**6. Conclusions**

The goal of our current research is to investigate the effects of the embedded parameters on different state variables, like velocity, temperature, and the behaviour of the modeled equations under these parameters. The modeled equations are tackled with an analytical approach "homotopy analysis method" HAM. The embedded parameters and their effects are investigated and studied graphically. The motion of the suspended particles always has a great physical significance in heat transfer processes. Physically, the larger the Brownian motion parameter, the greater is the heat induction. Similar results have been observed for temperature profile with the larger values of Brownian motion. The effect of increasing values of *Sc* on boundary-layer thickness show a decrease in the thickness of the layer. The surface temperature increases with the increasing values of *Pr*, while an opposite effect is observed for unsteadiness parameter *S*, i.e., large values of *S* reduce the surface temperature. It is also observed that the temperature profile falls with large numbers of thermophoresis parameter *Nt* and vice versa. It is further observed that larger values of *Nb* reduce the mass flux, where *Nt* increases the mass flux. Higher values of *Sc* reduce the flux of mass, while it increases with increasing values of *Sc*. On the other hand, skin friction *Cf* shows a decline in its behavior with larger values of the stretching parameters and *λ*, while the unsteadiness parameter *S* increases *Cf* . In the solution procedure, the implemented technique convergence under the variation of physical parameters is observed numerically, which shows the reliability of our technique.

The central concluded points are as follows:


**Author Contributions:** Conceptualization, S.I.; Data Curation, M.A.; Investigation, A.U., Z.S. and S.I.; Methodology, A.U., Z.S. and M.A.; Resources, A.U., Z.S. and M.A.; Software, A.U. and Z.S.; Supervision, M.A. and S.I.; Visualization, E.O.A. and M.A.; Writing–Original Draft, A.U.and Z.S.; Writing–Review and Editing, E.O.A. and S.I.

**Acknowledgments:** This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah. The authors, therefore, gratefully acknowledge DSR for technical and financial support.

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