W. D. McCormick

Pattern formation by interacting chemical fronts

Experiments on a bistable chemical reaction in a continuously fed thin gel layer reveal a new type of spatiotemporal pattern, one in which fronts propagate at a constant speed until they reach a critical separation (typically 0.4 millimeter) and stop. The resulting asymptotic state is a highly irregular stationary pattern that contrasts with the regular patterns such as hexagons, squares, and stripes that have been observed in many nonequilibrium systems. The observed patterns are initiated by a finite amplitude perturbation rather than through spontaneous symmetry breaking.

Long-wavelength surface-tension-driven Bénard convection: experiment and theory

Surface-tension-driven Benard (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 Benard. 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 cm 2 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.

Alternating periodic and chaotic regimes in a chemical reaction — Experiment and theory

Abstract Experiments on the Belousov-Zhabotinskii reaction in a stirred flow reactor reveal a sequence of alternating periodic and chaotic regimes as the residence time is increased. The same sequence is predicted by a numerical study of a four-variable model that describes the primary chemical steps of the reaction.

Universality, multiplicity, and the effect of iron impurities in the Belousov–Zhabotinskii reaction

In experiments on the Belousov–Zhabotinskii reaction in a flow reactor we have observed dynamical behavior that is described well by one‐dimensional maps with a single maximum. A sequence of period doubling bifurcations was observed as a parameter was varied, and beyond the accumulation point for the period doubling sequence there was a sequence of periodic states that has the same symbolic dynamics as the states of the U (universal) sequence of Metropolis, Stein, and Stein (1973). However, in another experiment with malonic acid from a different vendor, we found that some states with particular symbol sequences occurred in three different parameter ranges rather than in one range as in the U sequence. Analysis of the effect of impurities in the reagents showed that some impurities (e.g., Fe3+ and esters of malonic acid) at concentrations of only a few ppm produced dramatic changes in the dynamics; such impurities are contained in commercially available malonic acid. Experiments with purified malonic acid...

Time-independent square patterns in surface-tension-driven Bénard convection

The transition between hexagonal and square patterns is investigated in laboratory experiments on surface-tension-driven Benard (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.

Experimental Observations of Complex Dynamics in a Chemical Reaction

In experiments on a stirred flow chemical reactor we have observed a sequence of periodic and chaotic regimes that alternate as a function of the flow rate. The periodic regimes are characterized by power spectra consisting of a single fundamental frequency component and harmonics and by limit cycle attractors, while the chaotic regimes are characterized by broadband power spectra and strange attractors. One of the chaotic regimes has been studied in detail and has been found to correspond to a smooth one-dimensional map that has a single maximum and a positive Lyapunov exponent.

Turing patterns visualized by index of refraction variations

Gel pattern is visualized by the refractive index variations.The fossil patterns correspond to a spatial variation in the refractive index. (AIP)

Fluctuations in viscous fingering.

Our experiments on viscous (Saffman-Taylor) fingering in Hele-Shaw channels reveal finger width fluctuations that were not observed in previous experiments, which had lower aspect ratios and higher capillary numbers Ca. These fluctuations intermittently narrow the finger from its expected width. The magnitude of these fluctuations is described by a power law, Ca(-0.64), which holds for all aspect ratios studied up to the onset of tip instabilities. Further, for large aspect ratios, the mean finger width exhibits a maximum as Ca is decreased instead of the predicted monotonic increase.

TRAVELING WAVE INSTABILITY IN SUSTAINED DOUBLE-DIFFUSIVE CONVECTION

Experiments on buoyancy‐driven double‐diffusive convection sustained by imposed vertical concentration gradients (one stabilizing, the other destabilizing) have been conducted in a thin (Hele–Shaw) isothermal rectangular cell. Novel gel‐filled membranes were used to sustain the concentrations at the boundaries. When the destabilizing solute diffuses more rapidly than the stabilizing one, the primary instability leads to traveling waves with a high reflection coefficient at the ends of the cell. The measured critical Rayleigh numbers and frequencies are in reasonable accord with a stability analysis that includes corrections for the finite thickness of the cell and cross‐diffusion effects. The weakly nonlinear waves that appear at onset do not stabilize, even very close to the transition, but continue to evolve, eventually becoming a packet of large amplitude plumes. The packet travels back and forth along the cell in a nearly periodic manner. This behavior and the absence of measurable hysteresis are consistent with the present weakly nonlinear analysis which predicts tricritical scaling (∼e1/4 rather than the usual e1/2) all along the instability boundary. However, the range of this scaling in e was found to be less than 0.005, which is inaccessible in the present experiments.

Symmetry breaking in a chemical pinwheel

A transition that breaks the m‐fold rotational symmetry of a chemical pattern of traveling waves is reported. The chemical waves travel around a thin annular gel, the inner and outer edges of which are in contact with continuous flow chemical reservoirs. Observations of the dependence of the rotating wave pattern as a function of temperature reveal several distinct states, including a wavelength‐doubled state in which successive waves alternate in shape.