Newton-Krylov continuation of amplitude-modulated rotating waves in sheared annular electroconvection.

We present an approach for studying the primary, secondary, and tertiary flow transitions in sheared annular electroconvection. In particular, we describe a Newton-Krylov method based on time integration for the computation of rotating waves and amplitude-modulated rotating waves, and for the continuation of these flows as a parameter of the system is varied. The method exploits the rotational nature of the flows and requires only a time-stepping code of the model differential equations, i.e., it does not require an explicit code for the discretization of the linearized equations. The linear stability of the solutions is computed to identify the parameter values at which the transitions occur. We apply the method to a model of electroconvection that simulates the flow of a liquid crystal film in the smectic A phase suspended between two annular electrodes and subjected to an electric potential difference and a radial shear. Due to the layered structure of the smectic A phase, the fluid can be treated as two-dimensional (2D) and is modeled using the 2D incompressible Navier-Stokes equations coupled with an equation for charge continuity. The system is a close analog to laboratory-scale geophysical fluid experiments and thus represents an ideal system in which to apply the method before its application to these other systems that exhibit similar flow transitions. In the model for electroconvection, we identify the parameter values at which the primary transition from steady axisymmetric flow to rotating waves occurs, as well as at which the secondary transition from the rotating waves to amplitude-modulated rotating waves occurs. In addition, we locate the tertiary transition, which corresponds to a transition from the amplitude-modulated waves to a three-frequency flow. Of particular interest is that the method also finds a period-doubling bifurcation from the amplitude-modulated rotating waves and a subsequent transition from the flow resulting from this bifurcation.

Pattern of inclusions inside rippled icicles.

Icicles that have grown from slightly impure water develop ripples around their circumference. The ripples have a near-universal wavelength and are thought to be the result of a morphological instability. Using laboratory-grown icicles and various species of impurities, including fluorescent dye, we show that a certain fraction of the impurities remain trapped inside the icicle, forming inclusions within the ice. The inclusions are organized into chevron patterns aligned with the peaks of the ripples. Within the chevrons, a substructure of crescent-shaped structures is observed. We also examine the crystal grain structure of laboratory icicles with and without impurities. We present the first detailed study of these growth patterns in the interior of icicles and discuss their implications for the mechanism of the ripple-forming instability.

The heads and tails of buoyant autocatalytic balls.

Buoyancy produced by autocatalytic reaction fronts can produce fluid flows that advect the front position, giving rise to interesting feedback between chemical and hydrodynamic effects. In this paper, we numerically investigate the evolution of autocatalytic iodate-arsenous acid reaction fronts initialized in spherical configurations. Deformation of these "autocatalytic balls" is driven by buoyancy produced by the reaction. In our simulations, we have found that depending on the initial ball radius, the reaction front will develop in one of three different ways. In an intermediate range of ball size, the flow can evolve much like an autocatalytic plume: the ball develops a reacting head and tail that is akin to the head and conduit of an autocatalytic plume. In the limit of large autocatalytic balls, however, growth of a reacting tail is suppressed and the resemblance to plumes disappears. Conversely, very small balls of product solution fail to initiate sustained fronts and eventually disappear.

Frequency structure of the nonlinear instability of a dragged viscous thread.

A thread of viscous fluid falling onto a moving surface exhibits a spectacular variety of types of motion as the surface speed and nozzle height are varied. For modest nozzle heights, four clear regimes are observed. For large surface speed, the thread is dragged into a stretched centenary configuration which is confined to a plane. As the surface speed is lowered, the thread exhibits a supercritical bifurcation to a meandering state. At very low surface speeds, the state resembles the usual coiling motion of a viscous thread falling on a stationary surface. In between the meandering and coiling regimes, a window containing a novel multifrequency state, previously called "figures of eight," is found. Using an improved visualization technique and a fully automated apparatus, we made detailed measurements of the longitudinal and transverse motion of the thread in all these states. We found that the multifrequency state is characterized by a complex pattern of motion whose main frequencies are locked in a 3:2 ratio. This state appears and disappears with finite amplitude at sharp bifurcations without measurable hysteresis.

Lift and drag forces on an inclined plow moving over a granular surface.

We studied the drag and lift forces acting on an inclined plate while it is dragged on the surface of a granular media, both in experiment and in numerical simulation. In particular, we investigated the influence of the horizontal velocity of the plate and its angle of attack. We show that a steady wedge of grains is moved in front of the plow and that the lift and drag forces are proportional to the weight of this wedge. These constants of proportionality vary with the angle of attack but not (or only weakly) on the velocity. We found a universal effective friction law that accounts for the dependence on all the above-mentioned parameters. The stress and velocity fields are calculated from the numerical simulations and show the existence of a shear band under the wedge and that the pressure is nonhydrostatic. The strongest gradients in stress and shear occur at the base of the plow where the dissipation rate is therefore highest.

