Pattern formations of 2D Rayleigh-Benard convection with no-slip boundary conditions for the velocity at the critical length scales

ŞENGÜL M. T. , Shen J., Wang S.

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, vol.38, no.17, pp.3792-3806, 2015 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 38 Issue: 17
  • Publication Date: 2015
  • Doi Number: 10.1002/mma.3317
  • Page Numbers: pp.3792-3806


We study the Rayleigh-Benard 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 (C) 2014 JohnWiley & Sons, Ltd.