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

For any aspect ratio Ro/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.