Influence of the Prandtl number on the stability of convective flows between non-isothermal concentric spheres

This paper deals with a linear stability analysis of the convective flow of a Boussinesq fluid in a spherical gap where the inner shell is warmer than the outer one (T1 > T2). The numerical investigations are performed for the radii ratios η=R1R2=0.4,0.5 and 0.6. We show that the two-dimensional axisymmetric basic flow preferably loses its stability with respect to non-axisymmetric perturbations. The instability sets in either as a steady pitchfork or Hopf bifurcation. The corresponding stability diagrams, critical Grashof number Grc and wave number mc versus the Prandtl number Pr, are computed. The transitional Prandtl numbers at which the bifurcation type changes from Hopf to steady pitchfork bifurcation and vice versa, are indicated. The study is complemented by three-dimensional simulations of the main types of supercritical flows emerging from the instabilities detected.