Natural convection in horizontal annuli: Evaluation of the error for two approximations

We consider convection inside annuli, driven by a uniform temperature gap on the boundaries and gravitation as outer force. It takes place for any Rayleigh number while steady convective motions are observed only for small ones (but any Prandtl number and gap width). We provide estimates for the relative error of two popular approximations to the full Navier–Stokes–Fourier equations. For this we propose a new method. In particular we have to derive a lower bound for the norm of the velocity and the temperature both for steady nonlinear coupled and decoupled approximations in two space dimensions.