Natural convection in horizontal annuli: a lower bound for the energy

We study natural convection in the gap between two infinitely long horizontal coaxial cylindrical surfaces, each of which is maintained at constant temperature. If the inverse relative gap width $\cA$ is large, relevant steady convective flow and considerable heat transfer are observed, even for extremely small Rayleigh numbers $\Ra$. A lower bound for the norm of the velocity and of the temperature is rigorously found by studying an approximation problem, which is a good model in the parameter range where steady stable flow occurs. The lower bound depends on $\cA$ only, and is an increasing function of $\cA$. This means that the bound can be arbitrarily increased via the geometry only - no matter how small the temperature difference is, and independently of the Prandtl number $\Pr$.