The stability of natural convection in narrow-gap spherical annuli to axisymmetric disturbances

The linear stability with respect to axisymmetric disturbances of natural convection in narrow-gap, spherical annuli is investigated. The basic motion is an eight-order perturbation solution in the small parameter ε = 1-η, where η is the ratio of inner radius of the annulus to outer radius. The disturbance equations are reduced to a system of ordinary differential equations by means of a method of partial spectral expansions. These equations constitute an eigenvalue problem which is solved for the critical Rayleigh number as a function of η and Prandtl number, Pr. Cases considered are Pr = 0.1, 1, 10, and 100 for 0.900 ≤ η ≤ 0.995. A comparison with the experimental results found in the literature indicates that non-axisymmetric time periodic bifurcation will most likely take precedence over the case considered herein for Pr = 1,10. However, it appears that steady axisymmetric bifurcation is possible for Pr = 0.1.