The 336-year-old synchronization observations of Christiaan Huygens are re-examined in modern experiments. A simple model of synchronization is proposed.

Surface-tension-driven Bénard (Marangoni) convection in liquid layers heated from below can exhibit a long-wavelength primary instability that differs from the more familiar hexagonal instability associated with Bénard. This long-wavelength instability is predicted to be significant in microgravity and for thin liquid layers. The instability is studied experimentally in terrestrial gravity for silicone oil layers 0.007 to 0.027 cm thick on a conducting plate. For shallow liquid depths (<.017 cm for 0.102 cm2 s−1 viscosity liquid), the system evolves to a strongly deformed long-wavelength state which can take the form of a localized depression (‘dry spot’) or a localized elevation (‘high spot’), depending on the thickness and thermal conductivity of the gas layer above the liquid. For slightly thicker liquid depths (0.017–0.024 cm), the formation of a dry spot induces the formation of hexagons. For even thicker liquid depths (>0.024 cm), the system forms only the hexagonal convection cells. A two-layer nonlinear theory is developed to account properly for the effect of deformation on the interface temperature profile. Experimental results for the long-wavelength instability are compared to our two-layer theory and to a one-layer theory that accounts for the upper gas layer solely with a heat transfer coefficient. The two-layer model better describes the onset of instability and also predicts the formation of localized elevations, which the one-layer model does not predict. A weakly nonlinear analysis shows that the bifurcation is subcritical. Solving for steady states of the system shows that the subcritical pitchfork bifurcation curve never turns over to a stable branch. Numerical simulations also predict a subcritical instability and yield long-wavelength states that qualitatively agree with the experiments. The observations agree with the onset prediction of the two-layer model, except for very thin liquid layers; this deviation from theory may arise from small non-uniformities in the experiment. Theoretical analysis shows that a small non-uniformity in heating produces a large steady-state deformation (seen in the experiment) that becomes more pronounced with increasing temperature difference across the liquid. This steady-state deformation becomes unstable to the long-wavelength instability at a smaller temperature difference than that at which the undeformed state becomes unstable in the absence of non-uniformity.

This review summarizes recent experimental studies of instabilities in free-surface flows driven by thermocapillarity. Two broad classes are considered, depending upon whether the imposed temperature gradient is perpendicular (Marangoniconvection instability) or parallel (thermocapillary-convection instability) to the free surface. Both steady and time-dependent instabilites are reviewed in experiments employing both large-and small-aspect-ratio geometries of various symmetries.

Laboratory studies reveal a deformational instability that leads to a drained region (dry spot) in an initially flat liquid layer (with a free upper surface) heated uniformly from below. This long-wavelength instability supplants hexagonal convection cells as the primary instability in viscous liquid layers that are sufficiently thin or are in microgravity. The instability occurs at a temperature gradient 34% smaller than predicted by linear stability theory. Numerical simulations show a drained region qualitatively similar to that seen in the experiment.PACS numbers: 47.20.Dr, 47.20.Ky, 47.54.+r, 68.15.+e Bénard's observation in 1900 [1] of hexagonal convection patterns launched the modern study of convection, pattern formation, and instabilities; yet understanding of the surface-tension-driven regime in which Bénard performed his experiments is still far from complete. Block [2] and Pearson [3] first showed how temperature-induced surface tension gradients (thermocapillarity) caused the instability observed in Bénard's experiments. Pearson's linear stability analysis with a nondeformable liquid-gas interface yielded an instability at wavenumber q = 1.99 (scaled by the mean liquid depth d) and Marangoni number M c = 80, where M ≡ σ T △ T d/ρνκ (see Fig. 1) expresses the competition between the destabilization by thermocapillarity and the stabilization by diffusion.A deformable free surface allows a second type of primary instability in which a perturbation creates a nonuniform liquid depth and temperature profile [4,5]. Thermocapillarity causes cool, elevated regions to pull liquid from warm, depressed regions (see Fig. 1). The instability appears with a long wavelength since surface tension stabilizes short wavelengths. Linear stability analyses that allow for deformation reveal this instability (see Fig. 2) at zero wavenumber (q = 0) and M c = 2 3 G [5,6], where the Galileo number, G ≡ gd 3 /νκ (g is the acceleration of gravity), gives the relative strengths of the stabilizing mechanisms of diffusion and gravity and thus determines which instability will form. For sufficiently thin, viscous liquid layers or small g (e.g., microgravity), 2 3 G < 80, so the long-wavelength mode should become the primary instability (see Fig. 2). This longwavelength instability has not been experimentally investigated.In this Letter we describe experimental observations of the onset of the long-wavelength instability and compare these observations to linear stability theory. The instability leads to a large-scale drained region with diameter ∼ 100d. The qualitative features of the instability are compared to nonlinear theory based on a longwavelength evolution equation. We also explore the competition between the long-wavelength and hexagonal instabilities and study the physical mechanism that selects which pattern will appear. For a range of liquid depths, both patterns coexist.

Experiments with shadowgraph visualization reveal a subcritical transition to a hexagonal convection pattern in thin liquid layers that have a free upper surface and are heated from below. The measured critical Marangoni number (84) and observation of hysteresis (3%) agree with theory. In some experiments, imperfect bifurcation is observed and is attributed to deterministic forcing caused in part by the lateral boundaries in the experiment.