The Benard problem is concerned with a fluid heated from below between two infinite parallel planes. The analytical results of the critical Rayliegh number and cell patterns have been known for a long time. However, not many people have considered the effects of the side walls. In this paper, the Benard problem in a rectangular container of various ratios is considered. We study some properties of a linear operator whose eigenvalues determine the critical Rayleigh number. The case in which the container makes an angle with the horizontal is also considered.
Chiang and Wang showed that certain determinant condition must be satisfied in order that the solution of f'''+K1f'+(ez+K0)f = 0 has a finite exponent of convergence on its zero-sequence. In this paper, we raise a conjecture on the form of the determinant. In the case K1 = 0, our conjecture is consistent with the one raised by Chiang, Laine and Wang.