Natural convection of nanofluids in a shallow cavity heated from below

This paper 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 horizontal walls of the enclosure, while the two vertical ones are assumed insulated. 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. For convection in an infinite layer (A >> 1), analytical solutions for the stream function and temperature are obtained using a parallel flow approximation in the core region of the cavity and an integral form of the energy equation. The critical Rayleigh number for the onset of supercritical convection of nanofluids is predicted explicitly by the present model. Furthermore, a linear stability analysis of the parallel flow solution is studied and the threshold for Hopf bifurcation is determined. Also, results are obtained from the analytical model for finite amplitude convection for which the flow and heat transfer is presented in terms of the governing parameters of the problem. 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. (C) 2010 Elsevier Masson SAS. All rights reserved.