Stability of travelling fronts in a model for flame propagation part I: Linear analysis

We investigate the stability of travelling wave solutions of the multidimensional thermodiffusive model for flame propagation with unit Lewis number. This model consists in a system of two nonlinear parabolic equations posed in an infinite cylinder, with Neumann boundary conditions. In this paper, we prove that every travelling wave solution of this model is linearly stable. Our tools are exponential decay estimates for solutions of elliptic equations in a cylinder, and the Maximum Principle for parabolic equations.     

Bifurcations of travelling waves in the thermo-diffusive model for flame propagation

The main topic of this paper is the study of steady-state bifurcations occurring in the two-dimensional thermo-diffusive model in the framework of large activation energies.The physical situation is well established, due to the classical work of Sivashinsky. He derived a dispersion relation and observed that the planar waves bifurcated into stable multidimensional waves as the Lewis number crossed a critical value.The purpose of this paper is to give a mathematical basis to this theory, furthering a study of D. Terman. We then investigate the bifurcation in detail. Finally, we investigate the three-dimensional case, where a different bifurcation pattern may occur.     

A phase plane discussion of convergence to travelling fronts for nonlinear diffusion

The paper is concerned with the asymptotic behavior as t of solutions u(x,t) of the equation in the case f(0)=f(1)=0, with f(u) non-positive for u(>0) sufficiently close to zero and f(u) non-negative for u(<1) sufficiently close to 1. This guarantees the uniqueness (but not the existence) of a travelling front solution u;U(x–ct), U(–);0, U();, and it is shown in essence that solutions with monotonic initial data converge to a translate of this travelling front, if it exists, and to a stacked combination of travelling fronts if it does not. The approach is to use the monotonicity to take u and t as independent variables and p = u _x as the dependent variable, and to apply ideas of sub- and super-solutions to the diffusion equation for p.This research was sponsored by the United States Army under Contract No. DAAG29-75-C-0024.     

Nonlinear stability analysis of a disk brake model

It has become commonly accepted by scientists and engineers that brake squeal is generated by friction-induced self-excited vibrations of the brake system. The noise-free configuration of the brake system loses stability through a flutter-type instability and the system starts oscillating in a limit cycle. Usually, the stability analysis of disk brake models, both analytical as well as finite element based, investigates the linearized models, i.e. the eigenvalues of the linearized equations of motion. However, there are experimentally observed effects not covered by these analyses, even though the full nonlinear models include these effects in principle. The present paper describes the nonlinear stability analysis of a realistic disk brake model with 12 degrees of freedom. Using center manifold theory and artificially increasing the degree of degeneracy of the occurring bifurcation, an analytical expression for the turning points in the bifurcation diagram of the subcritical Hopf bifurcations is calculated. The parameter combination corresponding to the turning points is considered as the practical stability boundary of the system. Basic phenomena known from the operating experience of brake systems tending to squeal problems can be explained on the basis of the practical stability boundary.     

Stability of combustion of powder in a phenomenological model with nonadiabatic flame

A stability criterion for combustion of powder is obtained, taking into account the effect of the processes in the gas phase. It is shown that consideration of the effects of a nonadiabatic flame leads to the stability reserve of combustion being reduced and the natural frequency of vibrations being lowered. The effects thus found are physically explained by the radiation of a part of energy from the combustion zone with thermal and acoustic waves.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 82–87, November–December, 1973.Deceased.     

Wave propagation in periodic buckled beams. Part II: Experiments

Buckled periodic beams possess a geometrically nonlinear, load–deformation relation and intrinsic length scales such that stable, nonlinear waves are possible. In part I of this paper, a model has been derived that predict compressive/rarefaction, supersonic/supersonic solitary waves, varying the level of compression and the support type (guided or pinned). Although this work has been validated by simulating the structure with finite-elements, in the present paper, investigations are done experimentally, focusing on the guided-supported, slightly-buckled beam.     

Transient pressure effects in the evolution equation for premixed flame fronts

A nonlinear evolution equation for a scalar field G(x, t) is derived, whose level surface G ₀=const. represents the interface of a thin premixed flame propagating in a flow field. The derivation is an extended version of an equation already proposed by Markstein [1]. It was reconsidered by Williams [2] as a basis for theoretical and numerical analysis and takes, in addition to flame curvature and flame stretch time variations of the bulk pressure, heat loss and nonconstant transport coefficients into account. The equation is an extension of earlier analyses where a flame evolution equation was derived for slightly wrinkled flames such that the front can be described by a single-valued function of a normal coordinate. That formulation excluded situations where the mean flame front has an arbitrary shape in space. Here the more general situation is analysed by using a two-length-scale asymptotic analysis. The leading-order solution of this analysis is equivalent to the equation originally derived by Markstein [1]. In addition to nonconstant properties and heat-loss effects, that had already been considered by Clavin and Nicoli [3], the influence of transient changes of the bulk pressure is analysed. All these effects are combined into a unified formulation which will serve as a basis for a new flamelet concept for premixed turbulent combustion.     

