Stochastic geometry of polygonal networks: An alternative approach to the hexagon-square transition in Bénard convection

Abstract

We apply stochastic geometry to the transition from hexagonal to square cells recently observed in surface- tension-driven Benard convection. In particular we study the metric and topological evolution of Benard patterns as a function of the temperature difference, DT, across the layer. The preference of square Benard cells at higher DT is a consequence of both a higher efficiency in heat transfer and more favorable metric properties. Most notably, the perimeter-area ratio of a square cell exceeds that of a hexagonal cell by an unexpectedly high value. From a topological point of view, the Benard pattern obeys the Aboav-Weaire law at all times, even in the presence of threefold and fourfold vertices. The regimes above and below the transition are characterized by different topological correlations between neighboring cells. With the appearance of fourfold vertices, the topological correlation changes from negative to positive. @S1063-651X~98!14709-8#