Existence of travelling waves and high activation energy limits for a onedimensional thermo-diffusive lean spray flame model

TL;DR

This paper mathematically analyzes a thermo-diffusive lean spray flame model, proving the existence of travelling waves and identifying two distinct combustion regimes in the high activation energy limit.

Contribution

It introduces a rigorous mathematical framework for understanding combustion regimes in spray flames, including explicit profiles and transition conditions.

Findings

Existence of travelling wave solutions in the model

Identification of diffusion limited and vaporisation controlled regimes

Explicit characterization of combustion velocities and profiles

Abstract

We provide a mathematical analysis of a thermo-diffusive combustion model of lean spray flames, for which we prove the existence of travelling waves. In the high activation energy singular limit we show the existence of two distinct combustion regimes with a sharp transition -- the diffusion limited regime and the vaporisation controlled regime. The latter is specific to spray flames with slow enough vaporisation. We give a complete characterisation of these regimes, including explicit velocities, profiles, and upper estimate of the size of the internal combustion layer. Our model is on the one hand simple enough to allow for explicit asymptotic limits and on the other hand rich enough to capture some particular aspects of spray combustion. Finally, we briefly discuss the cases where the vaporisation is infinitely fast, or where the spray is polydisperse.

Equations

$$A\to P,$$

$${\partial }_t\rho +{\partial }_x(\rho v)=D_t\rho +\rho {\partial }_{x}v=0,$$

$$\rho D_{t}v+{\partial }_{x}p=\frac{4}{3}{\partial }_x(\kappa {\partial }_{x}v),$$

$$\rho c_p{D}_{t}T-{\partial }_x(\lambda {\partial }_{x}T)=Q\omega +D_{t}p+\frac{4}{3}\kappa ({\partial }_{x}v{)}^2,$$

$$\omega =B(T)\frac{\rho Y}{\mu }\exp \left(-\frac{E}{RT}\right),$$

$$\rho D_{t}Y-{\partial }_x(\gamma {\partial }_{x}Y)=-\mu \omega .$$

$${\mathrm{d}}_{t}M=-\varphi (T,M).$$

$$D_{t}M=-\varphi (T,M).$$

$$D_{t}n=0,$$

$$\rho D_{t}Y-{\partial }_x(\gamma {\partial }_{x}Y)=-\mu \omega -D_t{\rho }_l=-\mu \omega -n{D}_{t}M.$$

$$v_0{\rho }^{\prime }+(\rho v{)}^{\prime }=0,$$

$$\rho v_0{v}^{\prime }=0,$$

$$c_p(\rho v_0+\rho v)T^{\prime }-(\lambda T^{\prime }{)}^{\prime }=Q\omega ,$$

$$n^{\prime }=0,$$

$$(\rho v_0+\rho v)Y^{\prime }-(\gamma Y^{\prime }{)}^{\prime }=-\mu \omega -n(v_0+v)M^{\prime }.$$

$$c{c}_p{T}^{\prime }-(\lambda T^{\prime }{)}^{\prime }=Q\omega ,$$

$$c{Y}^{\prime }-(\gamma Y^{\prime }{)}^{\prime }=-\mu \omega -c\frac{n_0}{{\rho }_0}{M}^{\prime },$$

$$c\frac{n_0}{{\rho }_0}{M}^{\prime }=-n_0\varphi (T,M),$$

$$T(-\infty )=T_u,Y(-\infty )=Y_u,n_0=n_u,M(-\infty )=M_u,$$

$$Y(+\infty )=0,M(+\infty )=0.$$

$$\frac{c_p}{Q}T+\frac{1}{\mu }\left(Y+\frac{n_0}{{\rho }_0}M\right)$$

$$T_b=\frac{Q}{c_p\mu }\left(Y_u+\frac{n_0}{{\rho }_0}{M}_u\right).$$

$$u=\frac{T-T_u}{T_b-T},v=\frac{{\rho }_{0}Y}{{\rho }_0{Y}_u+n_0{M}_u},m=\frac{M}{{\rho }_0{Y}_u+n_0{M}_u},$$

$$\begin{array}[]{rll}-u^{\prime\prime}+cu^{\prime}&=&\tilde{f}(u)v\hskip 10.00002pt\mathrm{on}\hskip 5.0pt\mathbbm{R},\\ -\Lambda v^{\prime\prime}+cv^{\prime}&=&-\tilde{f}(u)v-cn_{0}m^{\prime}\hskip 10.00002pt\mathrm{on}\hskip 5.0pt\mathbbm{R},\\ cm^{\prime}&=&-\tilde{\phi}(u,m)\hskip 10.00002pt\mathrm{on}\hskip 5.0pt\mathbbm{R},\\ u(-\infty)=0,&&u(+\infty)=1,\\ v(-\infty)=v_{u},&&v(+\infty)=0,\\ m(-\infty)=m_{u},&&\end{array}$$

$$f_{\epsilon }(u):=\frac{1}{{\epsilon }^2}\exp \left(\frac{u-1}{\epsilon }\right),$$

$$\mu :=\underset{\epsilon \to 0}{\lim }{\int }_{{\theta }_i}^1{f}_{\epsilon }(s)(1-s)\mathrm{d}s.$$

$$\begin{array}[]{rcl}-u^{\prime\prime}+cu^{\prime}&=&c\delta_{x=\bar{x}},\\ -\Lambda v^{\prime\prime}+cv^{\prime}&=&-c\delta_{x=\bar{x}}-cn_{0}m^{\prime}\hskip 10.00002pt\mathrm{on}\hskip 5.0pt\mathbbm{R},\\ cm^{\prime}&=&-\phi(u,m)\hskip 10.00002pt\mathrm{on}\hskip 5.0pt\mathbbm{R},\\ u(-\infty)=0,&u(0)=\theta,&u(+\infty)=1,\\ v(-\infty)=v_{u},&&v(+\infty)=0,\\ m(-\infty)=m_{u}.&&\end{array}$$

$$c=\min \left\{\sqrt{2\mu /\Lambda },c_{\star }(m_u)\right\},$$

$$\begin{array}[]{rcl}-u^{\prime\prime}+cu^{\prime}&=&f(u)v\hskip 10.00002pt\mathrm{on}\hskip 5.0pt(-a,a),\\ -\Lambda v^{\prime\prime}+cv^{\prime}&=&-f(u)v-cn_{0}m^{\prime}\hskip 10.00002pt\mathrm{on}\hskip 5.0pt(-a,a),\\ cm^{\prime}&=&-\phi(u,m)\hskip 10.00002pt\mathrm{on}\hskip 5.0pt(-a,a),\\ -u^{\prime}(-a)+cu(-a)=0,&&u(a)=1,\\ -\Lambda v^{\prime}(-a)+cv(-a)=cv_{u},&&v(a)=0,\\ m(-a)=m_{u},&&\\ u(0)=\theta_{i}.&&\end{array}$$

$$c>0,$$

Full text

Our model is on the one hand simple enough to allow for explicit asymptotic limits and on the other hand rich enough to capture some particular aspects of spray combustion. Finally, we briefly discuss the cases where the vaporisation is infinitely fast, or where the spray is polydisperse.

Keywords: reaction-diffusion equations, spray flames, travelling waves, high activation energy, singular limits

1. Introduction

This paper provides a rigourous mathematical analysis of some aspects of spray combustion, including the analysis of the so-called high activation energy limit for a spray flame model. This notion of high activation energy limit was first introduced in the pioneering work of Zeldovich and Frank-Kamenetskii [42], and refers to the limit where the combustion rate is much faster than any other physical phenomenon, in particular diffusion. Since then, there have been many studies of these asymptotics and applications to gaseous flames. However, gas-vapor-droplets systems have many applications in industry or everyday’s life, such as diesel or propulsion engines. When trying to understand some of their specific features, it appears that the structure of these two-phase flames as well as their speed or stability are greatly affected by the presence of vaporising liquid droplets possibly interacting with the combustion zone.

The behaviour of spray flames has been investigated a lot in the literature and a wide variety of regimes were considered. Dating from the 70s and early 80s, we can quote the works of Polymeropoulos et al. [31][32], Mizutani et al. [25][26], Hayashi et al. [17][18]. Ballal et al. [2]. They present studies of the propagation of liquid fuel sprays, study the influence of the size of the droplets, the type and geometry of the spray flame as well as its structure. The investigation of those vapor–drop–air systems was continued in the 90s. To quote but a few works, we refer to the work of Aggarwal and Sirignano [1] as well as the papers of Greenberg, Tambour and Silverman [34][35], where the structure of spray flames as well as the influence of parameters such as droplet size, fuel volatility, or equivalence ratio are also investigated analytically. More elaborate situations appearing in propulsion engines are for example pulsating or acoustic instabilities. We refer to [16][15][11][12][13][7] and more recently [8][14][20][21][22] for studies in that direction.

