J. B. Swift

DETERMINING LYAPUNOV EXPONENTS FROM A TIME SERIES

We present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of nearby orbits in phase space. A system with one or more positive Lyapunov exponents is defined to be chaotic. Our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems: we monitor the long-term growth rate of small volume elements in an attractor. The method is tested on model systems with known Lyapunov spectra, and applied to data for the Belousov-Zhabotinskii reaction and Couette-Taylor flow.

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.

Mechanisms in the size segregation of a binary granular mixture

A granular mixture of particles of two sizes that is shaken vertically will in most cases segregate. If the larger particles accumulate at the top of the sample, this is called the Brazil-nut effect (BNE); if they accumulate at the bottom, it is called the reverse Brazil-nut effect (RBNE). While this process is of great industrial importance in the handling of bulk solids, it is not well understood. In recent years ten different mechanisms have been suggested to explain when each type of segregation is observed. However, the dependence of the mechanisms on driving conditions and material parameters and hence their relative importance is largely unknown. In this paper we present experiments and simulations where both types of particles are made from the same material and shaken under low air pressure, which reduces the number of mechanisms to be considered to seven. We observe both BNE and RBNE by varying systematically the driving frequency and amplitude, diameter ratio, ratio of total volume of small to large particles, and overall sample volume. All our results can be explained by a combination of three mechanisms: a geometrical mechanism called void filling, transport of particles in sidewall-driven convection rolls, and thermal diffusion, a mechanism predicted by kinetic theory.

Metastability in fluctuation-driven first-order transitions: Nucleation of lamellar phases.

The nucleation of a lamellar phase from a supercooled homogeneous phase in a fluctuation driven first-order transition is studied, based on a phenomenological free energy due to Brazovskii. The absence of phase coexistence in the corresponding mean-field approximation makes application of the standard droplet theory of homogeneous nucleation problematic. A self-consistent coarse-graining procedure is introduced to overcome this difficulty, and the barrier height for nucleation of a critical droplet is estimated in the weak-coupling limit. Contrary to earlier estimates the critical droplet shape is shown to be anisotropic in general. Some effects of distortions and defects in the lamellar structure are considered and are shown to affect the critical droplet only very near coexistence where the probability of nucleation vanishes. The coarse-graining procedure introduced here follows from a novel application of the momentum-shell renormalization group method to systems in the Brazovskii class. Possible applications of the theory to the microphase separation transition in diblock copolymers and to Rayleigh-Bénard convection are briefly discussed.

Shocks in vertically oscillated granular layers

We study shock formation in vertically oscillated granular layers, using both molecular dynamics simulations and numerical solutions of continuum equations to Navier-Stokes order. A flat layer of grains is thrown up from an oscillating plate during each oscillation cycle and collides with the plate later in the cycle. The collisions produce layer compaction near the plate and a high temperature shock front that rapidly propagates upward through the layer. The shock is highly time dependent, propagating through the layer in only a quarter of the cycle. We compare numerical solutions of the continuum equations to molecular dynamics simulations that assume binary, instantaneous collisions between frictionless, inelastic hard spheres. The two simulations yield results for the shock position, shape, and speed that agree well. An investigation of the effect of inelasticity shows that the shock velocity increases continuously with decreasing inelasticity; the elastic limit is not singular.

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.

Transport coefficients for granular media from molecular dynamics simulations

Under many conditions, macroscopic grains flow like a fluid; kinetic theory predicts continuum equations of motion for this granular fluid. In order to test the theory, we perform event-driven molecular simulations of a two-dimensional gas of inelastic hard disks, driven by contact with a heat bath. Even for strong dissipation, high densities, and small numbers of particles, we find that continuum theory describes the system well. With a bath that heats the gas homogeneously, strong velocity correlations produce a slightly smaller energy loss due to inelastic collisions than that predicted by kinetic theory. With an inhomogeneous heat bath, thermal or velocity gradients are induced. Determination of the resulting fluxes allows calculation of the thermal conductivity and shear viscosity, which are compared to the predictions of granular kinetic theory, and which can be used in continuum modeling of granular flows. The shear viscosity is close to the prediction of kinetic theory, while the thermal conductivity can be overestimated by a factor of 2; in each case, transport is lowered with increasing inelasticity.

Velocity distributions and correlations in homogeneously heated granular media.

We compare the steady state velocity distributions from our three-dimensional inelastic hard sphere molecular dynamics simulation for homogeneously heated granular media, with the predictions of a mean field-type Enskog-Boltzmann equation for inelastic hard spheres [T. P. C. van Noije and M. H. Ernst, Granular Matter 1, 57 (1998)]. Although we find qualitative agreement for all values of density and inelasticity, the quantitative disagreement approaches approximately 40% at high inelasticity or density. By contrast the predictions of the pseudo-Maxwell molecule model [J. A. Carrillo, C. Cercignani, and I. M. Gamba, Phys. Rev. E, 62, 7700 (2000)] are both qualitatively and quantitatively different from those of our simulation. We also measure short-range and long-range velocity correlations exhibiting nonzero correlations at contact before the collision, and being consistent with a slow algebraic decay over a decade in the unit of the diameter of the particle, proportional to r(-(1+alpha)), where 0.2<alpha<0.3. The existence of these correlations implies the failure of the molecular chaos assumption and the mean field approximation, which is responsible for the quantitative disagreement of the inelastic hard sphere kinetic theory.

Role of friction in pattern formation in oscillated granular layers.

Particles in granular flows are often modeled as frictionless (smooth) inelastic spheres; however, there exist no frictionless grains, just as there are no elastic grains. Our molecular dynamics simulations reveal that friction is essential for realistic modeling of vertically oscillated granular layers: simulations of frictionless particles yield patterns with an onset at a container acceleration about 30% smaller than that observed in experiments and simulations with friction. More importantly, even though square and hexagonal patterns form for a wide range of the oscillation parameters in experiments and in our simulations of frictional inelastic particles, only stripe patterns form in the simulations without friction, even if the inelasticity is increased to obtain as much dissipation as in frictional particles. We also consider the effect of particle friction on the shock wave that forms each time the granular layer strikes the container. While a shock wave still forms for frictionless particles, the spatial and temporal dependence of the hydrodynamic fields differ for the cases with and without friction.

Phase bubbles and spatiotemporal chaos in granular patterns

We use inelastic hard sphere molecular dynamics simulations and laboratory experiments to study patterns in vertically oscillated granular layers. The simulations and experiments reveal that phase bubbles spontaneously nucleate in the patterns when the container acceleration amplitude exceeds a critical value, about 7 g, where the pattern is approximately hexagonal, oscillating at one-fourth the driving frequency (f/4). A phase bubble is a localized region that oscillates with a phase opposite (differing by pi) to that of the surrounding pattern; a localized phase shift is often called an arching in studies of two-dimensional systems. The simulations show that the formation of phase bubbles is triggered by undulation at the bottom of the layer on a large length scale compared to the wavelength of the pattern. Once formed, a phase bubble shrinks as if it had a surface tension, and disappears in tens to hundreds of cycles. We find that there is an oscillatory momentum transfer across a kink, and the shrinking is caused by a net collisional momentum inward across the boundary enclosing the bubble. At increasing acceleration amplitudes, the patterns evolve into randomly moving labyrinthian kinks (spatiotemporal chaos). We observe in the simulations that f/3 and f/6 subharmonic patterns emerge as primary instabilities, but that they are unstable to the undulation of the layer. Our experiments confirm the existence of transient f/3 and f/6 patterns.