\makeatletter
\@ifundefined{HCode}
{\documentclass[screen,CRMECA,Unicode,published]{cedram}
\newenvironment{noXML}{}{}%
\def\tsup#1{$^{{#1}}$}
\def\tsub#1{$_{{#1}}$}
\def\sfrac#1#2{{#1}/{#2}}
\def\thead{\noalign{\relax}\hline}
\def\endthead{\noalign{\relax}\hline}
\def\tbody{\noalign{\relax}\hline}
\def\xtbody{\noalign{\relax}}
\def\tabnote#1{\vskip4pt\parbox{.51\linewidth}{#1}}
\def\xxtabnote#1{\vskip4pt\parbox{.65\linewidth}{#1}}
\def\mathbi#1{\boldsymbol{#1}}
\def\xmathop#1{\mathop{\text{#1}}}
\def\rmDelta{\Delta}
\usepackage{etoolbox}
\usepackage{upgreek}
\csdef{Seqnsplit}{\\}
\def\botline{\\\hline}
\newcounter{runlevel}
\usepackage{multirow} 
\def\nrow#1{\@tempcnta #1\relax%
\advance\@tempcnta by 1\relax%
\xdef\lenrow{\the\@tempcnta}}
\def\morerows#1#2{\nrow{#1}\multirow{\lenrow}{*}{#2}}
\def\xmorerows#1#2{#2}
\let\MakeYrStrItalic\relax
\def\jobid{crmeca20230756}
%\graphicspath{{/tmp/\jobid_figs/web/}} 
\graphicspath{{./figures/}}
\def\0{\phantom{0}}
\def\rmmu{\upmu}
\def\rmpi{\uppi}
\def\mn{\phantom{$-$}}
\def\refinput#1{}
\def\back#1{}
\newcommand{\bnabla}{\boldsymbol{\nabla}}
\newcommand{\bgamma}{\boldsymbol{\gamma}}
\newcommand{\bdelta}{\boldsymbol{\delta}}
\newenvironment{inftab}{}{}
\def\Custref#1#2{\hyperlink{#2}{#1}}
\def\og{\guillemotleft}
\def\fg{\guillemotright}
\def\tsqrt#1{\sqrt{\smash[b]{#1}}}
\makeatletter
\g@addto@macro{\UrlBreaks}{\UrlOrds}
\gappto{\UrlBreaks}{\UrlOrds}
\makeatother
\DOI{10.5802/crmeca.232}
\datereceived{2023-08-29}
\daterevised{2023-10-09}
\dateaccepted{2023-10-09}
\ItHasTeXPublished
}
{\documentclass[crmeca]{article}
\usepackage[T1]{fontenc} 
\usepackage{upgreek}
\def\CDRdoi{10.5802/crmeca.232}
\def\newline{\unskip\break}
\def\xmorerows#1#2{\morerows{#1}{#2}}
\def\sfrac#1#2{{#1}/{#2}}
\def\xxtabnote#1{\tabnote{#1}}
\def\xmathop#1{\mathop{{#1}}}
\def\xtbody{\tbody}
\def\hypertarget#1#2{\label{#1}}
\def\tsqrt{\sqrt}
}
\makeatother

\dateposted{2023-12-04}
\begin{document}

\begin{noXML}

\TopicEF{Research article}{Article de recherche}

\title{Intrinsic transition mechanism to detonation of gaseous laminar
flames in tubes}

\alttitle{M\'ecanisme intrins\`eque de la transition vers la
d\'etonation des flammes laminaires en phase gazeuse dans les tubes}

\author{\firstname{Paul} \lastname{Clavin}\CDRorcid{0000-0001-9614-3425}}
\address{Aix Marseille Universit\'e, CNRS, Centrale Marseille, IRPHE,
UMR7342, 49 rue Joliot Curie, BP 146, 13384 Marseille Cedex 13, France}
\email{paul.clavin@univ-amu.fr}

\thanks{Agence National de la Recherche (contract ANR-18-CE05-0030)}

\begin{abstract} 
The deflagration-to-detonation transition (DDT) on the tip of an
elongated flame in a tube is analyzed in the double limit of large
activation energy and small Mach number of laminar  flames.  A
spontaneous transition of a self-accelerated laminar flame taking the
form of a dynamical saddle-node bifurcation of the flow inside the
inner structure of the laminar flame is exhibited by the asymptotic
analysis. The predicted critical conditions for the finite-time 
pressure runaway are in good agreement with the experimental data of
the DDT onset in tubes.
\end{abstract}

\begin{altabstract} 
La transition d\'eflagration--d\'etonation (DDT) sur le bout arrondi
d'une flamme allong\'ee (en forme de doigt) dans un tube est analys\'ee
dans la double limite d'une grande \'energie d'activation et d'un petit
nombre de Mach des flammes laminaires.  L'analyse asymptotique met en
\'evidence une transition spontan\'ee de la flamme auto-acc\'el\'er\'ee
sous l'effet d'une bifurcation noeud-col de l'\'ecoulement gazeux à
l'int\'erieur de la structure interne de la flamme laminaire. 
L'analyse pr\'edit des conditions critiques d'emballement en temps fini
de la pression qui sont en bon accord avec les donn\'ees
exp\'erimentales de la DDT des flammes laminaires dans les tubes.
\end{altabstract}

\keywords{\kwd{Deflagration-to-detonation transition}
\kwd{Asymptotic analysis} 
\kwd{Finite time singularity}
\kwd{Dynamical saddle-node bifurcation}}

\altkeywords{\kwd{Transition d\'eflagration d\'etonation}
\kwd{Analyse asymptotique}
\kwd{Singularit\'e en temps fini}
\kwd{Bifurcation dynamique noeud-col}}

\maketitle

{\vspace*{-6pt}}

\end{noXML}

\section{Introduction}

The deflagration-to-detonation transition (DDT) is a
quasi-instantaneous transition (few microseconds) between two opposite
regimes of combustion wave.   The phenomenon was observed long
ago~\cite{shcheltrosh}. However, \pagebreak  after more than a century of
experimental works and decades of numerical studies reported in an
extensive literature, DDT is not yet
understood~\cite{lee2008,Clavin2016}. Gaseous detonations are
supersonic combustion waves involving a  pressure rise $\Delta p/p$
ranging from 20 to 50 and propagating with a velocity $\mathcal D$ 
between 2000 m/s  and 3500 m/s under normal conditions. By
contrast, laminar flames are quasi-isobaric reaction--diffusion waves 
characterized by a markedly subsonic velocity and a negligible pressure
drop $\Delta p/p\ll 1$. No other quasi-steady combustion waves exist
between these two extreme propagation regimes.  The abrupt transition
is explained here by a finite time singularity of the one-dimensional
reacting flow associated with a self-accelerated flame produced by an
extension increase in elongation of the multidimensional flame front.

Coupling reaction--diffusion waves with compressible flows is a
challenging problem that has been recently solved analytically by an
asymptotic analysis~\cite{Clavin2023}. For the sake of
self-consistency, this analysis is reproduced in 
Sections~\ref{formul}--\ref{QS}
with additional comments in Section~\ref{formul} on the
compatibility of the basic assumptions of the asymptotic analysis and
the conditions of the DDT experiments. Moreover, an original stability
analysis is presented in Section~\ref{stabanal0}. The 
Sections~\ref{formul}--\ref{QS}
are technical and non easy to follow by non specialists of
flame theory. However the conditions of the analysis discussed in
Section~\ref{formul} and their relevance for the experiments can be easily
understood as follows. The asymptotic analysis is based on two limits:
a large thermal sensitivity of the reaction rate $\beta \gg 1$ and a
small Mach number of laminar flames $\varepsilon \ll 1$. The first
limit leads to concentrate the chemical heat release in a reaction
sheet at high temperature while the second one leads to a multiple
scale flow; the length scale of the compressible  flows outside the
quasi-isobaric flame structure is much larger than the flame thickness
by a factor $1/\varepsilon$, so that, to leading order, the pressure is
quasi-uniform across the flame thickness and its compression-induced
time variation is mall of order $\varepsilon$. The key points of the
asymptotic analysis are a distinguished limit $\varepsilon \beta$ of
order unity $\varepsilon \beta=O(1)$ and an elongation rate smaller
than the inverse of the transit time across the flame, $\epsilon \ll
1$. The condition  $\varepsilon \beta=O(1)$ allows to study the whole
compression-induced dynamics for small variations of the flame
temperature, of order $1/\beta$. For a small elongation rate, assuming
the ordering $\varepsilon \ll \epsilon \ll 1$, the compression-induced
unsteadiness is described by a perturbation analysis keeping the
pressure uniform in the flame structure. Moreover assuming a flame
thickness $d$ smaller than the tube radius $r$, $d/r\ll 1$ of order
$\varepsilon$, $d/r=O(\varepsilon)$, the flame front around the point
at which the DDT onset occurs is quasi-planar because the pressure
propagates in the transverse direction faster than the longitudinal
dynamics. 

After reading the background in Section~\ref{backg} and a quick look to
the formulation of the problem in Sections~\ref{dislim}
and~\ref{V1dapprox}, the physics-oriented readers can skip
Sections~\ref{match}--\ref{stabanal0} to go directly to
Section~\ref{NLdyn} where the nonlinear dynamics is discussed. There, a
dynamical saddle-node bifurcation of the reacting flow is identified,
exhibiting  an intrinsic one-dimensional mechanism of  DDT. 

\section{Background}\label{backg}

The basic properties of combustion waves are well known for a long
time~\cite{Clavin2016}. They  are briefly recalled below with a
particular attention paid to the order of magnitude of the physical
mechanisms involved in the DDT. The existing experimental results and
the previous analyses that are relevant for the present analysis are
also recalled at the end of this section.

\subsection{Planar laminar flames}

 In gas, the laminar flame velocity relative to the unburned gas $U_L$
ranges from 10 cm/s in weakly energetic gaseous mixture to 10 m/s in
the most energetic ones. The flame temperature $T_b$ is in the range
1800--3100 K and the reaction rate in a flame is typically 
$1/\tau_{r}\approx 10^{6}~\mathrm{s}^{-1}$. The chemical kinetic network of
gaseous combustion is complex explaining why combustion cannot proceed 
below a crossover temperature $T_c\in$ [850--1200~K]~\cite{SaFaw}. 
However, at high temperature $T>1500~\mathrm{K}$ the flame dynamics can be
described with a good accuracy by a reaction rate governed by inelastic
binary collisions of molecules of reactant whose collision energy is
larger than an activation energy $\mathcal E$ much larger than the
thermal agitation $k_BT$.  Typically, the reduced activation
energy\footnote{In real combustible mixtures governed by a complex
chemical network of elementary reactions, the activation energy is
replaced by the thermal sensitivity of the exothermic reaction
rate~\cite{Clavin2021}.} $\beta\equiv \mathcal E/k_B T_b> 1$  is
between 2 in energetic mixtures (stoichiometric $\mathrm{H}_2/\mathrm{O}_2$ or
$\mathrm{C}_2\mathrm{H}_2/\mathrm{O}_2$ mixtures in which the flame temperature is large $T_b>
3000$~K)  and 8  in hydrocarbon--air flames ($T_b\approx 2000$~K).
According to the kinetic theory of gas, the reaction rate $1/\tau_r$ at
high temperature is expressed  in terms of the collision frequency
$1/\tau_{\mathrm{coll}}$ in the form of an Arrhenius law 
$\tau_{\mathrm{coll}}/\tau_r\propto  \beta\text{e}^{-\beta}$. The 
Zeldovich--Frank-Kamenetstskii (ZFK) analysis~\cite{ZFK} in the limit
$\beta\gg 1$ shows that  the laminar flame velocity relative to the
burned gas $U_b$ is expressed in terms of the molecular diffusivity
$D\approx a^2 \tau_{\mathrm{coll}}$ ($a$ is the sound speed) and the reaction
rate in the form $U_b\propto \beta^{-3/2}\sqrt{D_b/\tau_{rb}}\approx
a_b\text{e}^{-\beta/2}/\beta$ where the subscript b refers to the burned
gas, see~\cite{Clavin2016} for a didactic presentation. The flame
thickness $d$ and the transit time $t_{b}$ of a fluid particle across 
the flame are respectively $d=D_b/U_b\approx 2-4\times10^{-1}$~mm and
$t_{b}=d/U_b\propto \beta^{3}\tau_{rb} \approx 10^{-4}$~s under normal
conditions.  Due to a large activation energy, the flame Mach number 
$\varepsilon\equiv U_b/a_b \propto \text{e}^{-\beta/2}/\beta$ is small,
about $10^{-3}$ in hydrocarbon--air mixtures and $10^{-2}-10^{-1}$ in
the most energetic mixtures. In any cases, the   pressure drop across
the flame is negligible   $\Delta p/p\approx \varepsilon ^{2}$.  Planar
flames are unstable against transverse disturbances (Darrieus--Landau
instability), however, as shown by Zeldovich, the instability does not
concern the tip of a curved flames in tubes, see~\cite{Clavin2016} for
a review.

\subsection{Gaseous detonations} 

Detonations consist of a strong inert shock of few mean free paths
thick (${\approx} a \tau_{\mathrm{coll}}$) followed by a macroscopic zone of
reaction. The flow of compressed gas is subsonic relatively to the
shock but not smaller than $a_N/10$, the subscript $N$ denoting the
state of compressed gas adjacent to the shock (Neumann state).  The
diffusive transports are negligible  across  the reaction zone
(thickness $\approx a_N \tau_{rN}/10$) because the ratio
$\tau_{rN}/\tau_{\mathrm{coll}}$ is a large number, typically
$\tau_{rN}/\tau_{\mathrm{coll}}>10^3$.  A condition for the existence of a
detonation is that the inert shock is strong enough for making the
Neumann gas temperature $T_N$ larger than the crossover temperature,
$T_N>T_c$ so that  the detonation Mach number is always substantially
larger than unity $M\equiv \mathcal D/a_o> 4.5$, $a_o$ denoting the
sound speed ahead of the shock. Moreover, independently from chemical
kinetic considerations, the conservation of mass, momentum and energy
implies that the Mach number of a planar detonation in steady state
cannot be smaller than a lower bound called Chapman--Jouguet (CJ) $
M_{CJ}\approx 5$. The marginal CJ detonation is a self-propagating
regime characterized by a sonic condition in the burned gas. All the
other regimes $M>M_{CJ}$, called overdriven detonations,  are
piston-supported with a subsonic burned gas flow (relatively to the
lead shock).

\subsection{Experimental data. Unexplained results}

Several DDT experiments have been performed in long smooth-walled tubes
filled with  stoichiometric $\mathrm{H}_2/\mathrm{O}_2$ or
$\mathrm{C}_2\mathrm{H}_4/\mathrm{O}_2$ mixtures. We limit the
attention to those performed under the conditions considered in the
present analysis. The reactive mixture is ignited on a closed end of
the tube which is long enough for the DDT onset to occur before
reflected shocks interact with the flame. In micro-scale tubes (radius
$\approx$ few mms) the front of the laminar flame just before
transition is elongated and the flow is laminar~\cite{Wu10} see 
Figure~\ref{fig.1}. The DDT is also observed after the middle of the
channel in an open-ended tubes~\cite{Bikov22}, this case is not
considered here, even though the DDT mechanism is not so different.
The DDT is more difficult to analyze in larger tubes because the
wrinkled flame front can become turbulent~\cite{Opp66}, the problem
becoming stochastic in nature. Conclusions can nevertheless be drawn 
since  the salient features of the abrupt transition  in moderately
large tubes are not so different,  especially  as the flow stays
laminar ahead of the wrinkled flame front~\cite{Lieb10,Kuz10}.  Salient
features of DDT were pointed out in the sixties by the pioneering
experiments of Oppenheim and co-workers~\cite{Opp66}: the spontaneous
detonation onset is a local phenomenon concerning small explosion
centers. In some turbulent flames the explosion center is located in
the boundary layer~\cite{Opp66}. In such cases, the DDT is more likely
associated with the Zeldovich  gradient mechanism~\cite{Zeldo1980}
reinforced by compressible effects~\cite{Clavin2016}. This case is not
considered here. The attention is focused on the transition of laminar
flames accelerated by a self-generated convection flow as in  the 2010
experiments and numerical simulations~\cite{Lieb10,Kuz10} in which the
flow of unburned gas is laminar and the explosion center is located on
the flame front outside the boundary layer. The flow considered in the
analysis is sketched in Figure~\ref{fig.1}.

\begin{figure}[t!]
\includegraphics{fig01}
\caption{\label{fig.1}Sketch of an elongated flame in a smooth-walled
tube.}
\end{figure}

In  tubes of moderate transverse size (few cm in diameter) the
non-dimensional surface area (reduced by the cross section surface of
the tube) $\Sigma(t)>1$ of a elongated flame  first increases
exponentially up to  the transient formation of a tulip-shaped
flame~\cite{Searby96}. Then, the finger shaped front is restored
quickly  and $\Sigma(t)$ increases again but more slowly up to an
abrupt detonation onset~\cite{Lieb10,Kuz10}. In micro-scale tubes 
there is no transitory formation of tulip flame; the finger flame keeps
an elongated form with a growing length before an eventual DDT
onset~\cite{Wu10}.  Considering a quasi-steady flow of burned gas
sketched in Figure~\ref{fig.1} and assuming a laminar flame speed $U_b$
approximately constant all over the flame surface, the mass
conservation leads to a speed of the tip in the laboratory frame  in a
form extensively used in turbulent flame theory
{\begin{eqnarray}
 U_P\approx \Sigma U_b.
\label{UPSigUb}
\end{eqnarray}}\noindent
The increase in  surface-area $\Sigma >1$ produces a longitudinal flow
of burned gas with which the leading edge of the flame front is
convected at the velocity $U_P$ in (\ref{UPSigUb}), see
Figure~\ref{fig.1}. Acceleration of the convection flow by the rate of
surface-area increase  ($\text{d}\Sigma /\text{d}t>0$) accelerates the
flame on the tip. Generally speaking, the detonation  onset occurs
spontaneously when appropriate local conditions are attained in the
unburned gas adjacent to the leading edge of the self-accelerating
flame~\cite{lee2008}. This occurs in the 
experiments~\cite{Lieb10,Kuz10} when the overall speed of the flame on
the tip reaches a critical value $U_P^*=\Sigma^*U^*_b$ which is
comparable to the local sound speed
$a^*_b$~\cite{Bikov22,Lieb10,Kuz10}.\break The critical elongation $\Sigma^*$
is typically in the range 5--10.  Therefore, according to the
experimental data showing that $U_P^*$ is close to $a^*_b$, a large
laminar flame velocity is involved at the critical condition $U^*_b
\approx a^*_b/\Sigma^*$.  Such a large $U_b$ can be obtained  by the
small compressional heating of  downstream-running compression waves 
as long as the thermal sensitivity of the laminar flame  is large
enough.\footnote{In very energetic mixtures the thermal sensitivity
includes the temperature power laws in the pre-factor  of the Arrhenius
factor in the expression of the laminar flame
velocity~\cite{Clavin2021}.}  However,  as a rule of thumb, the
temperature of the compressed unburned gas stays well below the
crossover temperature $T_c$ so that the mixture just ahead of the flame
remains chemically frozen.  