Nonlinear dynamics and stability of microstructures in a lattice model for ferroelastic materials

With the view of understanding how precise macroscopic properties of a material emerge from the underlying physics of homogeneous microstructures, a lattice model which can describe complex non-linear patterns made of elastic domains and interfaces is proposed. On the basis of a two-dimensional lattice model involving non-linear and competing interactions the dynamics of microstructure formation is examined. The emphasis is placed especially on an instability mechanism of a strain band producing localized domains. The influence of applied forces and dissipative effects on the dynamics of two perpendicular strain bands is studied. The results are interpreted as a microtwinning in crystalline alloys. The physical conjectures are checked by means of numerical simulations performed directly on the microscopic system.

---

Sommario Si propone un modello reticolare che può descrivere complessi arrangiamenti fatti di domini elastici ed interfacce. Sulla base di un modello bidimensionale in cui sono presenti interazioni contrastanti e nonlineari si esamina la dinamica della formazione di microstrutture. L'accento è posto sui meccani'smi di instabilità che determinano bande di deformazione localizzata. Si studia l'influenza delle forze applicate e degli effetti dissipativi sulla dinamica di due bande perpendicolari e si interpretano i risultati come un microtwinning in leghe cristalline. Si verificano le congetture fisiche per mezzo di simulazioni numeriche del modello microscopico.

     

Existence of planar flame fronts in convective-diffusive periodic media

We prove the existence of planar travelling wave solutions in a reaction-diffusion-convection equation with combustion nonlinearity and self-adjoint linear part in R ⁿ, n1. The linear part involves diffusion-convection terms and periodic coefficients. These travelling waves have wrinkled flame fronts propagating with constant effective speeds in periodic inhomogeneous media. We use the method of continuation, spectral theory, and the maximum principle. Uniqueness and monotonicity properties of solutions follow from a previous paper. These properties are essential to overcoming the lack of compactness and the degeneracy in the problem.     

Stability of Relative Equilibria. Part II: Dumbell Satellites

In the second part a practically important problem, namely the stability of relative equilibria of a dumbell satellite on an orbit around the Earth is treated by means of the reduced energy-momentum method. The dumbell satellite is used to emphasize the advantages of the reduced energy-momentum method which did not become obvious in the simple example of the rotating pendulum treated in Part I, as well as, to discuss some of the finer technical details.     

Multi-mode wave propagation in ribbed plates. Part II: Predictions and comparisons

This paper addresses dispersion curve numerical computations for specific ribbed plates. Precisely, wave propagation in the direction parallel to the ribs is the main objective. First, analytical calculations are performed for such ribbed plates after which the results are compared to a more general numerical procedure. This procedure reuses a reduced finite element model of the ribbed plate and extracts guided multi-mode propagation parameters. Comparisons of analytical and numerical estimations show very good agreement. Finally, the experimental results obtained in the companion paper are considered. Specifically, experimental and numerical dispersion curves are compared over a wide frequency range. Close concordance is obtained allowing the dispersion curves to be fully interpreted.     

Nonlinear Dynamics of a Rigid Unbalanced Rotor in Journal Bearings. Part II: Experimental Analysis

In the first part of the present investigation [9], the dynamic behaviour of a rigid rotor supported on plain journal bearings was studied, focusing particular attention on its nonlinear aspects. In the present paper an experimental confirmation of the theoretical results is sought. The steel rotor of the experimental rig was given a constant circular cross section in order to fix in an easy way the two distances between supports corresponding, respectively, to the values of the parameter assigned in [9]. Two steel rings, each one with a series of holes and a clamping screw, were mounted onto the rotor with a small clearance. This arrangement made it possible to fix the positions of the rings and their holes respect to the rotor, so as to realize a pre-estabilished unbalance. The two bronze journal bearings were characterised by a relatively low length/diameter ratio, and a relatively high value of the radial clearance and were lubricated with oil delivered from a thermostatic tank. In this way, despite the relative lightness of the rotor, the dimensionless static eccentricity _s was given the high values that were apt to realize the operating conditions assumed in the theoretical analysis. The rotor was driven by means of a d.c. motor connected to a toothed belt-drive. Varying the rotor speed in the range 1000 ÷ 10000 r.p.m., made it possible to assign the values of the modified Sommerfeld number assumed in the theoretical analysis. Three pairs of eddy-current probes were mounted in order to detect the trajectories of three points (C₁, C and C₂) suitably fixed along the rotor axis. These orbits were finally put in comparison with the corresponding ones previously obtained through numerical analysis. The comparison pointed out that the experimental data were in good agreement with the theoretical predictions, despite the approximations that characterise the theoretical model and the unavoidable errors affecting measures in the course of the experimental test.     

