Stability of natural convection between spherical shells: energy theory

The energy stability problem with respect to axisymmetric disturbances of the natural convection in the narrow gap between two spherical shells under the earth gravity is discussed. The results are compared with the results of the linear stability analysis for the same problem. The problem is solved for different fluids with Pr=0–100 and different radius ratios η=0.9,0.925,0.95. With the aid of the variational principle Euler–Lagrange equations are received, which have the form of an eigenvalue problem, that is solved by means of Galerkin–Chebyshev spectral method. The convergence problem and the dependence of the critical stability parameter on Prandtl number are discussed. The calculations show that there is a big difference between critical numbers for energy and linear stability theories for the small Prandtl numbers. For large Prandtl numbers this difference is very small.