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首页> 外文期刊>Mathematical Methods in the Applied Sciences >Pattern formations of 2D Rayleigh-Benard convection with no-slip boundary conditions for the velocity at the critical length scales
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Pattern formations of 2D Rayleigh-Benard convection with no-slip boundary conditions for the velocity at the critical length scales

机译：临界长度尺度上具有无滑移边界条件的二维Rayleigh-Benard对流的模式形成

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摘要

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.
机译：我们研究了二维矩形域中的Rayleigh-Benard对流，该二维对流具有无滑移边界条件的速度。数学上的主要挑战是由于无滑移边界条件，因为在自由滑移情况下不再适用于线性特征值问题的变量分离。众所周知，当瑞利数越过临界阈值R-c时，系统分叉到一个吸引子，该吸引子是一个（m-1）维球体，其中m是特征值的数目，当R越过R-c时本征值越过零。本文的主要目的是在m = 2时得出该分叉吸引子的结构的完整分类。更精确地，我们严格证明当m = 2时，该分叉吸引子对于包含正则项的一维圆是同胚的四个或八个稳态及其相互连接的异斜轨道。此外，我们表明对于小Prandtl数，混合模式可以是稳定的稳态。版权所有（C）2014 JohnWiley＆Sons，Ltd.

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