Pattern selection in long-wavelength convection

Abstract

The long wavelength of the first instability in Rayleigh-Bénard convection between nearly thermally insulating horizontal plates is typical of a variety of physical systems. The evolution of such an instability is described by the equation α t = αα− μ▿ 2 α−▿ 4 α+ κ▿·‖▿ α‖ 2▿ α+ β▿·▿ 2 α▿ α− λ▿· α▿ α+ σ▿ 2‖▿ α‖ 2, where α is the planform function, μ is the scaled Rayleigh number and κ = ±1. The quantities α,β,λ represent the effects of finite Biot number, asymmetry in the boundary conditions at top and bottom, and departures from the Boussinesq approximation, respectively. The quantity σ= β when the original problem is self-adjoint, but σ ≠ β otherwise. Planform selection is studied for α < 0, κ = +1 using equivariant bifurcation theory. On the square lattice both rolls and squares can be stable, depending on the parameters β, λ and σ. The possible secondary bifurcations located near various codimension-two singularities are analyzed. On the hexagonal lattice the primary bifurcation is always degenerate when β= λ= σ=0. Of the six primary solution branches possible in this case the hexagon branch is stable. When β− σ= λ=0, both rolls and hexagons bifurcate supercritically, but rolls are stable. Finally, when β ≠ σ and/or ψ ≠ 0 a hysteretic transition to H + or H - occurs depending on sgn( β + λ k 2 c−σ ); if ‖ β‖, ‖ λ‖, ‖ σ‖ ⪡ stable H +, H - coexist at larger amplitudes, but if ‖β + λ k 2 c −σ‖ ⪡ ‖β‖, ‖λ‖, ‖σ‖ = 0(1) , a further hysteretic transition takes place with increasing amplitude in which the hexagons are replaced by stable rolls.