The objective of this work is the mathematical modelling and the numerical simulation of the stationary, laminar, and natural convection, in a confined square cavity (H = L) filled with two fluids (a mixture of nanoparticles of aluminum oxide and Al2O3 water) in one partition and pure water in the other partition. A porous conductive wall of thickness w (w = L/e) and thermal conductivity Keff constitutes the exchange surface between these two partitions. The fluid movement is modeled by the Navier-Stokes equations in the two partitions, while the porous medium is modelled by the Darcy–Brinkman equation. Comsol Multiphysics software is used to solve the system of differential equations that is based on the finite element method. The results are discussed with particular attention to the mean and local Nusselt number (Nu), streamlines and isotherms. A parametric study for Rayleigh number Ra (102 to 106), volume fraction j (0% to 10%), and Darcy number Da (10-7 to 10-2) is performed. The obtained findings show that the increase in Ra, Da, and j intensifies the flow and improves the thermal exchange on the cold wall. For Da £ 10-5, Nu remains practically low and the natural convection is being dominated by conduction. For Da > 10-5, an increase in Nu is observed and the flows tend towards a purely convective situation. Furthermore, an increase in the heat transfer coefficients is observed with the raise of the porous layer permeability, volume fraction and Rayleigh number.

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A numerical program is adapted for the solution of the 2D, unsteady state equations of coupling convective transport of Buongiorno model and surface radiation; computations are performed in a square cavity differentially heated filled up with an AL2O3-water magneto nanofluid. The governing equations in Helmholtz variables (ψ, ω) are solved using a method based on Gauss's theorem integral on triangles meshes. Effects of aiding buoyancy forces (Nr=0.1, 2, and 4), emissivity (ε=0, 0.2, 0.6 and 0.9) on flow structure and convective transport characteristics are investigated. Hartmann number (Ha=50), and Lewis number (Le=5), nanofluids parameter of Brownian motion (Nb=0.5), thermal Rayleigh number (Ra=105), H=0.0098, nanofluids parameter of thermophoretic (Nt=0.5), Prandtl number (Pr=10) are invariable.