Two-Dimensional Infinite Prandtl Number Convection: Structure of Bifurcated Solutions

This paper examines the bifurcation and structure of the bifur- cated solutions of the two-dimensional inflnite Prandtl number convection problem. The existence of a bifurcation from the trivial solution to an at- tractor §R was proved by Park (14). We prove in this paper that the bifur- cated attractor §R consists of only one cycle of steady state solutions and that it is homeomorphic to S1. By thoroughly investigating the structure and transitions the solutions of the inflnite Prandtl number convection problem in physical space, we conflrm that the bifurcated solutions are indeed struc- turally stable. In turn, this will corroborate and justify the suggested results with the physical flndings about the presence of the roll structure. This bi- furcation analysis is based on a new notion of bifurcation, called attractor bifurcation, and structural stability is derived using a new geometric theory of incompressible ∞ows. Both theories were developed by Ma and Wang; see (11, 12).