Pattern formation in Rayleigh–Bénard convection

Abstract

The main objective of this article is to study the three-dimensional Rayleigh-Bénard convection in a rectangular domain from a pattern formation perspective. It is well known that as the Rayleigh number crosses a critical threshold, the system undergoes a Type-I transition, characterized by an attractor bifurcation. The bifurcated attractor is an (m − 1)-dimensional homological sphere where m is the multiplicity of the first critical eigenvalue. When m = 1, the structure of this attractor is trivial. When m = 2, it is known that the bifurcated attractor consists of steady states and their connecting heteroclinic orbits. The main focus of this article is then on the pattern selection mechanism and stability of rolls, rectangles and mixed modes (including hexagons) for the case where m = 2. We derive in particular a complete classification of all transition scenarios, determining the patterns of the bifurcated steady states, their stabilities and the basin of attraction of the stable ones. The theoretical results lead to interesting physical conclusions, which are in agreement with known experimental results. For example, it is shown in this article that only the pure modes are stable whereas the mixed modes are unstable. 1 arXiv:1109.5655v1 [nlin.PS]

Bifurcation solutions of Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates

“…In nature, it is well-known that there are two different kinds of phase transitions including the dynamical phase transitions, and topological phase transitions (TPTs), also called the pattern formation transitions, for the details we refer to [22,30]. The rigorous theory of dynamical phase transitions in the dissipative system goes back to the pioneering work established by Ma and Wang in [28], many interesting results related to dynamical phase transitions refer to [12,18,22,23,24,25,26,27,28,36,37]. For the rigorous mathematical theory associated with topological phase transitions, we refer to [22] and the series works for TPTs in [30].…”

Pattern formations of 2D Rayleigh–Bénard convection with no‐slip boundary conditions for the velocity at the critical length scales

“…Let m be the number of modes, which become critical as the first Rayleigh number R c is crossed as given by (9). Ma and Wang [7,8] proved that under some general boundary conditions, the problem has an attractor † R , which bifurcates from .0, R c / as R crosses R c .…”

On the stability and transition for the Navier‐Stokes‐alpha model

“…The second one is using the technique that central unstable space can be expressed in a form of the complex combination of first eigenvector and its conjugate, to get a reduced equation, which is different from the method applied in previous studies. Through considering the bifurcation of the reduced equation by using the associated with bifurcation theorem developed by Ma and Wang, we prove that as Reynolds number R crosses a certain critical value bifurcates from the trivial solution to a periodic solution whose stability is completely determined by the sign of real part P given in . More precisely, if Re P <0, then bifurcates from the trivial solution to a stable periodic solution as Reynolds number R > R * .…”

“…Following this purpose, the dynamic transition theory shows that phase transitions are classified into three types: continuous, catastrophic, and random. Due to the classification of the phase transition is closely bound up with practical problems, the theory has been successfully applied in the study of a number of transition problems, including transitions of quasi-geostrophic channel flows, 9 instability and transitions of Rayleigh-Bénard convection, [12][13][14] dynamic transitions of Cahn-Hilliard equation, 15,16 and boundary layer separation, 17 to name a few. [18][19][20][21][22] Our present study relies on two methods.…”