Benard-Marangoni convection: planforms and related theoretical predictions

A derivation is given of the amplitude equations governing pattern formation in surface tension gradient-driven Benard-Marangoni convection. The amplitude equations are obtained from the continuity, the Navier-Stokes and the Fourier equations in the Boussinesq approximation neglecting surface deformation and buoyancy. The system is a shallow liquid layer heated from below, confined below by a rigid plane and above with a free surface whose surface tension linearly depends on temperature. The amplitude equations of the convective modes are equations of the Ginzburg-Landau type with resonant advective non-variational terms. Generally, and in agreement with experiment, above threshold solutions of the equations correspond to an hexagonal convective structure in which the fluid rises in the centre of the cells. We also analytically study the dynamics of pattern formation leading not only to hexagons but also to squares or rolls depending on the various dimensionless parameters like Prandtl number, and the Marangoni and Blot numbers at the boundaries. We show that a transition from an hexagonal structure to a square pattern is possible. We also determine conditions for alternating, oscillatory transition between hexagons and rolls. Moreover, we also show that as the system of these amplitude equations is non-variational the asymptotic behaviour (t --> infinity) may not correspond to a steady convective pattern. Finally, we have determined the Eckhaus band for hexagonal patterns and we show that the non-variational terms in the amplitude equations enlarge this band of allowable modes. The analytical results have been checked by numerical integration of the amplitude equations in a square container. Like in experiments, numerics shows the emergence of different hexagons, squares and rolls according to values given to the parameters of the system.