10,613 research outputs found
Pointwise Green function bounds and stability of combustion waves
Generalizing similar results for viscous shock and relaxation waves, we
establish sharp pointwise Green function bounds and linearized and nonlinear
stability for traveling wave solutions of an abstract viscous combustion model
including both Majda's model and the full reacting compressible Navier--Stokes
equations with artificial viscosity with general multi-species reaction and
reaction-dependent equation of state, % under the necessary conditions of
strong spectral stability, i.e., stable point spectrum of the linearized
operator about the wave, transversality of the profile as a connection in the
traveling-wave ODE, and hyperbolic stability of the associated Chapman--Jouguet
(square-wave) approximation. Notably, our results apply to combustion waves of
any type: weak or strong, detonations or deflagrations, reducing the study of
stability to verification of a readily numerically checkable Evans function
condition. Together with spectral results of Lyng and Zumbrun, this gives
immediately stability of small-amplitude strong detonations in the small
heat-release (i.e., fluid-dynamical) limit, simplifying and greatly extending
previous results obtained by energy methods by Liu--Ying and Tesei--Tan for
Majda's model and the reactive Navier--Stokes equations, respectively
A stability index for detonation waves in Majda's model for reacting flow
Using Evans function techniques, we develop a stability index for weak and
strong detonation waves analogous to that developed for shock waves in
[GZ,BSZ], yielding useful necessary conditions for stability. Here, we carry
out the analysis in the context of the Majda model, a simplified model for
reacting flow; the method is extended to the full Navier-Stokes equations of
reacting flow in [Ly,LyZ]. The resulting stability condition is satisfied for
all nondegenerate, i.e., spatially exponentially decaying, weak and strong
detonations of the Majda model in agreement with numerical experiments of [CMR]
and analytical results of [Sz,LY] for a related model of Majda and Rosales. We
discuss also the role in the ZND limit of degenerate, subalgebraically decaying
weak detonation and (for a modified, ``bump-type'' ignition function)
deflagration profiles, as discussed in [GS.1-2] for the full equations.Comment: 36 pages, 3 figure
Stability analysis for combustion fronts traveling in hydraulically resistant porous media
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.Comment: 21 pages, 4 figure
Combustion waves in a model with chain branching reaction and their stability
In this paper the travelling wave solutions in the adiabatic model with
two-step chain branching reaction mechanism are investigated both numerically
and analytically in the limit of equal diffusivity of reactant, radicals and
heat. The properties of these solutions and their stability are investigated in
detail. The behaviour of combustion waves are demonstrated to have similarities
with the properties of nonadiabatic one-step combustion waves in that there is
a residual amount of fuel left behind the travelling waves and the solutions
can exhibit extinction. The difference between the nonadiabatic one-step and
adiabatic two-step models is found in the behaviour of the combustion waves
near the extinction condition. It is shown that the flame velocity drops down
to zero and a standing combustion wave is formed as the extinction condition is
reached. Prospects of further work are also discussed.Comment: pages 32, figures 2
On the One-dimensional Stability of Viscous Strong Detonation Waves
Building on Evans function techniques developed to study the stability of
viscous shocks, we examine the stability of viscous strong detonation wave
solutions of the reacting Navier-Stokes equations. The primary result,
following the work of Alexander, Gardner & Jones and Gardner & Zumbrun, is the
calculation of a stability index whose sign determines a necessary condition
for spectral stability. We show that for an ideal gas this index can be
evaluated in the ZND limit of vanishing dissipative effects. Moreover, when the
heat of reaction is sufficiently small, we prove that strong detonations are
spectrally stable provided the underlying shock is stable. Finally, for
completeness, the stability index calculations for the nonreacting
Navier-Stokes equations are includedComment: 66 pages, 7 figure
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