Travelling waves in autocatalytic chemical systems with decay: bounds on existence

The existence of solutions to the travelling-wave equations governed by an autocatalytic reaction of order p (p ≥ 1) and an autocatalytic decay step of order q (q ≥ 1) are examined in the limit of large p. Two cases are treated, q of O(1) and . In the first case, an upper bound k _crit is found for k for the existence of a solution, where k is a dimensionless measure of the strength of the decay step. In the second case, an upper bound on k is also found when q < p. For q ≥ p, there is no upper bound on k and solutions exist for all (positive) values of k.     

Short-term instability in stochastic systems

Dynamic systems with lumped parameters which experience random temporal variations are considered in this paper. These variations may lead to “short-term” dynamic instability that is reflected in the system’s response as alternating periods of zero or almost zero response and rare short outbreaks. As long as it may be impractical to preclude completely such outbreaks for a designed system, the corresponding response should be analyzed to evaluate the system’s reliability.Results of such analyses are presented separately for cases of slow and rapid parameter variations. Linear models of the systems are studied in the former case using parabolic approximation (PA) for the variations in the vicinity of their peaks together with Krylov-Bogoliubov (KB) averaging for the transient response. This results in a solution for the response probability density function (PDF).The case of rapid broadband parameter variations is studied using theory of Markov processes. The system is assumed to operate beyond its stochastic instability threshold-although only slightly-and its nonlinear model is used accordingly. The analysis is based on solution of the Fokker-Planck-Kolmogorov (FPK) partial differential equation for stationary PDF of the response. Several such PDFs are analyzed; they are found to have integrable singularities at the origin indicating an intermittent nature of the response.     

Travelling waves in a differential flow reactor with simple autocatalytic kinetics

A simple prototype model for a differential flow reactor in which the possible initiation and propagation of a reaction-diffusion-convection travelling-wave solution (TWS) in the simple isothermal autocatalytic system A+mB (m+1)B, rate kab^m (m 1) is studied with special attention being paid to the most realistic cases (m=1,2). The physical problem considered is such that the reactant A (present initially at uniform concentration) is immobilised within the reactor. A reaction is then initiated by allowing the autocatalyst species to enter and to flow through the reaction region with a constant velocity. The structure of the permanent-form travelling waves supported by the system is considered and a solution obtained valid when the flow rate (of the autocatalyst) is very large. General properties of the corresponding initial-value problem (IVP) are derived and it is shown that the TWS are the only long-time solutions supported by the system. Finally, these results are complemented with numerical solutions of the IVP which confirm the analytical results and allow the influence of the parameters of the problem not accessible to the theoretical analysis to be determined.     

An A[r,R) instability criteria for a class of uncertain discrete systems

Abstract

In this paper, simple criteria for the instability of a class of uncertain discrete systems are derived. Such criteria can be easily checked by estimating the largest nonnegative zero of two specific polynomials. A numerical example is also provided to illustrate the main results.     

Frost heave and phase front instability in freezing soils

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.     

激光成丝不稳定性并行数值模拟研究

实现了三维非线性流体力学与激光传播耦合的并行数值模拟程序,用来研究高强度激光在稀疏等离子体中的激光成丝不稳定性.并行数值实验结果表明,该程序在64台处理机上取得了较好的并行加速比,且能定性模拟激光成丝现象.     

Computational simulation of chemical dissolution‐front instability in fluid‐saturated porous media under non‐isothermal conditions

This paper primarily deals with the computational aspects of chemical dissolution‐front instability problems in two‐dimensional fluid‐saturated porous media under non‐isothermal conditions. After the dimensionless governing partial differential equations of the non‐isothermal chemical dissolution‐front instability problem are briefly described, the formulation of a computational procedure, which contains a combination of using the finite difference and finite element method, is derived for simulating the morphological evolution of chemical dissolution fronts in the non‐isothermal chemical dissolution system within two‐dimensional fluid‐saturated porous media. To ensure the correctness and accuracy of the numerical solutions, the proposed computational procedure is verified through comparing the numerical solutions with the analytical solutions for a benchmark problem. As an application example, the verified computational procedure is then used to simulate the morphological evolution of chemical dissolution fronts in the supercritical non‐isothermal chemical dissolution system. The related numerical results have demonstrated the following: (1) the proposed computational procedure can produce accurate numerical solutions for the planar chemical dissolution‐front propagation problem in the non‐isothermal chemical dissolution system consisting of a fluid‐saturated porous medium; (2) the Zhao number has a significant effect not only on the dimensionless propagation speed of the chemical dissolution front but also on the distribution patterns of the dimensionless temperature, dimensionless pore‐fluid pressure, and dimensionless chemical‐species concentration in a non‐isothermal chemical dissolution system; (3) once the finger penetrates the whole computational domain, the dimensionless pore‐fluid pressure decreases drastically in the non‐isothermal chemical dissolution system.     

