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An ignition-temperature model with two free interfaces in premixed flames†

Published on Nov 1, 2016in Combustion Theory and Modelling1.654
· DOI :10.1080/13647830.2016.1220625
Claude-Michel Brauner9
Estimated H-index: 9
(University of Bordeaux),
Peter V. Gordon8
Estimated H-index: 8
(University of Akron),
Wen Zhang1
Estimated H-index: 1
(Ha Tai: Xiamen University)
Abstract
In this paper we consider an ignition-temperature zero-order reaction model of thermo-diffusive combustion. This model describes the dynamics of thick flames, which have recently received considerable attention in the physical and engineering literature. The model admits a unique (up to translations) planar travelling wave solution. This travelling wave solution is quite different from those usually studied in combustion theory. The main qualitative feature of this travelling wave is that it has two interfaces: the ignition interface where the ignition temperature is attained and the trailing interface where the concentration of deficient reactants reaches zero. We give a new mathematical framework for studying the cellular instability of such travelling front solutions. Our approach allows the analysis of a free boundary problem to be converted into the analysis of a boundary value problem having a fully nonlinear system of parabolic equations. The latter is very suitable for both mathematical and numeri...
  • References (20)
  • Citations (2)
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References20
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#1Lina Hu (Ha Tai: Xiamen University)H-Index: 2
#2Claude-Michel Brauner (University of Bordeaux)H-Index: 9
Last. Gregory I. Sivashinsky (TAU: Tel Aviv University)H-Index: 29
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We consider a model of gas-solid combustion with free interface proposed by L. Kagan and G.I. Sivashinsky. Our approach is twofold: (I) we eliminate the front and get to a fully nonlinear system with boundary conditions; (II) we use a fourth-order pseudo-differential equation for the front to achieve asymptotic regimes in rescaled variables. In both cases, we implement a numerical algorithm based on spectral method and represent numerically the evolution of the char. Fingering pattern formation ...
3 CitationsSource
#1Alvin Bayliss (NU: Northwestern University)H-Index: 25
#2Erin M. Lennon (NU: Northwestern University)H-Index: 2
Last. Vitaly Volpert (NU: Northwestern University)H-Index: 27
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We consider the use of step-functions to model Arrhenius reaction terms for traveling wave solutions to combustion problems. We develop a methodology by which the Arrhenius reaction rate term is replaced by a suitably normalized step-function. The resulting model introduces interior interfaces and allows the conservation equations for energy and species to be solved explicitly within the subdomains bounded by the interfaces. The problem can then be reduced to a small number of nonlinear algebrai...
7 CitationsSource
#1Claude-Michel Brauner (University of Bordeaux)H-Index: 9
#2Lina Hu (Ha Tai: Xiamen University)H-Index: 2
Last. Luca LorenziH-Index: 12
view all 3 authors...
The authors consider a free interface problem which stems from a gas-solid model in combustion with pattern formation. A third-order, fully nonlinear, self-consistent equation for the flame front is derived. Asymptotic methods reveal that the interface approaches a solution to the Kuramoto-Sivashinsky equation. Numerical results which illustrate the dynamics are presented.
3 CitationsSource
We consider a quasi-steady version of the κ-θ model of flame front dynamics introduced in [FGS03]. In this case the mathematical model reduces to a single integro-differential equation. We show that a periodic problem for the latter equation is globally well-posed in Sobolev spaces of periodic functions. We prove that near the instability threshold the solutions of the equation are arbitrarily close to these of the Kuramoto–Sivashinsky equation on a fixed time interval if the evolution starts fr...
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Abstract We deal with a free boundary problem, depending on a real parameter λ , in a infinite strip in R 2 , which admits a planar travelling wave solution for every λ∈ R . We prove existence, uniqueness and regularity results for the solutions near the travelling wave.
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#1Claude-Michel Brauner (University of Bordeaux)H-Index: 9
#2Josephus Hulshof (LEI: Leiden University)H-Index: 19
Last. Alessandra Lunardi (University of Parma)H-Index: 21
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Abstract Every solution of a linear elliptic equation on a bounded domain may be considered as an equilibrium of a free boundary problem. The free boundary problem consists of the corresponding parabolic equation on a variable unknown domain with free boundary conditions prescribing both Dirichlet and Neumann data. We establish a rigorous stability analysis of such equilibria, including the construction of stable and unstable manifolds. For this purpose we transform the free boundary problem to ...
39 CitationsSource
#1I. Brailovsky (TAU: Tel Aviv University)H-Index: 10
#2Gregory I. Sivashinsky (TAU: Tel Aviv University)H-Index: 29
A reduced model for premixed gas filtration combustion where the nonlinear effects are discarded everywhere but in the reaction rate term and where the only accounted for effect of the porous medium is its resistance to the gas flow is explored. While ruling out formation of shock waves the model appears rich enough to cover detonation-like phenomenon with barodiffusion acting as a driving agency. It is shown that depending on the initial conditions this creeping detonation mode is evoked either...
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#1Joel H. Ferziger (Stanford University)H-Index: 42
#2Tarek Echekki (Stanford University)H-Index: 20
Abstract A simple reaction rate model which eliminates much of the non-linearity associated with the Arrhenius model is suggested. The model is applied to one-dimensional flame problems: the unstrained flame at unity and non-unity Lewis numbers and the strained flame at unity Lewis number. Exact expressions for the temperature and species profiles and consumption rates are obtained; in thestrained flame, these require numerical evaluation. The solutions demonstrate that the model agrees in all e...
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Cited By2
Newest
#1Davide Addona (University of Milan)H-Index: 2
#2Claude-Michel Brauner (University of Bordeaux)H-Index: 9
Last. Wen Zhang (CUTe: China University of Technology)H-Index: 1
view all 4 authors...
Abstract We study in a strip of R 2 a combustion model of flame propagation with stepwise temperature kinetics and zero-order reaction, characterized by two free interfaces, respectively the ignition and the trailing fronts. The latter interface presents an additional difficulty because the non-degeneracy condition is not met. We turn the system to a fully nonlinear problem which is thoroughly investigated. When the width l of the strip is sufficiently large, we prove the existence of a critical...
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Abstract In this paper we analyze the stability of the traveling wave solution for an ignition-temperature, first-order reaction model of thermal-diffusional combustion, in the case of high Lewis numbers ( Le > 1 ). In contrast to conventional Arrhenius kinetics where the reaction zone is infinitely thin, the reaction zone for stepwise temperature kinetics is of order unity. The system of two parabolic PDEs is characterized by a free interface at which ignition temperature Θ i is reached. We tur...
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