Experiments on the morphology of icicles.

Icicles form when cool water drips from an overhanging support under ambient conditions which are below freezing. Ice growth is controlled by the removal of latent heat, which is transferred into the surrounding air via a thin film of water flowing over the ice surface. We describe laboratory experiments in which icicles were grown under controlled conditions. We used image analysis to probe the evolution of the icicle shape under various conditions. A recent asymptotic theory suggests that, overall, icicles converge to self-similar shapes which are predicted to be attractors. On the other hand, stability theory predicts that the ice-water interface can become unstable to form ripple patterns on the icicle surface. Our experimental results show that the predicted self-similar profile is only found in certain cases, and that icicles can also exhibit unpredicted nonuniformities such as branching near the tip. We find that pure water icicles are more likely to be self-similar than those grown from tap water. Ripples, which are also deviations from the self-similar profile, were observed to climb upward during icicle growth.

Autocatalytic plume pinch-off.

A localized source of buoyancy flux in a nonreactive fluid medium creates a plume. The flux can be provided by either heat, a compositional difference between the fluid comprising the plume and its surroundings, or a combination of both. For autocatalytic plumes produced by the iodate-arsenous acid reaction, however, buoyancy is produced along the entire reacting interface between the plume and its surroundings. Buoyancy production at the moving interface drives fluid motion, which in turn generates flow that advects the reaction front. As a consequence of this interplay between fluid flow and chemical reaction, autocatalytic plumes exhibit a rich dynamics during their ascent through the reactant medium. One of the more interesting dynamical features is the production of an accelerating vortical plume head that in certain cases pinches-off and detaches from the upwelling conduit. After pinch-off, a new plume head forms in the conduit below, and this can lead to multiple generations of plume heads for a single plume initiation. We investigated the pinch-off process using both experimentation and simulation. Experiments were performed using various concentrations of glycerol, in which it was found that repeated pinch-off occurs exclusively in a specific concentration range. Autocatalytic plume simulations revealed that pinch-off is triggered by the appearance of accelerating flow in the plume conduit.

Scaling and dynamics of washboard roads.

Granular surfaces subjected to forces due to rolling wheels develop ripples above a critical speed. The resulting pattern, known as washboard or corrugated road, is common on dry unpaved roads. We investigated this phenomenon theoretically and experimentally using laboratory-scale apparatus and beds of dry sand. A thick layer of sand on a circular track was forced by a rolling wheel on an arm whose weight and moment of inertia could be varied. We compared the ripples made by the rolling wheel to those made using a simple inclined plow blade. We investigated the dependence of the critical speed on various parameters and described a scaling argument that leads to a dimensionless ratio, analogous to the hydrodynamic Froude number, which controls the instability. This represents the crossover between conservative dynamic forces and dissipative static forces. Above onset wheel-driven ripples move in the direction of motion of the wheel, but plow-driven ripples move in the reverse direction for a narrow range of Froude numbers.

Nonequilibrium scale selection mechanism for columnar jointing.

Crack patterns in laboratory experiments on thick samples of drying cornstarch are geometrically similar to columnar joints in cooling lava found at geological sites such as the Giant's Causeway. We present measurements of the crack spacing from both laboratory and geological investigations of columnar jointing, and show how these data can be collapsed onto a single master scaling curve. This is due to the underlying mathematical similarity between theories for the cracking of solids induced by differential drying or by cooling. We use this theory to give a simple quantitative explanation of how these geometrically similar crack patterns arise from a single dynamical law rooted in the nonequilibrium nature of the phenomena. We also give scaling relations for the characteristic crack spacing in other limits consistent with our experiments and observations, and discuss the implications of our results for the control of crack patterns in thin and thick solid films.

Meandering instability of a viscous thread.