Strain localization in a continuum as an instability event

The paper presents a novel physical model, based on perturbation theory, to describe localization pattern formation in a solid material as a result of system instabilities. Such kind of approach has been inspired by the theory of population dynamics. In particular, the sinergetic phenomenon of strain localization into a stressed continuum, and its subsequent evolution to cohesive cracking, is obtained through the competition of an external source of energy (e.g., strain energy) and of the internal behavior of the material. The hypothesis of mobile energy entities within material bulk is put forward. These entities, which under low strain conditions are evenly distributed throughout the body, can be considered as strain quanta. The quantization of mechanical quantities is not new in continuum and fracture mechanics, [see, e.g., Novozhilov (1969, Prik Mat Mek 33:212–222)]. With increasing strain, a certain critical point is reached when the homogeneous situation becomes unstable and the strain quanta begin to aggregate into bands, leading to periodic strain localization patterns. The model, which is only theoretical at this stage, can be applied to the particular case of dry snow avalanches. In these cases, snow avalanche triggering is due toinstability (onset of sliding onto a weak plane) and is controlled by external loading (e.g., weight of the slope, load by skiers) and by internal factors (e.g., temperature changes, snow phase transformations etc.).     

Branching instability in expanding bacterial colonies

Self-organization in developing living organisms relies on the capability of cells to duplicate and perform a collective motion inside the surrounding environment. Chemical and mechanical interactions coordinate such a cooperative behaviour, driving the dynamical evolution of the macroscopic system. In this work, we perform an analytical and computational analysis to study pattern formation during the spreading of an initially circular bacterial colony on a Petri dish. The continuous mathematical model addresses the growth and the chemotactic migration of the living monolayer, together with the diffusion and consumption of nutrients in the agar. The governing equations contain four dimensionless parameters, accounting for the interplay among the chemotactic response, the bacteria–substrate interaction and the experimental geometry. The spreading colony is found to be always linearly unstable to perturbations of the interface, whereas branching instability arises in finite-element numerical simulations. The typical length scales of such fingers, which align in the radial direction and later undergo further branching, are controlled by the size parameters of the problem, whereas the emergence of branching is favoured if the diffusion is dominant on the chemotaxis. The model is able to predict the experimental morphologies, confirming that compact (resp. branched) patterns arise for fast (resp. slow) expanding colonies. Such results, while providing new insights into pattern selection in bacterial colonies, may finally have important applications for designing controlled patterns.     

Electronically driven short-range lattice instability: Cluster effects in superconductors

In the first part of this series [1] short- and medium-range interactions in superconductors were investigated. In the present paper a discussion is made on the cluster-mass dependence of the superconductive transition temperature and the relevant phenomenon of electron localization. A comparison with experiments is given; the predictions fit well to the observations.     

Factors affecting schedule instability in manufacturing companies

Schedule instability is a major issue in many manufacturing companies. It results in a low service level to customers, high inventory levels, and high costs associated with production changeovers. Much has been written on schedule instability in the literature, but most studies have used simulation and mathematical modelling to look at the internal operations of manufacturing companies. While schedule instability has always been very much a practical problem, there have been few empirical studies presented in the literature. The aim of this research was to obtain the opinions of practitioners on schedule instability, and to identify factors that affect the perceived level of instability experienced by manufacturing companies. An e-mail/web-based survey was administered to practitioners working in the planning and scheduling area. The results suggest that the majority of the respondents perceive schedule instability to be either an important or a very important issue in their operations. Analysis of the responses also shows that schedule instability is affected mainly by external factors, notably relationships with buyers and relationships with suppliers; internal operations have a lower impact. The research moves the debate about schedule instability beyond the current concentration on simulation and mathematical modelling, and from a focus on internal operations to a supply-chain context.     

Development of Richtmyer-Meshkov and Rayleigh-Taylor instability in the presence of magnetic field

Fluid instabilities like Rayleigh-Taylor (R-T), Richtmyer-Meshkov (R-M) and Kelvin-Helmholtz (K-H) instability can occur in a wide range of physical phenomenon from astrophysical context to Inertial Confinement Fusion (ICF). Using Layzer's potential flow model, we derive the analytical expressions of growth rate of bubble and spike for ideal magnetized fluid in R-T and R-M cases. In the presence of transverse magnetic field, the R-M and R-T instabilities are suppressed or enhanced depending on the direction of magnetic pressure and hydrodynamic pressure. Again the interface of two fluid may oscillate if both the fluids are conducting. However, it is observed that the magnetic field has no effect in linear case.     

Local lattice instability and ionic transport in high-temperature superconductors

Ionic transport, a characteristic feature of superionic conductors, is analyzed microscopically for the RBa₂Cu₃O{7-Σ} (R = rare earth) high-temperature superconductors. The electronic correlation effects are considered to be responsible for the formation of a local double-well potential for apical oxygen, which in its turn results in a decrease of the activation energy for interstitial chain oxygen defects. The behavior of the double well upon reduction of the oxygen content is discussed and its possible manifestation in the low-frequency dynamical response is considered.     

