Pattern Formation in Rayleigh Benard Convection

TL;DR

This paper investigates pattern formation in three-dimensional Rayleigh-Benard convection, classifying bifurcation scenarios, analyzing stability of various patterns, and confirming theoretical results with experimental observations.

Contribution

It provides a complete classification of transition scenarios and stability analysis for patterns in Rayleigh-Benard convection with m=2 eigenvalues.

Findings

Pure modes are stable, mixed modes are unstable.

Bifurcated attractor consists of steady states and heteroclinic orbits.

Results align with experimental observations.

Abstract

The main objective of this article is to study the three-dimensional Rayleigh-Benard convection in a rectangular domain from a pattern formation perspective. It is well known that as the Rayleigh number crosses a critical threshold, the system undergoes a Type-I transition, characterized by an attractor bifurcation. The bifurcated attractor is an (m-1)-dimensional homological sphere where m is the multiplicity of the first critical eigenvalue. When m=1, the structure of this attractor is trivial. When m=2, it is known that the bifurcated attractor consists of steady states and their connecting heteroclinic orbits. The main focus of this article is then on the pattern selection mechanism and stability of rolls, rectangles and mixed modes (including hexagons) for the case where m=2. We derive in particular a complete classification of all transition scenarios, determining the patterns of the bifurcated steady states, their stabilities and the basin of attraction of the stable ones. The theoretical results lead to interesting physical conclusions, which are in agreement with known experimental results. For example, it is shown in this article that only the pure modes are stable whereas the mixed modes are unstable.