Weakly nonlinear forced convection patterns in rectangular planform containers

This paper considers the structure of weakly nonlinear steady-state convection patterns in shallow rectangular planform containers heated from below. The lateral dimensions of the container are assumed to be much larger than the characteristic wavelength of convection, and the lateral boundaries are subject to forcing equivalent, for example, to imperfect thermal insulation in the Rayleigh-Benard problem. This has the effect of generating rolls parallel and perpendicular to the lateral boundaries. The resulting patterns are modelled by a coupled pair of nonlinear amplitude equations derived from a phenomenological model of convection introduced by Swift and Hohenberg [Phys. Rev. A15 (1977) 319]. These equations are applicable in the weakly nonlinear limit to a variety of pattern-forming systems such as the Rayleigh-Benard system. Solutions are found using both numerical and asymptotic methods. The boundary imperfection is shown to give rise to some novel effects, including the possibility of patterns containing square cells. More generally, patterns evolve that are dominated by rolls but with transitions to more complex bimodal forms near the edges of the container. The emergence and structure of transition lines, or grain boundaries, is analysed in detail.