Regular and chaotic Rayleigh-Bénard convective motions in methanol and water

Abstract

The study of regular and chaotic Rayleigh-Bénard convective motions in methanol and water is made. The stationary mode of convection is shown to be the preferred one at the onset of convection in the case of both the liquids. Using a higher-order truncated Fourier series representation we arrive at the energy-conserving penta-modal Lorenz model and then the tri-modal Lorenz model is obtained as a limiting case of it. To keep the study analytical the Ginzburg-Landau model is derived from the penta-modal Lorenz model. It is shown that the tri- and the penta-modal Lorenz models predict exactly the same results leading to the conclusion that the tri-modal Lorenz model is a good enough truncated model for a weakly nonlinear study of convection. The Rayleigh numbers at which the onset of regular convective and chaotic motions occur are reported for both methanol and water. The behavior of the dynamical system is studied using the spectrum of Lyapunov exponents, the maximum Lyapunov exponent, the bifurcation diagram and the phase-space plots. The Hopf bifurcation Rayleigh number is obtained analytically. It is shown that the thresholds for onset of regular and chaotic motions are smaller in the case of methanol compared to water. Another very important finding of the paper is to show the existence of a developing region for chaos before becoming fully-developed.