A viscous thread falling from a nozzle onto a surface exhibits the famous rope-coiling effect, in which the thread buckles to form loops. If the surface is replaced by a belt moving with speed U , the rotational symmetry of the buckling instability is broken and a wealth of interesting states are observed [see S. Chiu-Webster and J. R. Lister, J. Fluid Mech. 569, 89 (2006)]. We experimentally studied this "fluid-mechanical sewing machine" in a more precise apparatus. As U is reduced, the steady catenary thread bifurcates into a meandering state in which the thread displacements are only transverse to the motion of the belt. We measured the amplitude and frequency omega of the meandering close to the bifurcation. For smaller U , single-frequency meandering bifurcates to a two-frequency "figure-8" state, which contains a significant 2omega component and parallel as well as transverse displacements. This eventually reverts to single-frequency coiling at still smaller U . More complex, highly hysteretic states with additional frequencies are observed for larger nozzle heights. We propose to understand this zoology in terms of the generic amplitude equations appropriate for resonant interactions between two oscillatory modes with frequencies omega and 2omega . The form of the amplitude equations captures both the axisymmetry of the U=0 coiling state and the symmetry-breaking effects induced by the moving belt.

Conduits of steady-state autocatalytic plumes.

Plumes are typically formed when a continuous source of buoyancy is supplied at a localized source. We studied laminar plumes where buoyancy is supplied by an autocatalytic chemical reaction: The iodate-arsenous acid (IAA) reaction. The nonlinear kinetics of the IAA reaction produces a sharp propagating front at which buoyancy is produced by exothermicity and compositional change. When the reaction is initiated in an unconfined volume of reactant, a starting plume with a mushroom shaped head connected to the initiation point by a long conduit is formed. After the initial transient during the ascent of the head, we observed the emergence of a steady state in the conduit morphology and flow. Autocatalytic plumes were compared to nonreacting, compositionally buoyant plumes using the Gradient Echo Rapid Velocity and Acceleration Imaging Sequence (GERVAIS), an MRI velocimetric technique. Autocatalytic conduits had axisymmetric bimodal velocity profiles and cone-shaped morphologies, in contrast to the Gaussian profiles and cylindrical shapes of nonreacting conduits. The bimodal distribution for autocatalytic plumes is a consequence of the unique effect of entrainment in this system. Rather than the usual effect of entrainment in nonreacting plumes, where less buoyant fluid is incorporated into the plumes, entrainment in autocatalytic plumes provides a buoyancy flux along the entire conduit by means of chemical reaction, thereby delocalizing the buoyancy source.

Direct numerical simulation of supercritical annular electroconvection.

We use direct numerical simulation to study electrically driven convection in an annular thin film. The simulation models a laboratory experiment that consists of a weakly conducting, submicron thick liquid crystal film suspended between two concentric electrodes. The film is driven to convect by imposing a sufficiently large voltage across it. The flow is driven by a surface charge density inversion which is unstable to the imposed electrical force. This mechanism is closely analogous to the mass density inversion which is unstable to the buoyancy force in conventional, thermally driven Rayleigh-Bénard convection. The simulation uses a pseudospectral method with Chebyshev polynomials in the radial direction and Fourier modes in the azimuthal direction. The numerical results, which are in good agreement with previous experimental data and theoretical predictions, reveal several insights. The mode competition near a codimension-two point exhibits hysteresis. The primary bifurcation is supercritical for a broad range of fluid and geometrical parameters.

Washboard road: the dynamics of granular ripples formed by rolling wheels.

We report laboratory experiments on rippled granular surfaces formed under rolling wheels. Ripples appear above a critical speed and drift slowly in the driving direction. Ripples coarsen as they saturate and exhibit ripple creation and destruction events. All of these effects are captured qualitatively by 2D soft-particle simulations in which a disk rolls over smaller disks in a periodic box. The simulations show that compaction and segregation are inessential to the ripple phenomenon. We describe a simplified scaling model which gives some insight into the mechanism of the instability.

Experimental investigation of the scaling of columnar joints.

Columnar jointing is a fracture pattern common in igneous rocks in which cracks self-organize into a roughly hexagonal arrangement, leaving behind an ordered colonnade. We report observations of columnar jointing in a laboratory analog system, desiccated corn starch slurries. Using measurements of moisture density, evaporation rates, and fracture advance rates as evidence, we suggest that an advective-diffusive system is responsible for the rough scaling behavior of columnar joints. This theory explains the order of magnitude difference in scales between jointing in lavas and in starches. We investigated the scaling of average columnar cross-sectional areas due to the evaporation rate, the analog of the cooling rate of igneous columnar joints. We measured column areas in experiments where the evaporation rate depended on lamp height and time, in experiments where the evaporation rate was fixed using feedback methods, and in experiments where gelatin was added to vary the rheology of the starch. Our results suggest that the column area at a particular depth is related to both the current conditions, and hysteretically to the geometry of the pattern at previous depths. We argue that there exists a range of stable column scales allowed for any particular evaporation rate.

Charge transport scaling in turbulent electroconvection.

