Roll Patterns Research Articles

Localized radial solutions of the Swift–Hohenberg equation

Stationary localized solutions of the planar Swift–Hohenberg equation are investigated in the parameter region where the trivial solution is stable. In the parameter region where rolls bifurcate subcritically, localized radial ring-like pulses are shown to bifurcate from the trivial solution. Furthermore, radial spot-like pulses are shown to bifurcate from the trivial state, regardless of the criticality of roll patterns. These theoretical results apply also to general reaction–diffusion systems near Turing instabilities. Numerical computations show that planar radial pulses ‘snake’ near the Maxwell point where, by definition, the one-dimensional roll patterns have the same energy as the trivial state. These computations also reveal that spots, which are stable in a certain parameter region, become unstable with respect to hexagonal perturbations, leading to fully localized hexagon patterns.

Time reversal of the excitation wave form in a dissipative pattern-forming system

We discuss the consequences of time reversal of the excitation parameters on the stability of a periodically driven dynamic system. The example chosen, electrohydrodynamic convection of nematics, is a well-investigated dissipative pattern-forming system, usually driven with time-periodic external electric fields E(t) . The dynamic model that describes linear stability at onset can be represented by a set of two differential equations for the amplitudes of the dynamic variables. Floquet analysis reveals an astonishing dynamic property: Under time reversal of any periodic excitation function E(t) , the dynamic equations reproduce the same stability thresholds, pattern types, and critical pattern wavelengths, even when the dynamics of all involved system variables are essentially different for the excitations E(t) and E(-t) . Experimental confirmation of these mathematical predictions is provided by the measurement of threshold curves and the analysis of the time-resolved optical characteristics of the convection roll patterns.

Interaction between amino acid degradation, sensory attributes and lipid peroxidation in wheat rolls

AbstractThe influence of lipid peroxidation and heating duration on the amino acid pattern of wheat rolls was assessed in a model baking test. Therefore, three different types of dough were prepared including dough without any fat and two doughs with fat. To estimate the impact of lipid peroxidation products on the amino acid pattern, the two doughs containing fat were both made with rapeseed oil, one with fresh oil, the other with oxidised rapeseed oil. All kinds of dough were baked for three different duration times (18, 24 and 30 min). The lipid peroxidation status of both rapeseed oils was measured with the peroxide value and the p‐anisidine value of the unbaked fat. The amino acid content was analysed by HPLC after acid and alkaline hydrolysis. Besides determining water content and protein content, a sensory analysis was performed. Whereas the amino acid content in the dough without fat was only affected by the baking time, the doughs with rapeseed oil additionally showed a significant increase in amino acid degradation caused by lipid peroxidation products. Especially in the dough prepared with oxidised rapeseed oil, the amino acid content was partly decreased in raw dough without any heating influence. Confirming the influence of lipid peroxidation products in dough prepared with oxidised rapeseed oil, sensory analysis also showed the significantly worst overall score.

