Magnetohydrodynamic natural convection of TiO ₂ –Cu/water hybrid nanofluid in a partially porous square enclosure with thermal radiation and internal heat generation

A numerical investigation is carried out to examine magnetohydrodynamic natural convection of a TiO 2 –Cu/water hybrid nanofluid inside a square enclosure partially filled with a porous medium. The model incorporates the effects of thermal radiation, internal heat generation/absorption, and nonlinear temperature–buoyancy coupling. The governing dimensionless conservation equations are solved using an improved Marker-and-Cell (MAC) algorithm with second-order finite difference discretization. The influences of key parameters including the Hartmann number (0 ≤ Ha ≤ 30), heat generation parameter (–3 ≤ Q ≤ 3), Darcy number (10 −4 ≤ Da ≤ 10 −1 ), Rayleigh number (10 3 ≤ Ra ≤ 10 6 ), radiation parameter (0 ≤ Rd ≤ 4), and nonlinear temperature parameter (0 ≤ λ ≤ 3) are systematically examined. The results show that increasing Ra from 10 3 to 10 6 enhances the average Nusselt number by approximately 165%, indicating strong buoyancy-driven convection. The results reveal that thermal radiation substantially enhances heat-transfer performance; the average Nusselt number increases by approximately 246% as Rd rises from 0 to 4. In contrast, the application of a magnetic field suppresses convective flow, leading to a 35–40% reduction in the Nusselt number when Ha increases from 0 to 30. Increasing porous permeability (Da: 10 −4 –10 −1 ) strengthens circulation and improves heat transfer by nearly 30%, while internal heat generation significantly alters thermal stratification near the active boundaries. Overall, the study establishes that porous permeability, magnetic damping, nonlinear thermal buoyancy, and radiative heat flux strongly regulate convection strength and heat transfer performance in hybrid nanofluid-filled enclosures.