A theoretical study of the first transition for the non-linear Stokes problem in a horizontal annulus

• We theoretically address the natural convection of a fluid between coaxial cylinders whose axis is horizontal, when the temperatures on the surfaces are kept constant, the inner warmer than the outer one.
• We compare the outcomes of some mathematical models derived from the Oberbeck-Boussinesq approximation, having in common the existence of a steady solution different from zero for any curvature of the domain and arbitrary values of the Prandtl and Rayleigh numbers.
• The basic steady solution prove to be asymptotically stable for sufficiently small Rayleigh numbers. The critical values derived by the energy method depend on the curvature and their graphs converge in the region of the parameter space where the curvature is large.
• For large curvatures, we prove that the Non-Linear Stokes System exhibit a critical Rayleigh number which is mathematically well-defined and uniformly bounded from below. A numerical procedure to calculate it is suggested.

For any aspect ratio Ro/RiRo/Ri of the cylinder radii, the non-linear stability of the steady 2D-solutions of the non-linear Stokes system, which is an approximation of the Oberbeck–Boussinesq system, is theoretically studied. The sufficient condition for the stability shows a critical Ra which is a function of the aspect ratio. It is the same of the associated homogeneous linear problem and it can be found by looking for the largest eigenvalue of a suitable symmetric operator. The critical Ra so defined proves to be uniformly bounded from below in the space of dimensionless parameters, while it is non-uniformly bounded from above for the aspect ratio going to infinity. A scheme to evaluate it as a function of the aspect ratio is given. The results do not depend on the Prandtl number Pr.