Nonlinear analysis of coupled gravitational and capillary thermoconvention in thin fluid layers

A weakly nonlinear analysis of coupled surface-tension- and gravitational-driven instability in thin fluid layers is presented. The fluid is assumed to be Newtonian and incompressible and is heated from below. Newton's law of cooling is used to model the heat exchange at the upper surface. Ginzburg-Landau amplitude equations are established and the preferred mode of convection is obtained. The influence of the Prandtl and Biot numbers is emphasized. It is shown that hexagonal cells are the only stable configurations just above the threshold. Rolls are stable in a nonlinear regime at sufficiently large values of the thickness of the layer. A subcritical domain is also displayed. By increasing surface-tension effects one promotes the hexagonal pattern. In the limiting case of a negligible temperature dependence of the surface tension, only rolls are stable. Another interesting result is that, at small Prandtl numbers (Pr0.23), the direction of the flow may be downward at the center of the hexagonal cell, whatever the value of the buoyancy force. {} 1996 The American Physical Society.