The effect of distant sidewalls on the transition to finite amplitude Bénard convection—II

Abstract

A two-dimensional fluid layer of height d is confined laterally by rigid sidewalls distance 2Ld apart, where L, the semi-aspect ratio of the layer, is large. Constant temperatures are maintained at the upper and lower boundaries while at the sidewalls it is assumed that the horizontal heat flux has magnitude λ. If λ = 0 (perfect insulation) a finite amplitude motion sets in when the Rayleigh number R reaches a critical value R_c, but in part I (Daniels 1977) it was shown that if λ = O(L⁻¹) this bifurcation (in a state diagram of amplitude versus Rayleigh number) is displaced into a single stable solution in the region |R – R_c| = O(L⁻²), representing a smooth increase in amplitude of the cellular motion with Rayleigh number. All other solutions (or “secondary modes”) in this region were shown to be unstable. In the present paper an examination of the two intermediate regimes λ − O(L^−5/2) and λ = O(L⁻²) is carried out, to trace the location of an additional stable solution in the form of a secondary mode, which stems from R_c when λ = 0, and which in the limit as λL² → ∞ is shown to be removed from the region ߋR − Rc| = 0(L⁻²), consistent with the results of I.