Therefore, DDT can be explained neither by the Zeldovich gradient
mechanism~\cite{Zeldo1980} (since self-ignition  cannot occur) nor  by
a pressure pulse of downstream running compression waves in phase with
the rate of heat release. The latter mechanism would require a sonic
condition $U_b=a_b$  which can never be fulfilled in laminar
flames,\footnote{The sonic condition  defining the so-called \textit{CJ
deflagration} is sometimes referred to as a possible DDT mechanism of
turbulent flames~\cite{lee2008}. The author of the present article does
not share this point of view since a turbulent flame brush is
constituted by elements of laminar flames that are accelerated by a
self-generated convection flow.} no matter how large the surface area
of the elongated front $\Sigma$ and the tip velocity $U_P$.  The
objective of the present analysis is to explain the detonation onset in
the 2010 experiments and numerics~\cite{Lieb10,Kuz10} of laminar flames
in tubes.

\subsection{Previous analyses}

The previous one-dimensional analyses that are useful for the analysis
presented in this article are summarized below.

\subsubsection{First analyses (1989)--(2014): Self similar solutions and
1-D numerical simulations}

A key mechanism underlying the DDT was identified in 1989 by a
pioneering analysis~\cite{Joulin89} which was overlooked during  25
years. Treating a wrinkled flame as a planar discontinuity propagating
in a tube from a closed end at a velocity $U_P$ equal to the laminar
flame velocity multiplied by a wrinkling factor $s>1$,  $s$ being
equivalent to $\Sigma$ in (\ref{UPSigUb}), $U_P=s U_b$, Deshaies and
Joulin~\cite{Joulin89} considered the self-similar solutions
constituted by a lead shock followed by the wrinkled flame. In such
solutions the two planar discontinuities (flames and lead shock)
propagate at constant velocity (subsonic relatively to the gas for the
flame and supersonic for the shock). Deshaies and Joulin showed that
the self-similar solutions no longer exist above a critical value  of
the wrinkling factor $s^*$ close 10 in ordinary condition. This is due
to a nonlinear thermal feed back loop between the flame and the lead
shock: the laminar flame velocity $U_b$ increases strongly with the gas
temperature which increases with the strength of the shock, the later
increasing in turn  with $U_b$ through the flow induced upstream of the
flame by the density jump across the flame. The critical turning point
$s=s^*\approx 10$ corresponds to a gas temperature well below $T_c$. 
Nevertheless, unsteady numerical solutions of planar flames show a
runaway of the solution few time after a quasi-isobaric ignition on the
closed end of a tube provided the reaction rate is artificially
increased by a factor equal to (or larger than) $s^{*2}\approx
10^2$~\cite{Kagan2017}  so that the laminar flame velocity is ten times
larger than the usual one obtained with an Arrhenius law and the
molecular diffusive transport that are controlled by the kinetic theory
of gas. Even though the flame model is not satisfactory from a physical
point of view, these numerical solutions are instructive since they
illustrate an intrinsic DDT mechanism  of a subsonic combustion waves.

\subsubsection{Quasi-steady one-dimensional back-flow model. Recent analytical studies 2021--2023}

Motivated by the works~\cite{Joulin89,Kagan2017}, a one-dimensional
model~\cite{Clavin2021}  was recently developed and studied
analytically to elucidate an intrinsic DDT mechanism of planar flame
sustained by a reaction rate compatible with the  kinetic theory of
gas. The key mechanism is the longitudinal convective motion of the
flame by the self-induced  flow of burned gas, called back-flow
in\break \cite{Clavin2021,Clavin2022bis,Clavin2022}. The
latter results from the combustion of the lateral part of the flame
front quasi-parallel to the tube wall, see Figure~\ref{fig.1}. The tip of
the finger-shaped flame is  considered as a planar flame perpendicular
to the axial flow of burned gas impinging the flame from behind with
the velocity $u_b$. The resulting speed of the flame in the laboratory
frame is $U_P=u_b+U_b$, the subscript $P$ being for piston since the
flame acts as a semi-transparent piston. Following~\cite{Searby96} when
the gas  behind the foot of the finger flame  is at rest as in Figure~\ref{fig.1}
(closed end), the back-flow $u_b$ is proportional  to the surface area
$\mathcal S$ of the lateral part of the flame front. Considering the
elongated flame as quasi-cylindrical and introducing the elongation
parameter $S=\mathcal S/\mathcal S_0$ where  $\mathcal S_0$ is the
cross section area of the tube, one gets, using the same approximations
as in (\ref{UPSigUb}), $u_b \approx S U_b$, $S>0$, leading to
$U_P=u_b+U_b\approx \Sigma U_b$  with
{\begin{eqnarray}
\Sigma = S+1\geqslant 1, \quad U_P= \Sigma U_b, \quad u_b \approx SU_b.
\label{UbS}
\end{eqnarray}}\noindent
The key point is that the speed of the flame front $U_P$ (and thus the
flow $u_b$) is proportional to the laminar flame velocity $U_b$. This
is responsible for a thermal feed-back loop similar to that described
long ago~\cite{Joulin89}, leading to the same phenomenology discussed
now.  If the inner structure of the laminar flame  is in steady state,
the flow of unburned and burned gas adjacent to the flame, respectively
$u_u$ and $u_b$, are $u_u=U_P-U_L$ and $u_b=U_P-U_b$ where $U_L$ and
$U_b$ are the laminar flame velocity relative to the unburned and
burned gas respectively. According to the isobaric condition and the
mass conservation, the ratio $U_b/U_L=1/(1-q)$ is given by the reduced
heat release $q\equiv Q/(c_pT_b)<1$ that controls the increase of flow
velocity across the flame $u_u-u_b=U_b-U_L=qU_b$.  Considering the
flame as a semi-transparent piston ($u_u<U_P$), it was
shown~\cite{Clavin2021} that, as in the pioneering
analysis~\cite{Joulin89}, the self-similar solutions $U_P(\Sigma)$ have
a turning point namely a maximum of the curve $\Sigma(U_P)$ for a
critical elongation $\Sigma=\Sigma^*$  above which  there is no more
self similar solution and below which there are two branches of
self-similar solutions $\Sigma<\Sigma^*$:
$U^-_P(\Sigma)<U^+_P(\Sigma)$,
$U^+_P(\Sigma^*)=U^-_P(\Sigma^*)\equiv U^*_P$ such that $\text{d}
\Sigma/\text{d}U_P^->0$ (physical branch), $\text{d}\Sigma/\text{d}
U_P^+<0$ and $\text{d}\Sigma/\text{d}U_P^{\pm}\vert_{U_P^*}=0$. It was
also shown~\cite{Clavin2021} that the turning point ($\Sigma^*$,
$U^*_P$)  corresponds with a relatively good accuracy to the DDT onset
in the 2010 experiments~\cite{Lieb10,Kuz10}.

The basic assumption of self-similar solutions, namely a uniform flow
of unburned gas between the flame and the lead shock,  is a major
weakness of such solutions. The unsteady and non uniform compression
waves $u_{\mathrm{ext}+}(\text{x}, t)$ that are generated in the unburned gas by
the accelerating flame have been considered for a long time as an
essential ingredient of the gaseous DDT in tubes~\cite{lee2008}. Still
treating the flame as a semi-transparent piston, the problem was
extended recently to take into account the unsteady compression waves
in the unburned gas ahead of the
flame~\cite{Clavin2022bis,Clavin2022}. The so-obtained solutions
have the same structural form as the self-similar solutions: there is
two branches of solutions  $U_P^{\pm}(\Sigma)$ merging at a turning
point ($\Sigma^*$, $U^*_P$). However, the latter is now depending on
the initial condition  and presents an upper bound.  Starting from a
self-similar solution and assuming that the solution follows the
physical branch of the  self-similar solutions $U^-_P(\Sigma)$ when the
elongation  increases slowly $\Sigma(t)$, $(1/\Sigma)\text
d\Sigma/\text{d}t\ll 1/t_b$, the flame acceleration diverges at the
turning point  $\text{d}U^-_P/\text{d}
\Sigma\vert_{\Sigma=\Sigma^*}=\infty$. This structural property of the
turning point $\Sigma=\Sigma^*$ ($p=p^*$, $T_b=T_b^*$, $U_b=U^*_b$)
implies  that the gradient of the unsteady flow of unburned gas
diverges on the flame front when the critical elongation is
reached~\cite{Clavin2021,Clavin2022bis,Clavin2022}. Even
though no runaway  of  pressure and temperature is described for an
accelerating flame  considered as a semi-transparent piston, the finite
time singularity of the flow suggests the shock formation inside the
flame structure that could explain the detonation onset. 

The flame dynamics in~\cite{Clavin2022bis,Clavin2022} is based
on two assumptions: firstly, both burned gas flow and inner flame
structure are assumed in steady state and secondly, these quasi-steady
solutions are assumed stable. Here, the stability limits of the
quasi-steady solutions are analyzed. Moreover, removing the
quasi-steady assumptions, the non linear dynamics is studied in the
stable domain for an elongation increasing with the time. By contrast
with the previous results, a finite-time runaway of the pressure
responsible for the detonation onset is exhibited.

\section{Formulation}\label{formul}

We consider a quasi-planar geometry on the tip of the elongated flame
sketched in Figure~\ref{fig.1} and an axial back-flow $u_b$ on the hot
side of the flame as before.  We will not discuss the origin of the
increase in surface area $\Sigma(t)$ responsible for the acceleration
of the tip $\text{d}U_p/\text{d}t>0$. The elongation of a laminar flame
in a tube with stick walls is due to the stretching of the flame front 
which is anchored inside the boundary layer on the tube wall while the
flow in the bulk  is large. The elongation is governed by the
quasi-isobaric expansion of the gas across the flame front while the
abrupt DDT is a consequence of a quite different mechanism, namely the
compression waves in the external flows.  In the following, the
attention is focused on the second mechanism and the dynamics is
analyzed for a small growth rate of elongation  considered as a given
parameter. The initial condition (labelled $i$) is constituted by a
self-similar solution whose lead shock is at infinity; the initial back
flow $u_{bi}$, laminar flame speed $U_{bi}$, flame temperature $T_{bi}$
and flame speed in the laboratory frame $U_{Pi}=u_{bi}+U_{bi}$ are
constant before the elongation starts to increase, $t \leqslant 0$ and
the flow of unburned gas ahead of the flame  $u_{\mathrm{ext}+}$ is initially
constant and uniform  
{\begin{eqnarray}
t = 0: \quad u_b=u_{bi}, \quad U_b= U_{bi}, \quad U_P=U_{Pi}, \quad u_{\mathrm{ext}+}=u_{ui} 
\end{eqnarray}}\noindent
with, according to the mass conservation across a steady isobaric
flame, 
{\begin{eqnarray}
u_{ui}\equiv u_{bi}+qU_{bi}<U_{Pi}, \quad 0<U_{Pi}-u_{ui}=(1- q)U_{bi}<U_{bi} \quad \text{where }  q\equiv Q/c_pT_{bi}<1.
\label{uub}
\end{eqnarray}}\noindent
Denoting the initial elongation $S_i>0$,  we consider a time dependent
elongation in the form
{\begin{eqnarray}
S(t)=[1+\epsilon t/t_b]S_i,
\label{elongrate}
\end{eqnarray}}\noindent
the new parameter $\epsilon$ (not to be confused with
$\varepsilon\equiv U_b/a_b\ll1$) characterizing the growth rate of
elongation relative to the inverse of the transit time across the
planar flame $t_b$.  

\subsection{Distinguished limits}\label{dislim}

The analysis of the flow sketched in Figure~\ref{fig.1} is performed  in
the limit of small flame Mach number $\varepsilon\equiv U_b/a_b\ll1$
using the ZFK flame model~\cite{ZFK} in the limit $\beta\gg 1$ for
which the thickness of the reaction zone $d/\beta$ shrinks to zero. 
The same distinguished limit of large thermal sensitivity $\beta\gg 1$
and small Mach number of the laminar flame $\varepsilon\ll 1$ as  in
the previous analyses~\cite{Joulin89,Clavin2022} is used here
in the asymptotic study of the dynamics associated with
(\ref{elongrate})  
{\begin{eqnarray}
\varepsilon \to 0,\quad \beta\to \infty{:}\quad (\gamma-1)\,\beta \,\varepsilon\,S_i=O(1), \quad S^*/S_i>1, \quad S^*/S_i=O(1). 
\label{distlimit1}
\end{eqnarray}}\noindent
The critical elongation of the quasi-steady solutions, obtained  using
the ZFK expression\break $\varepsilon \equiv U_b/a_b\approx 
\text{e}^{-\beta/2}/\beta$, is not larger than $0.05$.  A key point in the
range of parameters (\ref{distlimit1}) is that  small variations of
acoustic pressure (of order $\varepsilon$) induce variations of back
flow $u_b$ and laminar flame speed $U_b$ of order unity.  The analysis
takes advantage of two simplifications: firstly, a negligible pressure
drop across the flame $\Delta p/p\approx \varepsilon^2 $  and secondly
a scale separation of the flow  discussed lengthly in
Section~\ref{2ls}. The acceleration-induced variation of pressure is of
order $\varepsilon$ so that the $\varepsilon$-terms are retained in the
perturbation analysis while the $\varepsilon^2$-terms are neglected.  

\subsection{Validity of the one-dimensional approximation}\label{V1dapprox}

The one-dimensional model is relevant under conditions that are
discussed now. For a curvature radius of the tip of the same order of
magnitude as the tube radius $r$, the transverse fluxes inside the
flame structure are negligible in front of the compressional heating
when the condition $D/r^2\approx U_b d/r^2\ll \varepsilon/t_b\approx
\varepsilon U_b/d$ is fulfilled  so that the curvature effects on the
flame are negligible in the perturbation analysis retaining
$\varepsilon$-terms. This leads to the condition $(d/r)^2\ll
\varepsilon$.

Another condition is required for making the the compression waves
quasi-planar and propagating in the longitudinal direction; the time to
make the pressure homogeneous in the transverse direction should be
smaller than the characteristic time of evolution. Using 
(\ref{elongrate}) $\text{d}S/\text{d}t= \epsilon S_i/t_b$, this
condition  can be written $ \epsilon S_i/S\ll (a/r)t_b =
(a/U_b)(d/r)=(1/\varepsilon) d/r$ yielding $\epsilon\ll (1/\varepsilon)
d/r$ for $S^*/S_i=O(1)$. Combined together the two conditions provide
the range of tube radius\break $\varepsilon \epsilon \ll d/r \ll \sqrt
\varepsilon$ requiring the ordering $\epsilon \ll 1/\sqrt \varepsilon$.
Being proportional to the time derivative of the  pressure in the
energy equation, see (\ref{enerz2}) below, the parameter measuring the
compressibility effect on the inner flame structure is 
$\epsilon\varepsilon$ which is larger than the neglected
$\varepsilon^2$-terms if $\varepsilon \ll \epsilon$. Therefore, the
non-dimensional elongation rate $\epsilon$ and the tube radius $r/d$
should be related to the Mach number of the laminar flame velocity
$\varepsilon$ by the following ordering
{\begin{eqnarray}
\varepsilon\ll \epsilon \ll 1/\sqrt \varepsilon, \quad \varepsilon \epsilon \ll d/r \ll \sqrt \varepsilon\, \Leftrightarrow \,\frac{1}{\sqrt{\varepsilon}}\ll \frac{r}{d} \ll \frac{1}{\varepsilon \epsilon}.
\label{cond0-}
\end{eqnarray}}\noindent
The second conditions in (\ref{cond0-}) is automatically satisfied if
$\varepsilon r/d$ is of order unity and $\epsilon \ll 1$. Therefore the
quasi-planar approximation on the tip is valid for
$r/d=O(1/\varepsilon)$ and $\epsilon \ll 1$ under a more restrictive
condition than in  (\ref{cond0-}) for the non-dimensional growth rate
$\epsilon$,
{\begin{eqnarray}
\varepsilon \ll \epsilon \ll 1, \quad \varepsilon (r/d)=O(1),
\label{cond0-bis}
\end{eqnarray}}\noindent
in  good agreement with the range of parameters characterizing the low
pressure flames before the DDT onset in the 2010
experiments~\cite{Lieb10,Kuz10} , $\varepsilon \approx 10^{-2} $,
$\epsilon \approx 10^{-1}$, $r/d\approx 50$.

To summarize a one-dimensional analysis performed in the distinguished
limit (\ref{distlimit1}) of large activation energy and small Mach
number of laminar flames is relevant under the condition
(\ref{cond0-bis}). 

