Natural convection heat transfer in the annular region between porous confocal ellipses

The Darcy-Boussinesq equations are solved in two dimensions and in elliptical cylindrical co-ordinates using a second-order-accurate finite difference code and a very fine grid. For the limiting case of a circular geometry, the results show that a hysteresis loop is possible for some values of the radius ratio, in agreement both with previous calculations using cylindrical co-ordinates and with the available experimental data. For the general case of an annulus of elliptical cross-section, two configurations, blunt or slender, are considered. When the major axes are horizontal (blunt case) a hysteresis loop appears for a certain range of Raleigh numbers. For the slender configuration, when the major axes are vertical, a transition from a steady to a periodic regime (Hopf bifurcation) has been evidenced. In all cases, the heat transfer rate from the slender geometry is greater than that obtained in the blunt case.