Only few of these studies of spray flames involve rigorous–in–the–mathematical–sense analysis of spray flames models where existence, uniqueness, or asymptotic limits are derived (see e.g. [23, 8]). This is in contrast with purely gaseous combustion, where a lot of results exist in the mathematical literature for various regimes and asymptotics. In particular, a complete study of thermo-diffusive lean gaseous flame fronts and the high activation energy limit is in the paper of Berestycki, Nicolaenko and Scheurer [4], and in the paper of Berestycki and Larrouturou [3]. The originality of the present work is to provide a complete mathematical analysis of the counterpart of those systems, namely a lean spray flame model that on the hand is simple enough to allow for explicit asymptotic limits, and on the other hand rich enough to capture some particular aspects of spray combustion. Those results should be compared with the very interesting work of Suard, Nicoli and Haldenwang [38]. Using numerical simulations, these authors investigate the scaling laws of the spray flame with respect to the vaporisation rate of the liquid phase. When the vaporisation is fast enough, the velocity of the spray flame is comparable to that of the gaseous flame where the whole reactant would be present only in a gaseous form. This is the diffusion controlled regime. On the contrary when the time for complete vaporisation exceeds a critical value, the velocity of the spray flame starts to decrease. This is the so–called vaporisation controlled regime. In this case, they also investigated the structure of the reaction zone, and showed that it has a more complex structure than a gaseous flame.

In this work, we prove the existence of spray flame travelling waves and the existence of two distinct combustion regimes with a sharp transition in the high activation energy (HAE). In this limit, we provide a complete characterisation of the profiles, that can be written explicitly provided simple analytic expressions of the vaporisation law. We also show that the internal combustion layer is likely to be much larger in the vaporisation controlled regime as compared to the size of the internal combustion layer of a comparable gaseous flame.

1.1. The equations of the lean spray flame model

The classical gaseous reactive flow.

A gas mixture of total mass density $\rho$ is considered to be made of $N$ fluids corresponding to the different species present in the mixture. If the mass density of those species are $(\rho_{i})_{i=1,N}$ then by definition $\rho=\sum_{i}\rho_{i}$, and the respective mass fractions $(Y_{i})_{i=1,N}$ of the species are defined by $\rho_{i}=\rho Y_{i}$. In the case of a single reactant $Y$, the chemical reaction writes as

[TABLE]

where $A$ denotes the reactant species and $P$ denotes one (or a linear combination) of the species produced by the reaction. Notice that the rate of variation of the single reactant $A$ is directly proportional to minus that of the products and it suffices to determine the mass fraction $Y$ of the reactant $A$ in order to determine the mass fractions of the other species.

The mass, momentum, energy equation for a reactive flow in one dimension without external forces write, setting $D_{t}:=\partial_{t}+v\partial_{x}$,

[TABLE]

[TABLE]

[TABLE]

where $v$ is the mass–average velocity of the mixture, $p$ the hydrostatic pressure, $\kappa$ the dynamic viscosity coefficient, $T$ the temperature, $c_{p}$ the specific heat at constant pressure, $\lambda:=\lambda(T)$ the thermal conductivity, $Q$ the chemical heat release of the reaction and $\omega$ is the rate at which the reaction occurs and has the form

[TABLE]

where $E$ is the activation energy of the single reaction, $R$ the perfect gas constant, $T_{A}=E/R$ the activation energy temperature, $\mu$ the molecular mass of the reactant $A$, and $B(T)$ a non–stiff prefactor. Finally the equation for the mass fraction $Y$ of the reactant $A$ obeys

[TABLE]

We refer the reader to [37, 41] for more details.

The liquid phase.

In the situation where the reactant $A$ is also present in the mixture in the form of vaporising liquid droplets that are well dispersed in the mixture, one can treat the liquid phase as a new continuous “species”, where the droplets are homogeneously spread inside the mixture, leading to both homogeneized vaporisation and combustion in the bulk of the gaseous phase. This is certainly not satisfied in practice, where spray combustion can typically involve flames surrounding individual droplets, with strong temperature gradients within the flame. However, we need this hypothesis for the derivation of our model. Also, ideally the continuity equation for the gaseous mixture should take into account the volume fraction occupied by the liquid phase, the momentum equation should incorporate the drag forces, the energy equation should include the loss of heat due to the latent heat of the liquid as well as the dissipation due to the drag forces, etc.

We are going to consider regimes in which we can discard those couplings. Since the mass density of the liquid reactant $A$ is in general several orders of magnitude larger than that of the gaseous species, on can assume that the volume fraction of the liquid phase is small. The drag forces are always present and can produce complex turbulent flows. However, the drag force is typically inversely proportional to the surface area of the droplets and proportional to the difference of velocities of the droplets and the surrounding gas. Small enough droplets behave as “passive scalars”, that is particles whose velocity can be set equal to that of the surrounding fluid. The drag forces are internal forces for the two-phase flow and do not affect the total momentum $(\rho_{l}v_{l}+\rho v)$ of the liquid–gas mixture. As a consequence, since $v_{l}\simeq v$, one can neglect the drag forces in the gaseous mixture if the mass density $\rho_{l}$ of the liquid phase is small enough compared to the mass density $\rho$ of the gas mixture. The same conclusion holds for the energy equation.

Let now $M$ denote the mass of an isolated vaporising droplet immersed in a gas. The vaporisation law of the droplet can be very complex (see e.g. [39, 40, 30, 41, 36]). As an approximation, we will assume that the vaporisation rate $\phi$ only depends upon the temperature $T$ of the surrounding gas and the mass $M$ of the droplet , i.e.

[TABLE]

We will be mainly interested in the monodisperse case, where all droplets in the unburnt gas have the same size. In the laminar flows with constant velocity that we will be considering, this implies that particles located at the same position $x$ at time $t$ all have the same size. We can therefore introduce the mass profile $M(t,x)$ which obeys,

[TABLE]

The situation for polydisperse sprays is briefly considered in the HAE limit in Section 4. Let further $n(t,x)$ be the number density of droplets, that is the number of droplets per unit volume. The mass density of the liquid phase writes $\rho_{l}(t,x)=n(t,x)M(t,x)$. In the laminar flows we are considering, the droplets do not coalesce nor break–up, hence

[TABLE]

that is the number density of the droplets is simply advected by the flow, and $D_{t}\rho_{l}=n(t,x)D_{t}m$. It follows that the continuity equation for the gaseous reactant $A$ now incorporates the vaporisation flux as a source term

[TABLE]

Finally, we assume that the effects of the latent heat are negligible. This is true when the mass density of the liquid phase is small enough compared to that of the gaseous mixture, which we assumed also in order to neglect the drag forces. The study of the influence of the latent heat is certainly an important and interesting feature since it can greatly influence the speed or existence of the flame. However this study is beyond the scope of the present work.

Two–phase system.

We are interested in laminar, low Mach number, regimes where one can neglect the pressure terms as well as the kinematic viscosity. We want to study the existence and properties of travelling waves moving at constant speed $v_{0}$ to the left. In the frame of reference of an observer moving at speed $v_{0}$ to the left, the profiles are steady, solution to the system

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

From the continuity equation, the mass flux $c:=\rho(v_{0}+v)$ is a constant. Moreover $v^{\prime}=0$ implies $v=0$ in the moving frame, and therefore also $\rho^{\prime}=0$, so that the gas density is constant, $\rho:=\rho_{0}$. Setting also $n:=n_{0}$, a constant, the system reduces to

[TABLE]

[TABLE]

[TABLE]

with prescribed values $(T_{u},Y_{u},n_{u},M_{u})$ in the unburnt region,

[TABLE]

and assuming complete reaction and complete vaporisation on the burnt region

[TABLE]

The temperature $T(+\infty):=T_{b}$ in the burnt region is obtained by integrating the equation for the quantity

[TABLE]

on $(-\infty,+\infty)$. It follows

[TABLE]

Normalized system.

The normalized variables are $(u,v,m)$ defined as

[TABLE]

solution to the system,

[TABLE]

where $\Lambda=\gamma c_{p}/\lambda$ is the reciprocal of the Lewis number, and with appropriate renormalized formulas for the reaction rate $\tilde{f}$ and the vaporisation rate $\tilde{\phi}$ (we omit the tildes below). Also $n_{0}\geqslant 0$, $v_{u}\geqslant 0$, $m_{u}\geqslant 0$ and $v_{u}+n_{0}m_{u}=1$.

Reaction and vaporisation laws.

In order to avoid the cold boundary effect, we assume that the reaction rate is zero below the normalized ignition temperature $0\char 60\relax\theta_{i}\char 60\relax 1$, namely $f:[0,1]\rightarrow\mathbbm{R}^{+}$, $f(u)=0$, for all $0\leqslant u\char 60\relax\theta_{i}$, $f$ positive on $(\theta_{i},1]$ and Lipschitz continuous on $[\theta_{i},1]$.

For the study of the high activation energy limit, we will use on $(\theta_{i},1]$ the normalized arrhenius law

[TABLE]

where $\varepsilon$ denotes the inverse of the (normalized) activation temperature. We also introduce $\mu$, $0\char 60\relax\mu\char 60\relax+\infty$, defined as

[TABLE]

We assume that vaporisation starts at temperature $\theta_{v}\char 62\relax 0$, i.e. $\phi(u,m)=0$ for all $0\leqslant u\char 60\relax\theta_{v}$ and we also impose $\phi(u,0)=0$ for all $0\leqslant u\leqslant 1$. It is also natural to assume that $\theta_{v}\char 60\relax\theta_{i}$ and to impose $\phi:[0,1]\times[0,1]\rightarrow\mathbbm{R}^{+}$ is an increasing function of $u$, a decreasing function of $m$, and is a positive Lipschitz function on $[\theta_{v},1]\times[0,1]$. Let be given any temperature $\theta$, with $1\char 62\relax\theta\char 62\relax\theta_{v}$, and any mass $m$. We finally make the natural assumption that the interval of time $\tau(\theta,m)$ needed for complete vaporisation of a droplet of initial mass $m$ at temperature $\theta$ is finite.