\subsection{Non-dimensional constitutive equations}
 
Choosing the burned gas of the initial flame as reference state and
denoting $T_{ui}$ the temperature of the unburned gas, $\rho_{bi}$ the
density of the burned gas in the initial state, the reference
temperature, velocity, density and pressure are 
{\begin{eqnarray}
T_{bi}\equiv T_{ui}+Q/c_p, \quad U_{bi}\equiv U_b(T_{bi}),
\quad \rho_{bi}, \quad p_{bi}=(c_p-c_v)\rho_{bi}T_{bi}.
\label{refst2}
\end{eqnarray}}\noindent
Using for unit length and time the initial flame thickness and transit
time $d=D_{bi}/U_{bi}$ and $t_{bi}=d_{bi}/U_{bi}=D_{bi}/U^2_{bi}$ 
where $D_{bi}$ is the molecular diffusion coefficient in the burned
gas, the non-dimensional variables are 
{\begin{eqnarray}
&& \quad  \quad  \quad  \quad \tau\equiv t/t_{bi}, \quad \xi\equiv (\text{x}-X_P)/d_{bi},
\label{rducedvarN} \Seqnsplit
&&
r\equiv {\rho}/{\rho_{bi}}, 
\quad \text{v}\equiv {u}/U_{bi}, 
\quad \rmpi\equiv {p}/p_{bi}, \quad \theta\equiv {T}/T_{bi}, 
\quad \text{v}_P\equiv {U_P}/U_{bi},\quad
\label{ndim} 
\end{eqnarray}}\noindent
where $\text{x}=X_P(t)$ is the instantaneous position of the reaction
sheet of the ZFK flame ($\xi=0$), $u(\text{x},t)$ is the flow  velocity
and  $U_P(t)\equiv \text{d}X_P/\text{d}t$ the propagation velocity of
the flame, both velocities being relative to the laboratory frame. A
perfect gas law is assumed in the following $r=\rmpi/\theta$.

As for the study of vibratory flames~\cite{PCPPLH1990},  the analysis
of the unsteady flame structure is performed with the mass-weighted
coordinate $z$, using  the change of variables $(\xi,\tau)\to (z,\tau)$
{\begin{eqnarray}
z\equiv\int_0^{\xi} r(\xi',\tau)\text{d}\xi',
\label{def z}  \quad 
\frac{\partial }{\partial \xi}=r \frac{\partial }{\partial
z}=\frac{\rmpi}{\theta} \frac{\partial }{\partial z},
\quad\left.\frac{\partial }{\partial \tau}\right
\vert_\xi+(\text{v}-\text{v}_P(\tau)) \left.\frac{\partial }{\partial
\xi} =\frac{\partial }{\partial \tau}\right\vert_z-m(\tau)\frac{\partial
}{\partial z}
\label{zxi}
\end{eqnarray}}\noindent
where $m(\tau)$ is the instantaneous reduced mass flux across the
reaction zone
{\begin{eqnarray}
m(\tau)= r(0,\tau) [\text{v}_P(\tau)-\text{v}(0,\tau) ]=\frac{\rmpi(0,\tau)}{\theta(0,\tau)} 
[\text{v}_P(\tau)-\text{v}(0,\tau)  ]>0.
\label{mwc}
\end{eqnarray}}\noindent
Introducing the Mach number of the laminar flame, the reduced heat
release and the reduced activation energy
{\begin{eqnarray}
\varepsilon\equiv  \frac{U_{bi}}{a_{bi}}\approx 10^{-2}, \quad q\equiv
\frac{Q}{c_pT_{bi}}\approx 0.7,\quad \beta\equiv \frac{\mathcal
{E}}{k_B T_{bi}}=4-8,
\label{varepsi}
\end{eqnarray}}\noindent
and assuming unity Lewis and Schmidt numbers with a diffusion
coefficient  verifying $\rho^2D=$ constant for simplicity, the
constitutive equations in a planar geometry  for reactive flows 
governed by a one-step chemical reaction  are\cite{PCPPLH1990}
{\begin{eqnarray}
\text{continuity }\frac{\partial \text{v}}{\partial z}&=&\left[\frac{\partial }{\partial \tau}-m(\tau)\frac{\partial }{\partial z} \right]\frac{\theta}{\rmpi}, \nonumber\\
&=&\frac{1}{\rmpi}\left[\frac{\partial }{\partial \tau}-m(\tau)\frac{\partial }{\partial z} \right]\theta-\frac{\theta}{\rmpi^2}\left[\frac{\partial }{\partial \tau}-m(\tau)\frac{\partial }{\partial z} \right]\rmpi \label{massz2bibis}\Seqnsplit
\text{momentum } \left[\frac{\partial \text{v}}{\partial \tau}-m (\tau) \frac{\partial\text{v}}{\partial z} -\frac{\partial^2 \text{v}}{\partial z^2}\right]&=&-\frac{1}{\gamma \varepsilon^2}\frac{\partial \rmpi}{\partial z} \label{mouvz2} \Seqnsplit
\text{species }\left[\frac{\partial Y}{\partial \tau}-m ( \tau)\frac{\partial Y}{\partial z} -\frac{\partial^2 \,Y}{\partial z^2}\right ]&=&w(\theta, Y),\quad Y(z, \tau)\in [0, 1] \label{speciesz}\Seqnsplit
\text{energy }\left[\frac{\partial \theta}{\partial \tau}-m ( \tau)\frac{\partial \theta}{\partial z}-\frac{\partial^2 \,\theta}{\partial z^2}\right ]&=& q w(\theta, Y)+\frac{(\gamma-1)}{\gamma}\frac{\theta}{\rmpi}\left[\frac{\partial \rmpi}{\partial\tau} -m(\tau) \frac{\partial \rmpi}{\partial z}\right] 
+ (\gamma-1)\varepsilon^2 \left(\frac{\partial \text{v}}{ \partial z}\right )^2  \label{enerz2}
\qquad
\end{eqnarray}}\noindent
where $1-Y$ is the reduced mass fraction of reactant ($Y=0$ in the
initial mixture) and $w(\theta, Y)$ is the non-dimensional reaction
rate. For a one-step reaction of order 2 governed by an Arrhenius law
with a large activation energy one has
{\begin{eqnarray}
w(\theta,\, Y)=\frac{(q\beta)^{3}}{8}\frac{\rmpi_b}{\theta_b}
(1-Y)^{2} \text{e}^{\beta ({\theta} -1)}.
\label{wred1}
\end{eqnarray}}\noindent
The perfect gas law $r=\rmpi/\theta$ has been used in (\ref{massz2bibis})
and (\ref{enerz2}) to eliminate the density. The main interest of the
mass-weighted coordinates is the simpler form of the Lagrangian
derivative $\partial /\partial \tau+m(\tau)\partial/\partial z$ instead
of $r(\xi,\tau)[\partial /\partial \tau+\text{v}(\xi,\tau)\partial
/\partial \xi$. When the dissipative terms (heat conduction, viscosity
and reaction rate) are neglected,  (\ref{enerz2}) takes the form of the
entropy wave  in an inert gas subjected to adiabatic compression
$\delta \theta/\theta=[(\gamma -1)/\gamma]\delta \rmpi/\rmpi$
{\begin{eqnarray}
\frac{1}{\theta}\left[\frac{\partial \theta}{\partial \tau}-m ( \tau)\frac{\partial \theta}{\partial z}
\right ] -\frac{(\gamma-1)}{\gamma}\frac{1}{\rmpi}\left[\frac{\partial \rmpi}{\partial\tau}
-m(\tau) \frac{\partial \rmpi}{\partial z}\right]=0.
 \label{entrop}
\end{eqnarray}}\noindent

\subsection{Limit of large activation energy $\beta\gg 1$. Boundary conditions on the reaction sheet }\label{limlact}

The ZFK analysis~\cite{ZFK} has been extended more than forty five
years ago to the unsteady flame structure~\cite{Joulin79}. In the limit
of large activation energy,  small temperature disturbances of order
$1/\beta$ have a large effect and the reaction is localized at $z=0$ in
a reaction sheet of thickness $1/\beta$ smaller than the flame
thickness. Jump conditions are obtained by integrating the balance
between diffusion and reaction rates across the thin reaction zone
{\begin{eqnarray}
z\leqslant 0{:}&& \quad  Y=1, \nonumber \\
z=0{:}&&\quad  Y=1, \quad  \theta=\theta_b(\tau),  \quad (\theta_b-1 )=O(1/\beta) , \quad \text{v}=\text{v}_b(\tau)\equiv \frac{u_b(\tau)}{U_{bi}} \label{bc1}\quad\Seqnsplit
\beta\gg 1, \quad \beta(\theta_b-1)=O(1){:}&&  \quad \left.\frac{\partial \theta}{\partial z}\right \vert_{z=0^+}=- q  \exp\left[\frac{\beta}{2}\left(\theta_b-1\right )\right]+O(1/\beta) \label{ZFKjump}\Seqnsplit 
&&\quad \left.\frac{\partial \theta}{\partial z}\right \vert_{z=0^-}= \left.\frac{\partial \theta}{\partial z}\right \vert_{z=0^+}-q \left.\frac{\partial Y}{\partial z}\right \vert_{z=0^+}+O(1/\beta^2), \label{ZFKjump2}
\end{eqnarray}}\noindent
$z=0^+$ and $z=0^-$ denoting respectively the preheated zone side of
the reaction zone and the exit on the burned-gas side. Equation 
(\ref{ZFKjump}) is a chemical-kinetics relation valid to order unity
expressing the balance between the heat flux entering the preheated
zone and the rate of chemical energy released in the reaction zone.
Equation  (\ref{ZFKjump2}) is an isobaric conservation of total energy
(thermal plus chemical), valid up to first order $1/\beta$ (included).
The flow velocity $u$ does not vary across the reaction zone in the
limit $\beta\ll 1$ and is equal to the back-flow $u=u_b$.  By
definition, the instantaneous laminar flame velocity is
$U_b(\tau)\equiv U_P(\tau)-u_b(\tau)$ where $U_P$ is the velocity of
the reaction sheet  in the laboratory frame.  Introducing the
non-dimensional back-flow, velocity of the reactive sheet and laminar
flame velocity
{\begin{eqnarray}
\text{v}_b(\tau)\equiv u_b(\tau)/U_{bi}, \quad \text{v}_P(\tau)\equiv  U_P(\tau)/U_{bi}, \quad \text{u}_{b}(\tau)\equiv U_b(\tau)/U_{bi},
\end{eqnarray}}\noindent
one has, by definition,
{\begin{eqnarray}
\text{u}_b(\tau) =\text{v}_P(\tau)-\text{v}_b(\tau), \quad \text{v}_{bi}=\text{v}_{Pi}-1
\label{vbub}
\end{eqnarray}}\noindent
where the subscript $i$ denotes the initial condition $\tau=0: 
\text{u}_b=\text{u}_{bi}=1$, $\text{v}_P=\text{v}_{Pi}$, $\text{v}_b=\text
v_{bi}$, $\theta_b=\theta_{bi}=1$. According to the conservation of
mass and energy across a quasi-isobaric flame, the initial
non-dimensional temperature and flow velocity of the unburned gas far
ahead of the flame are respectively $1-q$  and $\text{v}_{bi} +q$. 

To leading order in the distinguished limit (\ref{distlimit1}) for
$\beta\gg 1$ and $\varepsilon\ll 1$, $\left(\theta_b-1\right
)=O(1/\beta)$ and $\rmpi_b-1=O(\varepsilon)$, the non-dimensional mass
flux (\ref{mwc})  and laminar flame velocity  (\ref{vbub}) are equal
{\begin{eqnarray}
\beta\gg 1, \quad\varepsilon\ll 1:\quad  m ( \tau) \approx \text{u}_b(\tau).
\label{mub}
\end{eqnarray}}\noindent

\subsection{Limit of small flame Mach number $\varepsilon \ll 1$}

The coupling of reaction--diffusion mechanisms and compressible effects
in an accelerating flame gets simpler in the limit of small Mach number
of the laminar flame velocity.

\subsubsection{Two length-scales problem}\label{x2ls}

As soon as the flame accelerates ($\text{d}\text{v}_P/\text{d}\tau >0$)
downstream-running compression waves  are sent in the unburned gas
outside the flame structure ($z\gg 1$). They are denoted by the
subscript ${ext+}$ (short notation for upstream external flow).  The
problem being hyperbolic, the downstream boundary conditions far ahead
of the flame corresponds to the initial solution, 
{\begin{eqnarray}
 z \to \infty {:}\quad &&\rmpi \approx 1+O(\varepsilon^2),\quad Y=0, \quad \theta_{\mathrm{ext}+}\to  1-q+O(\varepsilon^2), \nonumber\\
&&\text{v}_{\mathrm{ext}+}\to(\text{v}_{\mathrm{ext}+})_i=\text{v}_{bi}+q.
\label{unbinfty}
\end{eqnarray}}\noindent
The neglected terms are of the same order of magnitude as the pressure
jump across a laminar flame  $\Delta p/p=O(\varepsilon^2)$ given by the
steady-state version of (\ref{mouvz2}).  According to the previous
result~\cite{Clavin2022}, the initial condition can be chosen not too
far from the critical condition so that the modification of the flow
ahead of the flame is not larger than the  laminar flame velocity
$\delta u_{\mathrm{ext}+}=O(U_b)$, $\delta \text{v}_{\mathrm{ext}+} \equiv\text
v_{\mathrm{ext}+}-(\text{v}_{\mathrm{ext}+})_i=O(1)$. Anticipating the same order of
magnitude of pressure variation as in acoustics (to be checked a
posteriori)  one gets $\delta p=O( \rho a U_b)$, $\delta \rmpi\equiv
1-\rmpi=O(\epsilon)$ 
{\begin{eqnarray}
\varepsilon\ll 1{:}\quad \rmpi=1+\varepsilon \,\rmpi_1, \quad
\theta_{\mathrm{ext}+}= 1-q+\varepsilon\,
(\theta_{\mathrm{ext}+})_1,\quad \rmpi_1=O(1), \quad
(\theta_{\mathrm{ext}+})_1=O(1).
\end{eqnarray}}\noindent
Therefore, the new shocks generated by the flame acceleration are weak
(intensity of order $\varepsilon$) with an entropy jump is of order
$\varepsilon^3$~\cite{Clavin2016}. Limiting the attention to the first
order in the limit of small Mach number, the jump across the
acceleration-induced shocks are negligible. Moreover, heat conduction
and viscosity involving terms of order $\varepsilon^2$, the entropy
production is also negligible upstream from the flame. The flame
running from left to right faster than the flow of unburned gases
$\text{v}_P>\text{v}_{\mathrm{ext}+}$, $m(\tau)>0$, the entropy wave
(\ref{entrop}) propagates from right to left in the reference frame
attached to the flame so that there is no downstream running entropy
flux coming from the flow ahead of 
the flame. Therefore the  acceleration-induced flow 
in the unburned gas is quasi-isentropic to first order in the limit
$\varepsilon  \ll 1$. This external flow of unburned gas is a
downstream-running acoustic wave,  namely a solution  to the
non-dissipative version of (\ref{massz2bibis})--(\ref{enerz2}), see
below~(\ref{energyred}),
{\begin{eqnarray}
\varepsilon \ll 1{:}\quad \frac{\partial
\text{v}_{\mathrm{ext}+}}{\partial
z}\approx-\frac{(1-q)}{\gamma}\varepsilon \frac{\partial
\rmpi_1}{\partial \tau}, \quad \frac{\partial
\text{v}_{\mathrm{ext}+}}{\partial \tau}\approx
-\frac{\varepsilon}{\gamma }\frac{\partial \rmpi_1}{\partial z_1}\;
\Rightarrow \;\frac{\partial^2 \rmpi_1}{\partial
\tau^2}=\frac{\varepsilon^2}{1-q}\frac{\partial^2 \rmpi_1}{\partial
z^2}+O(\varepsilon)
\label{Dal3+b}
\end{eqnarray}}\noindent
where, using the mass-weighted coordinates, the non-dimensional sound
speed in the external zone is $1/\sqrt{1-q}$ and, according to the
isentropic compression $\rmpi=\theta^{\,\gamma/(\gamma-1)}$, 
{\begin{eqnarray}
\theta_{\mathrm{ext}+}= (1-q)+(1-q)\frac{\gamma-1}{\gamma}\varepsilon
\rmpi_1+O(\varepsilon^2).
\label{Dal3T}
\end{eqnarray}}\noindent
The second equation in (\ref{Dal3+b}) shows that the pressure varies on
a length  larger than the flame thickness by a factor $1/\varepsilon$
provide the characteristic time is not smaller than the transit time
across the laminar flame. Then, it is convenient to introduce the
stretched coordinate $z_1\equiv \varepsilon z$, $\partial/\partial
z=\varepsilon \partial/\partial z_1$, writing the pressure field in the
form
{\begin{eqnarray}
\varepsilon\ll 1, \quad z_1\equiv \varepsilon z{:} \quad
\rmpi=1+\varepsilon \rmpi_1(z_1, \tau) + O(\varepsilon^2), \quad
\frac{\partial \rmpi_1(z_1,\tau)}{\partial
\tau}=-\frac{1}{\sqrt{1-q}}\frac{\partial \rmpi_1(z_1,\tau)}{\partial
z_1}+O(\varepsilon).
\label{pi1z1}
\end{eqnarray}}\noindent
This confirms that the pressure variation across the flame structure of
order $\varepsilon^2$ is negligible. Moreover, according to
(\ref{Dal3+b})--(\ref{Dal3T}), one has
{\begin{eqnarray}
\frac{\partial \text{v}_{\mathrm{ext}+}(z_1,\tau)}{\partial
z_1}&=&-\frac{(1-q)}{\gamma}\frac{\partial \rmpi_1(z_1,\tau)}{\partial
\tau}+O(\varepsilon)=\frac{\sqrt{1-q}}{\gamma}\frac{\partial
\rmpi_1(z_1,\tau)}{\partial z_1}+O(\varepsilon), 
\label{dvex/dz1}\Seqnsplit 
\theta_{\mathrm{ext}+}(z_1,\tau)&=&
(1-q)+(1-q)\frac{\gamma-1}{\gamma}\varepsilon
\rmpi_1(z_1,\tau)+O(\varepsilon^2).
\label{Text}
\end{eqnarray}}\noindent
Integrating (\ref{dvex/dz1}) from the leading edge of the compression
wave where, according to (\ref{unbinfty}), $\rmpi_1=0$ and $\text
v_{\mathrm{ext}+}= S_i+q$ leads to the external flow velocity in terms of the
pressure
{\begin{eqnarray}
\text{v}_{\mathrm{ext}+}(z_1,\tau)=S_i+q+\frac{\sqrt{1-q}}{\gamma} \rmpi_1(z_1,\tau)+O(\varepsilon).
\label{vexp}
\end{eqnarray}}\noindent

\subsubsection{Eigenvalue problem}\label{2ls}

Using (\ref{pi1z1}), Equations (\ref{massz2bibis})--(\ref{enerz2})
written outside the reaction zone takes the form 
{\begin{eqnarray}
\varepsilon\ll 1{:}\quad \frac{\partial \text{v}}{\partial z}&=& [1-\varepsilon \rmpi_1]\frac{\partial^2 \,\theta}{\partial z^2}-\varepsilon \frac{1}{\gamma}{\theta}\frac{\partial \rmpi_1}{\partial\tau}+O(\varepsilon^2) \label{massred}\Seqnsplit
\left[\frac{\partial \text{v}}{\partial \tau}-m(\tau) \frac{\partial\text{v}}{\partial z} -\frac{\partial^2 \text{v}}{\partial z^2}\right]&=&-\frac{1}{\gamma }\frac{\partial \rmpi_1}{\partial z_1}\label{momred}\Seqnsplit
\left[\frac{\partial Y}{\partial \tau}-m(\tau)\frac{\partial Y}{\partial z} -\frac{\partial^2 \,Y}{\partial z^2}\right ]&=&0\label{speciesred}\Seqnsplit
\left[\frac{\partial \theta}{\partial \tau}-m(\tau)\frac{\partial \theta}{\partial z} -\frac{\partial^2 \,\theta}{\partial z^2}\right ]&=&\varepsilon \frac{(\gamma-1)}{\gamma}{\theta}\frac{\partial \rmpi_1}{\partial\tau}+O(\varepsilon^2). \label{energyred} 
\end{eqnarray}}\noindent
Equation (\ref{massred}) is obtained from (\ref{massz2bibis}) by using
(\ref{energyred}) to eliminate ${\partial \theta}/{\partial
\tau}-m{\partial \theta}/{\partial z}$.  Equations
(\ref{Dal3+b})--(\ref{Dal3T}) for the external flow of unburned gas are
recovered from (\ref{massred}), (\ref{momred}) and (\ref{energyred})
when the second derivative with respect to $z$ (thermal and viscous
dissipation) are negligible, as it is the case to leading order in the
external flow.