Patterns on growing square domains via mode interactions

Numerical simulations of reaction-diffusion systems with Neumann boundary conditions (NBC) on growing square domains by Maini et al. exhibit square and stripe (or roll) patterns that are usually associated with bifurcations from a trivial equilibrium on a square lattice. However, these patterns change as the domain grows. In this article we discuss several of these transitions; namely, transitions between different types of squares and between squares and stripes (or rolls). We show that these transitions can be understood by tracing paths through the unfoldings of certain co-dimension two mode interactions. To understand these transitions, we need to discuss two issues: the speed at which the domain size changes and the relations between NBC and periodic boundary conditions (PBC) on a square. First, in the simulations, the domain growth takes place on a time scale that is longer than the one needed for pattern formation. Therefore, we can assume that the domain growth is identified with quasistatic variation of time; that is, the domain grows slowly enough that the PDE solution of the time-dependent system tracks equilibria of the reaction-diffusion systems posed on a fixed size domain. Second, as is well-known, NBC problems on a square of side length l can be embedded in PBC problems on a square with side length 2l. The PBC problem has translation symmetries that are not present in the NBC problem. These additional symmetries are called hidden symmetries in the literature. Moreover, solutions to PBC that restrict to the smaller square and satisfy NBC are just those solutions that satisfy certain symmetry constraints. We show further that the transitions between different patterns can be understood by considering relevant mode interaction bifurcation problems on the larger square and then restricting to the smaller square. We have found that a generic continuous transition can occur between two types of squares. Also, the transition between squares and stripes can generically occur either via steady states and time-periodic states (standing waves) or via a jump. Interestingly, in mode interactions, the symmetry constraints induced by NBC are important in understanding which solutions exist and which solutions are stable. For example, diagonal stripes cannot occur as a primary branch in the NBC problem but do in the PBC problem. Also, patterns can be stable in the NBC problem that are not stable in the PBC problem. As a consequence, in the NBC problem we see standing wave time-periodic solutions as stable patterns leading to stable stripes, whereas in the PBC problem we see wavy rolls steady states as stable patterns leading to stable stripes. In principle, a classification of all transitions in NBC mode interactions is possible. However, we concentrate only on those transitions that are relevant to the numerically observed transitions.

VCR and ASR technology for profile and flatness control in hot strip mills

The roll contour pattern and variety of work and backup rolls in service and its effect on profile and flatness control performance in 1 700 mm hot strip mill at Wuhan Iron and Steel (Group) Corporation were tested and analyzed by the developed finite element models of different typical roll contours configurations. A rather smooth local work roll contour near strip edges and an increase in rolled length can be obtained by application of long stroke work roll shifting system with conventional work roll contours that is incapable of the crown control. In comparison with the conventional backup and work roll contours configuration, the crown control range by the roll bending force enhances by 12.79% and the roll gap stiffness increases by 25.26% with the developed asymmetry self-compensating work rolls(ASR) and varying contact backup rolls(VCR). A better strip profile and flatness quality, an increase in coil numbers within the rolling campaign and a significant alleviated effect of severe work roll wear contours on performance of edge drop control are achieved by the application of ASR with crown control and wear control ability in downstream stand F5 and VCR in all stands of 1 700 mm hot strip mill.

Deformation analysis of surface defect on hot rolling by 3-D FEM simulation

The major objective of this research is to develop a 3-D FEM simulation technique in order to analyze the deformation behavior of surface defects on hot rolling. The developed technique will be applied to quantify the relationship between rolling condition and the process parameters, and the rolling pattern will be tested to verify and validate the simulation technique. This technique will allow us to quantify the shrinkage of slab width at the head and tail, width spread after the horizontal rolling, dog-bone shape, position change of the slab corner as the slab is being rolled, strain distributions. And the shape change of surface defects in hot rolling was investigated using the 3-D rolling simulation technique. The proposed prediction method of surface defect can be utilized to make defect free products in rolling.

Pattern Formation in a Model of a Vibrated Granular Layer

A phenomenological model for pattern formation in a vertically vibrated granular layer is examined in order to investigate its nonlinear dynamics. The model comprises two coupled partial differential equations: one describes the evolution of the short-scale pattern, while the other enforces conservation of granular material. In a layer of moderate horizontal extent, the model predicts that a variety of exotic regular patterns may be stable, according to the system parameters. The usual cubic-order amplitude equations are unable to determine the stable solution over a significant parameter range; we compute the corresponding fifth-order terms necessary to resolve this degeneracy. When spatial modulation of the pattern is taken into account, in a sufficiently wide layer, a stability analysis of regular one-dimensional roll and two-dimensional square patterns demonstrates that each may suffer a modulational instability, which tends to localize the pattern. The corresponding modulational stability boundaries,...

