Hexagon pattern convection for Benard-Marangoni problem

Abstract

Finite amplitude convection with hexagonal pattern is investigated in a horizontal fluid layer with poorly conducting boundaries. Effects of both buoyancy and surface tension are included. A perturbation technique based on a small parameter which is a measure of the ratio of the vertical to horizontal dimensions of the convective cells is developed which allows for the first time expansion of flow variables about many wave numbers of convective modes whose amplitudes can be of order unity. Stability of hexagon pattern convection is investigated with respect to three dimensional disturbances. Depending on the Marangoni number, up-hexagons (where motion is upward at the cells' centers) or down-hexagons (where motion is downward at the cells' centers) can be preferred. The wavelengths of the convective modes increase with their amplitudes, and there is a tendency for the preference of the larger wavelength modes. In particular, sufficiently long wavelength modes are always stable.