We describe a local-power-law scaling theory for the mean dimensionless electric current Nu in turbulent electroconvection. The experimental system consists of a weakly conducting, submicron-thick liquid-crystal film supported in the annulus between concentric circular electrodes. It is driven into electroconvection by an applied voltage between its inner and outer edges. At sufficiently large voltage differences, the flow is unsteady and electric charge is turbulently transported between the electrodes. Our theoretical development, which closely parallels the Grossmann-Lohse model for turbulent thermal convection, predicts the local-power law Nu approximately F(gamma)R(gamma)P(delta). R and P are dimensionless numbers that are similar to the Rayleigh and Prandtl numbers of thermal convection, respectively. The dimensionless function F(gamma), which is specified by the model, describes the dependence of Nu on the aspect ratio gamma. We find that measurements of Nu are consistent with the theoretical model.

Codimension-two points in annular electroconvection as a function of aspect ratio.

We rigorously derive from first principles the generic Landau amplitude equation that describes the primary bifurcation in electrically driven convection. Our model accurately represents the experimental system: a weakly conducting, submicron thick liquid crystal film suspended between concentric circular electrodes and driven by an applied voltage between its inner and outer edges. We explicitly calculate the coefficient g of the leading cubic nonlinearity and systematically study its dependence on the system's geometrical and material parameters. The radius ratio alpha quantifies the film's geometry while a dimensionless number P , similar to the Prandtl number, fixes the ratio of the fluid's electrical and viscous relaxation times. Our calculations show that for fixed alpha, g is a decreasing function of P , as P becomes smaller, and is nearly constant for P> or =1 . As P-->0, g-->infinity. We find that g is a nontrivial and discontinuous function of alpha. We show that the discontinuities occur at codimension-two points that are accessed by varying alpha.

Buoyant plumes and vortex rings in an autocatalytic chemical reaction.

Buoyant plumes, evolving free of boundary constraints, may develop well-defined mushroom-shaped heads. In conventional plumes, overturning flow in the head entrains less buoyant fluid from the surroundings as the head rises, robbing the plume of its driving force. We consider here a new type of plume in which the source of buoyancy is an autocatalytic chemical reaction. The reaction occurs at a sharp front which separates reactants from less dense products. In this type of autocatalytic plume, entrainment assists the reaction, producing new buoyancy which fuels an accelerating plume head. When the head has grown to a critical size, it detaches from the upwelling conduit, forming an accelerating, buoyant vortex ring. A second-generation head then develops at the point of detachment. Multiple generations of autocatalytic vortex rings can detach from a single triggering event.

Subdiffusive axial transport of granular materials in a long drum mixer.

Granular mixtures segregate radially by size when tumbled in a partially filled horizontal drum. The smaller component moves toward the axis of rotation and forms a buried core, which then splits into axial bands. Models have generally assumed that the axial segregation is opposed by diffusion. Using narrow pulses of the smaller component as initial conditions, we have characterized axial transport in the core. We find that the axial advance of the segregated core is well described by a self-similar concentration profile whose width scales as talpha, with alpha approximately 0.3<1/2. Thus, the process is subdiffusive rather than diffusive as previously assumed. We compare our results to two one-dimensional model equations which contain self-similarity and subdiffusion: a linear fractional diffusion model and the nonlinear porous medium equation.

Aspect-ratio dependence of charge transport in turbulent electroconvection.

We present measurements of the normalized charge transport or Nusselt number Nu as a function of the aspect ratio Gamma for turbulent convection in an electrically driven film. In analogy with turbulent Rayleigh-Bénard convection, we develop the relevant theoretical framework in which we discuss the local power-law scaling of Nu with a dimensionless electrical forcing parameter R. For these experiments where 10(4) less, similar R less, similar 2 x 10(5) we find that Nu approximately F(Gamma)Rgamma with either gamma=0.26 (+/-0.02) or gamma=0.20 (+/-0.03), in excellent agreement with the theoretical predictions of gamma=1/4 and 1/5. Our measurements of the aspect-ratio dependence of Nu for 0.3

Sequential bifurcations in sheared annular electroconvection.

A sequence of bifurcations is studied in a one-dimensional pattern forming system subject to the variation of two experimental control parameters: a dimensionless electrical forcing number R and a shear Reynolds number Re. The pattern is an azimuthally periodic array of traveling vortices with integer mode number m. Varying R and Re permits the passage through several codimension-two (CoD2) points. We find that the coefficients of the nonlinear terms in a generic Landau equation for the primary bifurcation are discontinuous at the CoD2 points. Further, we map the stability boundaries in the space of the two parameters by studying the subcritical secondary bifurcations in which m-->m+1 when R is increased at constant Re.