Natural convection in a shallow cavity filled with a micropolar fluid

This paper reports an analytical and numerical study of natural convection in a shallow rectangular cavity filled with a micropolar fluid. 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 various material parameter of the fluid. For convection in an infinite layer (A≫1A≫1), analytical solutions for the stream function temperature and angular velocity 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 numbers for the onset of supercritical convection are predicted explicitly by the present model. Furthermore, a linear stability analysis is conducted yielding numerically the critical Rayleigh numbers for the onset of motion. Also, results are obtained from the analytical model for finite-amplitude convection for which the flow and heat transfer are 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. The influence of the material parameters on the flow and heat transfer is demonstrated to be significant.