2013/02/14 by Taylan Şengül, Taylan Sengul, Sengul, Taylan +4
Engineering · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics and Turbulent Flows #Fluid Dynamics and Vibration Analysis #Pattern Formation and Solitons (nlin.PS) #Vibration and Dynamic Analysis #math.AP #nlin.PS
paper · pdf · doi:10.48550/arxiv.1302.3625
arxiv created 2013/02/14 · openalex publication_date 2013/02/14 · arxiv updated 2013/02/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the Rayleigh-Bénard convection in a 2-D rectangular domain with no-slip boundary conditions for the velocity. The main mathematical challenge is due to the no-slip boundary conditions, since 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 Rc, 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 Rc. 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.