Nonlinear thermal convection with finite conducting boundaries

Finite-amplitude thermal convection in a horizontal layer with finite conducting boundaries is investigated. The nonlinear steady problem is solved by a perturbation technique, and the preferred mode of convection is determined by a stability analysis. Square cells are found to be the preferred form of convection in a semi-infinite three-dimensional region Ω in the (γ b ,γ t , P )-space (γ b and γ t are the ratios of the thermal conductivities of the lower and upper boundaries to that of the fluid and P is the Prandtl number). Two-dimensional rolls are found to be the preferred convection pattern outside Ω. The dependence on γ b , γ t and P of the heat transported by convection is computed for the various solutions analysed in the paper.