1.2. Main results and summary

Section 2 is devoted to the proof of the existence of travelling waves solutions to system (2). Precisely,

Theorem 1**.**

**(existence of travelling waves) **System (2) admits a solution in $X=\mathcal{C}^{1}(\mathbbm{R})\times\mathcal{C}^{1}(\mathbbm{R})\times\mathcal{C}^{0}(\mathbbm{R})\times\mathbbm{R}$.

This result involves a priori bounds that are proved in Section 2.1, and where we emphasize the technical difficulties induced by the presence of vaporising droplets. This allows us to prove in Section 2.2 the existence of solutions to a system similar to (2) but defined on a bounded domain with appropriate boundary conditions. The proof of Theorem 1 follows in Section 2.3.

Section 3 is devoted to the asymptotic analysis of the system in the high activation energy limit. In Section 3.1, we prove the convergence of the system towards a Dirac model for spray flames:

Theorem 2**.**

**(the Dirac model for spray flames) **Let the reaction term $f_{\varepsilon}(\cdot)$ as in (3). There exists a decreasing sequence $(\varepsilon_{n})_{n\in\mathbbm{N}}$ such that $(u_{\varepsilon_{n}},v_{\varepsilon_{n}},m_{\varepsilon_{n}},c_{\varepsilon_{n}})$ solution to (2) converges in $H^{1}(\mathbbm{R})\times H^{1}(\mathbbm{R})\times H^{1}(\mathbbm{R})\times\mathbbm{R}$ to $(u,v,m,c)$, solution to the problem

[TABLE]

Here $\bar{x}=-\log\theta_{i}/c$.

For any given $c\char 62\relax 0$, the corresponding solution $(u,v,m)$ is uniquely determined and one can find explicit expressions for all the unknowns, provided the expression for $\phi$ can be explicitly integrated. The complete characterisation of the system in the HAE limit is therefore dictated by the exact value of the velocity $c\char 62\relax 0$ of the travelling wave. The latter is determined thanks to the analysis of the internal combustion layer, also proved in Section 3.1:

Theorem 3**.**

**(internal combustion layer analysis) **Let $(u,v,m,c)$ the limiting profile in the high activation energy limit. The velocity $c$ is given by

[TABLE]

where $c_{\star}(m_{u})$ is the unique velocity such that the droplets finish vaporising exactly at the position of the reaction zone.

This result provides a rigorous justification of the existence of a vaporisation controlled regime, i.e. a regime where $c=c_{\star}(m_{u})\char 60\relax\sqrt{2\mu/\Lambda}$ (see [38]). This regime appears when the time for complete vaporisation of the droplets is large enough, precisely when $m_{u}\char 62\relax m_{u}^{\star}$, where $m_{u}^{\star}$ is defined by $c_{\star}(m_{u}^{\star})=\sqrt{2\mu/\Lambda}$.

Section 3.2 provides more details about the internal combustion layer in the vaporisation controlled regime. An upper estimate of the size of the region where combustion and vaporisation overlap is given in Theorem 4, followed by an application to the so–called $d^{2}$–law for vaporisation.

Finally, Section 4 presents some immediate applications of our analysis to the case of polydisperse sprays, fast vaporisation, or radial geometry. We refer to [6] for a more detailed exposition of those results.

Sketch of typical profiles.

The pictures below sketch typical profiles for spray travelling waves moving from the right to the left. The temperature profile $u$ is in red, the gaseous reactant profile $v$ is in blue and the liquid phase density profile $n_{0}m$ is in green. Recall that $u(-\infty)=0$, $v(-\infty)=v_{u}$, $m(-\infty)=m_{u}$ and $u(+\infty)=1=v_{u}+n_{0}m_{u}$. We set $n_{0}=1$.

Figure 1 shows typical profiles for gas flames in the absence of droplets, for a Lewis number equal to unity. In Figure 2, droplets are present in the fresh gas with mass density $n_{0}m_{u}=0.4$. We therefore set $v_{u}=0.6$ so that the temperature in the burnt gas remains equal to unity. Notice that the gaseous reactant profile is no longer monotoneous, due to the vaporisation of the liquid phase.

Figures 3 & 4 are typical profiles in the high activation energy limit. The reaction zone is reduced to a point located at $\bar{x}=0$ in Figure 3 and at $\bar{x}=2.5$ in Figure 4. In both cases vaporisation starts at $x_{v}=-7.5$. However, in the first case the vaporisation front, i.e. the point $x_{vf}$ where vaporisation ends, is located before the reaction front, at $x_{v}=-2.5\char 60\relax 0$. Our analysis will show that in such a situation, the velocity of the spray flame in the HAE limit is equal to that of the purely gaseous flames with same temperature in the burnt gas. In the second case, the vaporisation ends at $x_{vf}=2.5$. Shall the temperature profile remain the same as in Figure 3, the vaporisation process would still occur after the reaction zone. Our analysis will show that this is an impossibility and that the preheating zone “has to” stretch so that the reaction front $\bar{x}=0$ from Figure 3 is “pushed” to the right until it coincides with the vaporisation front $\bar{x}=x_{vf}=2.5$ as in Figure 4. The stretching of the preheating zone is synonimous with a decrease of the velocity of the flame. This is the so–called vaporisation controlled regime (see [38]).

2. Existence of travelling waves

Let $[-a,a]$ an interval of $\mathbbm{R}$ for some $a\char 62\relax 0$. Let $X_{a}=\mathcal{C}^{1}([-a,a])\times\mathcal{C}^{1}([-a,a])\times\mathcal{C}^{0}([-a,a])\times\mathbbm{R}$. We consider system (2) restricted to the interval $[-a,a]$ with Dirichlet boundary conditions at $x=a$ and flux conditions at $x=-a$,

[TABLE]

The last equation allows one to break the translation invariance in the limit where $a$ goes to infinity. We first prove a priori estimates for solutions $(u,v,m,c)\in X_{a}$. This allows us to establish the existence of solutions in a bounded domain by using topological degree arguments. Finally, the existence is extended to the real line.

2.1. Preliminary estimates

The main result of this section consists of a priori estimates for solutions to problem (5) above that are uniform with respect to the size $a$ of the domain. We prove successively the following three results:

Proposition 1**.**

**(qualitative properties) **Let $(u,v,m,c)\in X_{a}$ a solution to (5) with $c\geqslant 0$. We have,

[TABLE]

Proposition 2**.**

**(a priori estimates) **Let $(u,v,m,c)$ a solution to (5) with $c\geqslant 0$. We have

[TABLE]

where $\beta_{1}(\Lambda):=\min(1,\Lambda^{-1})$ and $\beta_{2}(\Lambda):=\max(1,\Lambda^{-1})$.

In the following we note, for all $\theta_{i}\leqslant s\leqslant 1$,

[TABLE]

Proposition 3**.**

**(bounds on the velocity) **Let $\theta^{\star}:=(\theta_{v}+\theta_{i})/2$, $c_{\star}=\sqrt{\tau(m_{u},\theta^{\star})/\log(\theta_{i}/\theta^{\star})}$ and $a_{\star}=\log(\theta_{i}/\theta_{v})/c_{\star}$. For all $a\geqslant a_{\star}$, the solution $(u,v,m,c)$ to problem (5) with $c\geqslant 0$ obeys

[TABLE]

where we introduced $c_{1}=\sqrt{2\beta_{1}(\Lambda)G(1)}/\theta_{i}$, $c_{2}(a)=\sqrt{2\beta_{2}(\Lambda)(G(1)+u^{\prime}(a)^{2})}/\theta_{i}$, $M_{f}=\sup_{s\in[0,1]}f(s)$, $c_{3}=\max\left(M_{f},\log(4/\theta_{i})/a_{\star},2\sqrt{M_{f}/\theta_{i}}\right)$.

**Proof of Proposition 1 (qualitative properties)  ** The presence of vaporising fuel in the model does not allow one to recover all the qualitative properties found in the gaseous case. For example, an important difference is the non–monotonicity of the reactant profile $v(x)$. Also, we introduce below auxiliary functions (such as $v_{\ast}$, $\bar{w}$ or $\bar{v}$) that are specific to the analysis of spray flames, and specific steps such as the proof that $v\geqslant 0$ or the localisation of the vaporisation region.

Proof that $c\char 62\relax 0$.

Recall that we assumed $c\geqslant 0$. Assume by contradiction that $c=0$. Then the vaporisation term $cn_{0}m^{\prime}$ disappears in the equation of the reactant $v$. The function $y:=u+\Lambda v-1$ therefore solves $y^{\prime\prime}=0$ on $(-a,a)$, $y(-a)=y(a)=0$, hence $y=0$ on $[-a,a]$. It follows that $(u,c)$ solves the nonlinear Dirichlet–Neumann boundary value problem

[TABLE]

We then observe that $u\leqslant 1$ on $[-a,a]$ otherwise the maximum $u(x_{0})\char 62\relax 1\char 62\relax\theta$ of $u$ would be attained at $x_{0}\in[-a,a)$ with $f(u(x_{0}))\char 62\relax 0$. The equation above would imply $u^{\prime\prime}(x_{0})\char 62\relax 0$ in contradiction with $x_{0}$ being a maximum. It follows that $w:=u^{\prime}$ obeys $w^{\prime}\leqslant 0$ and $w(-a)=0$, hence $w\leqslant 0$ and $u$ decreasing on $[-a,a]$; a contradiction with $u(0)=\theta\char 60\relax 1=u(a)$.