The unsteady term $\varepsilon \,\theta \,{\partial
\rmpi_1}/{\partial\tau}$ on the right-hand side of (\ref{massred}) and
(\ref{energyred}) describes the small unsteady disturbances of pressure
that modify drastically the inner structure of the flame due to the
amplification by the large activation energy in the limit
(\ref{distlimit1}). The  time-dependent mass flux $m(\tau)$, namely the
unsteady laminar flame velocity $\text{u}_b(\tau)$ to leading order, is
an unknown function (eigenvalue of the problem) obtained by solving
(\ref{massred})--(\ref{energyred}) with the boundary conditions
(\ref{bc1})--(\ref{unbinfty}), in which the back flow $\text{v}_b(\tau)$
is a functional of $S(\tau)$ and $\text{u}_b(\tau)$. For example,
according to (\ref{UbS}), one has  simply $\text{v}_b(\tau)=S(\tau)
\text{u}_b(\tau)$ when the unsteadiness of the burned gas flow is
neglected.

\section{Matching the inner flame structure with the external flow of cold gas}\label{match}

Thanks to the two length-scale nature of the problem, the
downstream-running compression wave in the external flow of unburned
gas that is solution to the wave equation (\ref{Dal3+b}), can be
expressed in terms of the time dependent pressure on the front 
$1+\varepsilon \rmpi_{P}(\tau)$  
{\begin{eqnarray}
\rmpi_1(z_1, \tau)=\rmpi_{P} (\tau - {z_1}{\sqrt{1-q}} )\quad 
\text{where}\  \rmpi_{P}(\tau) \equiv \rmpi_1 \vert_{z_1=0}, 
\label{piB}
\end{eqnarray}}\noindent
and  $\theta_{\mathrm{ext}+}(z_1,\tau)$ and $\text{v}_{\mathrm{ext}+}(z_1, \tau)$  are
given by (\ref{Text}) and (\ref{vexp}). 

\subsection{Matching the temperature}

Denoting  the internal solution by the superscript $(i)$, the unsteady
structure of the preheated zone of the flame ($z=O(1)$) is governed by
(\ref{speciesred})--(\ref{energyred})  
{\begin{eqnarray}
\left[\frac{\partial Y}{\partial \tau}-m(\tau)\frac{\partial
Y}{\partial z} -\frac{\partial^2 \,Y}{\partial z^2}\right ]&=&0
\label{speciesredP} \Seqnsplit
z\geqslant0{:} \quad \left[\frac{\partial \theta^{(i)}}{\partial
\tau}-m(\tau)\frac{\partial \theta^{(i)}}{\partial z}-\frac{\partial^2
\,\theta^{(i)}}{\partial z^2}\right ]&=&\varepsilon
\frac{(\gamma-1)}{\gamma}{\theta^{(i)}(z,\tau)}\frac{\text{d}\rmpi_P(\tau)}{\text{d}\tau}
+O(\varepsilon^2). \label{energyredP} 
\end{eqnarray}}\noindent
The boundary condition at the exit of the preheated zone on the cold
gas side ($z=O(1)$, $z\to \infty$: $\lim_{z\to \infty} Y=0$) is
obtained by matching the internal temperature with the external
solution $\theta_{\mathrm{ext}_+}(z_1,\tau)$
{\begin{eqnarray}
\lim_{z\to \infty} \theta^{(i)}(z,\tau)
=\theta_{\mathrm{ext}_+}(z_1,\tau)\vert_{z_1=0}, \quad  \lim_{z\to
\infty} \partial \theta^{(i)}/\partial z=\varepsilon \partial
\theta_{\mathrm{ext}_+}/\partial z_1\vert_{z_1=0} =O(\varepsilon^2).
 \label{matchT}
\end{eqnarray}}\noindent
According to  (\ref{Text}), $\partial \theta_{\mathrm{ext}_+}/\partial
z_1=O(\varepsilon)$ so that $\lim_{z\to \infty} \partial
\theta^{(i)}/\partial z=O(\varepsilon^2)$ is negligible to first order
in a  perturbation analysis for small $\varepsilon$,
{\begin{eqnarray}
\lim_{z\to \infty} \theta^{(i)}(z,\tau)-
(1-q)\approx\varepsilon(1-q)\frac{\gamma-1}{\gamma}\rmpi_P(\tau),
\quad\lim_{z\to \infty} \partial \theta^{(i)}/\partial z \approx
0,\quad\lim_{z\to \infty} Y=0.
\label{limtheta2++}
\end{eqnarray}}\noindent
Equations (\ref{speciesredP})--(\ref{energyredP}) have to be solved
using (\ref{limtheta2++}) and the boundary conditions
(\ref{bc1})--(\ref{ZFKjump2})  on the  reaction sheet ($z=0$) in the
distinguished limit (\ref{distlimit1}). The fully unsteady problem
requires the solution in the burned gas side of the reaction sheet
$z<0$ to compute ${\partial \theta}/{\partial z} \vert_{z=0^-}$ in the
jump condition (\ref{ZFKjump2}). We will come back to this question
later.

\subsection{Matching the flow velocity. Master equation}\label{Mfv}

Matching the flow velocity in the preheated zone with the external flow
of cold gas (\ref{vexp}) yields
{\begin{eqnarray}
&\displaystyle \lim_{z\to \infty}\text{v}^{(i)}(z,
\tau)=\text{v}_{\mathrm{ext}_+}(z_1,\tau)\vert_{z_1=0}=S_i+q+\frac{\sqrt{1-q}}{\gamma }\rmpi_P(\tau)+O(\varepsilon) ,\label{vuvi--}\Seqnsplit
&\displaystyle\lim_{z\to \infty}\frac{ \partial \text{v}^{(i)}}{\partial
z}=\varepsilon \left.\frac{\partial \text{v}_{\mathrm{ext}_+}}{\partial
z_1}\right \vert_{z_1=0}=-\varepsilon \frac{(1-q)}{\gamma
}\frac{\text{d}\rmpi_P(\tau)}{\text{d}\tau}
+O(\varepsilon^2)\label{vuvi-} 
\end{eqnarray}}\noindent
where (\ref{vexp}) has been used. Equations
(\ref{vuvi--})--(\ref{vuvi-}) yield
{\begin{eqnarray}
&&\quad\lim_{z\to \infty}\text{v}^{(i)}(z,\tau)\to
S_i+q+\frac{\sqrt{1-q}}{\gamma }\rmpi_P(\tau)-\varepsilon
\,z\,\frac{(1-q)}{\gamma }\frac{\text{d}\rmpi_P(\tau)}{\text{d}\tau}
+O(\varepsilon^2).\label{vuvi}
\end{eqnarray}}\noindent
Integrating (\ref{massred}), written in the form
{$$
\frac{\partial \text{v}^{(i)}}{\partial z}= [1-\varepsilon
\rmpi_P(\tau) ]\frac{\partial^2 \,\theta^{(i)}}{\partial
z^2}-\varepsilon \frac{1}{\gamma}[\theta^{(i)}-(1-q) ]
\frac{\text{d}\rmpi_P(\tau)}{\text{d}\tau}- \varepsilon
(1-q)\frac{1}{\gamma}\frac{\text{d}\rmpi_P(\tau)}{\text{d}\tau}+O(\varepsilon^2), 
$$}\noindent
from the reaction sheet ($z=0: \text{v}^{(i)}=\text{v}_b$) yields the
flow velocity in the preheated zone
{\begin{eqnarray}
z=O(1){:}\quad \text{v}^{(i)}(z,\tau)-\text{v}_b(\tau) &=&[1-\varepsilon
\rmpi_P(\tau) ]\left(\frac{\partial \,\theta^{(i)}}{\partial
z}-\left.\frac{\partial \,\theta^{(i)}}{\partial z}\right \vert_{z=0^+}\right)
\label{intvb-va-} \nonumber\\
&&-\,\varepsilon
\frac{1}{\gamma}\frac{\text{d}\rmpi_P(\tau)}{\text{d}\tau}\int^{z}_0
[{\theta}^{(i)}-(1-q)]\text{d}z-\varepsilon\, z\,
(1-q)\frac{1}{\gamma}\frac{\text{d}\rmpi_u(\tau)}{\text{d}\tau}+O(\varepsilon^2).
\qquad
\end{eqnarray}}\noindent
Thanks to  (\ref{limtheta2++}) $\lim_{z\to \infty}
\theta^{(i)}(z,\tau)= (1-q)+O(\varepsilon)$, the leading order of the
integral on the right-hand side of (\ref{intvb-va-}) is well defined in
the limit $z\to \infty$ and is of order unity.  Then, using 
(\ref{vuvi}), the limit $z\to \infty$ of (\ref{intvb-va-}) yields
{\begin{eqnarray}
\text{v}_b(\tau)-\left[ S_i+q+\frac{\sqrt{1-q}}{\gamma }\rmpi_P(\tau)\right ] &=& [1-\varepsilon \rmpi_P(\tau) ]\left.\frac{\partial
\,\theta^{(i)}}{\partial z}\right \vert_{z=0^+}\nonumber \\
&&+\,\varepsilon
\frac{1}{\gamma}\frac{\text{d}\rmpi_P(\tau)}{\text{d}\tau}\int^{+\infty}_0[{\theta^{(i)}}-(1-q)]\text{d}z+O(\varepsilon^2).\label{vb-va}
\end{eqnarray}}\noindent
where the thermal flux out of the reaction sheet ${\partial
\,\theta^{(i)}}/{\partial z}\vert_{z=0^+}$ is obtained in terms of the
flame temperature   $\theta_b$ by the jump condition (\ref{ZFKjump}). 
The integral term on the right-hand side of (\ref{vb-va})  which is
meaningful as soon as  ${\text{d}\rmpi_P(\tau)}/{\text{d}
\tau}<\varepsilon$ is indeed not useful for the leading order of the
flow. Using the jump relation (\ref{ZFKjump}), the leading order of
Equation (\ref{vb-va}) takes the form 
{\begin{eqnarray}
\text{v}_b(\tau)-\left[ S_i+q+\frac{\sqrt{1-q}}{\gamma }\rmpi_P(\tau)
\right] =- q  \exp\left(\frac{\beta}{2}[\theta_b(\tau)-1]\right)+O(\varepsilon).\label{vb-va3+}
\end{eqnarray}}\noindent
valid for the ZFK model without restriction concerning the
characteristic time of the dynamics. When the inner structure of the
flame is in steady state, using the ZFK result for the ratio of laminar
flame velocity $\text{u}_b\equiv U_b(T_b)/U_b(T_{bi})=\text{e}^{\beta(
\theta_b-1)/2}$ on the right-hand side, Equation (\ref{vb-va3+}) 
reduces to the well known relation between the flows just upstream and
downstream a steady planar flame, $u_u$ in the unburned and $u_b$  in
the burned gas, $u_u-u_b=[T_b/T_{u}-1] U_b$. Equation~(\ref{vb-va3+})
is the \emph{master equation} used in the nonlinear dynamics of the
flame front. 

\section{Quasi-steady solutions}\label{QS}

Before considering the full dynamics, it is worth studying the dynamics
of the quasi-steady solutions (inner structure of the flame and burned
gas flow assumed in steady state). The results are not expected to be
different from those obtained when the flame was considered as a
discontinuity~\cite{Clavin2022}.  The flame structure of the
quasi-steady solutions denoted by an overbar, is obtained by neglecting
the unsteady terms in (\ref{speciesredP})--(\ref{energyredP}) 
{\begin{eqnarray}
-\overline m(\tau)\frac{\partial \overline Y}{\partial
z}-\frac{\partial^2 \,\overline Y}{\partial
z^2}&=&0\label{speciesredPss} \Seqnsplit
z\geqslant0 {:} \quad -\overline m(\tau)\frac{\partial \overline
\theta^{(i)}}{\partial z}-\frac{\partial^2 \,\overline
\theta^{(i)}}{\partial z^2}&=&O(\varepsilon^2). \label{energyredPss} 
\end{eqnarray}}\noindent
Introducing the short notation
{\begin{eqnarray}
\overline \theta_u(\tau)\equiv(1-q)\left[
1+\varepsilon\frac{\gamma-1}{\gamma}\overline \rmpi_P(\tau)\right], 
\label{shortnot}
\end{eqnarray}}\noindent
for the gas temperature just ahead of the flame as it is modified by
the downstream running acoustic wave in the unburned gas flow, the
quasi-steady solution satisfying the boundary condition
(\ref{limtheta2++}) reads
{\begin{eqnarray}
&& z\geqslant0{:}\quad \overline Y =\text{e}^{-\overline m(\tau)z},\quad 
\overline  \theta^{(i)}(z,\tau) =\left[\overline \theta_b(\tau)-
\overline \theta_{u}(\tau)\right ]\text{e}^{-\overline m\,z}+\overline
\theta_u(\tau), \nonumber \\
&& z\leqslant 0{:}\quad \overline Y =1,\quad  \quad\overline 
\theta^{(i)}=\overline \theta_b(\tau).\label{enerz2e2bibis+qs1}
\end{eqnarray}}\noindent
Then, the boundary conditions on the reaction sheet
(\ref{ZFKjump})--(\ref{ZFKjump2}) determine the solution. The flame
temperature $ \overline \theta_b(\tau)$ is obtained from
(\ref{ZFKjump2})  with ${\partial \theta}/{\partial z} \vert_{z=0^-}=0$
and the laminar flame velocity $\overline m(\tau)\approx \overline
{\text{u}}_b$ given by (\ref{ZFKjump})  corresponds to the ZFK solution
$ \overline {\text{u}}_b=\text{e}^{\beta(\overline \theta_b-1)/2}$. 
Expressed in terms of the pressure $\overline \rmpi_P(\tau)$, the steady
state solution reads
{\begin{eqnarray}
&\displaystyle\overline \theta_b(\tau)=\overline \theta_u(\tau)+q, \quad
\overline \theta_b(\tau)-1=
\varepsilon(1-q)\frac{\gamma-1}{\gamma}\overline
\rmpi_P(\tau)+O(\varepsilon^2), \label{expft}\Seqnsplit
&\displaystyle\beta(\overline \theta_b-1)/2={\beta}\left [\overline
\theta_u(\tau)+q-1\right ]/2=b\,\overline \rmpi_P(\tau),\quad \overline
m(\tau) =\text{e}^{b\,\overline \rmpi_P(\tau)}+O(1/\beta),\quad b\equiv
\frac{\beta\varepsilon}{2}\,
\frac{(\gamma-1)}{\gamma}(1-q)\label{expft+} \qquad
\end{eqnarray}}\noindent
where the parameter $b$ (not to be confused with the subscript $b$) is 
of order unity in the limit (\ref{distlimit1}).

