In this paper, numerical Galerkin Finite Element Method (GFEM) applies for natural convection heat transfer of U-shaped cavity filled by Fe3O4-water nanofluid under the magnetic field and including two baffles. The above boundaries of the cavity are at low temperatures and bottom boundary is in a variable function temperature. It is assumed that two baffles in the cavity make vortexes to enhance heat transfers. The dimensionless governing equations including velocity, pressure, and temperature

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... ormulation are solved by the Galerkin finite element method. The results are discussed based on the governing parameters such as a nanoparticle volume fraction, Hartmann and Rayleigh numbers, magnetic field angle and nanoparticles shapes. As a main result, increasing both Aspect Ratio (AR) and Ra numbers enhanced heat transfer process and improved the average Nusselt numbers, while increasing the Hartmann number decreased the Nusselt number. Furthermore, it concluded that AR=0.4 had the maximum ψ and Nusselt numbers among the other examined aspect rations. Also, platelet, cylindrical, brick and spherical shapes had the maximum Nusselt numbers in sequence.