Transition Front Research Articles

Asymptotic [formula omitted] stability for transition fronts in Cahn–Hilliard systems

We consider the asymptotic behavior of perturbations of transition front solutions arising in Cahn–Hilliard systems on R. Such equations arise naturally in the study of phase separation processes, and systems describe cases in which three or more phases are possible. When a Cahn–Hilliard system is linearized about a transition front solution, the linearized operator has an eigenvalue at 0 (due to shift invariance), which is not separated from essential spectrum. In cases such as this, nonlinear stability cannot be concluded from classical semigroup considerations and a more refined development is appropriate. Our main result asserts that if initial perturbations are small in L1∩L∞ then spectral stability—a necessary condition for stability, defined in terms of an appropriate Evans function—implies asymptotic nonlinear stability in Lp for all 1<p⩽∞.

Nonlinear problem of unsaturated frozen soil thawing in the presence of capillary forces

The problem of ice melting in unsaturated frozen soil in the presence of the capillary pressure in the water-air zone is formulated. The complete system of boundary conditions on the phase transition front is derived. For solving the nonlinear problem a numerical method is proposed. The dependence of the water saturation distribution on the form of the Leverett function, the capillary pressure, and the external pressure and temperature gradients is investigated. In the limiting case of saturated frozen soil the numerical and analytical solutions are compared.

Time Evolution of Power Spectrum Density in Spontaneous Transition in Cylindrical Magnetized Plasma

Preliminary observation results are reported for a new discharge regime in the Plasma Assembly for Nonlinear Turbulent Analysis (PANTA), where spontaneous transitions in the equilibrium profile and fluctuation spectra occur. Two different states are defined by using the mean density value. Axial and radial profiles are observed for the two states, and large profile changes are found. The spatiotemporal evolution of the transition front is measured. Changes in the fluctuation spectrum are evaluated using conditional average and lock-in average. c

Modelling of radiant-conductive heat transfer in the layer of semitransparent medium in approximation of the classical solution to the single-phase Stefan problem

The single-phase Stefan problem was modelled numerically in approximation of the classical solution in application to melting of a flat semitransparent sample by radiant-conductive technique in a wide range of emissivity of the phase transition front.

Spectral analysis for transition front solutions in Cahn-Hilliard systems

We consider the spectrum associated with the linear operatorobtained when a Cahn--Hilliard system on $\mathbb{R}$ is linearized abouta transition wave solution. In many cases it's possibleto show that the only non-negative eigenvalue is $\lambda = 0$,and so stability depends entirely on the nature of thisneutral eigenvalue. In such cases, we identify a stability conditionbased on an appropriate Evans function, and we verify thiscondition under strong structural conditions on our equations.More generally, we discuss and implement a straightforwardnumerical check of our condition, valid under mild structuralconditions.

Existence and Non-Existence of Fisher-KPP Transition Fronts

We consider Fisher-KPP-type reaction-diffusion equations with spatially inhomogeneous reaction rates. We show that a sufficiently strong localized inhomogeneity may prevent existence of transition-front-type global in time solutions while creating a global in time bump-like solution. This is the first example of a medium in which no reaction-diffusion transition front exists. A weaker localized inhomogeneity leads to existence of transition fronts but only in a finite range of speeds. These results are in contrast with both Fisher-KPP reactions in homogeneous media as well as ignition-type reactions in inhomogeneous media.

The concept of material forces in phase transition problems within the level-set framework

The dynamics of a phase transition front in solids using the level set method is examined in this paper. Introducing an implicit representation of singular surfaces, a regularized version of the sharp interface model arises. The interface transforms into a thin transition layer of nonzero thickness where all quantities take inhomogeneous expressions within the body. It is proved that the existence of an inhomogeneous energy of the material predicts inhomogeneity forces that drive the singularity. The driving force is a material force entering the canonical momentum equation (pseudo-momentum) in a natural way. The evolution problem requires a kinetic relation that determines the velocity of the phase transition as a function of the driving force. Here, the kinetic relation is produced by invoking relations that can be considered as the regularized versions of the Rankine–Hugoniot jump conditions. The effectiveness of the method is illustrated in a shape memory alloy bar.