\subsection{Turning point of the quasi-steady solutions (saddle-node bifurcation)}

In the quasi-steady state approximation, the back-flow is  (\ref{UbS}) 
$\text{v}_b(\tau)=S(\tau)\,\overline{ m}(\tau)$ to give  using
(\ref{expft+})
{\begin{eqnarray}
\overline{\text{v}}_b(\tau)=S(\tau)\,\overline
m(\tau)=S(\tau)\,\text{e}^{b\,\overline \rmpi_P(\tau)}, \quad b\equiv
\frac{\beta\varepsilon}{2}\, \frac{(\gamma-1)}{\gamma}\,S_i.  
\label{vb-va1}
\end{eqnarray}}\noindent
Introducing (\ref{vb-va1}) into the master Equation (\ref{vb-va3+}),
the same transcendental equation as in~\cite{Clavin2022} is obtained,
yielding the pressure $\overline \rmpi_P$ in terms of the flame
elongation $S(\tau)$
{\begin{eqnarray}
(S+q)\text{e}^{b\,\overline \rmpi_P}= (S_i+q)+\frac{\sqrt{1-q}}{\gamma }\,\overline \rmpi_P.
\label{vb-va5}
\end{eqnarray}}\noindent
The pressure and the flame temperature $\overline \rmpi_P$ and $\overline
\theta_b$ depend on the time only through the elongation $S$. 
Introducing the notation $\vartheta$ and $\zeta$ for the rescaled
pressure and elongation
{\begin{eqnarray}
&\vartheta\equiv b\rmpi_P=O(1), \quad \zeta\equiv S+q, \quad \zeta(\tau)=(1+\epsilon \tau)S_i+q,\label{newnot-}\Seqnsplit
&\text{b}\equiv b\gamma/\sqrt{1-q}=(\beta \varepsilon/2)(\gamma-1)\sqrt{1-q}=O(1/S_i),\label{newnot}
\end{eqnarray}}\noindent
Equation (\ref{vb-va5})  takes the  form 
{\begin{eqnarray}
\zeta\,\text{e}^{\overline \vartheta}-\zeta_i-\overline \vartheta
/\text{b}=0;  \quad \tau=0{:} \quad\zeta=\zeta_i\equiv S_i+q,
\quad \vartheta=0.\label{NLeq}
\end{eqnarray}}\noindent
The solution \emph{pressure versus elongation} $\overline
\vartheta(\zeta)$ depends on the initial elongation $\zeta_i$ and
involves a single parameter $\text{b}$ in (\ref{newnot}). The graph of
the inverse function $\zeta (\overline \vartheta)$ is a
bell-shaped curve sketched in thick black line in
Figure~\ref{fig:traj}.  The maximum $\zeta=\zeta^*$, $\overline
\vartheta=\vartheta^*$  corresponds to
{\begin{eqnarray}
\left.\frac{\text{d}\zeta}{\text{d}\overline
\vartheta}\right\vert_{\overline
\vartheta=\vartheta^*}=0{:} \quad \zeta^*
\text{e}^{\vartheta^*}=\frac{1}{\text{b}}, \quad
\vartheta^*=1-\zeta_i\,\text{b}>0, \quad
\frac{\zeta^*}{\zeta_i}=\frac{\text{e}^{ (\,\text{b}\zeta_i-1\;
)}}{\text{b}\zeta_i}\geqslant1,\label{elong*}
\end{eqnarray}}\noindent
the inequality ${\zeta^*}/{\zeta_i}\geqslant1$ following from the
ordering $0<\text{b}\zeta_i\leqslant 1$ valid for all reactive gaseous
mixtures~\cite{Clavin2022}.  The  dynamics of the flame is represented
by  the C-shaped curve $\overline \vartheta(\zeta)$  with a turning
point  at the critical elongation $S^*$, $\zeta^*=S^*+q$, $\text{d}
\overline \vartheta/\text{d}\zeta\vert_{\zeta=\zeta^*}=\infty $. There
is no more solution to (\ref{NLeq}) for $\zeta>\zeta^*$ and  there are
two branches of solutions for $\zeta<\zeta^*$, $\overline
\vartheta_\pm=\vartheta^*\pm\sqrt{2(\zeta^*-\zeta)/\zeta^*}$,
$\overline \vartheta_--\vartheta^*<0<\overline \vartheta_+-
\vartheta^*$, ${\text{d}\overline \vartheta_-}/{\text{d}\zeta}>0$ and
${\text{d}\overline \vartheta_+}/{\text{d}\zeta}<0$, see
Figure~\ref{fig:traj}. The physical branch is $\overline
\vartheta_-(\zeta)$ for which the flame temperature increases with the
flame acceleration on the tip ${\text{d}\zeta}/\text{d}\tau =\epsilon
S_i$ since ${\text{d}\overline \vartheta_-}/{\text{d}(\epsilon S_i)}=  
\tau{\text{d}\overline \vartheta_-}/{\text{d}\zeta}>0$.   As previously
mentioned~\cite{Clavin2022}, an initial elongation close to the upper
bound $\zeta_i=\zeta^*$,   $S=S_{\text{max}}^*$  corresponds to a universal
critical Mach number ${2}/[{\beta(\gamma-1)}]$ of the unburned gas flow
adjacent to the flame.  According to (\ref{NLeq})--(\ref{elong*}), this
critical flow Mach number is close to unity while the critical laminar
flame velocity remains markedly subsonic $U^*_b/a^*_b\approx 0.05$, in
good agreement with the pre-conditioned state in the DDT onset of the
2010 experiments and numerics~\cite{Lieb10,Kuz10}.
 
\subsection{Finite-time singularity of the quasi-steady dynamics}\label{Ftsqs}

Equations (\ref{vb-va5}) and (\ref{elong*}) are relevant if the
response of the flame structure to a change in elongation is
quasi-instantaneous. A necessary condition is that the rate of
elongation is smaller that the inverse of the transit time in
(\ref{elongrate})
{\begin{eqnarray}
\epsilon \ll 1.
\label{epsism}
\end{eqnarray}}\noindent
Then, according to (\ref{elongrate}) $\text{d}\zeta/\text{d}
\tau=\epsilon S_i$, $\epsilon\ll 1$, the laminar flame velocity of the
quasi-steady solutions $\overline m=\text{e}^{\overline \vartheta}$
($\overline \vartheta\equiv b\overline \rmpi_P$)  increase first  slowly
with the time $\text{d} \overline m/\text{d}\tau=O(\epsilon S_i)$, but,
according to (\ref{elong*}), the flame acceleration diverges abruptly 
when the elongation reaches its critical value $S^*$ $\text{d}\overline
m/\text{d}\tau\vert_{\tau=\tau^*}=\epsilon S_i\overline m^* {\text{d}
\overline \vartheta}/{\text{d}\zeta}\vert_{\zeta=\zeta^*}=\infty$ since
${\text{d}\overline \vartheta}/{\text{d}\zeta}\vert_{\zeta=\zeta^*}=
\infty$ (turning point). This finite-time singularity of the flame
acceleration occurs for a  critical laminar flame velocity $U^*_b$
markedly subsonic  $ \overline{m}^*=\text{e}^{ \vartheta^*}$ where
$0<\vartheta^*=(1-\zeta_i\text{b})<1$, $U^*_b/a^*_b=O(\varepsilon)$.  
As when the flame is treated as a discontinuity~\cite{Clavin2021},
Equation (\ref{NLeq}) takes a generic form near the critical point 
$\text{d}\zeta/\text{d}\overline \vartheta\vert_{\overline
\vartheta=\vartheta^*}=0$,
{\begin{eqnarray}
\left.\frac{\text{d}^2 \zeta}{\text{d}\vartheta^2}\right
\vert_{\vartheta=\vartheta^*}=-\zeta^*\quad \Rightarrow \quad
\frac{\zeta^*-\zeta}{\zeta^*}\ll
1{:}\quad\frac{\zeta^*-\zeta}{\zeta^*}\approx
\frac{1}{2}(\vartheta^*-\overline \vartheta)^2 \label{ufnc}
\end{eqnarray}}\noindent
obtained by a Taylor expansion.  Using (\ref{elongrate}), the
quasi-steady dynamics of the flame pressure near the critical condition
$\tau=\tau^*$ takes the form
{\begin{eqnarray}
\tau^*-\tau \to 0^+{:}\quad\vartheta^*-\overline
\vartheta_-(\tau)\approx\kappa\sqrt{\tau^*-\tau} \quad\text{where }
\kappa\equiv
\sqrt{2\epsilon\frac{S_i}{S^*+q}}=O(\sqrt{\epsilon}),\label{ufnc2}
\end{eqnarray}}\noindent
exhibiting the finite-time singularity of the flame acceleration of the
physical branch of solutions
{\begin{eqnarray}
\frac{\text{d}\overline \vartheta_-}{\text{d}\tau}\approx
\frac{\kappa/2}{\sqrt{\tau^*-\tau}}, \quad \frac{\text{d}\overline
\rmpi_P}{\text{d}\tau}\approx \frac{\kappa/2b}{\sqrt{\tau^*-\tau}}, \quad
\frac{1}{\overline m}\frac{\text{d}\overline m}{\text{d}\tau}\approx
\frac{\kappa/2}{\sqrt{\tau^*-\tau}}.
\label{fintimsing}
\end{eqnarray}}\noindent
According to  (\ref{pi1z1})--(\ref{dvex/dz1}) and (\ref{piB}), the
gradient of the external unburned flow also diverges on the flame,
{\begin{eqnarray}
\tau^*-\tau \to 0^+{:}\quad\left.\frac{\partial
\text{v}_{\mathrm{ext}_+}}{\partial z_1}\right \vert_{z_1=0}\approx
\frac{(1-q)}{\gamma}\frac{\kappa/2}{b}\frac{1}{\sqrt{\tau^*-\tau}},\quad 
\left.\frac{\partial \text{v}_{\mathrm{ext}_+}}{\partial \tau}\right \vert_{z_1=0}
\approx \frac{\sqrt{1-q}}{\gamma}\frac{\kappa/2}{b}\frac{1}{\sqrt{\tau^*-\tau}}.
\label{fintimsingext}
\end{eqnarray}}\noindent

The finite-time singularity (\ref{fintimsingext}) corresponds to a
break down of the quasi-steady dynamics so that the quasi-steady
solutions are no longer valid near the turning point even for a slow
elongation (\ref{epsism}). Moreover, the physical branch of
quasi-steady solutions $\overline \vartheta_-(\zeta)$ should be stable
to be meaningful for the unsteady solutions when the elongation
$\zeta(\tau)$ increases slowly $\epsilon \ll 1$. These problems are
addressed in the following sections.

\section{Linear stability of the quasi-steady solutions}\label{stabanal0}

The stability analysis is performed from the linearized version of the
master equation (\ref{vb-va3+}) yielding the relation between for
$\delta \rmpi_P\equiv \rmpi_P-\overline \rmpi_P$, $\delta \text{v}_b\equiv
\text{v}_b-\overline {\text{v}}_b$ and $\delta \theta_b\equiv \text
v_b-\overline \theta_b$
{\begin{eqnarray}
&\displaystyle\overline {\text{v}}_b-\left[ S_i+q+\frac{\sqrt{1-q}}{\gamma }\overline \rmpi_P(\tau) \right] =- q\overline m, \quad \text{where }\overline m=\exp\left(\frac{\beta}{2}\left[\overline \theta_b(\tau)-1\right ]\right) \text{ and } \overline {\text{v}}_b=S\,\overline m \label{vb-va3+lin-2}\Seqnsplit
&\displaystyle\delta \text{v}_b-\frac{\sqrt{1-q}}{\gamma }\delta \rmpi_P  =-\frac{q}{2}  \overline m\,\delta \theta_{b1}\quad \text{where } \theta_{b1}\equiv \beta(\theta_b-1)\mathop{]}=O(1).\label{vb-va3+lin-}
\end{eqnarray}}\noindent
The linear stability of the quasi-steady solutions involves the linear
variation of the flame temperature with the pressure for a fixed
elongation $S\leqslant S^*$  ($\overline m$ and $\overline \rmpi_P$
fixed) 
{\begin{eqnarray}
\delta \text{v}_b\vert_{S=\text{cst.}}-\frac{\sqrt{1-q}}{\gamma }\delta
\rmpi_P  =- \frac{q}{2} \overline m\, \delta
\theta_{b1}\vert_{S=\text{cst.}}. \quad \label{vb-va3+lin}
\end{eqnarray}}\noindent
Looking for a solution in the form 
{\begin{eqnarray}
\delta \rmpi_P (\tau)=\text{e}^{\sigma \tau}\tilde \rmpi_P(\sigma),
\quad\delta \text {v}_b\vert_{S=\text{cst.}} (\tau)=\text{e}^{\sigma
\tau}\tilde {\text {v}}_b(\sigma), \quad \delta
\theta_{b1}\vert_{S=\text{cst.}} (\tau)=\text{e}^{\sigma \tau}\tilde
\theta_{b1}(\sigma)
\end{eqnarray}}\noindent
an equation for the complex growth rate $\sigma$ is obtained when
$\tilde {\text {v}}_b(\sigma)$ and $\tilde \theta_{b1}(\sigma)$ are
expressed linearly in term of  $\tilde \rmpi_P(\sigma)$. The quasi-steady
solution is unstable (stable) if Re($\sigma$)>0 (Re($\sigma$)<0).  The
linear response of a steady planar flame  subjected to a uniform
pressure fluctuation should be performed in a first step.  

