The significance of fluid flow under hydrothermal conditions within a hexagonal enclosure spans across numerous fields, underlining its broad applicability. However, our understanding of the free convection flow in these geometries is still limited despite its potential importance in science and technology. Therefore, this study numerically examines the heat convection and entropy generation within a porous hexagonal cavity containing a heated obstacle while subjected to a static magnetic field of intensity B ₀ . Micropolar hybrid nanofluid, composed of TiO ₂ and graphene oxide nanoparticles, was used to fill the hexagonal cavity with water as the base fluid. The finite difference method is associated with successive over-relaxation, successive relaxation, and Gauss–Seidel techniques, which are used to solve the dimensionless governing partial differential equations. The desired outcomes are computed using in-house developed MATLAB codes. A specific result from prior research findings is used to validate the accuracy of these MATLAB codes. The outcomes demonstrate that an upsurge in Ra from 10 ⁴ to 10 ⁶ and ϕ hnf {\phi }_{{\rm{hnf}}} from 0 to 4% leads to an enhancement in Nu _ABW to 53.05 and 3.14%, respectively. However, Nu _ABW diminishes by approximately 0.797 and 4.135% as Ha increases from 0 to 20 and K ₀ increases from 2 to 7.5, respectively. The average Bejan number (Be _avg ) consistently decreases as Ra increases, but Be _avg improves as Ha, vortex viscosity parameter ( K ₀ ), and ϕ hnf {\phi }_{{\rm{hnf}}} increase. The most important finding of the work is that the position of the heated obstacle significantly influences both the heat convection and entropy generation processes.

The Darcy flow and heat convection in nanofluid through a cylinder filled with a foam porous medium subject to local non-thermal equilibrium (LNTE) condition and uniform radial injection on the outer wall of the cylinder is studied. The momentum and two-energy equations are solved by differential transformation method (DTM) in the form of stream function using similarity variables. The effect on flow and heat transfer of different types of nanofluids and involved physical parameters Prandtl number Pr, Reylond number Re, Darcy number Da, Biot number Bi, Ratio of thermal conductivities Rk, porosity parameter ε, solid volume fraction parameter φ and shape of nanoparticles are analyzed through graphs. The viscous drag force and heat convection at the wall of the cylinder is calculated in terms of non-dimensional skin-friction coefficient and Nusselt number respectively. Decreasing the porosity of foam porous medium causes increment in magnitude of heat transfer rate for both the phases. Spherical shape of nanoparticles transfers more heat in comparison of cylindrical shape nanoparticles. Amongst the nanofluid H₂O-Ag, H₂O-Cu and H₂O-Al₂O₃ the magnitude of heat transfer for fluid phase Nu_f is lowest for nanofluid H₂O-Al₂O₃.