Theoretical Results on Steady Convective Flows between Horizontal Coaxial Cylinders

For arbitrary Rayleigh number, 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 Ra<RaL, for some number RaL which is bounded from below (cf. Theorem 4.4). We prove that for large aspect ratio, RaL approaches the critical Rayleigh number for the stability of the rest state solution of a suitable linear problem.