Hydrodynamic stability of Rayleigh-Bénard convection with constant heat flux boundary condition

Abstract

We study the onset of thermal instability with the heat flux prescribed on the fluid boundaries. Assuming Boussinesq fluid, the Landau equation, which describes the evolution of the amplitude of the convection cells, is derived using the small amplitude expansion technique. For the case of a three-dimensional rectangular box with aspect ratio (8, 4, 1), the incipient convection cell is a two-dimensional one at p r = 0.72 pr = 0.72 , which is confirmed by the numerical solution of the three-dimensional Boussinesq equation with a Chebyshev-Fourier pseudospectral code. The secondary bifurcation gives rise to an oscillatory two-dimensional roll for the same Prandtl number at R = 2.0 R c R = 2.0{R_c} and the motion becomes three dimensional at R = 2.8 R c R = 2.8{R_c} .