Proof that $v\geqslant 0$.

Given a solution $(u,v,m,c)\in X_{a}$, introduce $v_{\ast}$ the auxiliary reactant profile solution to the linear elliptic boundary value problem

[TABLE]

The positive function $\phi\equiv 1$ obeys $\mathcal{L}\phi\geqslant 0$ on $(-a,a)$, $\mathcal{N}\phi(-a)=c\char 62\relax 0$. Therefore the problem has a unique solution, obviously $v_{\ast}\equiv 0$, and the generalized maximum principle implies, since $\mathcal{L}v=-cn_{0}m^{\prime}\geqslant 0$, $\mathcal{N}v(-a)=cv_{u}\geqslant 0$ and $v(a)\geqslant 0$, that $v\geqslant v_{\ast}\geqslant 0$ (see e.g. [33, Chapter 2]).

Bounds for $u$, $v$ and $m$.

The bound $0\leqslant m(x)\leqslant m_{u}$ is trivial. Let $w:=-\Lambda v^{\prime}+cv+cn_{0}m$. We have $w^{\prime}=-f(u)v\leqslant 0$, therefore $c=w(-a)\geqslant w(x)\geqslant w(a)=-\Lambda v^{\prime}(a)+cn_{0}m(a)\geqslant 0$. We used $v^{\prime}(a)\leqslant 0$, a consequence of $v\geqslant 0$ and $v(a)=0$. Integrating the equation $-u^{\prime\prime}+cu^{\prime}=-w^{\prime}$ on $[-a,x]$ yields $-u^{\prime}(x)+cu(x)=-e^{cx}(e^{-cx}u)^{\prime}=c-w(x)$. Therefore $-ce^{-cx}\leqslant(e^{-cx}u)^{\prime}\leqslant 0$ and integrating now on $[x,a]$ yields exactly $0\leqslant u(x)\leqslant 1$.

In order to bound $v$ from above, we introduce the auxiliary function $\bar{v}$, solving

[TABLE]

Then let $\bar{w}:=\bar{v}-v$, solution to $-\Lambda\bar{w}^{\prime\prime}+c\bar{w}^{\prime}=cn_{0}m^{\prime}$ on $(-a,a)$, $-\Lambda\bar{w}(-a)+c\bar{w}(-a)=cn_{0}m_{u}$ and $w(a)=0$. Integrating over $[-a,x]$ yields  $-\Lambda e^{cx/\Lambda}(e^{-cx/\Lambda}\bar{w})^{\prime}=cn_{0}m(x)$, and integrating now on $[x,a]$ yields

[TABLE]

obviously $\bar{w}\geqslant 0$, that is $v\leqslant\bar{v}$ and it remains to bound $\bar{v}$ from above. For that we observe that $-\Lambda\bar{v}^{\prime\prime}+c\bar{v}^{\prime}=+w^{\prime}$. Integrating on $[-a,x]$ yields $(-c/\Lambda)e^{-cx/\Lambda}\leqslant(e^{-cx/\Lambda}\bar{v})^{\prime}\leqslant 0$, and integrating now on $[x,a]$ yields exactly  $0\leqslant\bar{v}\leqslant 1$.

Bounds for $u^{\prime}$ and $v^{\prime}$.

We have from the previous step $u^{\prime}(x)=c(u(x)-1)+w(x)\leqslant w(x)\leqslant c$ since $u\leqslant 1$ and $w\leqslant c$. Also the equation for $u$ rewrites $-e^{cx}(e^{-cx}u^{\prime})^{\prime}=f(u)v\geqslant 0$ and integrating from $x$ to $a$ yields $e^{-ca}u^{\prime}(a)\leqslant e^{-cx}u^{\prime}(x)$. Integrating the equation $-(u+\Lambda\bar{v})^{\prime\prime}+c(u+\bar{v})^{\prime}=0$ on $(-a,a)$ allows us to estimate $u^{\prime}(a)=-\Lambda\bar{v}^{\prime}(a)\char 62\relax 0$, since $\bar{v}\geqslant v\geqslant 0$, $\bar{v}(a)=0$ and $\bar{v}$ is not identically zero. The monotonicity $u^{\prime}\char 62\relax 0$ follows. Let again $w:=-\Lambda v^{\prime}+cv+cn_{0}m$. The bounds on $0\leqslant w\leqslant c$, $0\leqslant v\leqslant 1$ and $0\leqslant m\leqslant m_{u}$ easily imply $-c\leqslant v^{\prime}\leqslant c(1+n_{0}m_{u})$.

This concludes the proof of of Proposition 1 **(qualitative properties) **. $\Box$

**Proof of Proposition 2 (a priori estimates)  ** In order to obtain a pointwise comparison of $v$ with $(1-u)$, we take a detour and compare rather $\bar{v}$ with $(1-u)$, where $0\leqslant\bar{v}\leqslant 1$ is the auxiliairy function defined in (10). Observe first that $\bar{v}$ is nonincreasing. Indeed from (10) we have $(e^{-cx/\Lambda}\bar{v}^{\prime})^{\prime}\geqslant 0$, and integrating on $[x,a]$ yields $e^{-cx/\Lambda}\bar{v}^{\prime}(x)\leqslant e^{-ca/\Lambda}\bar{v}^{\prime}(a)\leqslant 0$.

Following [4], introduce now the two auxiliary functions $z:=u+\bar{v}-1$ and $y:=u+\Lambda\bar{v}-1$ and the four relations

[TABLE]

Integrating these equations successively on $[-a,x]$ and $[x,a]$ yields

[TABLE]

Since $\bar{v}$ and $(1-u)$ are nonincreasing, it is not difficult to obtain

[TABLE]

which implies

[TABLE]

It remains to compare precisely $\bar{v}$ and $v=\bar{v}-\bar{w}$. But since $m$ is nonincreasing nonnegative, the expression (11) ensures pointwise $0\leqslant\bar{w}(x)\leqslant n_{0}m(x)$. The result announced follows.

This concludes the proof of Proposition 2 **(a priori estimates) **. $\Box$

**Proof of Proposition 3 (bounds on the velocity)  ** The nonlinear eigenvalue $c$ is estimated thanks to energy estimates.

Estimates on $[0,a]$.

Integrating successively the equation $f(u)v=-u^{\prime\prime}+cu^{\prime}$ against $1$, $u$ and $u^{\prime}$ on $[0,a]$ yields

[TABLE]

where we used $u(0)=\theta_{i}$, $u^{\prime}(0)=c\theta_{i}$. The combination $(\star\star\star)+c[(\star)-(\star\star)]$ reads

[TABLE]

Since $(1-u)\geqslant 0$, the two equations $(\star\star\star)$ and $(\star\star\star\star)$ imply

[TABLE]

It remains to estimate the contribution of the reaction term in the inequalities above.

First upper bound for $c$.

From Proposition 2, we have $v(x)\leqslant\beta_{2}(\Lambda)(1-u(x))$ on $[0,a]$, hence

[TABLE]

which provides the value of $c_{2}(a)$ in Proposition 3 such that

[TABLE]

Localisation of the vaporisation region.

We claim that for small enough velocities, the vaporisation region does not intersect the combustion zone. Recall that $\theta_{v}\char 60\relax\theta_{i}$ so that $\theta^{\star}:=(\theta_{v}+\theta_{i})/2\char 60\relax\theta_{i}$. Let $\tau(m_{u},\theta^{\star})$ the interval of time needed for complete vaporisation of a droplet of size $m_{u}$ at constant temperature $\theta^{\star}$. From the hypothesis on the vaporisation term, we have $\tau(m_{u},\theta^{\star})\char 60\relax\infty$. Given $c\char 62\relax 0$, let $x_{v}$ such that $u(x_{v})=\theta_{v}$, $x^{\star}$ such that $u(x^{\star})=\theta^{\star}$ and $x_{i}=0$ such that $u(x_{i})=\theta_{i}$. Since $u(x)=\theta_{i}e^{cx}$ on $[-a,0]$, we have for any given $c$ and $a$ large enough, $-a\leqslant x_{v}\char 60\relax x^{\star}\char 60\relax x_{i}$ and $x_{i}-x^{\star}=\log(\theta_{i}/\theta^{\star})/c$. The time $\tau^{\star}$ spent by the droplets advected at velocity $c$ inside the interval $[x^{\star},x_{i}]$ is $\tau^{\star}=(x_{i}-x^{\star})/c=\log(\theta_{i}/\theta^{\star})/c^{2}$. However, since $m(x^{\star})\leqslant m_{u}$, and $u\geqslant\theta^{\star}$ on $[x^{\star},x_{i}]$, the monotonicity properties of the vaporisation law imply that if $\tau^{\star}\geqslant\tau(m_{u},\theta^{\star})$ hence complete vaporisation occurs inside the intervall $[x^{\star},x_{i}]$. To prove our claim, it suffices to take $a\geqslant a_{\star}$ and $c\leqslant c_{\star}$, with

[TABLE]

Lower bound for $c$.

