Natural convection in the horizontal annulus: Critical Rayleigh number for the steady problem

For the 2D Oberbeck–Boussinesq system in an annulus, we are looking for the critical Rayleigh number for which the (non-zero) basic flow loses stability. For this, we consider the corresponding Euler–Lagrange equations and construct a precise functional analytical frame for the Laplace and the Stokes problem as well as the Bilaplacian operator in this domain. With this frame and the right set of basis functions, it is then possible to construct and apply a numerical scheme providing the critical Raleigh number.