\subsection{Linear response to a pressure fluctuation}\label{linresp}

Introducing the short notation $f(\tau)\equiv\beta \varepsilon
[(\gamma-1)/\gamma] \rmpi_{P}(\tau)$ for the reduced pressure
$p(\tau)/p_{i}=1+\varepsilon \rmpi_P(\tau)$ (forcing term) and working in
the limit $\beta \gg 1$ with the distinguished limit
(\ref{distlimit1}), $\beta\varepsilon=O(1)$, Equations
(\ref{speciesredP})--(\ref{energyredP}) governing the flame structure
outside the reaction sheet take the form
{\begin{eqnarray}
\left[\frac{\partial Y}{\partial \tau}-m ( \tau)\frac{\partial
Y}{\partial z}-\frac{\partial  \,Y}{\partial z^2}\right ]&=&0,\quad
Y(z, \tau)\in [0, 1]\label{speciesze2n+}\Seqnsplit
\left[\frac{\partial \theta}{\partial \tau}-m ( \tau)\frac{\partial
\theta}{\partial z}-\frac{\partial^2 \,\theta}{\partial z^2}\right
]&=&\frac{\theta}{\beta}\frac{\text{d}f}{\text{d}\tau}, \quad
f(\tau)\equiv\beta \varepsilon [(\gamma-1)/\gamma] \rmpi_{P}(\tau),
\label{enerz2e2n+}
\end{eqnarray}}\noindent
where the superscript (i) is suppressed for saving the notation. The
problem is solved in the limit $\beta \to \infty$ using  the  boundary
conditions (\ref{bc1})--(\ref{ZFKjump2}) on the reaction zone  and at
infinity in both preheated zone ($z\geqslant 0$) and  burned gas ($z<
0$)
{\begin{eqnarray}
z \to \infty {:}&& \quad \theta \to (1-q)+\frac{(1-q)}{\beta}f(\tau), \quad Y\to 0, \label{bc+}\Seqnsplit
z \leqslant 0{:} \quad Y=1, \quad z \to -\infty {:}&& \quad \theta \to 1+\frac{1}{\beta}f(\tau),\label{bc+<0}
\end{eqnarray}}\noindent
expressing that the compressional heating is adiabatic and uniform far
away from the flame  on both sides (hot burned gas and cold unburned
gas) $\delta T/\overline T=[(\gamma-1)/\gamma]\delta T/\overline T$.
The problem consists in determining the disturbances of reduced mass
flux  and flame temperature $\delta m(\tau)$ and $\delta (\beta
[\theta_b(\tau)-1])$ for a given time dependent pressure
fluctuation  $\delta f(\tau)$.   According to the asymptotic method for
large $\beta$, one introduces the decomposition,
{\begin{eqnarray}
\theta = \theta_0+\frac{\theta_1}{\beta},\quad \theta_b =
1+\frac{\theta_{b1}}{\beta},\quad Y = Y_0+\frac{Y_1}{\beta}, \quad
m=m_0+\frac{m_1}{\beta}
\end{eqnarray}}\noindent
and, according to (\ref{ZFKjump})--(\ref{ZFKjump2}) and
(\ref{bc+})--(\ref{bc+<0}),
{\begin{eqnarray}
&\displaystyle   z \to \infty {:} \quad \theta_0 \to (1-q); \quad z \to -\infty {:} \quad \theta_0 \to 1\Seqnsplit
&\displaystyle \left.\frac{\partial \theta_0}{\partial z}\right
\vert_{z=0^+}=-q\exp{(\theta_{b1} /2)}, \quad \left.\frac{\partial
\theta_{0,1}}{\partial z}\right \vert_{z=0^-}=\left.\frac{\partial
\theta_{0,1}}{\partial z}\right \vert_{z=0^+}-q \left.\frac{\partial
Y_{0,1}}{\partial z}\right \vert_{z=0^+},\label{jump01}
\end{eqnarray}}\noindent
and the temperature should be solved up to the first order in the
$1/\beta$ expansion. In the linear response one considers $\delta
f(\tau)\ll 1$ and looks for the linear perturbations $\delta \theta(z,
\tau)$ and $\delta Y(z, \tau)$
{\begin{eqnarray}
f=\overline f+\delta f, \quad \theta = \overline \theta+ \delta \theta, \quad Y=\overline Y+\delta Y.
\end{eqnarray}}\noindent
The unperturbed solution is a steady state solution in
(\ref{shortnot})--(\ref{expft+}) $\overline \beta[\overline
\theta_b-1]= (1-q)\overline f$, $\overline m=\text{e}^{(1-q)\overline
f/2}$
{\begin{eqnarray}
z>0{:}\quad \overline \theta = q \text{e}^{- \overline m
z}+(1-q)+\frac{(1-q)}{\beta}\overline f, \quad Y= \text{e}^{- \overline
m z};  \quad z<0{:} \quad \overline \theta=\overline
\theta_b=1+\frac{(1-q)}{\beta}\overline f, \quad\overline Y=1. \quad
\label{unpertsttsol}
\end{eqnarray}}\noindent
The linear Equations (\ref{speciesze2n+})--(\ref{enerz2e2n+}) in the
preheated zone read
{\begin{eqnarray}
z\geqslant 0{:}\quad&&\left[\frac{\partial \delta  Y}{\partial \tau}- \overline m\frac{\partial\delta Y}{\partial z}-\frac{\partial^2 \,\delta Y}{\partial z^2}\right ]=-\delta m\, \overline m\,\text{e}^{-\overline m \,z},\label{pertspeciesze2n+}\Seqnsplit
&&\left[\frac{\partial \delta  \theta}{\partial \tau}- \overline m\frac{\partial\delta \theta }{\partial z}-\frac{\partial^2 \,\delta \theta}{\partial z^2}\right ]=-q\,\delta m \,\overline m\,\text{e}^{-\overline m\,z}+\frac{\overline \theta (z)}{\beta}\frac{\text{d}\delta f}{\text{d}\tau},\label{pertenerz2e2n+}\Seqnsplit
&&z \to \infty {:}\quad \delta \theta=\frac{(1-q)}{\beta} \delta f,\quad 
\delta Y=0; \quad z=0{:}\quad \delta \theta=\frac{\delta \theta_{b1}}{\beta}, \quad \delta Y= 0.\quad \label{pertbc+}
\end{eqnarray}}\noindent
where the unknown fluctuations of flame temperature $\beta
(\theta_{b}-\overline \theta_{b})\equiv \delta \theta_{b1}(\tau)$ and
mass flux $\delta m(\tau)$ are eigenvalues.  The solution is obtained
in  Fourier transform. Considering a harmonic pressure disturbance
$\delta f(\tau)=\text{e}^{\mathrm{i}\omega \tau} \tilde f(\omega)$, the solution
is looked for in the form $\delta m(\tau)=\text{e}^{\mathrm{i}\omega \tau} \tilde
m(\omega)$, $\delta \theta(z,\tau)= \text{e}^{\mathrm{i}\omega \tau}\tilde
\theta(z,\omega)$, $\delta Y(z,\tau)= \text{e}^{\mathrm{i}\omega \tau}\tilde
Y(z,\omega)$. To leading  in the asymptotic limit $\beta\gg 1$, the
linear equations in the preheated zone take the form
{\begin{eqnarray}
z\geqslant 0{:}\quad&&\left[\mathrm{i}\omega \tilde Y_0- \overline m  \frac{\text{d}\tilde Y_0}{\text{d}z}-\frac{\text{d}^2\tilde Y_0}{\text{d}z^2}\right ]=-\tilde m_0 \, \overline m\,\text{e}^{-\overline m \,z},\label{pert0speciesze2n+}\Seqnsplit
&&\left[\mathrm{i}\omega \tilde \theta_0- \overline m  \frac{\text{d}\tilde \theta_0}{\text{d}z}-\frac{\text{d}^2\tilde \theta_0}{\text{d}z^2}\right ]=-q\,\tilde m_0 \, \overline m\,\text{e}^{-\overline m \,z},\label{pert0enerz2e2n+}\Seqnsplit
 &&z \to \infty {:}\quad \tilde \theta_0=0,\quad \tilde Y_0=0; \quad z=0{:}\quad \tilde \theta_0=0, \quad \tilde Y_0= 0. \label{pertbc0+}
\end{eqnarray}}\noindent
Introducing the notation $\kappa_\pm(\omega)\equiv \overline
m[-1\pm\sqrt{1+4\mathrm{i}\omega/\overline m^2}]/2$,
Re$(\kappa_+)>0$, Re$(\kappa_-)<0$, the general solutions of
(\ref{pert0speciesze2n+})--(\ref{pert0enerz2e2n+}) takes  the general
form
{\begin{eqnarray}
z \geqslant 0{:}\quad\tilde
\theta_0(z,\omega)=A_-\text{e}^{\kappa_-z}+A_+\text{e}^{\kappa_+z}+
\frac{1}{\kappa_--\kappa_+}\left[\frac{\text{e}^{-\overline m \,z}- 
\text{e}^{\kappa_+z}}{\overline m+\kappa_+}- \frac{\text{e}^{-\overline m\,z}- 
\text{e}^{\kappa_-z}}{\overline m+\kappa_-}\right]q\,\overline m\,\tilde m_0,
\end{eqnarray}}\noindent
where $A_\pm(\omega)$ are constants of integration determined by the
boundary conditions. According to the boundary condition in the
unburned gas, elimination of the divergence for $z\to \infty$ yields
$A_+=[(\kappa_--\kappa_+)(\overline m+\kappa_+)]^{-1}q\overline
m\,\tilde m_0$ and  $A_-=\tilde \theta_0(0)-A_+$ to satisfy $\tilde
\theta_0(0)=0$ ($\delta \theta=O(1/\beta$)), 
{\begin{eqnarray}
q\tilde Y_0= \tilde \theta_0=\frac{\text{e}^{-\overline m\,z}-
\text{e}^{\kappa_-z}}{(\overline m+\kappa_+)(\overline
m+\kappa_-)}q\,\overline m\,\tilde m_0, \quad
\left.\frac{\text{d}\tilde \theta_0}{\text{d}z} \right
\vert_{z=0^+}=-\frac{1}{\overline m+\kappa^+}q\,\overline m\,\tilde
m_0.
\end{eqnarray}}\noindent
The first jump condition (\ref{jump01}) ${\text{d}\tilde
\theta_0}/{\text{d}z}  \vert_{z=0^+}=-q \overline m\tilde
\theta_{b1}/2$ yields
{\begin{eqnarray}
\tilde m_0(\omega)=\frac{[\overline m+\kappa^+(\omega)]}{2}\tilde \theta_{b1}(\omega).\label{tim0}
\end{eqnarray}}\noindent
The $1/\beta$-modification to the flame temperature $\tilde
\theta_{b1}(\omega)$ is obtained by solving the first order in the
$1/\beta$ expansion of the linear Equations
(\ref{pertspeciesze2n+})--(\ref{pertbc+}) 
{\begin{eqnarray}
z\geqslant 0{:}\quad&&\left[\mathrm{i}\omega \tilde Y_1- \overline m  \frac{\text{d}\tilde Y_1}{\text{d}z}-\frac{\text{d}^2\tilde Y_1}{\text{d}z^2}\right ]=-\tilde m_1(\omega) \overline m\,\text{e}^{-\overline m \,z},\label{pert1speciesze2n+}\Seqnsplit
&&\left[\mathrm{i}\omega \tilde \theta_1- \overline m  \frac{\text{d}\tilde \theta_1}{\text{d}z}-\frac{\text{d}^2\tilde \theta_1}{\text{d}z^2}\right ]=-q\tilde m_1(\omega) \overline m\,\text{e}^{-\overline m \,z}+[q \text{e}^{- \overline m z}+(1-q)]\mathrm{i}\omega \tilde f,\label{pert1enerz2e2n+}\Seqnsplit
 &&z \to \infty {:}\quad \delta \theta_1=(1-q)\delta f,\quad\delta Y_1=0; 
 \quad z=0{:}\quad \delta \theta_1= \theta_{b1}, \quad\delta Y_1= 0. \label{pert1bc+}
\end{eqnarray}}\noindent
Elimination of $\tilde m_1(\omega)$ by considering $\tilde
Z_1(z,\omega)\equiv \tilde \theta_1(z,\omega)-(1-q)\tilde f-q \tilde
Y_1(z,\omega)$ yields
{\begin{eqnarray}
&&\frac{\text{d}^2\tilde Z_1}{\text{d}z^2}+\overline m\frac{\text{d}\tilde Z_1}{\text{d}z}-\mathrm{i}\omega \tilde Z_1=-q\text{e}^{-\overline m z}\mathrm{i}\omega \tilde f\label{pert1enerz2e2n+b+}\Seqnsplit
&& z \to \infty {:}\quad \tilde Z_1=0; \quad z=0{:}\quad \tilde Z_1= \tilde \theta_{b1}(\omega)-(1-q)\tilde f.\label{pert1bc+b+}
\end{eqnarray}}\noindent
The solution of (\ref{pert1enerz2e2n+b+}) satisfying the boundary
condition (\ref{pert1bc+b+}) is expressed in terms of $\tilde
\theta_{b1}(\omega)$
{\begin{eqnarray}
z \geqslant 0{:}\quad &&\tilde Z_1=[\tilde \theta_{b1}-(1-q)\tilde f]\text{e}^{\kappa_-z}-\frac{\text{e}^{-\overline m\,z}- \text{e}^{\kappa_-z}}{(\overline m+\kappa_+)(\overline m+\kappa_-)}q\,\mathrm{i}\omega \tilde f\Seqnsplit
&&\left.\frac{\text{d}\tilde Z_1}{\text{d}z} \right \vert_{z=0^+}=\left.\frac{\text{d}(\tilde \theta_{1}-q\tilde Y_1)}{\text{d}z} \right \vert_{z=0^+}=\kappa_-[\tilde \theta_{b1}-(1-q)\tilde f]+\frac{q\,\mathrm{i}\omega \tilde f}{\overline m+\kappa_+}.
\end{eqnarray}}\noindent
The second jump condition in (\ref{jump01}), 
{\begin{eqnarray}
\left.\frac{\text{d}\tilde \theta_{1}}{\text{d}z} \right \vert_{z=0^-}=\kappa_-[\tilde \theta_{b1}-(1-q)\tilde f]+\frac{q\,\mathrm{i}\omega \tilde f}{\overline m+\kappa_+}\label{dt-}
\end{eqnarray}}\noindent
yields the perturbation of the flame temperature $\tilde
\theta_{b1}(\omega)$ in terms of the pressure fluctuation $\tilde
f(\omega)$ when the solution for $\tilde \theta_{1}(z,\omega)$  is
known in the burned gas flow ($z<0$). Using (\ref{unpertsttsol})
$\overline \theta_0=1$ ($\tilde \theta_0=0$) in the burned gas,  the
linearized version of Equation (\ref{enerz2e2n+}) and the boundary
condition (\ref{bc+<0}) read 
{\begin{eqnarray}
z\leqslant 0{:}\quad  \frac{\text{d}^2 \tilde \theta_1}{\text{d} z^2}+\overline m\frac{\text{d} \tilde \theta_1}{\text{d} z}-\mathrm{i}\omega \tilde \theta_1=-\mathrm{i}\omega \tilde f \quad z \to -\infty : \quad\tilde \theta_1\to \tilde f; \quad z=0: \quad\tilde \theta_1= \tilde \theta_{b1}. 
\label{pert1bc++bis}
\end{eqnarray}}\noindent
The solution yields the temperature fluctuation in the burned gas (of
order $1/\beta$)
{\begin{eqnarray}
z\leqslant 0{:}\quad\tilde \theta_1(z,\omega)=\tilde f(\omega)+(\tilde \theta_{b1}-\tilde f)\,\text{e}^{\kappa_+z}, \quad \left.\frac{\text{d}\tilde \theta_{1}}{\text{d}z} \right \vert_{z=0^-}=\kappa_+[\tilde \theta_{b1}-\tilde f].
\end{eqnarray}}\noindent
The fluctuation of flame temperature is then given in terms of the
pressure fluctuation by (\ref{dt-})
{\begin{eqnarray}
&&(\kappa_+-\kappa_-)[\tilde \theta_{b1}-\tilde f]= q\kappa_-\tilde f+q\frac{\mathrm{i}\omega \tilde f}{\overline m+\kappa_+}\Seqnsplit
\tilde \theta_{b1}(\omega)&=&\tilde f(\omega)+\frac{q/\overline m}{\sqrt{1+\frac{4\mathrm{i}\omega}{\overline m^2}}}\left [-\frac{1+ \sqrt{1+\frac{4\mathrm{i}\omega}{\overline m^2}}}{2}\overline m+\frac{2/\overline m}{1+ \sqrt{1+\frac{4\mathrm{i}\omega}{\overline m^2}}}\mathrm{i}\omega\right]\tilde f(\omega) \label{tildethetab1}\Seqnsplit
\lim_{\omega\to 0}\tilde \theta_{b1}&=&(1-q)\tilde f+2q \frac{\mathrm{i}\omega}{\overline m^2}\tilde f+O(\omega^2).\label{lflT}
\end{eqnarray}}\noindent
According to  (\ref{tildethetab1}), the fluctuation of flame
temperature is proportional to the pressure if the heat release is zero
(no gas expansion across the flame)  $\lim_{q \to 0}\delta
\theta_{b}=\delta f$ as it should be  (uniform compressional heating). 
The fluctuation of the laminar flame velocity  obtained from
(\ref{tim0})  is 
{\begin{eqnarray}
\tilde m_0(\omega)=[\overline m+\kappa_+(\omega)]\frac{\tilde \theta_{b1}(\omega)}{2}, \quad \lim_{\omega \to 0}\tilde m_0(\omega)=\frac{1-q}{2}\overline m\left [\tilde f+\frac{1+q}{1-q}\frac{\mathrm{i}\omega}{\overline m^2} \tilde f+O(\omega^2)\right]\label{lflm0}
\end{eqnarray}}\noindent
in agreement with (4.7) in~\cite{PCPPLH1990}. Back to the original
variable, the solution for a slowly varying pressure obtained from
(\ref{lflT})--(\ref{lflm0}) takes the form
{\begin{eqnarray}
\frac{1}{\overline f}\frac{\text{d} f}{\text{d}\tau}\ll 1{:}\quad&&{\delta \theta_{b1}(\tau)}=({1-q})\left[f(\tau)+\Delta \tau_\theta\frac{\partial f}{\partial \tau}+\cdots\right ], \quad\Delta \tau_\theta=\frac{2q}{(1-q)}\frac{1}{\overline m^2},\label{thetab1vsfbi-}\Seqnsplit
 &&\frac{\delta m_0(\tau)}{\overline m}=\frac{1-q}{2}\left[f(\tau)+\Delta \tau_m\frac{\partial f}{\partial \tau}+\cdots\right ], \quad\Delta \tau_m= \frac{(1+q)}{(1-q)}\frac{1}{\overline m^2}. \label{thetab1vsfbi}
\end{eqnarray}}\noindent
According to the ZFK solution (\ref{expft+}) $\delta \overline
m/\overline m=\delta \overline \theta_{b1}/2$, the first terms  on the
right-hand side of (\ref{thetab1vsfbi-}) and (\ref{thetab1vsfbi})
correspond to the change in flame temperature and laminar flame
velocity on the branch of the physical quasi-steady solutions when the
pressure is modified, $ \delta \overline{\theta}_{b1}=(1-q)\delta f$,
$\delta \overline m/\overline m=(1-q)\delta f/2$. Unsteady
modifications to the inner flame structure  are described by the
following terms that are proportional to the time derivative of the
pressure. The variation of flame temperature $\beta(\theta_b-\overline
\theta_b)$ in (\ref{tildethetab1}) in the first Equation (\ref{lflm0}),
relative to its quasi-steady modification, can be conveniently
rewritten as
{\begin{eqnarray}
\tilde \theta_{b1}(\omega)-(1-q)\tilde f = q\tilde {\mathcal{Y}}(\mathrm{i}\omega)\frac{4 \mathrm{i}\omega}{\overline m^2}\tilde f, \quad \tilde {\mathcal{Y}}(\mathrm{i}\omega)\equiv \frac{1}{\sqrt{1+{4\mathrm{i}\omega}/{\overline m^2}}}\frac{1}{\left[1+\sqrt{1+{4\mathrm{i}\omega}/{\overline m^2}}\right]}, \label{causal}
\end{eqnarray}}\noindent
where $ \lim_{\omega \to 0}\tilde {\mathcal{Y}}={1}/{2}$ to give the
time dependent flame temperature in the form
{\begin{eqnarray}
\delta\theta_{b1}(\tau)-(1-q)\delta f(\tau)=\frac{4}{\overline m^2}q\int_{-\infty}^{+\infty}\text{d}\tau'\mathcal{Y}(\tau-\tau')\frac{\partial \delta f(\tau')}{\partial \tau'}.\label{causal 3}
\end{eqnarray}}\noindent
Equation (\ref{causal 3}) satisfies the principe of causality for a
linear response to the time derivative of the pressure (forcing term).
This is not the case for the link of the flame temperature (or the
laminar flame velocity) with the instantaneous pressure since Equations
(\ref{thetab1vsfbi-})--(\ref{thetab1vsfbi}) can be interpreted as the
first term of an expansion  in powers  of $(1/f)\text{d}f(\tau)/\text{d}
\tau$ of the flame temperature and laminar flame velocity, $\delta
\theta(\tau)\approx (1-q)f(\tau+\Delta \tau_\theta)+\cdots $, ${\delta
m(\tau)}/{\overline m}\approx [{(1-q)}/{2}]f(\tau+\Delta \tau_m)+\cdots$
at a later time ($\Delta \tau_\theta>0$ for the flame temperature and 
$\Delta \tau_m>0$ for the laminar flame velocity).  This promotes an
instability of the physical branch of quasi-steady solutions presented
now.

