Hexagonal Marangoni Convection in a Rectangular Box With Slippery Walls

A linear and nonlinear study of surface-tension-driven instability in a rectangular box with slippery lateral walls is presented. Particular attention is devoted to steady convection with hexagonal structure. It is shown that, even in very small boxes, convection can set in in the form of hexagons more or less distorted according to the aspect ratios of the box. The distorted hexagons appear generally as subcritical solutions; the depth of the subcritical domain is determined as a function of the Prandtl number. In particular, it is found that, at small Prandtl numbers (Pr less-than-or-equal-to 0.23), the direction of the flow may be downwards at the cell centre. For medium to large values of the Prandtl number, the fluid rises at the centre of the hexagons, as is observed in most experiments.