Take $a\geqslant a_{\star}$. If $c\geqslant c_{\star}$ there is nothing to prove. Otherwise, we have $c\leqslant c_{\star}$, $m(x)=0$ on $[0,a]$ so that from Proposition 2 we have $v(x)\geqslant\beta_{1}(\Lambda)(1-u(x))$, therefore

[TABLE]

which allows us to define the value $c_{1}$ in Proposition 3 thanks to the relation $c_{1}^{2}\theta_{i}/2:=\beta_{1}(\Lambda)G(1)$.

Second upper bound for $c$.

Let $a_{\star}$ defined above. Let $M:=\sup_{s\in[0,1]}f(s)$. Let $\bar{u}$ be the solution to $-\bar{u}^{\prime\prime}+c\bar{u}^{\prime}=M\mathbbm{1}_{]0,a[}$ on $]-a,a[$, $-\bar{u}^{\prime}(-a)+c\bar{u}(-a)=0$ and $\bar{u}(a)=1$. Since $0\leqslant v\leqslant 1$, the maximum principle asserts that $u\leqslant\bar{u}$ on $[-a,a]$, in particular $\theta_{i}\leqslant\bar{\theta}:=\bar{u}(0)$. Solving explicitly for $\bar{u}$ yields

[TABLE]

Assume $a\geqslant a_{\star}\char 62\relax 0$, then the right hand side tends towards zero as $c$ goes to infinity. More precisely, if both $c\geqslant M$, $e^{-ca_{\star}}\leqslant\theta_{i}/4$ and $M/c^{2}\leqslant\theta_{i}/4$ one has $\bar{u}(0)=\bar{\theta}\leqslant\theta_{i}/2$, a contradiction. It follows that

[TABLE]

is an upper–bound for $c$.

This concludes the proof of Proposition 3 **(bounds on the velocity) **. $\Box$

2.2. Existence of a solution on bounded domains

The proof of the existence of a topological degree relies on the existence of the continuous map $K_{\tau}$ below that is not continuous if the reaction or vaporisation terms have discontinuities at $\theta_{i}$ or $\theta_{v}$ respectively. Following Berestycki–Nicolaenko–Scheurer (see [4, footnote p. 1225]), we assume below that $f$ and $g$ are continuous. The case where a discontinuity is present can be attained by a standard smoothing procedure.

The main point when dealing with a spray flame model is to correctly account for the vaporisation terms in the homotopy argument in order to preserve the a priori estimates of the previous section.

Proposition 4**.**

**(existence of a solution on bounded domains) **There exists $a_{0}\char 62\relax 0$ such that for $a\geqslant a_{0}$ system (5) admits a solution in $X_{a}=\mathcal{C}^{1}([-a,a])\times\mathcal{C}^{1}([-a,a])\times\mathcal{C}^{0}([-a,a])\times\mathbbm{R}$.

**Proof. ** We use the Leray–Schauder topological degree argument. Let $\tau\in[0,1]$ the homotopy parameter and consider the solutions $(u_{\tau},v_{\tau},m_{\tau},c_{\tau})$ to the new system

[TABLE]

This is system (5) where the condition $u(0)=\theta_{i}$ has been rewritten as a fixed point for the velocity $c$ and where the homotopy parmeter appears in the right–hand–side of the system. It is important to notice that this system can be obtained by replacing the vaporisation terms $f(u)$ and $\phi(u,m)$ in (5) by $f_{\tau}:=\tau f$ and $\phi_{\tau}:=\tau\phi$. It follows that all a priori estimates from the previous sections hold.

Fixed point formulation.

We look for solutions $(u_{\tau},v_{\tau},m_{\tau},c_{\tau})$ defined as a fixed point of the mapping $K_{\tau}:X_{a}\rightarrow X_{a}$ that maps $(u,v,m,c)$ to $(\hat{u},\hat{v},\hat{m},\hat{c})$ solution to the linear boundary value problem

[TABLE]

Since $H^{2}(]-a,a[)$ embeds compactly in $\mathcal{C}^{1}([-a,a])$ and $H^{1}(]-a,a[)$ embeds compactly in $\mathcal{C}([-a,a])$, it follows that $K_{\tau}$ is a compact mapping and uniformly continuous with respect to $\tau$. Let $F_{\tau}:=\mathrm{Id}\hskip 5.0pt-K_{\tau}$. A solution $(u_{\tau},v_{\tau},m_{\tau},c_{\tau})$ to system (12) is a fixed point of $F_{\tau}$, i.e. $F_{\tau}(u_{\tau},v_{\tau},m_{\tau},c_{\tau})=0$.

Existence of degree of $F_{\tau}$.

A solution to (12) for $0\leqslant\tau\leqslant 1$ exists as soon as the degree $F_{\tau}$ is well defined and non zero,  Let $\Omega\subset X_{a}$ the open set

[TABLE]

for some positive constants $M$, $\underline{c}$, $\bar{c}$. These constants can be chosen so that for all $0\leqslant\tau\leqslant 1$, $F_{\tau}\neq 0$ on $\partial\Omega$. Indeed, Proposition 2 provides estimates for the case $\tau=1$, namely $0\char 60\relax c_{\star}/2\char 60\relax c\char 60\relax 2c^{\star}$, $\|u\|_{\mathcal{C}^{1}(\bar{I}_{a})}\leqslant 1+c\leqslant 1+c^{\star}$, $\|v\|_{\mathcal{C}^{1}(\bar{I}_{a})}\leqslant 1+\beta_{2}(\Lambda)c^{\star}(1+n_{0}m_{u})$, $\|m\|_{\mathcal{C}(\bar{I}_{a})}\leqslant m_{u}$. Setting $M:=\max(1+c^{\star},1+\beta_{2}(\Lambda)c^{\star}(1+n_{0}m_{u}),m_{u})$, $\underline{c}:=c_{\star}/2$, $\bar{c}:=2c^{\star}$ ensures $F_{1}(u,v,m,c)\neq 0$ for any $(u,v,m,c)\subset\partial\Omega$. Notice now that rescaling $f_{\tau}:=\tau f$ and $\phi_{\tau}=\tau\phi$ in (12), leads to a fixed point problem similar to $F_{1}$ but with rescaled reaction and vaporisation terms. The conclusions of Proposition 2 still hold with the same bounds, ensuring that for all $0\leqslant\tau\leqslant 1$, $F_{\tau}(u,v,m,c)\neq 0$ for any $(u,v,m,c)\subset\partial\Omega$.

Calculation of the degree of $F_{\tau}$.

Thanks to the properties of $K_{\tau}$, and the homotopy invariance of the degree we have $\deg(F_{\tau},\Omega,0)=\deg(F_{0},\Omega,0)$ for all $0\leqslant\tau\leqslant 1$ and it suffices to compute the degree of $F_{0}$. Given $(u,v,m,c)$, the solution $(\hat{u},\hat{v},\hat{m},\hat{c})$ to (13) with $\tau=0$ is

[TABLE]

The mapping

[TABLE]

is homotopic to

[TABLE]

This linear system has a unique solution in $X_{a}$ therefore its degree is different from [math]. The same result holds for $F_{1}$ which therefore admits at least one solution in $X_{a}$. $\Box$

2.3. Existence of travelling waves

We are now able to prove the first main result of the paper, that is the existence of a travelling wave for system (2) on the real line.

**Proof of Theorem 1 ** Let $(u_{a},v_{a},m_{a},c_{a})$ the solution to (5) for $a\geqslant a_{\star}$. We proved that $(u_{a},v_{a},m_{a},c_{a})$ is bounded in $W^{1,\infty}(-a,a)\times W^{1,\infty}(-a,a)\times L^{\infty}(-a,a)\times\mathbbm{R}$ uniformly with respect to $a\geqslant a_{\star}$. Using system (5), the uniform boundedness also holds in $W^{2,\infty}(-a,a)\times W^{2,\infty}(-a,a)\times L^{\infty}(-a,a)\times\mathbbm{R}$. Let $(a_{n})_{n\geqslant 0}$, with $a_{n}\geqslant a_{\star}$ an increasing sequence tending to infinity. We can extract a sequence $(u_{a_{n}},v_{a_{n}},m_{a_{n}},c_{a_{n}})$ converging in $\mathcal{C}^{1}_{\mathrm{loc}\hskip 3.5pt}(\mathbbm{R})\times\mathcal{C}^{1}_{\mathrm{loc}\hskip 3.5pt}(\mathbbm{R})\times\mathcal{C}^{0}_{\mathrm{loc}\hskip 3.5pt}(\mathbbm{R})\times\mathbbm{R}$ solution to

[TABLE]

