Onset of convection in shallow cylindrical containers

This paper considers the onset of convection in shallow cylindrical contain ers heated from below where the boundary which forms the cross-section of t he cylinder is of arbitrary shape. The lateral dimensions of the boundary a re assumed to be much larger than the characteristic wavelength of convecti on, allowing use of the methods of multiple scales and matched asymptotic e xpansions. A theory is constructed which describes a local onset of convect ion consisting of rolls confined to the neighbourhood of the diameter of th e boundary (that is, the chord of maximum length that spans the boundary). Use of a Fourier transformation allows the curvature of both the boundary a nd the resulting roll pattern to be tal;en into account within a weakly non linear framework. Explicit solutions are obtained for the limiting case whe re the boundary curvature is large. Certain features of the nonlinear devel opment of the roll pattern and the manner in which it spreads to other part s of the domain are discussed with reference to particular geometries. The results suggest that; the pattern evolves in a way that maximizes the area of convection. One implication of the theory is that the onset of convectio n in shallow rectangular containers does not occur in the form of rolls par allel to the shorter sides.