Rayleigh–Benard Instability: a Study by the Methods of Cahn–Hillard Theory of Nonequilibrium Phase Transitions

Abstract

This article is an attempt to study the process of Rayleigh–Benard convective instability by the methods used for mathematical modeling of critical phenomena as nonequilibrium phase transitions in their initial stages of spinodal decomposition. We show that it is possible to extend the formalism adopted in the Cahn–Hillard theory of nonequilibrium phase transitions and perfected on problems of highgradient crystallization to other types of problems, in particular, those pertaining to the Rayleigh–Benard convective instability. For the initial stage of instability, a model is constructed that represents it as a nonequilibrium phase transition due to diffusive stratification. It is shown that the Gibbs free energy of deviation from the homogeneous state (with respect to the instability under consideration) is an analogue of the Ginsburg–Landau potential. Numerical experiments, by means of boundary temperature control, have been conducted with regard to self-excitation of the homogeneous state. Numerical analysis shows that convective flows may appear and proceed from regular forms (the so-called regular structures) to nonregular flows through a chaotization of the process. External factors, such as temperature growth, may lead to chaos via period doubling bifurcations.