Spatiotemporal Antiphase Dynamics in Coupled Extended Optical Media

Experimental evidence of spatiotemporal antiphase dynamics is given for an extended system made of two liquid crystal slices that are optically coupled by two equal amplitude counterpropagating pumping beams. Theory and experiments carried out in a transverse one-dimensional configuration show that roll patterns are generated in each slice. These rolls are spatially in-phase or antiphase for a focusing or a defocusing nonlinearity type, respectively. These in-phase or antiphase dynamics remain robust even for complex spatiotemporal regimes such as dislocation regimes.

Subharmonic Pattern Formation in a Nematic Cell – A 1D Flexoelectric Model Approach

The application of an AC electric field to a nematic liquid crystal cell produces two distinct instability pattern regimes, conductive or dielectric. Recently, asymmetric electric field waveforms have been shown to cause a period-doubling subharmonic bifurcation. We extend these calculations to include flexoelectricity in the one-dimensional case. We calculate the threshold values for the electric field strength determining the optimum oblique angle at which the roll patterns form. We find, in some areas of parameter space, the formation of oblique rolls is favoured over normal rolls. In addition we find cross-over points from normal to oblique rolls called Lifshitz points. These have been found in the subharmonic and dielectric regions of the linear stability diagram.

Stripe patterns in degenerate optical parametric oscillators

We experimentally demonstrate the stripe (or roll) patterns in a broad-aperture degenerate optical parametric oscillator in a plane-mirror minicavity. The stabilization of stripes is achieved by seed injection at a subharmonic frequency. We measure the temporal spectra of the stripe pattern and obtain the 1/f-like noise spectra.

Global feedback control for pattern-forming systems

Global feedback control of pattern formation in a wide class of systems described by the Swift-Hohenberg (SH) equation is investigated theoretically, by means of stability analysis and numerical simulations. Two cases are considered: (i) feedback control of the competition between hexagon and roll patterns described by a supercritical SH equation, and (ii) the use of feedback control to suppress the blowup in a system described by a subcritical SH equation. In case (i), it is shown that feedback control can change the hexagon and roll stability regions in the parameter space as well as cause a transition from up to down hexagons and stabilize a skewed (mixed-mode) hexagonal pattern. In case (ii), it is demonstrated that feedback control can suppress blowup and lead to the formation of spatially localized patterns in the weakly nonlinear regime. The effects of a delayed feedback are also investigated for both cases, and it is shown that delay can induce temporal oscillations as well as blowup.

Pattern Formation in the Rotating Cylindrical Annulus with an Azimuthal Magnetic Field at low Prandtl Numbers

Patterns are theoretically formed in the frame of a hydromagnetic convection induced by radial buoyancy in an electrically conducting fluid contained by a rotating cylindrical annulus with a homogeneous magnetic field ( B) in the azimuthal direction. The annulus is assumed to rotate with an angular frequency, Ω under the small gap approximation with rigid cylindrical boundaries. The onset of convection is found in the form of axial, axisymmetric or oblique rolls with a broken symmetry. The roll angle Ψ depends on the ratio between the Chandrasekhar number, Q ~ B 2, and the Coriolis number, τ ~ Ω. In addition to fully three-dimensional (3D) numerical simulations, weakly nonlinear and Galerkin analyses for roll patterns are performed for Prandtl number P = 0.1. At finite amplitudes, secondary instabilities are encountered in the form of longwave and shortwave.

The [formula omitted] mode interaction and heteroclinic networks in Boussinesq convection

Methods of equivariant bifurcation theory are applied to Boussinesq convection in a plane layer with stress-free horizontal boundaries and an imposed square lattice periodicity in the horizontal directions. We consider the problem near the onset of instability of the uniform conducting state where spatial roll patterns with two different wavelengths in the ratio 1 : 2 become simultaneously unstable at a mode interaction. Center manifold reduction yields a normal form on C 4 with very rich dynamical behavior. For a fixed Prandtl number P the mode interaction occurs at an isolated point in the parameter plane ( R , L ) (where R is the Rayleigh number and L the length of the horizontal periodicities) and acts as an organizing center for many nearby bifurcations. The normal form predicts the appearance of many steady states and travelling waves, which are classified by their symmetries. It also predicts the appearance of robust heteroclinic networks involving steady states with several different symmetries, and robust attractors of generalized heteroclinic type that include connections from equilibria to subcycles. This is the first example of a heteroclinic network in a fluid dynamical system that has ‘depth’ greater than one. The normal form dynamics is in good correspondence (both quantitatively and qualitatively) with direct numerical simulations of the full convection equations.

