Theoretical Results on Steady Convective Flows between Horizontal Coaxial Cylinders

For arbitrary Rayleigh number, $\mathrm{Ra}$, Prandtl number, and any ratio of the cylindrical radii, existence of steady solutions of the two-dimensional Oberbeck–Boussinesq system is proved in Theorem 3.5. We show nonlinear stability for large values of the aspect ratio and $\mathrm{Ra}<\mathrm{Ra}_L$, for some number $\mathrm{Ra}_L$ which is bounded from below (cf. Theorem 4.4). We prove that for large aspect ratio, $\mathrm{Ra}_L$ approaches the critical Rayleigh number for the stability of the rest state solution of a suitable linear problem.