**4. Results and Discussion**

This section uses plotted figures to discuss the physical aspects of flow, thermal field changes, and concentration profile, and to explain physical interpretations triggered by the dominant dimensionless factors. The HAM is used to solve shortened ODEs numerically. Two diverse cases, namely, the *AA*7072 alloy and the *AA*7075 alloy, are well-thought-out in this analysis. Graphs are used to explain the effects of various specifications on *f* (*η*), *θ*1(*η*), and *χ*1(*η*), such as the Ree-Eyring fluid parameter, the ferromagnetic interaction parameter, the Schmidt number, the Prandtl number, and the reaction rate parameter. Additionally, skin friction and the Nusselt number are illustrated graphically.

Figure 5 shows the change in *f* (*η*) of both alloys *AA*7072 and *AA*7075 as *β* changes. In this case, increasing *β* lowers the *f* (*η*) of both alloys. This means that a large mass flux can reduce the velocity of the liquid on the surface. The occurrence of *β* and Curie temperature in this circumstance is crucial to consider the ferromagnetic stimulus on the flow, which upsurges liquid viscosity and diminishes the velocity gradient. Physically, when the magnetic influence is absent, the fluid velocity upsurges. The magnetic dipole effect, on the other hand, causes the fluid velocity to decrease. Furthermore, when comparing the *AA*7072 alloy to the *AA*7075 alloy, fluid velocity is quite slow. Figure 6 depicts the oscillation in *f* (*η*) with various values of *φ* for both alloys. The increase in *φ* lowers the *f* (*η*). The velocity *f* (*η*) of both alloys decreases as the volume fraction rises. Furthermore, when compared to the *AA*7075 alloy, the velocity of the *AA*7072 alloy is strongly motivated by the volume fraction and falls faster. Figure 7 depicts the behaviour of *f* (*η*) in relation to the Weissenberg number *We*. The velocity of the liquid is observed to be reduced across the entire domain as the Weissenberg number rises. Furthermore, when the Weissenberg number increases, the velocity layer thickness decreases. Mathematically, the Weissenberg number *We* is used in the investigation of viscoelastic flows. It is the ratio of viscous and elastic forces. As a result, as the Weissenberg number rises, the viscous forces diminish, and the velocity profile rises. Figure 8 indicates the effects of *β* on *θ*1(*η*) for both alloys. This indicates that an increase in *β* values significantly improves the temperature profile *θ*1(*η*). This is due to the fact that as the tension between the fluid particles boosts, too much heat is produced, resulting in higher fluid temperatures. Furthermore, for both *AA*7072 and *AA*7075 alloys, the inter-relevance thickness of the thermal layer is increased. Additionally, in *AA*7075 and when treated with *AA*7072 alloy, the closeness of the thermal layer further improves. Figure 9 shows that as *Pr* increases, so does the temperature of the fluid *θ*1(*η*). According to the observations, the thickness of the boundary layer appears to decrease as *Pr* increases. As a result, as the Prandtl number upsurges, so does the rate of thermal conductivity. *Pr* is the ratio of thermal diffusivity and momentum diffusivity. As a result, with a higher *Pr*, heat will dissipate from the sheet more quickly. Fluids with a higher *Pr* have a lower thermal conduction value. As a result, the *Pr* attempts to improve the cooling behaviour of the flows. The effect of *λ* on the *θ*1(*η*) profile is portrayed in Figure 10. It shows that as the value of *λ* increases, the temperature field decays. Additionally, for booming values of *λ*, the inter-relevance thickness is reduced for both alloys. Furthermore, heat abatement is enhanced in *AA*7072 alloy when treated with *AA*7075 alloy. The fluctuation in the thermal gradient for various values of *δ<sup>e</sup>* for both alloys is shown in Figure 11. The thermal distribution is enhanced when the values of the thermal relaxation parameters are increased. The heat flow relaxation time causes this parameter to emerge physically. The higher the *δ<sup>e</sup>* value, the longer it takes for the liquid particles to exchange heat with nearby particles, resulting in a decrease in temperature but an improvement in the temperature gradient. Figure 12 describes the outcome of volume fraction *φ* on heat transport in both alloys. The heat transmission of both alloys is improved as the volume fraction increases. Furthermore, in *AA*7075 and when treated with *AA*7072 alloy, the closeness of the thermal layer further improves. Figure 13 depicts the effect of *σ* on *χ*1(*η*) in both alloys. This figure confirms that *χ*1(*η*) has a decreasing nature for various *σ* values, and an increase in the reaction rate parameter *σ* diminishes the concentration of the liquids. In fact, as the reaction rate parameter values increase, the concentration field and related boundary layer thickness decreases. According to Figure 14, a higher Schmidt number corresponds to a lower solute diffusivity, allowing for a shallower penetration of the solute effect. As a result, as *Sc* rises, *χ*1(*η*) falls. Thus, with lower concentrations of *Sc*, the solute boundary layer is thicker, and vice versa.

