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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 ͑PrϽ0.23͒, the direction of the flow may be downward at the center of the hexagonal cell, whatever the value of the buoyancy force.

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Linear coupled buoyancy and thermocapillary instabilities (the Benard-Marangoni problem) in a Maxwell viscoelastic fluid layer heated from below are studied. As the principle of exchange of stability is no longer valid, both stationary and oscillatory solutions are considered. Beyond a critical value of the relaxation time, the instability appearing in a fluid layer with a free upper surface subject to a temperature-dependent surface tension takes the form of oscillations. The critical temperature difference between the lower and upper surfaces is determined as a function of the Prandtl number and the relaxation time. The instability thresholds are graphically represented on 'Nield's diagrams' where the critical Marangoni number is given versus the Rayleigh number. At high Prandtl numbers discontinuities in the solutions are displayed for some specific values of the fluid layer thickness. For some range of variation of the parameters, thermocapillarity has an unusual stabilizing effect.