This study aims to investigate the effect of a constant magnetic field on heat transfer, flow of fluid, and entropy generation of mixed convection in a lid-driven porous medium enclosure filled with nanofluids (TiO₂-water). Uniform constant heat fluxes are partially applied to the bottom wall of the enclosure, while the remaining parts of the bottom wall are considered to be adiabatic. The vertical walls are maintained at a constant cold temperature and move with a fixed velocity. A sinusoidal wall is assumed to be fixed and kept adiabatic at the top enclosure. Three scenarios are considered corresponding to different directions of the moving isothermal vertical wall (±1). The influence of pertinent parameters on the heat transfer, flow of fluid, and entropy generation in an enclosure are deliberated. The parameters are the Richardson number (${\tilde{R}}_i$ = 1, 10, and 100), the Hartmann number (0 ≤ $\tilde{H}$a ≤ 75 with a 25 step), and the solid volume fraction of nanoparticles (0 ≤ $\tilde{\Phi }$ ≤ 0.15 with a 0.05 step). The Grashof and Darcy numbers are assumed to be constant at 10⁴ and 10⁻³, respectively. The finite element method, utilizing the variational formulation/weak form, is applied to discretize the main governor equations. Triangular elements have been employed within the studied envelope, with the elements adapting as needed. The results showed that the streamfunction and fluid temperature decreased as the solid volume fraction increased. The local $\tilde{\mathrm{N}}\mathrm{u}$ number increased by more than 50% at low values of $\tilde{\mathsf{\Phi }}$ (up to 0.1). This percentage decreases between 25% and 40% when $\tilde{\mathsf{\Phi }}$ is in the range of 0.1 to 0.15. As $\tilde{\mathrm{H}}$a increases from 0 to 75, these percentages increase at low values of the value of $\tilde{\mathrm{R}}\mathrm{i}=1$ and 10. These variations are primarily dependent on the value of the Richardson number.

The optimization of heat transfer in various engineering applications, such as thermal management systems and energy storage devices, remains a crucial challenge. This study aims to investigate the potential of Casson-based Cu-H₂O nanofluids in enhancing free convection heat transfer within complex geometries. The research examines the free convection heat transfer and fluid flow characteristics of a Casson-based Cu-H₂O nanofluid within a semi-parabolic enclosure that includes a wavy corrugated cylinder. Utilizing numerical simulations based on the Galerkin Finite Element Method, the study investigates the impact of different factors, including the Rayleigh number (10³ ≤ Ra ≤ 10⁶), Casson fluid parameter (0.1 ≤ γ ≤ 1), corrugation number (3 ≤ N ≤ 10), nanoparticle volume fraction (0 ≤ ϕ ≤ 0.15), and enclosure inclination angle (0° ≤ ζ ≤ 60°), on both heat transfer efficiency and flow patterns. The results reveal that increasing the Rayleigh number and Casson fluid parameter enhances heat transfer performance, with the average Nusselt number increasing by up to 165 % as Ra increases from 10³ to 10⁶. An optimal range of corrugation numbers is identified for maximizing heat transfer at higher Rayleigh numbers. The addition of nanoparticles significantly improves heat transfer, with a 20 % increase in the average Nusselt number observed at Ra = 10⁵ when the nanoparticle volume fraction increases from 0 to 0.15. These findings provide valuable insights for designing more efficient thermal management systems in applications such as electronics cooling, solar thermal collectors, and heat exchangers, potentially leading to improved energy efficiency and performance in various industrial and technological sectors.