The transition between hexagonal and square patterns is investigated in laboratory experiments on surface-tension-driven Bénard (Marangoni) convection in a fluid of Prandtl number 81. As the Marangoni number M is increased, an ideal hexagonal pattern is supplanted by a defect-free square pattern; the transition occurs gradually with patterns of mixed hexagonal, pentagonal, and square symmetry arising at intermediate values of M. An elementary topological process associated with two-dimensional patterns governs local changes in morphology; the dynamics are relaxational with all patterns becoming stationary with M fixed for a sufficiently long time. The transition is hysteretic and depends strongly on the pattern wave number.

The classical theory of surface-tension-driven convection by Pearson [J. Fluid Mech. 4, 489 (1958)] has been challenged by Rabin [Phys. Rev. E 53, R2057 (1996)] on the grounds that Pearson used an improper thermal boundary condition at the upper surface of the Liquid. We show that Pearson's theory is correct.

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.

Our experiments in thin liquid layers (approximately 0.1 mm thick) heated from below reveal a well-defined long-wavelength instability: at a critical temperature difference across the layer, the depth of the layer in the center of the cell spontaneously decreases until the liquid-air interface ruptures and a dry spot forms. The onset of this critical instability occurs at a temperature difference across the liquid layer that is 35% smaller than that predicted in earlier theoretical studies of a single layer model. Our analysis of a two-layer model yields predictions in accord with the observations for liquid layer depths greater than or equal to 0.15 mm, but for smaller depths there is an increasing difference between our predictions and observations (the difference is 25% for a layer 0.06 mm thick). In microgravity environments the long-wavelength instability observed in our terrestrial experiments is expected to replace cellular convection as the primary instability in thick as well as thin liquid layers heated quasistatically from below.

We demonstrate the stabilization of unstable periodic orbits whose trajectories in phase space are distant from the unperturbed dynamics in a convective flow experiment. A model independent, nonlinear control algorithm uses temperature measurements near the free surface of a convecting liquid bridge to compute control perturbations which are applied by a thermoelectric element. The algorithm employs a time series reconstruction of a nonlinear control surface to alter the system dynamics.

Electric fields produced by coplanar point charges have often been represented by field line diagrams that depict two-dimensional slices of the three-dimensional field. Serious problems with these ‘‘conventional’’ field line diagrams (CFLDs) have been overlooked. Two of these problems, ‘‘equatorial clumping’’ and ‘‘false monopole moment,’’ occur because a two-dimensional slice lacks information vital to the accurate representation of an inherently three-dimensional field. Equatorial clumping causes most CFLDs to exhibit unphysical behavior such as irregular spacing between field lines terminating on negative charges. CFLDs can also mistakenly indicate that a neutral charge distribution has a significant monopole moment. Such phenomena make the visual estimation of local field strengths impossible and render CFLDs of little utility for representing three-dimensional fields. While these ‘‘projection’’ problems can be avoided by using two-dimensional field line diagrams to represent two-dimensional (1/r) electric fields, or by using three-dimensional field line diagrams to represent three-dimensional fields, other forms of distortion generally remain.

High resolution laboratory experiments with large aspect ratio are being conducted for thin fluid layers heated from below and bounded from above by a free surface. The fluid depths are chosen sufficiently small (less than 0.06 cm) so that surface tension is the dominant driving mechanisms; the Rayleigh number is less than 5 for the results reported here. Shadowgraph visualization reveals that the primary instability leading to hexagons is slightly hysteretic (approximately 1 percent). Preliminary measurements of the convection amplitude using infrared imaging are also presented.