Amplitude equations and nonlinear dynamics of surface-tension and buoyancy-driven convective instabilities

This work is a theoretical contribution to the study of thermo-hydrodynamic instabilities in fluids submitted to surface-tension (Marangoni) and buoyancy (Rayleigh) effects in layered (Benard) configurations. The driving constraint consists in a thermal (or a concentrational) gradient orthogonal to the plane of the layer(s).

Linear, weakly nonlinear as well as strongly nonlinear analyses are carried out, with emphasis on high Prandtl (or Schmidt) number fluids, although some results are also given for low-Prandtl number liquid metals. Attention is mostly devoted to the mechanisms responsible for the onset of complex spatio-temporal behaviours in these systems, as well as to the theoretical explanation of some existing experimental results.

As far as linear stability analyses (of the diffusive reference state) are concerned, a number of different effects are studied, such as Benard convection in two layers coupled at an interface (for which a general classification of instability modes is proposed), surface deformation effects and phase-change effects (non-equilibrium evaporation). Moreover, a number of different monotonous and oscillatory instability modes (leading respectively to patterns and waves in the nonlinear regime) are identified. In the case of oscillatory modes in a liquid layer with deformable interface heated from above, our analysis generalises and clarifies earlier works on the subject. A new Rayleigh-Marangoni oscillatory mode is also described for a liquid layer with an undeformable interface heated from above (coupling between internal and surface waves).

Weakly nonlinear analyses are then presented, first for monotonous modes in a 3D system. Emphasis is placed on the derivation of amplitude (Ginzburg-Landau) equations, with universal structure determined by the general symmetry properties of the physical system considered. These equations are thus valid outside the context of hydrodynamic instabilities, although they generally depend on a certain number of numerical coefficients which are calculated for the specific convective systems studied. The nonlinear competitions of patterns such as convective rolls, hexagons and squares is studied, showing the preference for hexagons with upflow at the centre in the surface-tension-driven case (and moderate Prandtl number), and of rolls in the buoyancy-induced case.

A transition to square patterns recently observed in experiments is also explained by amplitude equation analysis. The role of several fluid properties and of heat transfer conditions at the free interface is examined, for one-layer and two-layer systems. We also analyse modulation effects (spatial variation of the envelope of the patterns) in hexagonal patterns, leading to the description of secondary instabilities of supercritical hexagons (Busse balloon) in terms of phase diffusion equations, and of pentagon-heptagon defects in the hexagonal structures. In the frame of a general non-variational system of amplitude equations, we show that the pentagon-heptagon defects are generally not motionless, and may even lead to complex spatio-temporal dynamics (via a process of multiplication of defects in hexagonal structures).

The onset of waves is also studied in weakly nonlinear 2D situations. The competition between travelling and standing waves is first analysed in a two-layer Rayleigh-Benard system (competition between thermal and mechanical coupling of the layers), in the vicinity of special values of the parameters for which a multiple (Takens-Bogdanov) bifurcation occurs. The behaviours in the vicinity of this point are numerically explored. Then, the interaction between waves and steady patterns with different wavenumbers is analysed. Spatially quasiperiodic (mixed) states are found to be stable in some range when the interaction between waves and patterns is non-resonant, while several transitions to chaotic dynamics (among which an infinite sequence of homoclinic bifurcations) occur when it is resonant. Some of these results have quite general validity, because they are shown to be entirely determined by quadratic interactions in amplitude equations.

Finally, models of strongly nonlinear surface-tension-driven convection are derived and analysed, which are thought to be representative of the transitions to thermal turbulence occurring at very high driving gradient. The role of the fastest growing modes (intrinsic length scale) is discussed, as well as scalings of steady regimes and their secondary instabilities (due to instability of the thermal boundary layer), leading to chaotic spatio-temporal dynamics whose preliminary analysis (energy spectrum) reveals features characteristic of hydrodynamic turbulence. Some of the (2D and 3D) results presented are in qualitative agreement with experiments (interfacial turbulence).