Convective flows in closed cavities with variable fluid properties

A compressible vector potential-vorticity formulation of the steady-state equations of motion with variable properties is developed for a Newtonian fluid. These equations include an equation for calculating the pressure. The two-dimensional form of the equations together with the appropriate boundary conditions is solved for vertical and inclined rectangular cavities differentially heated on the sides. It is shown that the conventional parameters, the Rayleigh and Prandtl numbers, aspect ratio and angle of inclination do not fully specify a problem and that an additional six parameters are necessary. Solutions are presented in the form of flow and temperature fields for air at various Rayleigh numbers, aspect ratios, and angles of inclination. Although the streamline and isotherm patterns are substantially different from those obtained using the Boussinesq approximation, the Nusselt number is not significantly affected.