Communications in Mathematical Sciences

Volume 11 (2013)

Number 1

Pattern formation in Rayleigh–Bénard convection

Pages: 315 – 343



Taylan Sengul (Department of Mathematics, Indiana University, Bloomington, In., U.S.A.)

Shouhong Wang (Department of Mathematics, Indiana University, Bloomington, In., U.S.A.)


The main objective of this article is to study the three-dimensional Rayleigh-Bénard 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.


Rayleigh-Bénard convection, pattern formation, rolls, rectangles, hexagons, dynamic transitions

2010 Mathematics Subject Classification

35Qxx, 76-xx, 86-xx

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