## 6 Conclusions  
This study is a continuous study that investigates the influence of magnetic fields on heat transfer, fluid flow, and entropy generation in a corrugated lid-driven enclosure with a constant heat flux source. The study employs the finite element method to discrete the governor equations. It presents the results in graphical and tabular formats for various parameters and discusses them. The numerical results describe heat via isotherms and heatlines, fluid via streamlines, and irreversibility via entropy lines and Bejan number. Based on the study, the following conclusions can be drawn:  
1. Case-3 is the most efficient scenario among others with similar physical parameters.
2. The maximum streamfunction value decreases as \(\widetilde{\mathrm{Ha}}\) and \(\widetilde{\Phi}\) increase, except for low \(\widetilde{\mathrm{Ri}}\) where the maximum streamfunction increases as \(\widetilde{\Phi}\) increases (\(\widetilde{\Phi}=0.15\)).
3. The increase in nanoparticle volume fraction will reduce the maximum temperature, and this reduction increases as \(\widetilde{\mathrm{Ha}}\) and \(\widetilde{\mathrm{Ri}}\) increase.
4. At low \(\widetilde{\mathrm{Ri}}\), circular heatlines appear at the top portion near the vertical wall, which has a negative sign (V = \(-1\)), and at the lower portion near the vertical wall, which has a positive sign (V = \(+1\)). These circular lines disappear at high \(\widetilde{\mathrm{Ri}}\).
5. The dense entropy generation is located in the center and near the heat sources. At high values of \(\widetilde{\mathrm{Ri}}\) = 100 and \(\widetilde{\mathrm{Ha}}\), the entropy generation lines are distributed horizontally (parallel to the base wall) in the middle of the cavity.  
The local Nusselt number increases as \(\widetilde{\Phi}\) and \(\widetilde{\text{Ri}}\) decrease, and \(\widetilde{\text{Ha}}\) increases. The effect of \(\widetilde{\Phi}\) on the local Nusselt number above the heat sources decreases at a high value of \(\widetilde{\text{Ri}}=100\).
* The \(\widetilde{\text{Be}}\) number decreases as \(\widetilde{\Phi}\) and \(\widetilde{\text{Ha}}\) increase. The effect of \(\widetilde{\Phi}\) on the \(\widetilde{\text{Be}}\) increases significantly at high values of \(\widetilde{\text{Ri}}\) and \(\widetilde{\text{Ha}}\) for all cases. The direction of the vertical wall movement (positive or negative) does not noticeably affect the \(\widetilde{\text{Be}}\) number. The variation of the average \(\widetilde{\text{Nu}}\) Number and \(\widetilde{\text{Be}}\) Number is linear with \(\widetilde{\text{Ha}}\) for all volume fractions of nanoparticles.  
Conceptualization, R.N.A.-K.; methodology, R.N.A.-K.; software, R.N.A.-K.; validation, S.M.A. and R.H.H.; formal analysis, R.N.A.-K. and A.Y.; data curation, S.M.A.; writing--original draft preparation, R.N.A.-K., S.M.A. and R.H.H.; writing--review and editing, R.H.H. and A.Y.; visualization, S.M.A.; supervision, R.N.A.-K.; project administration, R.N.A.-K. and A.Y. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors are grateful and sincerely thank the editors and anonymous referees for strictly reading the paper and according to useful suggestions to progress the paper.  
The authors declare no conflicts of interest.