CXLII. On the application of eigenfunction expansions to the problem of the thermal instability of a fluid sphere heated within

Abstract

The eigenvalue problem which describes the onset of convection in a fluid sphere heated within is solved by means of expansions of the convective velocity field and temperature in spherical Bessel functions. The Rayleigh number C_l,n , at which there first appears the nth radial convection pattern with the spherical symmetry of the lth spherical harmonic, can be computed for any l and n by this device, and bracketed to arbitrary accuracy. A table of C_l,n for Ifrom 1 through 6 and n=1 and 2 is compared with Chandrasekhar's values of C _{l 1} obtained by a variational method. A similar attack is shown to succeed on the classical Rayleigh- Benard problem of convection in a horizontal layer of fluid. Finally, the class of problems which should be amenable to such an attack is delineated.