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

Stability of a premixed flame in stagnation-point flow against general disturbances

Previously, the stability of a premixed flame in a stagnation flow was discussed for a restricted class of disturbances that are self-similar to the basic undisturbed flow; thus, flame fronts with corrugations only in the cross stream direction were considered. Here, we consider a more general class of three-dimensional flame front perturbations which also permits corrugations in the streamwise direction. It is shown that, because of the stretch experienced by the flame, the hydrodynamic instability is limited only to disturbances of short wavelength. If in addition diffusion effects have a stabilizing influence, as would be the case of mixtures with Lewis number greater than one, a stretched flame could be absolutely stable. Instabilities occur when the Lewis number is below some critical value less than one. Neutral stability boundaries are presented in terms of the Lewis number, the strain rate, and the appropriate wavenumbers. Beyond the stability threshold, the two-dimensional self-similar modes always grow first. However, if disturbances of long wavelength are excluded, it is possible for the three-dimensional modes to be the least stable one. Accordingly, the pattern that will be observed on the flame front, at the onset of instability, will consist of either ridges in the direction of stretch or the more common three-dimensional cellular structure

Phenylketonuria

Genome research is emerging as a new and important tool in biology used to obtain information on gene sequences, genomic interaction, and how genes work in concert to produce the final syndrome or phenotype. Defect in phenylalanine hydroxylase (PAH) gene result in Phenylketonuria (PKU). Molecular studies using the brain of the mouse model for PKU (PAHenu2) showed altered expression of several genes including upregulation of orexin A and a low activity of branched chain aminotransferase. These studies suggest that a single gene (PAH) defect is associated with altered expression, transcription and translation of other genes. It is the combination of the primary gene defect, the altered expression of other genes, and the new metabolic environment that is created, which lead to the phenotype

Analytical treatment of 2D steady flames anchored in high-velocity streams

The problem of burning of high-velocity gas streams in channels is revisited. Previous treatments of this issue are found to be incomplete. It is shown that despite relative smallness of the transversal gas velocity, it plays crucial role in determining flame structure. In particular, it is necessary in formulating boundary conditions near the flame anchor, and for the proper account of the flame propagation law. Using the on-shell description of steady anchored flames, a consistent solution of the problem is given. Equations for the flame front position and gas-velocity at the front are obtained. It is demonstrated that they reduce to a second-order differential equation for the front position. Numerical solutions of the derived equations are found.Comment: 15 pages, 6 figure

The effect of thermal expansion on diffusion flame instabilities

In this paper we examine the effect of thermal expansion on the stability of a planar unstrained diffusion flame and provide a comprehensive characterization of diffusive-thermal instabilities while realistically accounting for density variations. The possible patterns that are likely to be observed as a result of differential and preferential diffusion are identified for a whole range of parameters including the distinct Lewis numbers associated with the fuel and oxidizer, the initial mixture strength and the flow conditions. Although we find that thermal expansion has a marked influence on flame instability, it does not play a crucial role as it does in premixed combustion. It primarily affects the parameter regime associated with the onset of the instabilities and the growth rate of the unstable modes. Perhaps the most surprising result is that its has a different influence on the various modes of instability - a destabilizing influence on the formation of cellular structures and a stabilizing influence on the onset of oscillation

Hydrodynamic and thermodiffusive instability effects on the evolution of laminar planar lean premixed hydrogen flames

Numerical simulations with single-step chemistry and detailed transport are used to study premixed hydrogen/air flames in two-dimensional channel-like domains with periodic boundary conditions along the horizontal boundaries as a function of the domain height. Both unity Lewis number, where only hydrodynamic instability appears, and subunity Lewis number, where the flame propagation is strongly affected by the combined effect of hydrodynamic and thermodiffusive instabilities are considered. The simulations aim at studying the initial linear growth of perturbations superimposed on the planar flame front as well as the long-term nonlinear evolution. The dispersion relation between the growth rate and the wavelength of the perturbation characterizing the linear regime is extracted from the simulations and compared with linear stability theory. The dynamics observed during the nonlinear evolution depend strongly on the domain size and on the Lewis number. As predicted by the theory, unity Lewis number flames are found to form a single cusp structure which propagates unchanged with constant speed. The long-term dynamics of the subunity Lewis number flames include steady cell propagation, lateral flame movement, oscillations and regular as well as chaotic cell splitting and mergin

Formation of Liesegang Patterns

It has been recently shown that precipitation bands characteristic of Liesegang patterns emerge from spinodal decomposition of reaction products in the wake of moving reaction fronts. This mechanism explains the geometric sequence of band positions x_n ~ Q(1+p)^n and, furthermore, it yields a spacing coefficient, p, that is in agreement with the experimentally observed Matalon-Packter law. Here I examine the assumptions underlying this theory and discuss the choice of input parameters that leads to experimentally observable patterns. I also show that the so called width law relating the position and the width of the bands w_n ~ x_n follows naturally from this theory.Comment: Talk presented at NATO Advanced Workshop on Statistical Physics Applied to Practical Problems (Budapest, May 1999); to appear in Physica A. 6 pages, 1 jpeg and 3 ps figure