Rayleigh-Bénard convection in a newtonian liquid bounded by rigid isothermal boundaries

Abstract

A non-linear stability analysis of Rayleigh-Bénard convection is reported in the paper for the case of rigid isothermal boundaries. The Ginzburg-Landau equation with cubic non-linearity is obtained. Out of the three fixed points of the equation, one arises in the pre-onset regime and two during post-onset. The pre-onset one is unstable at post-onset times and hence we have a pitchfork bifurcation point. Comparing heat transports it is found that the Nusselt number of the rigid isothermal boundary condition is less than that of the free isothermal case. Prandtl number influence on the Nusselt number is negligible in very high Prandtl number liquids at all times and at long times this is true for all liquids. Using vital information from the procedure leading to the Ginzburg-Landau equation a minimal Fourier-Galerkin expansion is proposed to derive a generalized Lorenz model for rigid-isothermal boundaries. The Hopf bifurcation point signaling onset of chaos is obtained by transforming the Lorenz model of rigid isothermal boundaries to a standard form resembling the classical one. The possibility of regular, chaotic, periodic and mildly chaotic motions in the Lorenz system is shown through the plots of the maximum Lyapunov exponent and the bifurcation diagram.