Two Dimensional MHD Free Convection with Internal Heating in a Square Cavity

The laminar steady magnetohydrodynamics free convection in a liquid filled square enclosure with internal heat generation is investigated by two-dimensional numerical simulation. The enclosure is heated by a uniform volumetric heat density and walls have constant temperature. A fixed magnetic field is applied to the enclosure. The dimensionless governing equations are solved numerically for the stream function, vorticity and temperature using finite difference method for various Rayleigh (Ra) and Hartmann (Ha) numbers with MATLAB software. The stream function equation is solved using fast Poisson's equation solver on a rectangular grid (poicalc function in MATLAB), voricity and temperature equations are solved using red-black Gauss-Seidel and bi-conjugate gradient stabilized (BiCGSTAB) methods respectively. The proposed method is fast and there is no need to the under-relaxation factors for variables. The results show that the strength of the magnetic field has significant effects on the flow and temperature fields. For the square cavity, the maximum temperature reduces with increasing Ra number. It is also observed that at low Ra number, the maximum temperature is at the centre of the cavity and it shifts upwards with increase in Ra number. Circulation inside the enclosure and therefore the convection become stronger as the Ra number increases while the magnetic field suppresses the convective flow and the heat transfer rate.