Dynamic Bifurcation and Transition in the Rayleigh-Bénard Convection with Internal Heating and Varying Gravity
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 S1, 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.
D. Han et al., "Dynamic Bifurcation and Transition in the Rayleigh-Bénard Convection with Internal Heating and Varying Gravity," Communications in Mathematical Sciences, vol. 17, no. 1, pp. 175-195, International Press of Boston, Inc., Jan 2019.
The definitive version is available at https://doi.org/10.4310/cms.2019.v17.n1.a7
Mathematics and Statistics
Center for High Performance Computing Research
Keywords and Phrases
Continuous and catastrophic transition; Dynamic transition; Internal heating sources; Rayleigh-BÃ©nard convection; Reduced equation
International Standard Serial Number (ISSN)
Article - Journal
© 2019 International Press of Boston, Inc., All rights reserved.
01 Jan 2019