Pattern formation in square planform containers

Insight is gained into the onset and non-linear development of convection in square planform containers heated from below through a study of the 2D Swift–Hohenberg equation with no-slip boundary conditions. Numerical computation is used to identify the bifurcation structure of steady-state solutions and to track their non-linear development. For planforms whose dimensions are much greater than the characteristic wavelength of convection, a weakly non-linear theory based on the use of multiple-scale matched asymptotic expansions is used to describe the onset of convection in the form of rolls confined to the neighbourhood of the diagonals of the square. Use of Fourier transforms allows roll curvature induced by the corners of the container to be taken into account and a local cross-roll structure is found to occur near the corners. Theoretical predictions are compared with experimental results for Rayleigh–Bénard convection.