61 research outputs found
Stability of planar flames as gasdynamic discontinuities
No full textThe stability of a steadily propagating planar premixed flame has been the subject of numerous studies since Darrieus and Landau showed that in their model flames are unstable to perturbations of any wavelength. Moreover, the instability was shown to persist even for very small wavelengths, i.e. there was no high-wavenumber cutoff of the instability. In addition to the Darrieus-Landau instability, which results from thermal expansion, analysis of the diffusional thermal model indicates that premixed flames may exhibit cellular and pulsating instabilities as a consequence of preferential diffusion. However, no previous theory captured all the instabilities including a high-wavenumber cutoff for each. In Class, Matkowsky & Klimenko (2003) a unified theory is proposed which, in appropriate limits and under appropriate assumptions, recovers all the relevant previous theories. It also includes additional new terms, not present in previous theories. In the present paper we consider the stability of a uniformly propagating planar flame as a solution of the unified model. The results are then compared to those based on the models of Darrieus-Landau, Sivashinsky and Matalon-Matkowsky. In particular, it is shown that the unified model is the only model to capture the Darrieus-Landau, cellular and pulsating instabilities including a high-wavenumber cutoff for each
A Hopf bifurcation for a reaction-diffusion equation with concentrated chemical kinetics
No full textAbstractA reaction-diffusion equation related to some mathematical models of gasless combustion of solid fuel is studied. A formal bifurcation analysis by B. J. Matkowsky and G. I. Sivashinsky (SIAM J. Appl. Math. 35 (1978), 465–478) shows that solutions demonstrate behavior typical for the Hopf bifurcation. A rigorous treatment of this phenomenon is developed. The problem is considered as an evolution equation in a Banach space. To circumvent difficulties involving a possible resonance with the continuous spectrum, appropriate weighted norms are introduced. A suitable version of the Hopf bifurcation theorem is developed and the existence of time periodic solutions is proved for values of the parameter near some critical value
An adaptive pseudo-spectral method for reaction diffusion problems
No full textThe spectral interpolation error was considered for both the Chebyshev pseudo-spectral and Galerkin approximations. A family of functionals I sub r (u), with the property that the maximum norm of the error is bounded by I sub r (u)/J sub r, where r is an integer and J is the degree of the polynomial approximation, was developed. These functionals are used in the adaptive procedure whereby the problem is dynamically transformed to minimize I sub r (u). The number of collocation points is then chosen to maintain a prescribed error bound. The method is illustrated by various examples from combustion problems in one and two dimensions
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Studies in nonlinear problems of energy. Progress report, October 1, 1993--September 30, 1994
No full textThe authors concentrate on modeling, analysis and large scale scientific computation of combustion and flame propagation phenomena, with emphasis on the transition from laminar to turbulent combustion. In the transition process a flame passed through a stages exhibiting increasingly complex spatial and temporal patterns which serve as signatures identifying each stage. Often the transitions arise via bifurcation. The authors investigate nonlinear dynamics, bifurcation and pattern formation in the successive stage of transition. They describe the stability of combustion waves, and transitions to combustion waves exhibiting progressively higher degrees of spatio-temporal complexity. One aspect of this research program is the systematic derivation of appropriate, approximate models from the original models governing combustion. The approximate models are then analyzed. The authors are particularly interested in understanding the basic mechanisms affecting combustion, which is a prerequisite to effective control of the process. They are interested in determining the effects of varying various control parameters, such as Nusselt number, Lewis number, heat release, activation energy, Damkohler number, Reynolds number, Prandtl number, Peclet number, etc. The authors have also considered a number of problems in self-propagating high-temperature synthesis (SHS), in which combustion waves are employed to synthesize advanced materials. Efforts are directed toward understanding fundamental mechanisms. 167 refs
Singular perturbations in noisy dynamical systems
No full textConsider a deterministic dynamical system in a domain containing a stable equilibrium, e.g., a particle in a potential well. The particle, independent of initial conditions, eventually reaches the bottom of the well. If however, the particle is subjected to white noise, due, e.g., to collisions with a population of smaller, lighter particles comprising the medium through which the particle travels, a dramatic difference in the behaviour of the Brownian particle occurs. The particle will exit the well. The natural questions then are how long will it take for it to exit and from where on the boundary of the domain of attraction of the deterministic equilibrium (the rim of the well) will it exit. We compute the mean first passage time to the boundary and the mean probabilities of the exit positions. When the noise is small each quantity satisfies a singularly perturbed deterministic boundary value problem. We treat the problem by the method of matched asymptotic expansions (MAE) and generalizations thereof. MAE has been used successfully to solve problems in many applications. However, there exist problems for which MAE does not suffice. Among these are problems exhibiting boundary layer resonance, i.e., the problem of ‘spurious solutions’, which led some to conclude that this was ‘the failure of MAE’. We present a physical argument and four mathematical arguments to modify or augment MAE to make it successful. Finally, we discuss applications of the theory.</jats:p
Bifurcation and Stability Theory with Application to Problems of Combustion and Flame Propagation.
No full textA Variational Approach to Singularly Perturbed Boundary Value Problems for Ordinary and Partial Differential Equations with Turning Points
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Studies in nonlinear problems of energy. Progress report, January 1, 1992--December 31, 1992
No full textEmphasis has been on combustion and flame propagation. The research program was on modeling, analysis and computation of combustion phenomena, with emphasis on transition from laminar to turbulent combustion. Nonlinear dynamics and pattern formation were investigated in the transition. Stability of combustion waves, and transitions to complex waves are described. Combustion waves possess large activation energies, so that chemical reactions are significant only in thin layers, or reaction zones. In limit of infinite activation energy, the zones shrink to moving surfaces, (fronts) which must be found during the analysis, so that (moving free boundary problems). The studies are carried out for limiting case with fronts, while the numerical studies are carried out for finite, though large, activation energy. Accurate resolution of the solution in the reaction zones is essential, otherwise false predictions of dynamics are possible. Since the the reaction zones move, adaptive pseudo-spectral methods were developed. The approach is based on a synergism of analytical and computational methods. The numerical computations build on and extend the analytical information. Furthermore, analytical solutions serve as benchmarks for testing the accuracy of the computation. Finally, ideas from analysis (singular perturbation theory) have induced new approaches to computations. The computational results suggest new analysis to be considered. Among the recent interesting results, was spatio-temporal chaos in combustion. One goal is extension of the adaptive pseudo-spectral methods to adaptive domain decomposition methods. Efforts have begun to develop such methods for problems with multiple reaction zones, corresponding to problems with more complex, and more realistic chemistry. Other topics included stochastics, oscillators, Rysteretic Josephson junctions, DC SQUID, Markov jumps, laser with saturable absorber, chemical physics, Brownian movement, combustion synthesis, etc
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