\subsection{Instability mechanism}

The instability mechanism of the quasi-steady solutions for $S$ fixed
is illustrated when the unsteadiness of the burned gas flow is
neglected leading to the same expression for the instantaneous
back-flow as in (\ref{vb-va1}) for the quasi-steady solutions  
{\begin{eqnarray}
{\text{v}}_b=S\,m, \quad \delta {\text{v}}_b\vert_{S=\text{cst.}}=S\,\delta m\vert_{S=\text{cst.}}\label{vbSqs}.
\end{eqnarray}}\noindent
Using (\ref{vbSqs}) and the same notation as in (\ref{enerz2e2n+})
$\delta f =\beta \varepsilon [(\gamma-1)/\gamma] \delta \rmpi_{P}(\tau)$,
Equation (\ref{vb-va3+lin}) yields
{\begin{eqnarray}
S\tilde m_0-\frac{\sqrt{1-q}}{\beta\varepsilon(\gamma-1) }\tilde f =- \frac{q}{2} \overline m \,\tilde \theta_{b1}.\label{vb-va3+lin+-}
\end{eqnarray}}\noindent
Using (\ref{lflm0}) $\tilde m_0=(\overline m+\kappa^+)\tilde
\theta_{b1}/2$ 
{\begin{eqnarray}
S\,(\overline m+\kappa^+)\frac{\tilde \theta_{b1}}{2}-\frac{\sqrt{1-q}}{\beta\varepsilon(\gamma-1) }\tilde f &=&- q  \overline m\ \frac{\tilde \theta_{b1}}{2},\label{linSfi3}\Seqnsplit
(S+q)\overline m \frac{\tilde \theta_{b1}}{2}-\frac{\sqrt{1-q}}{\beta\varepsilon(\gamma-1) }\tilde f  &=&-S\kappa^+\frac{\tilde \theta_{b1}}{2}, \quad \kappa^+\equiv \frac{\overline m}{2}\left [-1+\sqrt{1+4\mathrm{i}\omega/\overline m^2}\right ].\label{linSfi4}
\end{eqnarray}}\noindent
After substitution $\mathrm{i}\omega \to \sigma$, an equation for the linear
growth rate $\sigma$ (dispersion relation) is obtained when the
expressions for $ \tilde \theta_{b1}$ and $\tilde m_0$ in
(\ref{causal})  are introduced into (\ref{linSfi4})
{\begin{eqnarray}
\hspace*{-2em}(S+q)\overline m\frac{(1-q)}{2}-\frac{\sqrt{1-q}}{\beta\varepsilon(\gamma-1) }  &=&-(S+q)\overline m\,\frac{q}{2}\mathcal Z(\sigma)-S\frac{\overline m}{4}(\sqrt{1+{4\sigma}/{\overline m^2}}-1)[ (1-q)+q\,\mathcal Z(\sigma)]\Seqnsplit
\vert\sigma\vert \ll 1{:} \quad &\approx& -\left [(S+q)\overline m\frac{q}{4}+S\frac{\overline m}{8}(1-q)\right ]\,\frac{4}{\overline m^2}\sigma +O(\sigma^2)\nonumber\\
&\approx& -\frac{1}{2\overline m}[S(1+q)+2q^2]\sigma=-\frac{\overline m}{2}(1-q)[S\Delta\tau_m+q\Delta\tau_\theta]\sigma.\label{tildethetab1bi++}
\end{eqnarray}}\noindent
The left-hand side of (\ref{tildethetab1bi++}) is zero at the critical
elongation  and negative  on the physical branch of the quasi-steady
solutions as noticed by the derivative of (\ref{vb-va5}) with respect
to the pressure
{\begin{eqnarray}
(S+q)\text{e}^{(1-q)\,\overline f/2}&=& (S_i+q)+\frac{\sqrt{1-q}}{\beta\varepsilon(\gamma-1) }\,\overline f, \quad f\equiv\beta \varepsilon [(\gamma-1)/\gamma] \rmpi_{P}, \quad \overline m=\text{e}^{(1-q)\,\overline f/2}\label{vb-va5++}\Seqnsplit
\frac{\text{d}S}{\text{d}\overline f}\mathrm{e}^{(1-q)\overline f/2}&=&-(S+q)\text{e}^{(1-q)\overline f/2}\frac{(1-q)}{2} +\frac{\sqrt{1-q}}{\beta\varepsilon(\gamma-1) }. \label{dS(f)}
\end{eqnarray}}\noindent
The physical branch being characterized by ${\text{d}S}/{\text{d}
\overline f}>0$, the left-hand side of (\ref{tildethetab1bi++}) is
negative so that this branch is unstable since the growth rate is
positive $\sigma>0$ and goes to zero on the turning point
$\lim_{(S^*-S)\to 0^+}\sigma \to 0$, it should be for the exchange of
stability between two branches of quasi-steady solutions merging on a
turning point (saddle-node bifurcation).

To conclude this section, the physical branch of quasi-steady solutions
associated with the back-flow (\ref{vbSqs}) ${\text{v}}_b=S\,m$ is
unstable. The instability of the self-accelerating flame is inherent to
the thermal feedback loop responsible for the turning point; the flame
temperature increases with the pressure of the external flow which
increases in turn with the laminar flame velocity and thus with the
flame temperature.

\subsection{Stabilisation by unsteadiness effects in the burned gas flow}

The instability concerns flames subjected to a back-flow proportional
to the laminar flame velocity ${\text{v}}_b=S\,\text{u}_b$ corresponding
to a burned gas flow in quasi-steady state when the pressure varies
while the unsteady effects have been retained in the cold flow ahead of
the flame. A more consistent stability analysis should take into
account the unsteadiness of the burned gas flow. Unfortunately the
unsteady flow of burned gas in a finger  flame is too complicated to be
studied analytically. Hopefully a detailed study is not needed  and a
phenomenological modeling is sufficient to improve our understanding of
the DDT.

\subsubsection{Delayed back-flow model}

Following~\cite{Searby96}, the longitudinal gradient of burned gas flow
on the tube axis $u(\text{x}, t)$ is roughly modeled by a source term of
mass whose origin is the burning of the lateral flame parallel to the
wall.  Denoting the laminar flame velocity (relative to the burned gas)
of the lateral flame $U_{bw}(\text{x},t)$, the gradient of the flow on
the tube axis $u(\text{x}, t)$ is approximated by a one-dimensional 
mass conservation in a one-dimensional incompressible flow 
{\begin{eqnarray}
\frac{\partial u}{\partial \text{x}}= 
\frac{2}{R}U_{bw}(\text{x},t)
  \label{elonglawdelayed1}
\end{eqnarray}}\noindent
where $R$ is the radius of the tube. The longitudinal back-flow
$u_b(t)$ impinging the tip from behind is obtained by integration along
the tube axis. For a closed-end tube on the burned gas side, assuming
an incompressible flow of burned gas at rest behind the  foot of the
finger-flame as in Figure~\ref{fig.1}, one gets
{\begin{eqnarray}
 u_b(t)\equiv u(\text{x}=\text{X}_P(t),t)=\frac{2}{R}\int_{\text{X}_P-L}^{X_P}U_{bw}(\text{x},t)\text{d}\text{x}\label{elonglawdelayed2}
\end{eqnarray}}\noindent
where $\text{X}_P(t)$ is the position of the tip and $L(t)$ the length
of the elongated flame.  The preceding model $u_b(t)=S(t)\,U_b(t)$
($S=2L/R$ in cylindrical geometry) is obtained by neglecting both heat
loss on the wall and unsteadiness  of the burned gas flow so that
$U_{bw}$ was considered as uniform along the lateral flame front and
equal to the laminar flame velocity on the tip at the same time
$U_{bw}(t)=U_b(t)$.  Taking advantage of the planar approximation valid
under the condition described in Section~\ref{V1dapprox}, the unsteady
effects of the burned gas flow  can be modeled by a delay  $\Delta
t(\text{X}_P-\text{x})$ introduced by the quasi-planar downstream-running
compression waves between the state of the gas controlling the laminar
flame velocity $U_b(t)$ on the tip at time $t$  and on the lateral wall
at a distance $X_P-\text{x}$ from the tip $U_{bw}(\text{x},t)\approx
U_{b} (t-\Delta t(\text{X}_P-\text{x}) )$,
{\begin{eqnarray}
 u_b(t)\approx\frac{2}{R}\int_{\text{X}_P-L}^{X_P}U_{b} (t-\Delta t(\text{X}_P-\text{x}) )\text{d}\text{x}.\label{elonglawdelayed33}
\end{eqnarray}}\noindent
Assuming short delays compared to the characteristic time  on the tip
$U_b(t)$, $ U_b/(\text{d}U_b/\text{d}t) \gg \Delta t(\text{X}_P-\text{x})$
and neglecting second order effects, a Taylor expansion yields
{\begin{eqnarray}
U_{b} (t-\Delta t)\approx U_{b} (t)+ \Delta t\,\text{d}U_b/\text{d}t \quad \Rightarrow  \quad u_b(t)\approx\frac{2L(t)}{R}U_b(t) -\frac{2}{R}\frac{\text{d}U_b}{\text{d}t}\int_{\text{X}_P-L}^{X_P}\Delta t(\text{X}_P-\text{x})\text{d}\text{x}.\label{elonglawdelayed34-}
\end{eqnarray}}\noindent
According to the downstream-running acoustic wave in the burned gas 
with  a quasi-constant sound speed $a_b$, $\Delta t(\text{X}_P-\text
x)\approx (\text{X}_P-\text{x})/a$,   $\Delta t_w\equiv \int_{\text
X_P-L}^{X_P}\Delta t(\text{X}_P-\text{x})\text{d}\text{x}\approx
L^2/2a_b,$ Equation (\ref{elonglawdelayed34-}) yields, after
introducing the overall delay $\Delta t_w$,
{\begin{eqnarray}
\text{delayed back-flow: } \quad u_b(t)\approx S(t)\left[ U_b(t)-\Delta t_w\frac{\text{d}U_b}{\text{d}t}\right], \quad \Delta t_w\equiv \frac{L}{2a_b}.\label{delayedmod}
\end{eqnarray}}\noindent
It is worth stressing that, due to the thermal amplification by a large
activation energy,  the modification to $u_b$ is mainly due to the
change in burned gas flow $U_b$ issued from the lateral flames, the
effect of the flow velocity of the acoustic wave being negligible.

\subsubsection{Stability of the quasi-steady solutions}\label{stabanal}

The additional delay introduced by the delayed back-flow counteracts
the unstable mechanism associated with the delays
(\ref{thetab1vsfbi-})--(\ref{thetab1vsfbi}) introduced by the response
of the inner flame structure. Introducing (\ref{delayedmod}) written in
non-dimensional form 
{\begin{eqnarray}
\text{v}_b(t)\approx \left[ m_{o}-\Delta \tau_w\frac{\text{d}
m_{o}}{\text{d}\tau}\right]S, \quad \tilde {\text{v}}_b \approx (1-
\Delta \tau_w \sigma)\tilde m_0(\sigma)S\quad \Delta \tau_w\approx
\frac{L/ t_{bi}}{2a_b}=\frac{\varepsilon}{2}
\frac{L}{d}\label{delayedmod2}
\end{eqnarray}}\noindent
into the linearized form (\ref{vb-va3+lin-}) of the master equation
yields the dispersion relation in the form
{\begin{eqnarray}
(1- \Delta \tau_w \sigma)\tilde m_0S-\frac{\sqrt{1-q}}{\beta\varepsilon(\gamma-1) }\tilde f =- \frac{q}{2} \overline m \,\tilde \theta_{b1}\label{vb-va3+lin+-+}
\end{eqnarray}}\noindent
where the linear expressions of $ \tilde \theta_{b1}$ and $\tilde m_0$
proportional to $\tilde f$ are given by the response to pressure
fluctuations in (\ref{thetab1vsfbi-}) and  (\ref{thetab1vsfbi}).
Equation (\ref{vb-va3+lin+-+}) which differs from  (\ref{vb-va3+lin+-})
by the term $- \Delta \tau_w \tilde m_0S\sigma=- \Delta \tau_w\overline
m(1-q)S\sigma/2+O(\sigma^2)$ yields 
{\begin{eqnarray}
\vert\sigma\vert \ll 1:\quad (S+q)\overline
m\frac{(1-q)}{2}-\frac{\sqrt{1-q}}{\beta\varepsilon(\gamma-1) } 
&=&-\frac{\overline m}{2}(1-q)[S(\Delta\tau_m-\Delta
\tau_w)+q\Delta\tau_\theta]\sigma+O(\sigma^2)
\label{tildethetab1bi++new} \qquad
\end{eqnarray}}\noindent
instead of (\ref{tildethetab1bi++}). Using the same arguments as in the
text below (\ref{tildethetab1bi++}), Equation
(\ref{tildethetab1bi++new}) shows that the physical branch is stable
under the condition 
{\begin{eqnarray}
\Delta \tau_w>\Delta\tau_m+(q/S)\Delta\tau_\theta \, \Leftrightarrow
\,\varepsilon\frac{L}{d}>\frac{2}{(1-q)\overline
m^2}\left[(1+q)+\frac{2q}{S}\right] \label{condstab}
\end{eqnarray}}\noindent
which is easily fulfilled  under the condition (\ref{cond0-bis}) for
the validity of the asymptotic analysis in the limit (\ref{distlimit1})
since the parameter  $2[(1+q)+2q/S]/[(1-q)\overline m^2]$ is of order
unity, see the text below  (\ref{cond0-}). Moreover the linear growth
rate still vanishes at the critical condition and changes sign on the
nonphysical branch branch $\overline \vartheta_+(\zeta)$ which becomes
unstable as it should be, see Figure~\ref{fig.1}.  

\section{Nonlinear dynamics. Dynamical saddle-node bifurcation}\label{NLdyn}

The nonlinear dynamics takes the form of a differential equation for
the pressure on the flame\break $1+\varepsilon\rmpi_P(\tau)$ obtained by the
asymptotic analysis of the unsteady flame structure leading to the
master equation (\ref{vb-va3+}) linking the velocity jump across the
flame structure $u_b-u_u$ to the instantaneous flame temperature
$T_b(t)$, see Figure~\ref{flamestructure}. Using the notation
$\theta_{b1}\equiv \beta[T_b(\tau)-T_b(0)]/T_b(0)=O(1)$ and the delayed
back-flow (\ref{delayedmod2}) one gets 
{\begin{eqnarray}
\left[ m_{o}(\rmpi_P)-\Delta \tau_w\frac{\text{d} m_{o}}{\text{d}
\tau}\right](1+\epsilon \tau)S_i-\left[ S_i+q+\frac{\sqrt{1-q}}{\gamma
}\rmpi_P(\tau) \right]  =- q  
\text{e}^{\theta_{b1}(\rmpi_P)/2}+O(\varepsilon),
\label{vb-va3+bibi++}
\end{eqnarray}}\noindent
where the first and the second term on the left-hand side are
respectively the instantaneous back-flow $u_b(t)$ and the flow velocity
just ahead of the flame $u_u(t)$ as it is modified from its initial
value $\rmpi_P(0)=0$, $u_u(0)=S_i+q$ by the downstream running
compression waves in the external flow ahead of the flame, see
(\ref{vuvi--}) and Figure~\ref{flamestructure}. For a weakly perturbed
inner structure of the flame, the flame temperature
$\theta_{b1}(\rmpi_P)$ and the laminar flame velocity $m_o(\rmpi_P)$,
expressed in terms of the instantaneous flame pressure $\rmpi_P(t)$, to
be introduced into (\ref{vb-va3+bibi++}), are given by the linear
response of the flame to pressure fluctuations studied in
Section~\ref{linresp}. 

\begin{figure}[t!]
\includegraphics{fig02}
\caption{\label{flamestructure}Sketch of the unsteady flame structure
in the framework of the reaction sheet ($x=0$). The problem being
hyperbolic, the boundary condition at $x\to \pm \infty$ is the initial
condition in the burned  and unburned gas, $\lim_{x\to -\infty}
u_b(t)=u_b(0)$ and $\lim_{x\to +\infty} u_u(t)=u_u(0)$ respectively.}
\end{figure}

\begin{figure}[t!]
 \includegraphics{fig03}
\caption{\label{fig:traj}Sketch of the solutions ``elongation $\zeta$
versus pressure $\vartheta$'' (not to scale). The two branches
$\overline \vartheta_\pm (\zeta)$ of the quasi-steady solutions of
(\ref{NLeq}) $\zeta=F(\vartheta)$  in thick line show the critical
elongation $\zeta^*$, above which there is no quasi-steady solutions
(saddle-node bifurcation); $\overline \vartheta_-$ in solid line is the
physical solution which is stable under the condition (\ref{condstab}).
The thin red curve represents the finite-time singularity of the
solution of the dynamical equation in (\ref{NLeqdyn4}). The sudden
increase in pressure and flame temperature $\vartheta$ leads to a
spontaneous DDT at finite time for an elongation $\zeta=\zeta_c$
slightly larger than the critical elongation $\zeta^*$. }
\end{figure}

\subsection{Nonlinear dynamics for a quasi-steady inner structure of the flame}

To enlighten the essential features of the nonlinear dynamics nearby
the critical conditions, it is worth starting by assuming a
quasi-steady inner structure of the flame so that the expressions of
$\theta_{b1}/2$ and $m_o$ in terms of $\rmpi_P$ are the same as
(\ref{expft+}) for the steady-state solution $\theta_{b1}/2 =b \rmpi_P$,
$m_o =\text{e}^{b\,\rmpi_P}$ yielding $\text{d}m_o/\text{d}\tau=\text{e}^{b\,
\rmpi_P}b\text{d}\rmpi_P/\text{d}\tau$. Introducing these  expressions  into
(\ref{vb-va3+bibi++}) yields
{\begin{eqnarray}
\left [S_i(1+\epsilon \tau)+q\right ]\text{e}^{b\rmpi_P(\tau)}-\left[ S_i+q+\frac{\sqrt{1-q}}{\gamma }\rmpi_P(\tau) \right] 
&=&(1+\epsilon \tau)S_i\Delta \tau_w\text{e}^{b\rmpi_P}\,b\,\frac{\text{d}\rmpi_P(\tau)}{\text{d}\tau}.
\label{vb-va6bis}
\end{eqnarray}}\noindent

