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Pattern formations of 2D Rayleigh–Bénard convection with no‐slip boundary conditions for the velocity at the critical length scales
Author(s) -
Sengul Taylan,
Shen Jie,
Wang Shouhong
Publication year - 2014
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3317
Subject(s) - attractor , prandtl number , mathematics , eigenvalues and eigenvectors , mathematical analysis , rayleigh number , boundary value problem , boundary (topology) , convection , geometry , physics , mechanics , natural convection , quantum mechanics
We study the Rayleigh–Bénard convection in a 2D rectangular domain with no‐slip boundary conditions for the velocity. The main mathematical challenge is due to the no‐slip boundary conditions, because the separation of variables for the linear eigenvalue problem, which works in the free‐slip case, is no longer possible. It is well known that as the Rayleigh number crosses a critical threshold R c , the system bifurcates to an attractor, which is an ( m − 1)‐dimensional sphere, where m is the number of eigenvalues, which cross zero as R crosses R c . The main objective of this article is to derive a full classification of the structure of this bifurcated attractor when m = 2. More precisely, we rigorously prove that when m = 2, the bifurcated attractor is homeomorphic to a one‐dimensional circle consisting of exactly four or eight steady states and their connecting heteroclinic orbits. In addition, we show that the mixed modes can be stable steady states for small Prandtl numbers. Copyright © 2014 John Wiley & Sons, Ltd.