Local wave director analysis of domain chaos in Rayleigh–Bénard convection

We present new results both from numerical simulations using the Swift–Hohenberg modeland from experiment for domain chaos near the onset of rotating Rayleigh–Bénardconvection. Using both global Fourier analysis and local wave director (LWD) analysis, thedomain chaos patterns were characterized. Several image analysis techniques are discussedand were applied to the director field obtained from LWD analysis. They yielded statisticalparameters, averaged over long time series, that characterize several aspects of the chaoticpatterns. The effect of the finite image size on the determination of the domain switchingfrequency obtained with Fourier analysis, as well as with LWD analysis, was investigated.Using the LWD method, we studiedξθ, a correlation length obtained from the time-averaged autocorrelation of the local angle fieldθ. The defects and domains in the pattern were examined locally and their average areafraction, average area, and average major axis length were obtained. A periodic componentwas observed in the time series of the instantaneous correlation length and of the defect area, but was found not to be present in the time series of the domainarea. We determined a parameter , defined as the average dot product of the wave director and the major domain axis, thatcharacterizes the orientation of the domains relative to the roll patterns within them. Theresults for were independent of the control parameterε and of the image size. Simulations and experiment gave remarkably similar results for.

The effect of rolling and sintering steps on microstructure andJc properties of Bi2223 wire sheathed with Ag–Mg alloy

The use of an Ag alloy sheath is effective for improving themechanical strength of Bi2223 wires although it may easily decreaseJc through undesired reactions with Bi2223 superconductor. In this study, Bi2223tapes sheathed with Ag–Mg alloy, which is less reactive with the superconductingphase, were fabricated, and the effect of processing on the microstructure andJc properties was investigated. The Ag–Mg alloy sheathed Bi2223 tapes showedmany Bi2223 crystal outgrowths when they were fabricated with thesame pattern as was used for fabricating pure Ag sheathed tape, and theJc value of the tapesbecame poorer: about 9 × 103 A cm−2. This degradation of Jc is attributed to the poorer crystal alignment seen as many outgrowths, which would be causedby the reduction of the conversion rate of Bi2223. To improve the crystal alignment andJc values, the rolling and sintering patterns were improved. It was found that theaddition of intermediate rolling and careful selection of the sintering time could beused to suppress the formation of outgrowths effectively and improve theJc valueto 1.4 × 104 A cm−2 through the improvement of the crystal alignment. This means that the timing of therolling step(s) should be selected at effective point(s) in the phase development of Bi2223crystal.

Effects of Bottleneck on Granular Convection Cells and Segregation

Dry glass granular material confined to a 2D-chamber convects when it is subjected to vertical sinusoidal vibrations of sufficient intensity. Effects of the container geometry on convection pattern and segregation process are studied experimentally. Here we introduce a bottleneck into the ordinary rectangular chamber, with one sidewall bended inward, characterized by λ being the ratio of the length of the bottleneck to the length of the chamber’s base. The convection roll and segregation pattern are significantly affected by λ. For λ = 0.9, two different stabilized patterns co-exist, depending on initial granular distribution. The sloping angle of the free surface to the horizontal increases with increasing λ, and reaches its saturation at λ = 0.9. The angle of the interface of the segregation region to the horizontal also increases with increasing λ.