Travelling waves in a reaction-diffusion model for electrodeposition

In this paper we consider an analytical and numerical study of a reaction-diffusion system for describing the formation of transition front waves in some electrodeposition (ECD) experiments. Towards this aim, a model accounting for the coupling between morphology and composition of one chemical species adsorbed at the surface of the growing cathode is addressed. Through a phase-space analysis we prove the existence of travelling waves, moving with specific wave speed. The numerical approximation of the PDE system is performed by the Method of Lines (MOL) based on high order space semi-discretization by means of the Extended Central Difference Formulae (D2ECDF) introduced in [1]. First of all, to show the advantage of the proposed schemes, we solve the well-known Fisher scalar equation, focusing on the accurate approximation of the wave profile and of its speed. Hence, we provide numerical simulations for the electrochemical reaction-diffusion system and we show that the results obtained are qualitatively in good agreement with experiments for the electrodeposition of Au–Cu alloys.

Model for unstable global modes in the rotating-disk boundary layer

Recent simulations and experiments appear to show that the rotating-disk boundary layer is linearly globally stable despite the existence of local absolute instability. However, we argue that linear global instability can be created by local absolute instability at the edge of the disk. This argument is based on investigations of the linearized complex Ginzburg–Landau equation with weakly spatially varying coefficients to model the propagation of a wavepacket through a weakly inhomogeneous unstable medium. Therefore, this mechanism could operate in a variety of locally absolutely unstable flows. The corresponding nonlinear global mode has a front at the radius of onset of absolute instability when the disk edge is far from the front. This front moves radially outwards when the Reynolds number at the disk edge is reduced. It is shown that the laminar–turbulent transition front also behaves in this manner, possibly explaining the scatter in observed transitional Reynolds numbers.

Frost heave and phase front instability in freezing soils

Abstract The main equations and conditions at the phase transition front are presented for a generalized model of secondary frost heave in freezing fine-grained soils. The analytical criterion for the stability/instability of the freezing phase front in porous media is derived. This criterion is obtained for the occurrence of the frost heave process by using the perturbation method in a two-dimensional, coupled heat and mass transfer model. This model assumes that the non-instantaneous crystallization process takes place in the kinetic zone, and that the rate of crystallization is a function of supercooling. This corresponds to the Arrhenius form equation and agrees with experimental investigations. The perturbation analysis of the freezing front shows that the stability criterion depends upon 1) the Stefan and Peclet numbers, 2) a parameter describing the phase transition kinetics and also 3) dimensionless parameters which characterize the frost heave process. Employing Fourier synthesis, actual front shape evolution is calculated. It is seen that the front displays a periodic morphology whose scale is essentially unrelated to that of the initial (starting) perturbation. The effect of the non-instantaneous kinetics on the front shape evolution is described. As is shown in results, the kinetics has a stabilizing effect and, in this case, the perturbations grow more slowly. The theoretical stability/instability conditions as predicted from the derived criterion were found to be in agreement with experimental investigations of the formation of soil cryogenic structure in the freezing process. On the basis of the asymptotic solution the engineering approach for the calculation of the heave rate and maximal frost penetration depth values — main characteristics for design and construction in cold regions, is presented. The good agreement between calculated values and experimental data is observed.

Dispersion of Nanoparticles in Liquid Crystals by Sputtering and Its Effect on the Electrooptic Properties

The authors describe a method to produce gold nanoparticle-dispersed liquid crystals by means of sputtering, and discuss how the presence of gold nanoparticles affect the electro-optic response of the host liquid crystal. The method exploits the fact that liquid crystals possess low vapor pressures which allow them to undergo the sputtering process, and the target material is sputtered directly on the liquid crystal in a reduced air pressure environment. The sample attained a red-brownish color after sputtering, but no aggregations were observed in the samples kept in the liquid crystal phase. Polarization optical microscopy of the sample placed in a conventional sandwich cell revealed that the phase transition behaviour is affected by the presence of the nanoparticles and that the onset of the nematic phase is observed in the form of bubble-like domains whereas in the pure sample the nematic phase appears after the passing of a phase transition front. Transmission electron microscopy confirmed the presence of single nano-sized particles that were dispersed without forming aggregates in the material. The electro-optic properties of the nanoparticle-dispersed liquid crystal was investigated by measuring the threshold voltage for a twisted-nematic cell. The threshold voltage was found to depend on the frequency of the applied rectangular voltage, and at frequencies higher than 200Hz, the threshold became lower than the pure samples.

