Pattern formation in fluids heated from below is examined in the presence of a free flat surface. On that surface, the Marangoni effect is taken into account as a second instability mechanism. It is shown that phase instabilities already known from Rayleigh-Bénard convection confine the region of stable hexagons and shrink the band of stable wavelengths considerably. The problem is attacked from two sides: amplitude equations are derived explicitly from the basic hydrodynamic equations and are analyzed with regard to secondary instabilities. This results in stability diagrams analogous to earlier calculations obtained for parallel rolls in simple Rayleigh-Bénard convection. On the other hand, we developed a numerical scheme that allows for a direct integration of the fully-three-dimensional hydrodynamic equations. This method is described in detail and time series for pattern evolution are presented, showing phase and amplitude instabilities as expected from the formalism of amplitude equations. Finally we show the connection between amplitude equations and two-dimensional generalized Ginzburg-Landau equations. These models may reproduce pattern formation near convective threshold in a quantitative way. They have the advantage of being more general than the basic equations and they can be treated numerically in a much easier way. This allows the computation of pattern evolution in very-large-aspect-ratio systems.

• Received 26 April 1993

DOI:https://doi.org/10.1103/PhysRevE.48.3622