**4. Conclusions**

In this study, the heated couple stress bi-phase fluid with spherical particles of metal Hafnium is investigated. The flow bounded by two parallel plates is caused by solely the influence of pressure gradient in an axial direction. A uniform and constant magnetic field of strength *B*<sup>0</sup> snags the flow, transversely acting across the channel. Extra effects of lubrication are applied on both walls to minimize the surface roughness and to attain a smooth flow. The significant outcomes of the analysis are:


**Author Contributions:** Conceptualization, R.E.; Writing—Review and Editing, A.Z.; Methodology, F.H.; Investigation, T.A.

**Funding:** This research received no external funding.

**Acknowledgments:** Rahmat Ellahi thanks the Director Office of Research Chair Professors, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia, to honor him with the Chair Professor at KFUPM. Farooq Hussain also acknowledges the Higher Education Commission Pakistan to provide him the indigenous scholarship for the pursuance of his Ph.D. studies.

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