A surface instability of granules under vibration in partitioned containers

This paper describes a surface instability of vibrated granules contained in rectangular vessels. The vessels are partitioned in two equal sections by a vertical plate whose lower end is placed at a certain distance from the vessel base. The free surfaces of granules in two sections, initially at an equal level, start to move when submitted to vertical vibrations, and stabilize at different levels. Measurements are made of the pressure at the bottom of each of the partitioned beds, which reveals the underlying mechanism of the surface instability. Computer simulations are performed to show microscopic structures of this surface instability. Received: 25 February 2000     

The influence of combined alignments on lateral acceleration on mountainous freeways: a driving simulator study

Combined horizontal and vertical alignments are frequently used in mountainous freeways in China; however, design guidelines that consider the safety impact of combined alignments are not currently available. Past field studies have provided some data on the relationship between road alignment and safety, but the effects of differing combined alignments on either lateral acceleration or safety have not systematically examined. The primary reason for this void in past research is that most of the prior studies used observational methods that did not permit control of the key variables. A controlled parametric study is needed that examines lateral acceleration as drivers adjust their speeds across a range of combined horizontal and vertical alignments. Such a study was conducted in Tongji University’s eight-degree-of-freedom driving simulator by replicating the full range of combined alignments used on a mountainous freeway in China. Multiple linear regression models were developed to estimate the effects of the combined alignments on lateral acceleration. Based on these models, domains were calculated to illustrate the results and to assist engineers to design safer mountainous freeways.     

Modeling lateral control in driving studies

In driving studies based on simulators and instrumented vehicles, specific models are needed to capture key aspects of driving data such as lateral control. We propose a model that uses weighted polynomial projections to predict each data point from the previous three time points, and accommodates the attempts of the drivers to re-center the vehicle before crossing the borders of the traffic lane. Our model also allows the possibility that average position within the lane may vary from driver to driver. We demonstrate how to fit the model using standard statistical procedures available in software packages such as SAS. We used a fixed-base driving simulator to obtain data from 67 drivers with Alzheimer's disease and 128 elderly drivers without dementia. Using these data, we estimated the subject-specific parameters of our model, and we compared the two groups with respect to these parameters. We found that the parameters based on our model were able to distinguish between the groups in an interpretable manner. Hence, this model may be a useful tool to define outcome measures for observational and interventional driving studies.     

On hydrodynamic instability of the flame-wedge

The hydrodynamic stability of the premixed wedge-shaped flame involving a burned-gas stagnation zone is considered. It is shown that the Darrieus-Landau instability of the flame interface is reinforced by the Kelvin-Helmholtz instability of the stagnation-zone boundary.     

Modulation instability scenario in negative index materials

We present an investigation of the critical frequency windows permitting modulation instability in negative index materials. The principal motivation for our analysis stems from the impact of the inevitable presence of the effective dispersive magnetic permeability in addition to the effective dielectric permittivity determining the propagation model for ultrashort pulses in negative index materials. We emphasize the influence of nonlinear dispersion terms, arising out of the combinatorial effect of the dispersive permeability with the nonlinear polarization, over the MI phenomena, the outcome of its development achieved by using linear stability analysis. Gain spectrum investigation has been carried out for both anomalous and normal dispersion regime in the focusing and defocusing cases of nonlinearity and near zero dispersion regime where higher order linear dispersive effects emerge. The results of linear stability analysis have been validated by direct numerical simulation of the governing equation using the split-step Fourier transform method. We show that second-order nonlinear dispersion opens up distinct MI windows with their appropriate conditions and unlike the first-order nonlinear dispersion term, the sign of it has got deep consequences in the modulation instability scenario.     

Influence of vibrations on convective instability of polymerization fronts

Propagation of polymerization fronts with liquid monomer and solid polymer is considered, and the influence of vibrations on critical conditions of convective instability is studied. The model includes the heat equation, the equation for the concentration and the Navier-Stokes equations considered under the Boussinesq approximation. Linear stability analysis of the problem is fulfilled, and the convective instability boundary is found depending on the amplitude and on the frequency of vibrations.     

Stable supercontinuum pulse generated by modulation instability in a dispersion-managed fiber

We study on non-linear localized waves on continuous wave background in a dispersion and non-linearity management fibre. We find a stable supercontinuum pulse can be generated from a small modulation on continuous wave in a proper management way, for which the pulse spectrum width and its growth rate can be controlled well by the management parameters. Additionally, we demonstrate a Kuznetsov–Ma breather like non-linear localized wave can exist in a periodic dispersion management fibre, and its spectrum evolution is distinctive from the Kuznetsov–Ma breather’s.