From the explicit expressions of $u_{a_{n}}$ and $v_{a_{n}}$ on $(-a_{n},0)$ it follows that $u(-\infty)=0$ and $v(-\infty)=v_{u}$. Moreover, since the velocity $c_{a_{n}}$ is uniformly bounded below by a certain $\underline{c}\char 62\relax 0$, the vaporisation point $x_{v}^{a_{n}}$ such that $u(x_{v}^{a_{n}})=\theta_{v}$ is bounded below by $x_{v}^{\star}=-\log(\theta_{i}/\theta_{v})/\underline{c}$ and $m_{a_{n}}=m_{u}$ on $(-a_{n},x_{v}^{\star})$. It follows $m(-\infty)=m_{u}$. Now, $v$ and $v^{\prime}$ inherit the uniform bounds of $v_{a_{n}}$ and $v_{a_{n}}^{\prime}$, so that $v^{\prime}(+\infty)$ is bounded, therefore $v^{\prime}(+\infty)=0$ since $v$ is bounded. From the equation satisfied by $v$, since $\phi(u,m)$ and $f(u)v$ are bounded, we have also that $v^{\prime\prime}$ is bounded, therefore $v^{\prime\prime}(+\infty)=0$. Let $x_{vf}^{a_{n}}$ the vaporisation front, that is the first point where $m_{a_{n}}(x_{vf}^{a_{n}})=0$. Since $c_{a_{n}}$ is bounded above by some $\bar{c}\char 62\relax 0$, and since by hypothesis the droplets vaporise in finite time, one can bound uniformly $x_{vf}^{a_{n}}\char 60\relax x_{vf}^{\star}$ for some $x_{vf}^{\star}\char 62\relax 0$, and for all $a_{n}$, $m_{a_{n}}(x)=0$ if $x\geqslant x_{vf}^{\star}$. It follows $m(\infty)=0$, and the equation for $v$ implies $f(u(+\infty))v(+\infty)=0$. The monotonicity of $u$ holds for the same reason as the monotinicity of $u_{a_{n}}$ holds, therefore $u(+\infty)\char 62\relax\theta_{i}$ and $f(u(+\infty))\char 62\relax 0$, hence $v(+\infty)=0$. Finally, since $m(x)=0$ on $(x_{vf},+\infty)$, we have from Proposition 2 that $v\geqslant\beta_{1}(\Lambda)(1-u)\geqslant 0$ on $(x_{vf},+\infty)$ hence $u(+\infty)=1$. $\Box$

3. High activation energy limit

3.1. Characterisation of the limiting profiles

In this section we assume a reaction term $f_{\varepsilon}(u)$ obeying the Arrhenius law (3) on $[\theta_{i},1]$. Set $\theta_{\varepsilon}:=1+A\varepsilon\ln\varepsilon\char 60\relax 1$ with $A=100$ and $\varepsilon$ small enough. Notice $\theta_{i}\char 60\relax\theta_{\varepsilon}$ for $\varepsilon$ small enough. We have immediately

[TABLE]

[TABLE]

We follow and adapt the high energy activation analyses and tools from [4, 9, 10].

**Proof of Theorem 2 ** **(limiting system in the HAE limit) **Choose $\varepsilon\char 62\relax 0$ small enough so that $\mu/2\leqslant G_{\varepsilon}(1)\leqslant 2\mu$. The bounds (9) for the velocity with $a=+\infty$ and $u_{\varepsilon}^{\prime}(+\infty)=0$ allow us to estimate independently of $\varepsilon$

[TABLE]

with $\underline{c}:=\min(c_{\star},c_{1})$, $c_{\star}$ as in Proposition 3,  $c_{1}=\sqrt{2\beta_{1}(\Lambda)\mu}/\theta_{i}$, and $\bar{c}:=\sqrt{2\beta_{2}(\Lambda)\mu}/\theta_{i}$.

We can therefore find a decreasing sequence $(\varepsilon_{n})_{n\in\mathbbm{N}}$ such that $c_{\varepsilon_{n}}$ converges

[TABLE]

For $\varepsilon$ small enough, let $x_{\varepsilon}\char 62\relax 0$ be the unique point satisfying $u_{\varepsilon}(x_{\varepsilon})=1+A\varepsilon\ln\varepsilon=:\theta_{\varepsilon}$ <1, where $A$ is a fixed sufficiently large constant (take $A=100$). Let $x_{v}^{\varepsilon}$ be the point where vaporisation starts, i.e. $u_{\varepsilon}(x_{v}^{\varepsilon}):=\theta_{v}$. Let $x_{v}^{\varepsilon}\leqslant x\leqslant x_{\varepsilon}$. Rewriting equation $(\star\star\star)$ on $[x_{v}^{\varepsilon},x]$ and comparing $v_{\varepsilon}$ with $u_{\varepsilon}$ yields

[TABLE]

where the last integral is $\cal{O}(\varepsilon)$ thanks to the properties of $f_{\varepsilon}$. Hence there exist constants $\alpha\char 62\relax 0$ and $\varepsilon_{0}\char 62\relax 0$ such that for any $\varepsilon\leqslant\varepsilon_{0}$ and $x\in[x_{v}^{\varepsilon},x_{\varepsilon}]$, we have $u^{\prime}(x)\geqslant\alpha$. Since $u_{\varepsilon}(0)=\theta_{i}\char 60\relax\theta_{\varepsilon}=u_{\varepsilon}(x_{\varepsilon})$, it follows $0\char 60\relax x_{\varepsilon}\leqslant(\theta_{\varepsilon}-\theta_{v})/\alpha\char 60\relax(1-\theta_{v})/\alpha=:x_{0}\char 60\relax+\infty$. The uniform boundedness of $x_{\varepsilon}$ implies, up to extracting a subsequence,

[TABLE]

Now for the point $x_{v}^{\varepsilon}$ where vaporisation starts, we have $\char 124\relax x_{v}^{\varepsilon}-0\char 124\relax\leqslant(\theta_{v}-\theta_{i})/\alpha$, hence $x_{v}^{\varepsilon}$ is bounded and converges, after extraction of a subsequence,

[TABLE]

Convergence of $u_{\varepsilon}$.

The convergence of $c_{\varepsilon}$ together with the explicit expressions of the exponential profile $u_{\varepsilon}$ on $(-\infty,0)$ implies the convergence in $H^{1}(-\infty,0)$ of $u_{\varepsilon}$ towards $u$. Inspecting the $L^{2}$ estimates in the proof of Proposition 3 yields the uniform $H^{1}(0,x_{0})$ boundedness of $u_{\varepsilon}$, therefore convergence in $\mathcal{C}^{0}((0,x_{0}))$. Thanks to the monotonicity of $u_{\varepsilon}$, we have that $u_{\varepsilon}\rightarrow 1$ uniformly on $(\bar{x},+\infty)$. Finally, the reaction term is uniformly bounded on $(-\infty,x_{\varepsilon})$ by a multiple (14), which tends to [math] by hypothesis. This implies that the reaction term tends uniformly towards zero on any compact subset of $(-\infty,\bar{x})$. To summarize, $u_{\varepsilon}$ converges in $H^{1}(\mathbbm{R})$ towards a continuous profile $u$ solution to $-u^{\prime\prime}+cu^{\prime}=0$ on $(-\infty,\bar{x})$, $u=1$ on $(\bar{x},+\infty)$.

Convergence of $m_{\varepsilon}$.

Since $\theta_{v}\char 60\relax 1$, we have for $\varepsilon$ small enough that $x^{\varepsilon}_{v}\char 60\relax x_{\varepsilon}$, and $\theta_{\varepsilon}\geqslant(1+\theta_{v})/2\char 62\relax\theta_{v}$. From the hypothesis on the vaporisation law, it follows that complete vaporisation occurs on a finite intervall. Let $x_{{vf}}^{\varepsilon}$ denote the unique position of the vaporisation front. The uniform boundedness of $x_{{vf}}^{\varepsilon}$ implies the convergence

[TABLE]

Moreover, from the $\mathcal{C}^{0}$ convergence of $u_{\varepsilon}$ towards $u$ on the compact $[x_{v}-1,x_{{vf}}+1]$, together with the Lipshitz properties of the vaporisation terms, it follows that $m_{\varepsilon}$ converges towards $m$ on $[x_{v}-1,x_{{vf}}+1]$ in $\mathcal{C}^{0}([x_{v}-1,x_{{vf}}+1])$ solution to $cm^{\prime}=-g(u,m)$. Since $m_{\varepsilon}^{\prime}=0$ outside of $[x_{v}^{\varepsilon},x_{{vf}}^{\varepsilon}]$, the convergence holds on $\mathcal{C}^{0}(\mathbbm{R})$ with $m$ solving $m(-\infty)=m_{u}$ and $cm^{\prime}=-g(u,m)$ on $\mathbbm{R}$.

Convergence of $v_{\varepsilon}$.

The convergence of $v_{\varepsilon}$ follows the same line as that of $u_{\varepsilon}$ and uses the convergence of $m_{\varepsilon}$. The limiting solution $v\in\mathcal{C}^{0}(\mathbbm{R})$ satisfies $-\Lambda v^{\prime\prime}+cv^{\prime}=-cn_{0}m^{\prime}$on $(-\infty,\bar{x})$, $v=0$ on $(\bar{x},+\infty)$.

Convergence on $(\bar{x},+\infty)$.

