Stability of pole solutions for planar propagating flames: II. Properties of eigenvalues/eigenfunctions and implications to stability

In a previous paper (Part I) we focused our attention on pole solutions that arise in the context of flame propagation. The nonlinear development that follows after a planar flame front becomes unstable is described by a single nonlinear PDE which admits pole solutions as equilibrium states. Specifically, we were concerned with coalescent steady states, which correspond to steadily propagating single-peak structures extended periodically over the infinite domain. This pattern is one that is commonly observed in experiments. In order to examine the linear stability of these equilibrium solutions, we formulated in Part I the corresponding eigenvalue problem and derived exact analytical expressions for the spectrum and the corresponding eigenfunctions. In this paper, we examine their properties as they relate to the stability issue. Being based on analytical expressions, our results resolve earlier controversies that resulted from numerical investigations of the stability problem. We show that, for any period 2L, there always exists one and only one stable steady coalescent pole solution. We also examine the dependence of the eigenvalues and eigenfunctions on L which provides insight into the behavior of the nonlinear PDE and, consequently, on the nonlinear dynamics of the flame front

On a viscous critical-stress model of martensitic phase transitions

The solid-to-solid phase transitions that result from shock loading of certain materials, such as the graphite-to-diamond transition and the alpha-epsilon transition in iron, have long been subjects of a substantial theoretical and experimental literature. Recently a model for such transitions was introduced which, based on a CS condition (CS) and without use of fitting parameters, accounts quantitatively for existing observations in a number of systems [Bruno and Vaynblat, Proc. R. Soc. London, Ser. A 457, 2871 (2001)]. While the results of the CS model match the main features of the available experimental data, disagreements in some details between the predictions of this model and experiment, attributable to an ideal character of the CS model, do exist. In this article we present a version of the CS model, the viscous CS model (vCS), as well as a numerical method for its solution. This model and the corresponding solver results in a much improved overall CS modeling capability. The innovations we introduce include: (1) Enhancement of the model by inclusion of viscous phase-transition effects; as well as a numerical solver that allows for a fully rigorous treatment of both, the (2) Rarefaction fans (which had previously been approximated by “rarefaction discontinuities”), and (3) viscous phase-transition effects, that are part of the vCS model. In particular we show that the vCS model accounts accurately for well known “gradual” rises in the alpha-epsilon transition which, in the original CS model, were somewhat crudely approximated as jump discontinuities

The strongly attracting character of large amplitude nonlinear resonant acoustic waves without shocks : a numerical study

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1996.Includes bibliographical references (p. 133-135).by Dimitri Vaynblat.Ph.D

Stability of pole solutions for planar propagating flames: I. Exact eigenvalues and eigenfunctions

It is well known that the nonlinear PDE describing the dynamics of a hydrodynamically unstable planar flame front admits exact pole solutions as equilibrium states. Such a solution corresponds to a steadily propagating cusp-like structure commonly observed in experiments. In this work we investigate the linear stability of these equilibrium states-the steady coalescent pole solutions. In previous similar studies, either a truncated linear system was numerically solved for the eigenvalues or the initial value problem for the linearized PDE was numerically integrated in order to examine the evolution of initially small disturbances in time. In contrast, our results are based on the exact analytical expressions for the eigenvalues and corresponding eigenfunctions. In this paper we derive the expressions for the eigenvalues and eigenfunctions. Their properties and the implication on the stability of pole solutions is discussed in a paper which will appear later

Bubble growth in a two-dimensional viscoelastic foam

The effects of viscoelasticity on the expansion of gas bubbles arranged in a hexagonal array in a polymeric fluid are investigated. The expansion is driven by the diffusion of a soluble gas from the liquid phase, and the rate of expansion is controlled by a combination of gas diffusion, fluid rheology and surface tension. In the diffusion limited case, the initial growth rate is slow due to small surface area, whereas at high diffusivity initial growth is rapid and resisted only by background solvent viscosity. In this high Deborah number limit, we see a two stage expansion in which there is an initial rapid expansion up to the size at which the elastic stresses balance the pressure difference. Beyond this time the bubble expansion is controlled by the relaxation of the polymer. We also illustrate how viscoelasticity affects the shape of the bubble. In addition to a full finite element calculation of the two-dimensional flow, two one-dimensional approximations valid in the limits of small and large gas area fractions are presented. We show that these approximations give accurate predictions of the evolution of the bubble area, but give less accurate predictions of the bubble shape

Shock–induced martensitic phase transitions: critical stresses, Riemann problems and applications

In this paper we develop a theory for phase transitions in solids under shock loading. This theory applies, in particular, to a class of experiments which, as a result of solid–to–solid phase transitions, give rise to certain characteristic patterns consisting of two shock–like waves. We show that the single assumption that stresses in a phase cannot lie beyond its transition boundaries leads to a complete model for the physical systems at hand. This model is different from others proposed in the literature: it does not make use of kinetic relations and it accounts for the observed wave histories without parameter fitting. The first part of this paper focuses on the basic mathematical description of our model and it presents solutions to the complete set of Riemann problems which could arise as a result of dynamic interactions, including the basic two–wave structures mentioned above. In the second part of the paper we use our Riemann solver to construct general solutions for the piecewise constant initial–value problems usually arising in experiment, and we specialize our solutions to two widely studied polymorphic phase changes: the graphite–diamond transition and the α–∈ transition in iron. We show that, in the presence of well–accepted quations of state for the pure phases, our model leads to close quantitative agreement with a wide range of experimental results; possible sources of disagreement in certain fine features are also discussed. Interestingly, in some cases our theory predicts sequences of events which differ from those generally accepted. In particular, our model predicts a variety of regimes for the iron experiments which previous theoretical investigations had not surmised, and it indicates the existence of certain unexpected transformation domains in the graphite systems