## 1 Introduction
The effect of magnetic field on heat transfer rate was maximum and minimum for a Rayleigh number in the order of \(10^{4}\) and \(10^{3}\), respectively. The effect of Rayleigh number on the heat transfer rate was maximum for a null Hartmann number whatever the value of the solid volume fraction. [PERSON] and [PERSON] [9] numerically investigated a 2D laminar MHD natural convection within a heated rectangular enclosure, subjected to uniform horizontal magnetic field, with insulated square block. Simulations of heat transfer performance within the nanofluid were carried out for various Rayleigh number and Hartmann number with fixed Prandtl number. Results show that the heat transfer rate was reduced when the magnetic field intensity increased. Simulations show that the incorporation of insulated block, with a certain size range under a certain Hartman number range permits to improve the heat transfer rate within the nanofluid. [PERSON] et al. [10] numerically studied the MHD natural convection in a 2D and 3D cavity subjected to thermal radiation under different plank number and Hartmann number. The 3D Navier Stokes equations were solved implementing a combined spectral collocation method-artificial compressibility method. The thermal radiation effect on the heat transfer was comparatively analyzed between 2D and 3D cavity under identical parameters. Results show that the heat transfer and the flow were enhanced by the thermal radiation while they were reduced by the magnetic field in the spanwise direction of the 3D cavity. The flow and the temperature distribution difference between 2D and 3D cavity were more pronounced for strong thermal radiation and weak magnetic field. [PERSON] et al. [11] numerically investigated an unsteady natural convective heat transfer flow, within a square enclosure filled with nanofluids, without considering magnetic field effect. Uniform hot and cold temperatures were imposed in bottom and upper walls, respectively while the left and right walls were insulated. The implementation of the Galerkin weighted residual finite element method permits to obtain numerical simulations. Four types of nanoparticles and 3 types of based fluid were considered. Numerical results show that the increase of Rayleigh number and nanoparticles volume fraction permits to enhance the average Nusselt number. In addition, results show that the Nusselt number was higher for the nanoparticles with blade shape and for the Mn-ZnFe\({}_{2}\)O\({}_{4}\)-kerosene nanofluid. [PERSON] et al. [12] numerically studied the influence of nanofluid and hot isothermal obstacle inside a \(\Pi\)-shaped enclosure. Uniform cold temperature was imposed in the interior walls of the enclosure and the other walls were adiabatic. The hot obstacle was located on the upper side of the enclosure. The implementation of the lattice Boltzmann method permits to obtain numerical simulations. Numerical results show that the average Nusselt number was enhanced when the Rayleigh number, the aspect ratio and the nanoparticles solid volume fraction increase. Results show, also, that the heat transfer was more effective when the hot obstacle was located on the center of upper wall. [PERSON] and [PERSON] [13] numerically investigated the natural convection heat transfer within an annulus between a wavy circular cylinder and a rhombus enclosure. Uniform cold and hot temperatures were imposed in the annulus wall and the rhombus enclosure, respectively. A novel viscosity model named magnetic field dependent was applied. The control volume finite element method was implemented. The effect of physical parameters (such as Rayleigh number, Hartmann number, aspect ratio, solid volume fraction,..., etc.) was investigated. Numerical results show that the local heat transfer rate was improved with the decrease of aspect ratio when the Hartmann number was absent. This work aims to numerically investigate the impact of the Rayleigh number, the Hartmann number and the round corner radius on the Fe\({}_{3}\)O\({}_{4}\)-water nanofluid flow and heat transfer.