Pattern Formations of 2D Rayleigh-B\`enard Convection with No-Slip Boundary Conditions for the Velocity at the Critical Length Scales

TL;DR

This paper rigorously classifies the bifurcation structure in 2D Rayleigh-Bénard convection with no-slip boundary conditions, revealing a circle of steady states and heteroclinic connections at critical length scales.

Contribution

It provides a full mathematical classification of the bifurcated attractor for the case when two eigenvalues cross zero, including stability of mixed modes.

Findings

Bifurcated attractor is homeomorphic to a circle with four or eight steady states.

Existence of heteroclinic orbits connecting steady states.

Mixed modes can be stable for small Prandtl numbers.

Abstract

We study the Rayleigh-B{\'e}nard 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 $R_c$, 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.