The onset of convection in horizontal cylinders

The convective instability of a fluid that fills a horizontal cylindrical cavity is considered. The boundaries of the cavity are conducting, and the driving force is a linear temperature gradient far from the cylinder. Critical Rayleigh numbers governing the onset of convection are determined for neutral stability. The results show that the critical Rayleigh number depends on the ratio of thermal conductivities of the solid to the fluid $\left ( \lambda \right )$, and a wavenumber. Both two- and three-dimensional disturbances are included. The disturbances are separated into even modes and odd modes. The most unstable odd modes have been found to be two-dimensional, while the most unstable even modes are three-dimensional. The two-dimensional odd modes are most unstable in the vicinity of $\lambda = 1$. The three-dimensional even modes are more unstable for other values of $\lambda$. The results are compared with the previous results of Gershuni and Zhukhovitskii.