Soft-Mode Turbulence in Electroconvection of Nematics

Soft-mode turbulence (SMT) observed in the electroconvection of a nematic liquid crystal is a first experimental example of spatiotemporal chaos (STC) due to an additional Goldstone mode. In a nematic liquid crystal with homeotropic alignment a continuous rotational symmetry is broken by a kind of buckling instability of the alignment by applying voltage (control parameter), and this spontaneous symmetry breaking leads to a Goldstone mode. Further increasing the control parameter, electroconvective instability sets in and a roll pattern could be expected to appear. SMT however ensues by the coupling of the roll pattern with the Goldstone mode. SMT has the following properties: (i) SMT is one of the rare examples of STC appearing via single supercritical bifurcation directly from a quiescent state. With decreasing the control parameter, the correlation time of the chaotic fluctuation becomes longer and diverges at the bifurcation point, that is, the fluctuations become soft. (ii) SMT is the STC in respect to the directions of convective wavevector, and patches which are the areas with the same directions appear. (iii) The averaged size of the patch, as well as the the correlation length, becomes larger with decreasing the control parameter from the above of threshold. Furthermore we have investigated transport properties in SMT by observing non-thermal Brownian motion in order to elucidate a fluctuation theorem in dissipative systems. This study gives us important information about SMT from a viewpoint of the Lagrangian picture which is different from the properties of the Euler picture (pattern dynamics). We presented the time-scale dependence of diffusion constants obtained by a coarse-graining. The obtained result reflects that SMT pattern has two different length scales, rolls and patches. The diffusion of particles inside a patch is anomalous, i.e. Levy diffusion, while the diffusion of inter-patches is normal. Thus the transport property of SMT is closely related with its spatial hierarchy.

Unordered Roll Flow Patterns of Interfacial Turbulence and Its Influence on Mass Transfer

By utilizing the optical Schlieren method, Marangoni convection produced in different gas–liquid mass transfer processes was observed and recorded. Different liquid solvent mixtures were selected to investigate the flow patterns of the Marangoni convection as well as the driven mechanism of the surface tension gradient. Unordered flow patterns were observed frequently in experiments, and it is different from ordinary orderly Marangoni flow patterns such as regular roll and cellular flow patterns. Qualitative analysis of flow patterns in mass transfer was made through the Schlieren images observed, and in consequence a scaling analysis was made to discuss the relationship between the Marangoni effect and the unusual mass transfer performance. By combining the Marangoni hydraulic boundary condition with the penetration theory of mass transfer, the Marangoni number characterizing the intensity of Marangoni effect enters the conventional mass transfer correlation. The quantitative result successfully explains these different correlations between Sh and Ma obtained in mass transfer processes accompanied by the Marangoni effect.

An improved MPS method for numerical simulations of convective heat transfer problems

AbstractAn improved moving‐particle semi‐implicit (MPS) method was developed for numerical simulations of convective heat transfer problems. The MPS method, which is based on particles and their interactions, is a fully Lagrangian particle method for incompressible flows. A new Laplacian model and a new method for treating boundary conditions were proposed to solve numerical difficulties resulting from the original MPS method. Results of several numerical tests show the validity of the improved MPS method with the proposed model and method.The application of the present MPS method to Rayleigh–Benard convection phenomena demonstrated the effectiveness of the proposed model and method on the numerical simulation of convective heat transfer problems. The dependence of the Nusselt number on the Rayleigh number was in good agreement with an empirical formula. The temperature contour and velocity distribution also agree well with the simulation results obtained with other methods. The roll pattern developed in the horizontal fluid layer as well as the convective heat transfer was successfully simulated with three‐dimensional MPS calculations. Copyright © 2005 John Wiley & Sons, Ltd.

Ring-like size segregation in vibrated cylinder with a bottleneck

In this Letter, a ring-like segregation pattern of bi-dispersed granular material in a vibrated bottleneck-cylinder is presented. The driving frequency can greatly affect the strength and structure of the convection roll and segregation pattern. The position and height of the ring (cluster of big beads) can be adjusted by altering the vibration frequency. And a heuristic theory is developed to interpret the ring's position dependence on driving frequency.