一个端部开口短管气体爆燃外场火焰传播模型

针对端部开口短管气体爆燃火焰传播问题,通过借鉴Clanet等和Bychkov等提出的火焰传播模型,在假设绝热、不可压缩的条件下,得到了可燃气体分布与火焰锋面传播的数学模型。以汽油蒸气为实验工质,在全透明实验管道上进行了爆燃实验。通过高速摄影及纹影图像,对所提出的模型进行了验证。结果表明,该模型能够较准确地预测长径比4：1至10：1的端部开口短管气体爆燃外场可燃气体界面与火焰锋面位置。上述成果在可燃气防爆安全领域具有一定应用价值。

     

Nonlinear travelling waves in a hyperelastic rod composed of a compressible Mooney-Rivlin material

In this paper, we study strongly nonlinear axisymmetric waves in a circular cylindrical rod composed of a compressible Mooney-Rivlin material. To consider the travelling wave solutions for the governing partial differential system, we first reduce it to a nonlinear ordinary differential equation. By using the bifurcation theory of planar dynamical systems, we show that the reduced system has seven periodic annuluses with different boundaries which depend on four parameters. We further consider the bifurcation behavior of the phase portraits for the reduced one-parameter vector fields when other three parameters are fixed. Corresponding to seven different periodic annuluses, we obtain seven types of travelling wave solutions, including solitary waves of radial contraction, solitary waves of radial expansion, solitary shock waves of radial contraction, solitary shock waves of radial expansion, periodic waves and two types of periodic shock waves. These are physically acceptable solutions by the governing partial differential system. The rigorous parameter conditions for the existence of these waves are given.     

Hydrodynamic stability of evaporation fronts in porous media

The stability of phase transition fronts in water flows through porous media is considered. In the short-wave approximation a linear stability analysis is carried out and a sufficient condition of hydrodynamic instability of the phase discontinuity is proposed. The problem of injection of a water-vapor mixture into a two-dimensional mixture-saturated formation is solved and its numerical solution is compared with an exact solution of the corresponding one-dimensional self-similar problem. It is discovered that, instead of the unstable discontinuities in the one-dimensional formulation, in the two-dimensional case a lengthy mixing zone with a characteristic scale that increases self-similarly with time is formed.     

Theory of flame propagation in a gas and drop mixture

The equations of a reacting multiphase continuous medium [1] are used to investigate the problem of steady-state flame front propagation in a gas mixture with evaporating drops. A simple model for ignition of the liquid drops is proposed which is based on the application of the method of equally accessible surfaces [2] to the heat and mass exchange processes between the microflames surrounding the separate drops, the drops, and the carrying gas medium. The parameter distributions in the macroscopic flame front as well as the dependences of the flame propagation velocity in the gas suspension on a number of parameters governing the process under investigation are represented.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 101–108, July–August, 1973.     

Experimental Investigation of the Nonlinear Response of a Hanging Cable. Part II: Global Analysis

The nonlinear dynamics of the same experimental model of an internally resonant hanging elastic cable considered in Part I [1] are addressed here from the point of view of the global system behaviour in the control parameter space. Synthetic results of systematic response measurements, made at different amplitudes of the support motion in frequency ranges including meaningful external resonance conditions, are reported and discussed. Attention is devoted to the detection of the most robust classes of motion. Quite complicated overall pictures of regular response regions with variable contributions from different planar and nonplanar cable modes are observed, as well as several regions of quasiperiodic and chaotic responses. Sample quantitative characterizations of nonregular motions are presented. Some experimental results are also observed against the background of the nonlinear dynamic phenomena exhibited by a theoretical model of a continuous cable with four-degrees-of-freedom.     

Theoretical investigation of an elliptical crack in thermopiezoelectric material. Part II: Crack propagation

Propagation behavior of an elliptical crack in thermopiezoelectric material subjected to a uniform temperature is investigated in this paper. The three-dimensional strain energy density formulation is used to determine the direction of crack propagation and the shape of the initial fracture increment. It is found that the elliptical crack grows coplanarly under this particular load case but not normal to the crack front. The elliptical crack tends to become a circular one when thermal loading is applied.