The term on the right-hand side is the dynamical effect of the
unsteadiness of the downstream running compression waves in the burned
gas, leading to the delay $\Delta \tau_w>0$ in the back-flow. If this
delay is neglected $\Delta \tau_w=0$, Equation  (\ref{vb-va6bis})
reduces to the nonlinear equation for the quasi-steady solutions
characterized by a turning point.   Using the same notation as in
(\ref{newnot-})--(\ref{newnot}), $\vartheta\equiv b\rmpi_P$ for the
pressure (or the flame temperature when the flame structure is in
steady state), $\zeta\equiv (1+\epsilon \tau)S_i+q$ for the elongation
(or the time since the time derivative $\text{d}\zeta/\text{d}\tau=
\epsilon S_i$ is a constant) and a parameter of order unity $\text
b\equiv (\beta \varepsilon/2)(\gamma-1)\sqrt{1-q}$, Equation
(\ref{vb-va6bis}), after simplification by $\text{e}^\vartheta$, takes
the form of a non-autonomous (and nonlinear) ordinary differential
equation for $\vartheta(\zeta)$, namely the function ``pressure versus
elongation'',
{\begin{eqnarray}
\zeta-\left[\zeta_i+\frac{\vartheta }{\text{b}}\right ]\text{e}^{-\vartheta}=\epsilon (1+\epsilon \tau)S_i^2\Delta \tau_w\frac{\text{d}\vartheta}{\text{d}\zeta}.
\label{NLeqdyn}
\end{eqnarray}}\noindent
For clarity it is worth introducing the function $F(\vartheta) =
\left[\zeta_i+{\vartheta }/{\text{b}}\right ]\text{e}^{- \vartheta}$
which presents a maximum at the critical point  $\zeta^*=(1/\text{b})
\text{e}^{-\vartheta^{*}}$\unskip, $\vartheta^*=1-\zeta_i\text{b}$, see
(\ref{elong*}). Neglecting terms of order $\epsilon^2$, 
Equation~(\ref{NLeqdyn}) can then be written 
{\begin{eqnarray}
\zeta-F(\vartheta)=\epsilon S_i^2\Delta \tau_w\frac{\text{d}\vartheta}{\text{d}\zeta}.
\label{NLeqdyn1}
\end{eqnarray}}\noindent
The quasi-steady solutions $\overline \vartheta(\zeta)$ are represented
by the two real roots of the left-hand side of (\ref{NLeqdyn1}) $\zeta
=F(\overline \vartheta)$ that exist for $\zeta \leqslant \zeta^*$. The
two branches $\overline\vartheta_\pm (\zeta)$ merge at $\zeta=\zeta^*$
and disappear for  $\zeta 
> \zeta^*$, see Figure~\ref{fig:traj} 
and the text below (\ref{elong*}). The physical branch of the
quasi-steady solutions is $\overline \vartheta_- (\zeta)\leqslant
\vartheta^*$ for which $\text{d}\overline \vartheta_-/\text{d}\zeta>0$.
The solution of (\ref{NLeqdyn1}) around a point  $(\vartheta_o,\,
\zeta_o)$ on the physical branch of quasi-steady solutions $
\zeta_o=F(\vartheta_o)$ is obtained by a power expansion in the form
$\delta \vartheta=\vartheta-\vartheta_o$, $\delta \zeta=\zeta-\zeta_o$
{\begin{eqnarray}
\delta \zeta-\delta \vartheta\left.\frac{\partial F}{\partial \vartheta}\right \vert_{\vartheta=\vartheta_o}-\frac{1}{2}(\delta \vartheta)^2\left.\frac{\partial^2 F}{\partial \vartheta^2}\right \vert_{\vartheta=\vartheta_o}=\epsilon S_i^2\Delta \tau_w\frac{\text{d}\delta \vartheta}{\text{d}\delta \zeta},
\label{NLeqdyn2}
\end{eqnarray}}\noindent
where $\vartheta_o<\vartheta^*$ :  $\partial F/\partial
\vartheta\vert_{\vartheta=\vartheta_o}>0$ and $\partial F/\partial
\vartheta\vert_{\vartheta=\vartheta^*}=0$, $\partial^2 F/\partial
\vartheta^2\vert_{\vartheta=\vartheta^*}=-\zeta^*<0$. The two first
terms on the left-hand side of (\ref{NLeqdyn2}) represent the linear
dynamics of the quasi-steady solutions on the physical branch. The
stability of the steady solution at $\zeta=\zeta_0$ on the physical
branch is obtained from the second term  on the left-hand side using
$\delta \zeta=0$ in the first term.  For a positive delay $\Delta
\tau_w>0$ introduced by the downstream running acoustic waves in the
burned gas, the physical branch of quasi-steady solutions $\overline
\vartheta_-(\zeta)\leqslant\zeta^*$ is stable, $\text{d}F/\text{d}\zeta
> 0$. The third term on the left-hand side of (\ref{NLeqdyn2}) becomes
dominant near the critical point $\partial F/\partial
\vartheta\vert_{\vartheta=\vartheta^*}=0$ indicating how the critical
dynamics is strongly modified by the nonlinear feedback loop of
temperature in the compressible flow ahead of the flame.  Focusing the
attention on the vicinity of the critical point $\delta
\vartheta=\vartheta-\vartheta^*$, $\delta \zeta=\zeta-\zeta^*$, the
second term on the left-hand side of (\ref{NLeqdyn2})  vanishes and we
are left with 
{\begin{eqnarray}
\frac{\zeta-\zeta^*}{\zeta^*}+\frac{1}{2}(\vartheta-\vartheta^*)^2=\tilde \epsilon \frac{\text{d}(\vartheta-\theta^*)}{\text{d}(\zeta/\zeta^*)}
\quad \text{where }\tilde \epsilon=\epsilon \frac{S_i^2}{\zeta^{*2}}\Delta \tau_w>0.
\label{NLeqdyn3ibis}
\end{eqnarray}}\noindent
The two roots of the left-hand side $\zeta<\zeta^*$: $\overline
\vartheta_\pm=\vartheta^*\pm\sqrt{2(\zeta^*-\zeta)/\zeta^*}$ are the
quasi-steady solutions (flame pressure or temperature versus
elongation) resulting from the compressional heating by the downstream
running compression waves in the unburned gas ahead of the flame that
are generated by the back-flow without delay, $u_b=S(\tau)U_b(\tau)$. 
However, the dynamics is drastically modified near the critical
condition by the  quadratic term on the left-hand side of
(\ref{NLeqdyn3ibis}). Generally speaking, Equation (\ref{NLeqdyn3ibis})
describes the  dynamics nearby a saddle-node bifurcation and was
extensively used for sharp transitions in different problems of physics
or biophysics.

\subsection{Dynamical saddle-node bifurcation}\label{dybsanobif}

Conveniently rescaled
{\begin{eqnarray}
(1/2^{2/3})(1/\tilde \epsilon)^{1/3}(\vartheta-\vartheta^*) \to y', \quad (1/2^{1/3})(1/\tilde \epsilon)^{2/3}(\zeta-\zeta^*)/\zeta^* \to t',
\end{eqnarray}}\noindent
Equation (\ref{NLeqdyn3ibis})  takes,  after  multiplication by
$(1/\sqrt 2\tilde \epsilon)^{2/3}$, the generic normal form 
{\begin{eqnarray}
\frac{\text{d}y'(t')}{\text{d}t' }=t' + y'^{2}
\label{snbif2b}
\end{eqnarray}}\noindent
which was extensively used in the theory of catastrophic events
recently revisited and extended  in~\cite{Pomeau2012}. For $t'<0$, the
solution of (\ref{snbif2b}) has two fixed points collapsing at $t'=0$,
the stable one corresponding to the negative root $y'=-\sqrt{-t'}$
(physical branch of solutions).   Considering an initial condition on
the stable branch $t'=t'_i<0: y'=-\tsqrt{-t_i'}$  for $-t_i'/t'_c
={y'}^2_i/t'_c$ large enough, the asymptotic solution of
(\ref{snbif2b}) is obtained in terms of the Airy function to give
{\begin{eqnarray}
\lim_{t'\to t'_c}y'(t')=\frac{1}{t'_c-t'}-\frac{t'_c}{3}(t'_c-t')+\cdots \quad \text{where } t'_c\approx 2.338\ldots,
\label{snbif3b}
\end{eqnarray}}\noindent
see the references in~\cite{Pomeau2012}.   The finite-time singularity
of the solutions to (\ref{snbif3b}) is of the same type as obtained
with the Riccati equation ${\text{d}y'}/{\text{d}t' }=y'^{\,2}$.  The
pressure and  flame temperature $\vartheta\equiv b\rmpi_P(\tau)$ blow up
at finite time $\tau=\tau_c$ for a finite elongation 
$\zeta_c=S_i(1+\epsilon\tau_c)+q$,
$(\zeta_c-\zeta^*)=(\tau_c-\tau^*)\epsilon S_i$
{\begin{eqnarray}
\frac{(\zeta_c-\zeta^*)/\zeta^*}{( 2\tilde \epsilon^2)^{1/3}}= 2.338\ldots, \quad b(\rmpi_u-\rmpi_u^*)\equiv \vartheta-\vartheta^*\approx 2\tilde \epsilon\frac{\zeta^*}{\zeta_c-\zeta}=2\frac{S_i}{\zeta^*}\frac{\Delta \tau_w}{\tau_c-\tau}. 
\label{asymbe}
\end{eqnarray}}\noindent
The solution $\vartheta(\tau)$ still exists above the critical
elongation $\zeta^*$ and  diverges at $\zeta=\zeta_c>\zeta^*$ 
($\tau_c>\tau^*$) like $\Delta \tau_w/(\tau_c-\tau)$ where
$\tau_c-\tau^*\propto 1/\epsilon ^{1/3}$, see Figure~\ref{fig:traj}.
The smaller the elongation rate $\epsilon$, the closer  to the critical
elongation  $S^*$ the divergence of the pressure, $\lim_{\epsilon \to
0}(S_c-S^*)/S^*\propto \epsilon ^{2/3}$. In contrast to 
Section~\ref{Ftsqs} where the  compression waves in the burned gas are
neglected ($\Delta \tau_w=0$)  a violent increase in pressure and flame
temperature develops suddenly for an elongation $S=S_c$ slightly larger
than $S^*$. The existence of a finite-time singularity is not limited
to the quadratic expansion on the left-hand side of
(\ref{NLeqdyn3ibis}); a similar singularity  is exhibited by the
solution of (\ref{NLeqdyn}) which is not limited to small values of
$\vert\vartheta-\vartheta^*\vert$.

Notice that Equation (\ref{NLeqdyn3ibis}) for $\tilde \epsilon <0$
would lead to $-{\text{d}y'(t')}{\text{d}t' }=t' + y'^{\,2}$ showing
that the stable branch would be $y'=\sqrt{-t'}$ and the finite-time
singularity would correspond to $\lim_{t'\to t'_c}y'=-\infty$.
Therefore the sign $\tilde \epsilon >0$ in (\ref{NLeqdyn3ibis}) is
essential for relevant physical insights into the DDT of
self-accelerated laminar flames in tubes. 

\subsection{Effects of the unsteadiness of the inner structure of the flame}\label{Eusof}

A small unsteadiness of the inner structure of the flame does not
modify qualitatively the nonlinear dynamics provided the quasi-steady
solutions are stable. Considering a characteristic time scale of the
dynamics larger than the transit time $t_b$ (slow dynamics generated by
a slow increase in elongation (\ref{elongrate}) for $\epsilon\ll1$),
the disturbances of the flame temperature and of the laminar flame
velocity due to the time derivative of the pressure on the right-hand
side of  (\ref{energyred}) (energy conservation) are accurately modeled
by the linear expressions (\ref{thetab1vsfbi-})--(\ref{thetab1vsfbi}). 
Introducing the latter into (\ref{vb-va3+bibi++}) and following the
same development as in Section~\ref{stabanal} one gets neglecting
$\epsilon^2$-terms
{\begin{eqnarray}
\zeta-F(\vartheta)=\epsilon S_i^2\left(\Delta \tau_w-\Delta \tau_m-\frac{q}{S_i}\Delta \tau_\theta\right )\frac{\text{d}\vartheta}{\text{d}\zeta}
\label{NLeqdyn4}
\end{eqnarray}}\noindent
leading to the same equation as (\ref{NLeqdyn3ibis}) but with a
different expression for $\tilde \epsilon$
{\begin{eqnarray}
\tilde \epsilon=\epsilon \frac{S_i^2}{\zeta^{*2}}\left(\Delta \tau_w-\Delta \tau_m-\frac{q}{S_i}\Delta \tau_\theta\right )
\label{NLeqdyn3ibisnew}
\end{eqnarray}}\noindent
where  $\tilde \epsilon>0$ in the stable case, see 
Section~\ref{stabanal}.  The results presented in the preceding section
are still valid ($\Delta \tau_w\to\Delta \tau_w-\Delta \tau_m-{q}\Delta
\tau_\theta/{S_i}>0$) as long as $\tilde \epsilon {\text{d}
\vartheta}/\text{d}\zeta$ remains not much smaller than unity  that is
until the slope ${\text{d}\zeta}/\text{d}\vartheta$ of the curve in red
in Figure~\ref{fig:traj} is larger than $\tilde \epsilon$. According to
(\ref{asymbe}), this is the case for $\zeta_c-\zeta>\tilde \epsilon$,
namely until a time $\tau$ relatively close to $\tau_c$,
$(\tau_c-\tau)\approx \tilde \epsilon^{1/3}(\tau_c-\tau^*)$ for which
the relative increase in flame temperature is of order unity
$(\vartheta-\vartheta^*)/\vartheta^*=O(1)$.  However the results of
Section~\ref{dybsanobif} are no longer accurate for
$(\vartheta-\vartheta^*)/\vartheta^*>1$  because of the break down of
the slow timescale assumption  used in
(\ref{thetab1vsfbi-})--(\ref{thetab1vsfbi}). 

However, the existence of a finite-time singularity is not doubtful, as
discussed in the next section.  A simple explanation can be given for
the finite time singularity sketched by the red curve in
Figure~\ref{fig:traj}. The generic divergence of the flame acceleration
in the quasi-steady solutions $\overline \vartheta_-(\zeta)$ at the
bifurcation $\zeta=\zeta^*$, $\text{d}\vartheta_-/\text
d\zeta\vert_{\zeta=\zeta^*}=\infty$ is delayed to
$\zeta=\zeta_c>\zeta^*$ by the delay $\Delta \tau_w$ of the back-flow
appearing on the left-hand side of (\ref{vb-va3+bibi++}) and the
solution $\vartheta(\zeta)$ must go to infinity since there is no
steady state solution above $\zeta^*$ but the expression (\ref{asymbe})
of $\zeta_c$, obtained with the quadratic approximation,  may not be
accurate. The singularity  is thus expected to appear systematically
whatever the elongation rate and the delay in the back-flow, as small
as they may be.

\section{Discussion of the results and conclusion}\label{Conclusion} 

The nonlinear dynamics of the one-dimensional model for a
self-accelerated flame on the tip of an elongated flame front has been
investigated in the double limit of large activation energy and small
Mach number of laminar flames.  Starting with a small growth rate of
elongation from a self-similar solution (constant elongation), the
downstream running compressible waves coupled with the quasi-isobaric
flame produces a sudden transition. The role of the compressible waves
is different in the unburned and burned gas flows. In the unburned gas,
the flame acts as an accelerating  semi-transparent piston and the
compressional heating produces a nonlinear thermal feed back loop
responsible for a saddle-node bifurcation similar to that 
characterizing the  self-similar solutions. The compressible waves  in
the burned gas flow introduce unsteadiness in the back-flow. The latter
is the mechanism through which the flame on the tip is accelerated when
the elongation of the finger like front increases (increase in surface
area of the finger-shaped flame front).  The joined effect of  these
two unsteady mechanisms lead to a finite-time singularity of the flame
temperature and pressure in the form of a saddle-node bifurcation. 

 This was proved in a first step, assuming a flame structure in steady
state. Clearly, the assumption of a inner structure of the flame in
quasi-steady state is no longer valid as soon as the timescale of the
dynamics becomes smaller than  the transit time across the laminar
flame structure as it is the case for a finite-time singularity of the
flame temperature. When the flames associated with a constant
elongation are stable against pressure disturbances, the qualitative 
feature of the finite-time singularity is shown by a perturbation
analysis to not be modified by the unsteadiness of the inner structure.
This analysis is valid until the slow time scale assumption in the
dynamics of the inner structure of the flame breaks down in the
ultimate divergence of the flame temperature and pressure. In addition
to the comment at the end of Section~\ref{Eusof}, there are two other
good reasons to conjecture that a finite-time singularity characterizes
effectively the unsteady inner solution of the reactive flow of a
self-accelerated laminar flame: firstly, flame quenching cannot occur
with an increase in flame temperature, secondly there is no other
\mbox{combustion} waves in a one-dimensional geometry than laminar flames and
detonations. Nevertheless, transient regimes similar to intermittent
quenching and re-ignition of the detonation cannot be completely ruled
out, even though they are not likely in the absence of oscillatory
flames (unity Lewis number) and  galloping detonations.  Some work
remains to be done in this direction.

A strong shock should be generated by the pressure runaway leading
quasi-instantaneously to the DDT onset. The pre-conditioned  state of
unburned gas just ahead of the flame and just before the abrupt
transition  is characterized  by a universal critical Mach number of
the induced flow of unburned gas which is close to unity, in agreements
with experiments and direct numerical simulations. This critical
condition is all the easier to achieve in very energetic mixtures for
an elongation which is not so large. This could well be the case for
the cellular structure of Rayleigh--Taylor unstable flame fronts of very
energetic mixtures as those involved in supernovae SNIa. On the
contrary, the transition to detonation of laminar flames in tubes
filled with weakly energetic mixtures is not possible because the
predicted critical elongation is too large to be observed. 

To summarize, the DDT mechanism  presented in this article concerns the
one-dimensional dynamics of reacting flows characterized by a  rate of
heat release highly sensitive to the temperature. Although the origin
of the self-induced flow responsible for the flame acceleration is
multidimensional (increase in surface area  of the  elongated flame
front), the DDT onset is a local process of a one-dimensional nature.
This mechanism of transition concerns also turbulent wrinkled flames
and/or unconfined cellular flames, the flame brush being considered as
a chaotic array of elongated flames the tip of which is accelerated by
the self-induced flow associated with an increase in surface area of
the wrinkled flame front. In that sense, the DDT mechanism described
here could have a  certain degree of universality. This should be
confirmed by direct numerical simulations. 

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

\section*{Acknowledgments}

Yves Pomeau is acknowledged for enlightening discussion on the
dynamical saddle-node bifurcation. I am grateful to Grisha Sivashinsky
for drawing my attention few years ago on the work of Joulin and
Deshaies. I thank Misha Liberman for fruitful discussions on DDT and
comments on the manuscript. I thank also Professors Bruno Denet, Guido
Lodato, Luc Vervisch and the Phd students Ra\'ul~Hern\'andez-S\'anchez
and Hassan Tofaili for lively interactions during their numerical
activities.  Partial financial support of \textit{Agence National de la
Recherche} (contract ANR-18-CE05-0030)  is acknowledged.

\CDRGrant[ANR]{ANR-18-CE05-0030}

\back{}

\bibliographystyle{crunsrt}
\bibliography{crmeca20230756}
\refinput{crmeca20230756-reference.tex} 

\end{document}
