Nonlinear analysis of the coupling between interface deflection and hexagonal patterns in Rayleigh-Bénard-Marangoni convection.

An analytical study is conducted to examine the coupling between buoyancy and surface tension forces in a shallow pool of a Boussinesq fluid. The liquid layer is bounded below by a rigid and poorly conducting plate, and is open to the air at its upper deformable free surface. At the onset of convection, the coupling between buoyancy and surface-tension-induced instabilities is represented by a single parameter m . A pattern selection study predicts the existence of a critical value for the parameter m , mc , that separates subcritical hexagons with upflow at their centers from subcritical hexagons with downflow at their centers. Several experimental situations are identified in terms of the Prandtl, Galileo, and capillary numbers, and the dependence of mc on these parameters is analyzed. Further, expressions for the interface deflection that accompanies the onset of Rayleigh-Bénard-Marangoni convection are derived. These expressions show explicitly the dependence of the interface morphology on the coupling parameter m and on the Prandtl, Galileo, and capillary numbers. It is also noted that, for a specific set of experimental parameters, spatial resonance will occur which leads to a situation in which the interfacial deformations do not conform to the flow pattern. @S1063-651X~96!10906-5#