Since $u_{\varepsilon}$ is increasing, we have for any $x_{\varepsilon}\leqslant x$, $u(x_{\varepsilon_{n}})=\theta_{\varepsilon_{n}}\leqslant u_{\varepsilon_{n}}(x)\leqslant 1$ with $\theta_{\varepsilon_{n}}\rightarrow 1$. This implies easily $u_{\varepsilon}(x)\rightarrow 1$ on $(\bar{x},+\infty)$. The upper bound $v_{\varepsilon}(x)\leqslant\beta_{2}(\Lambda)(1-u_{\varepsilon}(x))$ ensures $v_{\varepsilon}(x)\rightarrow 0$ on $(\bar{x},+\infty)$. Now, the auxiliary function $y_{\varepsilon}:=u_{\varepsilon}+\Lambda v_{\varepsilon}-1$ satisfies $y_{\varepsilon}(x_{\varepsilon})=\theta_{\varepsilon}+\Lambda v_{\varepsilon}(x_{\varepsilon})-1$, $y_{\varepsilon}(+\infty)=0$, and $-y_{\varepsilon}^{\prime\prime}+c_{\varepsilon}y_{\varepsilon}^{\prime}=(\Lambda-1)c_{\varepsilon}v^{\prime}_{\varepsilon}-\Lambda c_{\varepsilon}n_{0}m^{\prime}_{\varepsilon}$. Therefore the $H^{1}$–limit $y$ satisfies $y=u+\Lambda v-1=0$ on $(\bar{x},+\infty)$, together with $-y^{\prime\prime}+cy^{\prime}=(\Lambda-1)cv^{\prime}-\Lambda cn_{0}m^{\prime}$. This implies $m^{\prime}=0$ on $(\bar{x},+\infty)$ therefore also $m(x)=0$ on $(\bar{x},+\infty)$. Obviously, $u$ and $v$ are $\mathcal{C}^{2}$ on $(\bar{x},+\infty)$.

Estimate of $x_{{vf}}$.

The fact that $m(x)=0$ for $x\geqslant\bar{x}$ is equivalent to

[TABLE]

that is in the HAE limit, the vaporisation region ends before or at the reaction zone.

Convergence on $(-\infty,\bar{x})$.

Since $u_{\varepsilon}$ is increasing and $u_{\varepsilon}(0)=\theta_{i}$, the reaction term is zero on $(-\infty,0)$. From property (14) together with the upper estimate $v_{\varepsilon}\leqslant\beta_{2}(\Lambda)(1-u_{\varepsilon})$, we deduce that $f_{\varepsilon}(u_{\varepsilon})v_{\varepsilon}$ converges uniformly to zero on any compact set of $[0,\bar{x})$, therefore also on any compact set of $(-\infty,\bar{x})$. This implies that the limiting functions $u$, $v$ and $m$ satisfiy $-u^{\prime\prime}+cu^{\prime}=0$, $-\Lambda v^{\prime\prime}+cv^{\prime}=-cn_{0}m^{\prime}$, and $cn_{0}m^{\prime}=-g(u,m)$ on $(-\infty,\bar{x})$, with $m^{\prime}\in L^{2}((-\infty,\bar{x}))$, $v\in H^{2}((-\infty,\bar{x}))\subset\mathcal{C}^{1}((-\infty,\bar{x}))$ and $u(x)=\theta_{i}e^{c(x-\bar{x})}\in\mathcal{C}^{2}((-\infty,\bar{x}))$.

This concludes the proof of Theorem 2 **(limiting system in the HAE limit) ** $\Box$

**Proof of Theorem 3 ** **(internal layer analysis) **Set now $u_{\varepsilon}(0)=1+A\varepsilon\ln\varepsilon=:\theta_{\varepsilon}$ at $x=0$. Let then $x_{i}^{\varepsilon}\leqslant 0$ the ignition point where $u_{\varepsilon}(x_{i}^{\varepsilon})=\theta_{i}$. We have seen in the proof of Theorem 2 above that $\char 124\relax x_{i}^{\varepsilon}-0\char 124\relax$ is bounded independently of $\varepsilon$. From property (14), we have thereofre $f_{\varepsilon}(u_{\varepsilon})v_{\varepsilon}=\mathcal{O}(\varepsilon^{A/2})$ uniformly on $(x_{i}^{\varepsilon},0)$ and is zero on $(-\infty,x_{i}^{\varepsilon})$. Integrating the equation for $u_{\varepsilon}$ between $-\infty$ and [math], this implies

[TABLE]

where $\theta_{\varepsilon}=1+\mathcal{O}(\varepsilon\ln\varepsilon)$. It follows that

[TABLE]

Rescaled system.

Let $\xi:=x/\varepsilon$.

[TABLE]

together with the usual auxiliary functions

[TABLE]

[TABLE]

These functions obey the system of equations:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Notice that we have $u_{\varepsilon}^{\prime}(0)=\hat{u}_{\varepsilon}^{\prime}(0)$, therefore also $c_{\varepsilon}=\hat{u}^{\prime}_{\varepsilon}(0)+\mathcal{O}(\varepsilon\ln\varepsilon)$. Our goal is to estimate $\hat{u}^{\prime}_{\varepsilon}(0)$ in the limit $\varepsilon\rightarrow 0$.

Approximate system.

Let $\tilde{u}_{\varepsilon}$ and $\tilde{y}_{\varepsilon}$ the solution to the linear system

[TABLE]

[TABLE]

with the same boundary conditions as $\hat{u}_{\varepsilon}$ and $\hat{y}_{\varepsilon}$, and where we omitted the terms involving the factors $\varepsilon c_{\varepsilon}$. Classical results in elliptic regularity ensure that $(\hat{u}_{\varepsilon},\hat{y}_{\varepsilon})-(\tilde{u}_{\varepsilon},\tilde{y}_{\varepsilon})$ is of order $\mathcal{O}(\varepsilon)$ on $[0,\infty)$ in $\mathcal{C}^{1}$. Therefore $c_{\varepsilon}=\tilde{u}^{\prime}_{\varepsilon}(0)+\mathcal{O}(\varepsilon\ln\varepsilon)$.

Case $x_{{vf}}\char 60\relax 0$.

Let $x_{{vf}}^{\varepsilon}$ the position of the vaporisation front such that $\hat{m}_{\varepsilon}=0$ on $(x_{{vf}}^{\varepsilon},+\infty)$. If $x_{{vf}}\char 60\relax 0$, then for $\varepsilon$ small enough, $\hat{m}_{\varepsilon}=0$ and $\tilde{y}_{\varepsilon}=0$ on $(0,+\infty)$ and

[TABLE]

Omitting the last term in the equation above, it remains to study the solution to the new approximate system with the same boundary conditions

[TABLE]

Integrating against $\check{u}_{\varepsilon}^{\prime}$ on $(0,+\infty)$ yields

[TABLE]

where the integral in the right hand side tends to one as $\varepsilon$ goes to zero. Hence

[TABLE]

and the limiting velocity is $c=\sqrt{2\mu/\Lambda}$ when $x_{{vf}}=0$.

Case $x_{{vf}}=0$.

In the case where the vaporisation front is located in the limit at the position $\bar{x}=0$ of the reaction zone, the velocity $c$ is imposed by the condition $m(x)=0$. Indeed, we have in the limit $u(x)=e^{cx}$ for $x\leqslant 0$. Assume $0\char 60\relax c_{1}\char 60\relax c_{2}$. The corresponding profiles satisfy $u_{1}\char 62\relax u_{2}$ on $(-\infty,0)$ and the corresponding positions where vaporisation starts satisfy $x_{v}^{1}\char 60\relax x_{v}^{2}$. The monotonicity properties of the vaporisation law imply $x_{{vf}}^{1}\char 60\relax x_{{vf}}^{2}$, where those are the vaporisation fronts associated to the velocities $c_{1}$ and $c_{2}$ respectively. Consequently, the vaporisation front $x_{{vf}}$ is an increasing function $x_{{vf}}:=x_{{vf}}(c)$ of the velocity $c$. With similar arguments, it is also an increasing function of $m_{u}$. Moreover, since the vaporisation law $\phi$ is Lipschitz on $(\theta_{v},1)$, it follows that $x_{{vf}}(c)$ is continuous and we can define in a unique manner $c_{\star}(m_{u})$ to be the velocity such that $x_{{vf}}(c_{\star}(m_{u}))=0$.

As a conclusion, when $x_{{vf}}=0$, we have $c=c_{\star}(m_{u})$, whereas $c=\sqrt{2\mu/\Lambda}$ for $x_{{vf}}\char 60\relax 0$. The monotonicity of $x_{{vf}}:c\mapsto x_{{vf}}(c)$ implies

[TABLE]

[TABLE]

[TABLE]

This concludes the proof of Theorem 3. $\Box$

3.2. Internal combustion layer in the vaporisation controlled

regime

We are interested in the case where the droplets finish vaporising inside the combustion layer. In order to measure the quantity of droplets still present in the combustion region $[0,+\infty)$, it is convenient to introduce $M_{\varepsilon}(x)$ the following primitive of the mass of liquid

[TABLE]

Theorem 4**.**

**(estimate of the overlapping region) **Let $M_{\varepsilon}$ defined above. We have

[TABLE]

**Proof. ** Consider the approximate system on the intervall $(0,+\infty)$

[TABLE]

[TABLE]

such that $\check{u}_{\varepsilon}-\hat{u}_{\varepsilon}=\mathcal{O}(\varepsilon)$ and $\check{y}_{\varepsilon}-\hat{y}_{\varepsilon}=\mathcal{O}(\varepsilon)$. We also rescale $M_{\varepsilon}(x)$,

[TABLE]

Let $\xi_{{vf}}^{\varepsilon}$ the position of the vaporisation front and assume $\xi_{{vf}}^{\varepsilon}\char 62\relax 0$, that is some vaporisation occurs after $\xi=0$. We have $\hat{m}^{\prime}_{\varepsilon}=0$ on $(\xi^{\varepsilon}_{{vf}},+\infty)$, hence $\check{y}_{\varepsilon}$ is linear with limit zero at infinity, therefore $\check{y}_{\varepsilon}=0$ on $(\xi^{\varepsilon}_{{vf}},+\infty)$. It follows that

[TABLE]

On the other hand, we had for the original rescaled unknown

[TABLE]

because $\hat{u}_{\varepsilon}(0)=A\ln\varepsilon$, and (Proposition 1 & 2), $0\leqslant\hat{v}_{\varepsilon}\leqslant\beta_{2}(\Lambda)\char 124\relax\hat{u}_{\varepsilon}\char 124\relax$. It follows

[TABLE]

Since $\hat{M}_{\varepsilon}(0)=M_{\varepsilon}(0)/\varepsilon$, we proved

[TABLE]

This concludes the proof of Theorem 4. $\Box$

Application to the $d^{2}$–law.

