Natural convection of nanofluids in a shallow rectangular enclosure heated from the side

This article reports an analytical and numerical study of natural convection in a shallow rectangular cavity filled with nanofluids. Neumann boundary conditions for temperature are applied to the vertical walls of the enclosure while the two horizontal ones are assumed insulated. Three types of nanoparticles are taken into consideration: Cu, Al2O3 and TiO2. Various models are used for calculating the effective viscosity and thermal conductivity of nanofluids. The governing parameters for the problem are the thermal Rayleigh number, Ra, the Prandtl number Pr, the aspect ratio of the cavity, A and the solid volume fraction of nanoparticles, Phi. In the first part of the analytical study a scale analysis of the present problem is made. In the second part analytical solutions are obtained, in the limit of an infinite layer (A >> 1), for the stream function and temperature fields. The analytical model relies on the assumption of a parallel flow approximation in the core region of the cavity and an integral form of the energy equation in the end regions. In the high Rayleigh number regime a good agreement is obtained between the predictions of the scale analysis and those of the analytical solution. Numerical solutions of the full governing equations are obtained for a wide range of the governing parameters. A good agreement is observed between the analytical model and the numerical simulations based on the lattice Boltzmann method.