Communications in Mathematical Sciences

Volume 17 (2019)

Number 1

Dynamic bifurcation and transition in the Rayleigh–Bénard convection with internal heating and varying gravity

Pages: 175 – 192

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n1.a7

Authors

Daozhi Han (Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, Mo., U.S.A.)

Marco Hernandez (Department of Mathematics, Indiana University, Bloomington, Ind., U.S.A.)

Quan Wang (Department of mathematics, Sichuan University, Chengdu, China)

Abstract

In this article, we study the dynamic transition of the Rayleigh–Bénard convection with internal heating and varying gravity. We show that this problem can only undergo a continuous or catastrophic transition, and the specific type is completely determined by the sign of a parameter—referred to as the transition number—that depends on the aspect ratio. Through numerical simulations we compute, for six qualitatively different heating sources, the corresponding value of the transition number, and find that the transition is always continuous. In particular, after transition, the system bifurcates from a basic steady state to a family of stable steady states, homeomorphic to $\mathbb{S}^1$, that describe the heating convection. Furthermore, upon varying the aspect ratio immediately after the first transition has occurred, we find the existence of a second transition, which is always catastrophic. More precisely, there exists a family of discrete values of the aspect ratio, which are the discontinuity points of the transition number, at which the transition is catastrophic and the number of convection rolls changes.

Keywords

Rayleigh–Bénard convection, internal heating sources, dynamic transition, continuous and catastrophic transition, reduced equation

2010 Mathematics Subject Classification

37L15, 37L99, 76D10, 76E06, 76E20

Received 10 February 2018

Accepted 3 November 2018

Published 30 May 2019