Assume a vaporisation term of the form

[TABLE]

Integrating twice on $[x_{\varepsilon},x^{\varepsilon}_{{vf}}]$ yields

[TABLE]

[TABLE]

The estimate (3.2) above implies, as soon as $x_{{vf}}^{\varepsilon}\char 62\relax 0$ that

[TABLE]

The so–called “$d^{2}$–law” states that the rate of variation of the surface area of a vaporising droplet is approximately constant and corresponds to the case $\delta=1/3$. We let the reader check that it satisfies the hypothesis imposed on the vaporisation term, in particular that complete vaporisation occurs in finite time. The equation above reads

[TABLE]

This should be compared with the size of the reaction zone in the case $x_{{vf}}\char 60\relax 0$, where $x_{{vf}}^{\varepsilon}\leqslant\gamma\char 60\relax 0$ for some $\gamma$ and $\varepsilon$ small enough. In that case, the spray travelling wave has the same velocity as the purely gaseous flame with same temperature in the burnt gas. Also, the analysis of the internal combustion layer is similar to the gaseous case and provides an estimate of $\mathcal{O(\varepsilon\ln\varepsilon)}$ for the size of the reaction zone. This analysis suggests that the internal combustion layer in the presence of droplets might be considerably stretched and provides a quantitative upper estimate in the high activation energy limit. This is in accordance with many observations in the literature. See in particular the numerical experiments in [38] that illustrate the fact that the spray flame consists of a sharp deflagration front followed by a longer afterburn region where the droplets continue to be slowly vaporised.

4. Extensions

In this section, we state without proofs some easy extensions of our analysis. We consider (i) the case where vaporisation is instantaneous, (ii) the case where the spray is polydisperse, and (iii) other geometries. We refer to [6] for more details and extensions.

Heuristic analysis in the high activation energy limit.

A feature specific to our model is the existence of a simple alternative in the high activation energy limit, where it is easy to predict whether the system is in the diffusion or vaporisation controlled regime. Assume the system is in the diffusion controlled regime. This implies that the velocity of the flame is equal to that of the equivalent gaseous flame with same temperature in the burnt gas. The temperature profile in that case is known, hence also the corresponding liquid phase profile, from which one deduces the position $x_{{vf}}$ where the vaporisation ends. If the vaporisation ends in the preheating zone before the reaction front, i.e. $x_{{vf}}\leqslant\bar{x}$, then we are indeed in the diffusion controlled regime. If on the contrary $x_{{vf}}\char 62\relax\bar{x}$, this is a contradiction with our analysis and the spray flame is in the vaporisation controlled regime. In that case the position $\bar{x}$ of the reaction front obeys the constraint  $\bar{x}=x_{{vf}}$, which corresponds to a unique value of $c$.

Notice also that thanks to the monotonicity of the vaporisation law we have seen that the transition from diffusion to vaporisation controlled regime is sharp and occurs at a certain $m_{u}=m_{u}^{\star}$. As an experience of thought, fix $\rho_{l}$, $0\char 60\relax\rho_{l}\char 60\relax 1$, and vary $m_{u}$ from zero to infinity and $n_{0}$ accordingly so that $n_{0}m_{u}=\rho_{l}$ for all values of $m_{u}$. Set $v_{u}=1-\rho_{l}$, and solve for the system in the HAE limit. Our analysis shows that $c=\sqrt{2\mu/\Lambda}$ for $m_{u}\in(0,m_{u}^{\star})$ and $c:=c_{\star}(m)$ for $m\char 62\relax m_{u}^{\star}$, a decreasing function of $m$. This is in accordance with the numerical experiments shown in [38].

Fast vaporisation.

The situation where the vaporisation is very rapid can be modeled by an instantaneous vaporisation (see [7]) leading to a Dirac model for the vaporisation term

[TABLE]

Since $\theta_{v}\char 60\relax\theta_{i}$, vaporisation ends before the reaction zone at $x_{vf}=x_{v}\char 60\relax\bar{x}$. Explicit expressions of the profiles can be found.

Polydisperse sprays.

All the previous results can be extended in the case of polydisperse sprays, that is the situation where droplets of possibly different sizes are present at a given position. In that case, we need a statistical description of the distribution of size of the droplets as pointed out by Williams [39, 41]. See also O’Rourke [30]. For our problem, we assume that the distribution of size of droplets is known in the fresh gases.

Let $\mathrm{d}n(t,x,m)=\nu(t,x,m)\mathrm{d}m$ denote the density number of droplets at $(t,x)$ with size in the intervall $[m,m+\mathrm{d}m]$. The total mass density $\rho_{l}(t,x)$ of the liquid phase is

[TABLE]

and the function $\nu(t,x,m)$ obeys the conservation equation

[TABLE]

The total vaporisation rate of the liquid phase is

[TABLE]

and the corresponding coupled system writes,

[TABLE]

where $\nu_{u}(m)$ is the distribution profile of mass of droplets in the fresh gas.

This polydisperse model admits travelling waves (see also [24]). In the HAE limit, the regime of the flame is determined by the size of the largest droplet in the fresh gas. This result is a consequence of the simplicity of our vaporisation law, where no collective effect, nor influence of the density of the gaseous reactant, nor the fine geometry of the system at small scales were taken into account.

Other geometries.

The large activation energy limit analysis can be carried out in other onedimensionnal geometries, such as the anchored flame on the half-line, counter-flow-like configurations, or finally radial geometries in dimension $2$ or $3$. In each case, on can solve explicitly the problem (provided the expression of the vaporisation law is relatively simple), and it is possible to study the influence of different parameters on the existence of a profile or on the value of the burning rate. Relevant parameters are the typical velocities of injection of the gas and/or droplets, the value of the vaporisation rate, or the space dimension. We refer the reader to [6] for more details.

5. Perspectives and Concluding Remarks

This paper considered the existence of travelling fronts for a simple onedimensionnal thermo-diffusive lean spray flame model. We proved the existence of travelling waves for a class of combustion and vaporisation laws. As far as qualitative results are concerned, the most important part of the paper is the study of the high activation energy limit for the system. The limiting problem involves simple explicit profiles and preserves some important features of the dynamics. Extensions of these results to the cases of fast vaporisation, polydisperse sprays or other onedimensionnal geometries were briefly mentionned.

The present work is a first a step towards rigorous derivation of new asymptotic models for spray flames. However, much remains to be done in order to understand the effect of droplets on the dynamics of the flame structure, as observed by many physicists and experimentalists (see [16] and the references therein). Indeed, our model omits among other things the effects of the latent heat or more complex couplings in the vaporisation law (see [24] for such considerations). It would be very interesting to derive some high activation energy models incorporating these effects, in the hope of deriving explicit expressions of the limiting profiles and combustion rates, as well as extinction limits. Also, a striking consequence of the high activation energy asymptotics is that droplets cannot cross the flame front in that limit, but may enter the reaction zone only for large but not infinite values of the activation energy. It would be interesting to derive intermediate asymptotic models providing a precise understanding of the structure and thickening of the combustion region in the presence of vaporising droplets. Ideally, one would hope to recover some of the features of spray flames described in [12, 28, 19, 21, 29].

As far as dynamical phenomena are concerned, the problem of the stability of spray flame systems is crucial. The work [8] presents a mathematical analysis of spray pulsating waves. See also [27, 6, 5]. As a second step we have in mind the problem of acoustic instabilities in spray flame systems. In this direction, let us mention the work of Clavin an Sun [7]. Their mathematical analysis relies on the possibility of deriving explicit expressions of the solutions to the problem. This is only possible in certain asymptotic limits. For that, the authors consider the large activation energy limit for combustion phenomena and assume an instantaneous vaporisation of the droplets. As a consequence, both the combustion and the vaporisation zones reduce to infinitely small regions, whose internal layer is analysed.

However, assuming that the vaporisation zone is very small compared to the preheating zone is a very restrictive assumption in many applications where the droplets can spread into the preheating zone, approach the combustion zone, or enter the combustion zone. We have shown in the present paper that it is possible to analyse such situations where the droplets approach or reach the reaction front, and where the slowly vaporising droplets induce a dramatic change of the combustion rate. We hope that our work will motivate new studies carrying out rigorous mathematical analysis of asymptotic models for spray flames.

Acknowledgments

It is our pleasure to thank J.-M. Roquejoffre for suggesting the study and many helpful advices, as well as P. Haldenwang for very interesting and stimulating discussions.