In the present work, for the first time, laminar natural convection of water-TiO 2 nanofluid around a hot obstacle in a square cavity is simulated by Lattice Boltzmann Method (LBM). The effect of Rayleigh number, obstacle dimension, volume fraction of nanofluid, cavity dimensions, surface ratio and various models for computing of thermal conductivity coefficient and viscosity coefficient on Nusselt number and heat transfer around hot obstacle in enclosure has been investigated. The results show that by increase in Rayleigh number and volume fraction, average Nusselt number will increase. The average Nusselt number will increase when obstacle dimensions increase to 0.5 L but it will decrease when the obstacle dimensions increase to 0.7 L. Vortexes will create in 0.8 L and it causes to increase heat transfer. By reduplicating the obstacle width, heat transfer is better than reduplicating the obstacle length. The average Nusselt number increases by increase in cavity length and it will decrease by increase in cavity wide. All results are equal in Hamilton-Crosser and Maxwell-Garnett model when the surface ratio is one. But heat transfer will increase by decreasing surface ratio. The average Nusselt number in Wang model is less than Nusselt number in Brinkman model. Ó 2017 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

In the present research, the behavior of a Newtonian nanofluid (water-Al 2 O 3 ) in a mixture phase model approach is numerically examined. The process of heating is done in two different ways. Deterioration was found in the mean Nusselt number of a nanofluid in the mixture-phase model approach when compared to the mean Nusselt number of pure water. However, in the single-phase model there was an increase in the Nusselt number when compared to the Nusselt number of pure water.

The present study aims to identify heat transfer and flow characteristic due to buoyancy forces in a heated enclosure using nanofluid and their behaviors under natural convective heat transfer condition. In the present work nanofluids with water based containing Al 2 O 3 nanoparticle numerically investigated. Numerical works are done on the use of the stable nanofluids under natural convective heat transfer conditions. Process of heating is done in two different ways: in first process the heater mounted to the down wall and in second way it mounted to the left vertical wall with a finite length, also heated and cooled walls keep in a constant temperature. Our numerical simulation has been undertaken incorporating a homogenous solid-liquid mixture. In particular this study deals with Al 2 O 3 nanofluids with Newtonian behavior. Simulation have been carried out in the ranges Ra=10 3-10 6. Our volumetric fraction of nanoparticles was 1.3%. It was shown the Nusselt-Rayleigh number relation and then nanofluid Nu-Ra number diagrams based on found is plotted. Results showed an increasing in Nusselt-Rayleigh number at nanofluids diagrams as compared to Nusselt-Rayleigh relations of pure water. Increase in the average Nusselt number plays a significant role in heat transfer applications. Due to our numerical investigations vertical cavities with nanofluid were better than horizontal cavities. Also the cavities, which we used nanofluid, had better efficiency in natural convection numerical modeling for both horizontal and vertical fluid layer.