Figure 15 depicts the variants in surface drag force *Cfx* versus *We* for various *φ* values for both alloys. It has been discovered that significantly greater values of *φ* enhance the surface drag force, whereas contrasting actions are observed for growing values of *β*; see Figure 16. Figure 17 shows the outcome of *δ<sup>e</sup>* on the rate of heat transfer versus *We* for both alloys *AA*7072 and *AA*7075. In both *AA*7072 and *AA*7075 alloys, an increase in *δ<sup>e</sup>* degrades the Nusselt number. Figure 18 illustrates the importance of *φ* on *Rex* <sup>−</sup>1/2*Nux* versus *We* for both *AA*7072 and *AA*7075 alloys. For both alloys, boosting the *φ* values improves the heat transmission rate. Figure 19 depicts the variation in *Rex* <sup>−</sup>1/2*Nux* versus *We* for various *β* values. It can be observed that significantly higher values of *β* enhance the heat transmission rate. Figure 20 depicts the variation in *Rex* <sup>−</sup>1/2*Shx* versus *Sc* for various *φ* values. It can also be observed that significantly higher values of *φ* enhance the concentration rate. A comparison between previous and present works for the validation of the results for skin friction is presented in Table 5.


**Table 5.** Comparison of −*f* (0) with literature.

**Figure 5.** Influence of ferromagnetic interaction parameter *β* on velocity profile.

**Figure 6.** Influence of volume fraction *φ* on velocity profile.

**Figure 7.** Influence of Weissenberg number We on velocity profile.

**Figure 8.** Influence of ferromagnetic interaction parameter *β* on temperature profile.

**Figure 9.** Influence of Prandtl number Pr on temperature profile.

**Figure 10.** Influence of viscous dissipation parameter *λ* on temperature profile.

**Figure 11.** Influence of thermal relaxation parameter *δ*e on temperature profile.

**Figure 12.** Influence of volume fraction *φ* on temperature profile.

**Figure 13.** Influence of reaction rate parameter *σ* on concentration profile.

**Figure 14.** Influence of Schmidt number Sc on concentration profile.

**Figure 15.** Various values of *φ* versus We for skin friction.

**Figure 16.** Various values of *β* versus We for skin friction.

**Figure 17.** Various values of *δ*e versus We.

**Figure 18.** Various values of *φ* versus We for nusselt number.

**Figure 19.** Various values of *β* versus We for nusselt number.

**Figure 20.** Various values of *φ* versus Sc.

#### **5. Conclusions**

Using the influence of magnetic dipoles and the Koo-Kleinstreuer model, the momentum, heat transfer, and mass transfer behavioru of Ree-Eyring nanoliquids through a stretching surface are investigated in this research. Moreover, the heat transmission is described by the Cattaneo-Christov heat flux model, and viscous dissipation is taken into account. Finally, the constructed governing equations related to the momentum, thermal, and mass distributions are converted to ODEs and solved with the HAM. The following are the results of the present analysis:


**Author Contributions:** Conceptualization, Z.S.; Data curation, M.R., W.D. and M.S.; Formal analysis, Z.S., M.R. and W.D.; Funding acquisition, N.V.; Investigation, Z.S.; Methodology, Z.S., M.R. and M.S.; Project administration, Z.S. and M.S.; Resources, Z.S. and N.V.; Software, Z.S., M.R., W.D. and M.S.; Supervision, Z.S.; Validation, Z.S., N.V., W.D. and M.S.; Visualization, Z.S., N.V., M.R. and M.S.; Writing—original draft, Z.S. and W.D.; Writing—review & editing, Z.S. and N.V. All authors have read and agreed to the published version of the manuscript.

**Funding:** The project was financed by Lucian Blaga University of Sibiu and Hasso Plattner Foundation research Grants LBUS-IRG-2021-07.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data that support the findings of this study are available from the corresponding author upon reasonable request.

**Conflicts of Interest:** The authors declare no conflict of interest.