Geophysical characterization of bottom simulating reflectors in the Fairway Basin (off New Caledonia, Southwest Pacific), based on high resolution seismic profiles and heat flow data

High-resolution reflection and refraction seismic data were collected in 2004 to investigate, in further detail than allowed by pre-existing low resolution seismic data, the nature of a Bottom Simulating Reflector (BSR) that extends over a broad area of the Fairway Basin, a rifted, continental structure located on the eastern flank of the Lord Howe Rise, to the southwest of New Caledonia. Two main reflectors are documented: the shallower (RN) mimics the seafloor and has a negative polarity while the deeper (RP) does not always mimic the seafloor and has a positive polarity. Using the existing regional seismic lines, we can show that reflector RN can be continuously followed up to DSDP 208 drill hole site. Reflector RP is discontinuous and cannot be traced to DSDP 208. Based on DSDP 208 stratigraphic data, Reflector RN is assigned to the Eocene/Oligocene regional unconformity; reflector RP is interpreted in terms of a diagenetic BSR, likely related to an Opal-A/Opal-CT transition front. Heat flow data collected in 2006 suggest that reflector RP lies too deep to be related to methane hydrates, strengthening our interpretation that RP is of diagenetic origin.

Atomistic examinations of the solid-phase epitaxial growth of silicon

The low-temperature vapor deposition of silicon thin films and the ion implantation of silicon can result in the formation of amorphous silicon layers on a crystalline silicon substrate. These amorphous layers can be crystallized by a thermally activated solid-phase epitaxial (SPE) growth process. The transformations are rapid and initiate at the buried amorphous to crystalline interface within the film. The initial stages of the transformation are investigated here using a molecular dynamics simulation approach based upon a recently proposed bond order potential for silicon. The method is used first to predict an amorphous structure for a rapidly cooled silicon melt. The radial distribution function of this structure is shown to be similar to that observed experimentally. Molecular dynamics simulations of its subsequent crystallization indicate that the early stage, rate limiting mechanism appears to be removal of tetrahedrally coordinated interstitial defects in the nominally crystalline region just behind the advancing amorphous to crystalline transition front. The activation barriers for this interstitials migration within the bulk crystal lattice are calculated and are found to be comparable to the activation energy of the overall solid-phase epitaxial growth process simulated here.

Response of the magnetosphere‐ionosphere system to a sudden southward turning of interplanetary magnetic field

A sudden southward turning of interplanetary magnetic field (IMF) is simulated by the University of Michigan's BATS‐R‐US model. The main goal of this study is to determine the time delay as well as physical processes between when an IMF discontinuity reaches the bow shock, when reconnection is initiated at the magnetopause, and when the ionosphere starts to react. While observations or empirical models might give an estimate of the time delay for the propagation of the discontinuity from the bow shock to the magnetopause, the global MHD simulation provides a more comprehensive insight of responses of the magnetosphere‐ionosphere system. An idealized north‐to‐south IMF transition is modeled, using a solar wind velocity of 400 km/s. After the southward IMF encounters the bow shock, it takes about 6 min for the north‐to‐south IMF transition front to arrive at the subsolar geomagnetic field. The ionospheric response to this sudden southward IMF turning is delayed by another ∼4 min, during which the magnetosphere undergoes a conversion from cusp reconnection to subsolar reconnection and the Alfvén wave propagation to the ionosphere takes place. Thereafter, changes in the ionosphere and ground magnetic perturbations associated with the southward IMF are observed. These responses appear to be globally onset as described in many other studies. The time it takes from the encounter of the IMF transition with the bow shock to when the ionospheric reaction takes place varies with the solar wind speed, ranging from nearly 15 min for a solar wind speed of 300 km/s to just over 6 min for solar wind speeds of 600 km/s.

