when *λ* = 0.1.

**Table 7.** The numerical values of Re−0.5 *<sup>x</sup> Nu<sup>x</sup>* when *λ* = 0.1.



**Table 7.** *Cont.*

In addition, bar diagrams are also shown in Figures 17–19.

It is concluded from these observations that the larger values of *M* subdued the friction factor in both nanofluids. The major reason is that MF capitulates the flow of nanofluids through the surface of the sheet owing to the prominent magnetic impact, which subdues the friction factor. In addition, the friction factor increases owing to the *φ* in both nanofluids. In the water-based γ − Al2O<sup>3</sup> nanofluid, the values of the skin factor are greater compared to the ethylene-based γ − Al2O<sup>3</sup> nanofluid, due to the superior thermal diffusivity. Moreover, the Nusselt number increases with the radiation due to fact that the radiation generates superior molecular force in the flow, while the opposite trend is explored due to the Eckert number. Both the Nusselt number and the friction factor increase due to the time-dependent parameter. The streamlines and isotherms are plotted in Figure 20a,b and Figure 21a,b.
