Spectral Behavior of Combustion Fronts with High Exothermicity

Ghazaryan, Anna; Humpherys, Jeffrey; Lytle, Joshua

doi:10.1137/120864891

Cited by 9 publications

(13 citation statements)

References 43 publications

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“…The Evans function is an analytic function of the spectral parameter defined in some region of the complex plane. It was used for example in the context of nerve impulse equations [1,9,20,39], pulse solutions to the generalized Korteveg-de Vries, Benjamin-Bona-Mahoney and Boussinesq equations [28], multi-pulse solutions to reaction-diffusion equations [2,3], solutions to perturbed nonlinear Schrödinger equations [21,22,24], near integrable systems [25], traveling hole solutions of the one-dimensional complex Ginzburg-Landau equation near the Nonlinear-Schrödinger limit [23], and in the context of combustion problems [4,12,18]. The Evans function can be expressed in several different ways depending on the size of the stable and unstable manifolds of the linear system.…”

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confidence: 99%

Stability Analysis for Combustion Fronts Traveling in Hydraulically Resistant Porous Media

Ghazaryan¹,

Lafortune²,

McLarnan³

2015

SIAM J. Appl. Math.

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Abstract. We study front solutions of a system that models combustion in highly hydraulically resistant porous media. The spectral stability of the fronts is tackled by a combination of energy estimates and numerical Evans function computations. Our results suggest that there is a parameter regime for which there are no unstable eigenvalues. We use recent works about partially parabolic systems to prove that in the absence of unstable eigenvalues the fronts are convectively stable.Key words. traveling waves, combustion fronts, Evans function, spectral stability, nonlinear stability, partly parabolic systems 1. Introduction. This paper is devoted to the stability analysis of a traveling front in a combustion model. A combustion front is a coherent flame structure that propagates with a constant velocity. In a physical system it represents an intermediate asymptotic behavior of a nonstationary combustion wave [5]. To capture combustion waves, mathematical models are often cast as systems of partial differential equations (PDEs) posed on infinite domains, with multiple spatially homogeneous equilibria that correspond to cold states and completely burned states. Traveling fronts then are exhibited as a result of a competition of two different states of the system. To know the stability stability of a combustion front is of importance for the general understanding of the underlying phenomena, and has implications, for example, for chemical technology and fire and explosion safety. From the mathematical point of view, combustion models are of interest because they are formulated as systems of coupled nonlinear PDEs of high complexity and as such require refined mathematical techniques for their treatment.We consider here a system that is obtained from a model for combustion in highly hydraulically resistant porous media [35]. The original model [6,35] is considered to adequately capture the rich physical dynamics of the combustion in hydraulically resistant porous media and at the same time be approachable from the mathematical point of view. The model consists of a partly parabolic system of two PDEs coupled through a nonlinearity. Partly parabolic systems are systems where some but not all quantities diffuse. A reduced system can be obtained from the porous media combustion model suggested in [6,35] for a special value of the ratio of pressure and molecular diffusivities or a special value of the Lewis number, which is the ratio of the thermal diffusivity to mass diffusivity [15].The existence and uniqueness of combustion front solutions of the system have been studied extensively [8,11,16,17]. The stability of the fronts has not been studied yet to the authors' knowledge. We present here an attempt to provide the fullest stability analysis possible that would allow for physical interpretation of the results. Our strategy is to perform the stability study for the reduced system. This has the advantage of simplifying the computations. We then show that when the fronts are

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confidence: 99%

Stability Analysis for Combustion Fronts Traveling in Hydraulically Resistant Porous Media

Ghazaryan¹,

Lafortune²,

McLarnan³

2015

SIAM J. Appl. Math.

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“…However, we have the lingering issue of translation invariance: that the equation (2.2) is translation invariant and that the above projective boundary conditions do not fix a particular translate. To resolve this, we use a trick from [12] that allows us to specify a particular function value in the "middle" of the region in exchange for doubling the dimension of the problem. Specifically, that if (2.2) when written as a system is ∂ z y = F (y), then we introduce the variableỹ with ∂ zỹ = −F (ỹ) so rather than working on the region [−L, L], y is the solution on [0, L] andỹ is the solution running backwards on [−L, 0].…”

Section: Numerical Existence Studymentioning

confidence: 99%

On the dynamics of traveling fronts arising in nanoscale pattern formation

Johnson

Lyng

Smith

2020

Physica D: Nonlinear Phenomena

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We study the stability and dynamics of traveling-front solutions of a modified Kuramoto-Sivashinsky equation arising in the modeling of nanoscale ripple patterns that form when a nominally flat solid surface is bombarded with a broad ion beam at an oblique angle of incidence. Structurally, the linearized operators associated with these fronts have unstable essential spectrum-corresponding to instability of the spatially asymptotic states-and stable point spectrum-corresponding to stability of the transition profile of the front. We show that these waves are linearly orbitally asymptotically stable in appropriate exponentially weighted spaces. While the technical device of exponential weights allows us to accommodate the unstable essential spectrum of individual waves in our linear analysis, it does not shed light on the long-time pattern formation that is observed experimentally and in numerical simulations. To begin to address this issue, we consider a periodic array of unstable front and back solutions. While not an exact solution of the governing equation, this periodic pattern mimics experimentally observed phenomena. Our numerical experiments suggest that the convecting instabilities associated with each individual wave are damped as they pass through transition layers and that this stabilization mechanism underlies the pattern formation seen in experiments.

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“…We numerically compute these integrals to locate zeroes of the Evans function in a given domain. Generally, this technique is known as the method of moments [15,41,42]. However, since in our case we are only looking for one root at a time only, the use of the higher moments (where z is replaced by z n ) is not necessary.…”

Section: Linear Stabilitymentioning

confidence: 99%

“…Front solution of the system(15) in the case c = 1.8588, ε = 0.1, h = 0.3, σ = 0.25, δ = 0.0005, and T ign = 0.01, and Z = 6. The solid line corresponds to u, the dashed line to y, and the dotted line to z.…”

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confidence: 99%

Combustion waves in hydraulically resistant porous media in a special parameter regime

Ghazaryan

Lafortune

McLarnan

2016

Physica D: Nonlinear Phenomena

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In this paper we study the stability of fronts in a reduction of a well-known PDE system that is used to model the combustion in hydraulically resistant porous media. More precisely, we consider the original PDE system under the assumption that one of the parameters of the model, the Lewis number, is chosen in a specific way and with initial conditions of a specific form. For a class of initial conditions, then the number of unknown functions is reduced from three to two. For the reduced system, the existence of combustion fronts follows from the existence results for the original system. The stability of these fronts is studied here by a combination of energy estimates and numerical Evans function computations and nonlinear analysis when applicable. We then lift the restriction on the initial conditions and show that the stability results obtained for the reduced system extend to the fronts in the full system considered for that specific value of the Lewis number. The fronts that we investigate are proved to be either absolutely unstable or convectively unstable on the nonlinear level.

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Spectral Behavior of Combustion Fronts with High Exothermicity

Cited by 9 publications

References 43 publications

Stability Analysis for Combustion Fronts Traveling in Hydraulically Resistant Porous Media

Stability Analysis for Combustion Fronts Traveling in Hydraulically Resistant Porous Media

On the dynamics of traveling fronts arising in nanoscale pattern formation

Combustion waves in hydraulically resistant porous media in a special parameter regime

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