Spectral analysis of planar transition fronts for the Cahn–Hilliard equation

We consider the spectrum of the linear operator that arises upon linearization of the Cahn–Hilliard equation in dimensions d ⩾ 2 about a planar transition front (a solution that depends on only one distinguished space variable and that has different values at ±∞). In previous work the author has established conditions on this spectrum under which such planar transition fronts are asymptotically stable, and we verify here that those conditions hold for all such waves arising in a general form of the Cahn–Hilliard equation.

Infrared thermography investigations in transitional supersonic boundary layers

The study of boundary-layer transition in supersonic flows is conducted employing infrared thermography (IRT). Several models of swept wings are tested in a blow-down facility at Mach number 2.4. The effects of wing sweep and other parameters (angle of attack, leading-edge contour, presence/absence of surface roughness) are successfully observed. The transition front is clearly identified, demonstrating the utility of IRT for this type of study. The technique is particularly indicated for flows that are sensitive to surface alterations (roughness), such as transitional boundary layers, because it does not require interaction with the model or the flow under investigation. The additional advantage of no need for special apparatus, except for the infrared camera, makes IRT well suited for both wind-tunnel and in-flight testing. Practical problems and limitations encountered when dealing with IRT in high-speed flows are also discussed.

Conversion of a neutron star to a strange star: A two-step process

The conversion of a neutron star to a strange star is studied. Such a transition may be viewed as a two-step process in which the hadronic matter first gets converted to two-flavor quark matter, which, in turn, converts to strange quark matter in the second step of the process. Relativistic hydrodynamical equations are employed to obtain the velocity of propagation of the first conversion front. The second transition front, arising from the conversion of two-flavor to three-flavor quark matter, is studied by using an appropriate weak interaction rate. The propagation velocity of the first conversion front initially shoots up near the core of the star to eventually saturate to some ultrarelativistic value. The first conversion takes about a millisecond, during which the second conversion front is likely to be generated. The second process takes about a hundred seconds to convert the whole quark star into a strange star.

Boundary-Layer Transition on an Axisymmetric Body at Incidence at Mach 1.2

Experimental investigation of the boundary-layer transition on an axisymmetric nose model at 1- and 2-deg incidence was conducted at Mach 1.2. The configuration of the model is the forward part of a Sears-Haack body defined to have minimum wave drag caused by volume at 0-deg incidence in supersonic flow. Transition locations were obtained with small surface roughness on the order of 0.1 μ using an infrared camera. An unsteady pressure transducer was applied to investigate instability mechanisms that lead to transition. These results show that the most aft transition occurred on the leeward ray rather than on the windward rays as observed on sharp cones. Tollmien-Schlichting instability dominated the transition process on the windward ray, and crossflow instability was assumed to dominate on the side similar to the transition on the sharp cones. However, transition occurred more aft on the leeward ray than for the case of the sharp cones, and the transition front was determined by turbulent wedges formed as a consequence of the more forward transition on the side. The disturbance believed to be a traveling crossflow wave was observed at 1-deg incidence, and its measured frequency was in good agreement with the maximum amplified frequency, which was calculated using a compressible linear e N code.

On the heat and mass transfer at the initial stage of ice formation in the sea

A mathematical description of the processes of diffusion heat and mass transfer in a freezing seminfinite layer of seawater is presented. The mathematical formulation of a version of Stefan’s problem accounts for the possibility of the formation of a zone of phase transitions beyond which no phase transitions take place and the phase composition is known. Within the frameworks of a one-dimensional case and under a series of simplifying assumptions, self-similar solutions of the problem were obtained and plotted, such as the temperature and salinity distributions in the ice and the sub-ice layer, the phase composition of the sea ice, the heat flux across its upper boundary, and the position of the phase transition front. A qualitative comparison of these results with the experimental data available is presented.

Dynamics of poloidal flows in enhanced reverse shear bifurcation

A simple reduced enhanced reverse shear (RERS) model is constructed to study the dynamics of poloidal flows during the ERS transition. This model predicts that a reversal of poloidal flow shear occurs just prior to the transition, as seen in experiment [R. E. Bell et al., Phys. Rev. Lett. 81, 1429 (1998)]. This transition front propagates until the radial location where the safety factor (q) is minimum and becomes locked there due to insufficient input power to overcome the threshold requirement for the bifurcation. This study also reveals that there can be many routes to ERS transition depending upon various tunable parameters.