Planform selection in Bénard-Marangoni convection: l hexagons versus g hexagons.

The planform of weakly nonlinear B\'enard-Marangoni convection in a horizontally unbounded layer is analyzed using a combination of direct numerical simulation, amplitude equations, and qualitative discussion. It is demonstrated that there exists a critical Prandtl number ${\mathrm{Pr}}_{\mathit{c}}$ such that in fluids with Pr${\mathrm{Pr}}_{\mathit{c}}$ convection sets in as a pattern of hexagonal cells with downward motion in the center (g hexagons), while for fluids with Prg${\mathrm{Pr}}_{\mathit{c}}$ conventional hexagonal cells with upward motion in the center (l hexagons) appear at the onset of instability. For fluids with Marangoni and Prandtl numbers in the vicinity of the bicritical point (${\mathrm{Ma}}_{\mathit{c}}$,${\mathrm{Pr}}_{\mathit{c}}$) the hexagonal patterns undergo a secondary instability, leading to stationary rolls. The stability domain of stationary rolls increases with the distance from the critical Marangoni number. (c) 1995 The American Physical Society