Pattern Formation in Rectangular Planform Containers

Abstract

Insight is gained into the onset and nonlinear development of convection in rectangular planform containers uniformly heated from below through a study of the two-dimensional 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 nonlinear development, a key element of which is the formation of a centrosymmetric roll pattern via a secondary bifurcation. The notion that in such systems, the pattern at onset always consists of rolls parallel to the shorter lateral boundaries is shown to be false. For planforms whose dimensions are much greater than the characteristic wavelength of convection, a weakly nonlinear 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 rectangle. For large planforms of a given diagonal length, an aspect ratio is determined at which the earliest onset of convection occurs.