Evolution and Stability Limits of Patterns in Surface-Tension-Driven-Bénard Convection

Abstract

Experimental results on the evolution of patterns in surface-tension driven Bénard convection are reported. Experiments were performed in a hexagonal container with an aspect ratio Г = 65 for different distances (ε) to convection threshold and two Prandtl numbers (Pr₁ = 440 and Pr₂ = 880). Regular hexagonal patterns are initially imposed that are then allowed to relax. A stability diagram of the spatial modes, the variation of the stable modes band and the selected mean wavenumber were established as functions of ε and Pr. For the mean wavenumber, an increase followed by a decrease, is observed for Pr₁; and an opposite behavior is observed for Pr₂. The bandwidth of stable modes broadens with ε, the broadening being larger for Pr₂ than for Pr₁. Using the phenomenological model of Swift-Hohenberg, developed for roll patterns and adapted to hexagonal patterns, a potential function is defined, which allows the study of the time evolution of such arrays. This function is minimized during the transition of the structure from the initial to the steady (asymptotic) state. The various wavenumber selection mechanisms are considered: bulk contribution (wavenumber disparity, cell row bending), topological defects and sidewalls. The total potential is mainly influenced by the wavenumber variation and by defects. Owing to the special geometry of the container, sidewall and bulk effects related to the local wave vector divergence (due to the disalignment and the bending of the cell rows), play a less important role, at least in such medium size containers. From the experiment, it follows that the model is valid up to an ε of about 9 for both Pr. Beyond that limit, the potential fluctuates probably due to the spatiotemporal mobility of the structure and the nucleation of defects, whose number increases with increasing ε.