Riemann solutions for counterflow combustion in light porous foam

Abstract

The paper is motivated by a model for the injection of air into a porous medium that contains a solid fuel. In previous work, a system of three evolutionary partial differential equations that models combustion of light porous foam under air injection was considered. The existence and uniqueness of traveling waves were studied and the wave sequences appearing in Riemann solutions were identified. This analysis was done under assumption that the combustion wave velocity is positive. In the present work, this hypothesis was neglected and the results generalized including the case of negative combustion wave speed. The existence of such waves was proved and the uniqueness investigated for some particular cases using Melnikov’s integral. The wave sequences appearing in Riemann solutions were identified and the numerical examples using finite difference scheme were presented.

Introduction

The paper is motivated by a model for the injection of air into a porous medium that contains a solid fuel. This process appears in many applications as oil recovery through in situ combustion, foam combustion, self-propagating high-temperature synthesis, etc.

This paper is part of long-term research project the purpose of which is to identify waves that arise in one-dimensional models of combustion in porous media, and to understand how the waves fit together in solutions of Riemann problems; see Chapiro et al. (2012, 2014), Chapiro and de Souza (2016), Mailybaev et al. (2011), Marchesin and Schecter (2003), Mota and Schecter (2006), Ozbag (2016), Ozbag et al. (2017), and Schecter and Marchesin (2001), and references therein.

There are two basic configurations for one dimensional combustion models: in the co-flow combustion the combustion front is assumed to move in the direction of the injected gas flow; during the counterflow combustion these directions are opposed and the combustion wave has negative speed.

The model addressed here was proposed in Akkutlu and Yortsos (2003) and further studied in Chapiro et al. (2012). It was simplified in Chapiro et al. (2014) in order to (1) reproduce the variety of phenomena observed when air is injected into a porous medium containing a solid fuel, yet (2) to be simple enough to permit a rigorous investigation. This simplification allowed proofs of existence of traveling waves by phase plane analysis. In Chapiro et al. (2015) the stability for traveling wave solution appearing in this model was addressed numerically. All these papers addressed the co-flow combustion, in this sense the present work extend what was done in Chapiro et al. (2014) to the counterflow case.

Other works studied counterflow combustion. For example, in Schult et al. (1998), the authors explore strong nonlinearity of the Arrhenius factor in the reaction rate, which allows neglecting the reaction rate as soon as the temperature decreases (Zeldovich et al. 1985). In Souza (2008) Riemann problem was investigated and some necessary conditions for the counterflow traveling wave were obtained. This work was further extended in Chapiro and de Souza (2016), where the traveling wave solution for counterflow combustion was investigated using singular perturbation analysis.

The model derived in Chapiro et al. (2014) is reviewed in Sect. 2. It consists of three equations that express energy, oxygen, and fuel balance laws. We use a shifted reaction rate Arrhenius law for which combustion begins at a threshold temperature. We analyze the simple case in which the thermal capacity of the medium is negligible compared to that of the air. A consequence is that oxygen and heat are both transported at the velocity of the moving gas. The negligible thermal capacity assumption is not correct for oil recovery, but is approximately valid for polyurethane foam smoldering such as that used in furniture. Possible way to circumvent this problem for the co-flow combustion was proposed in Ozbag (2016); Ozbag et al. (2017). Application of these techniques to the counterflow case is left for future work.

In Sect. 3, the previous results are summarized (Chapiro et al. 2014). Section 4 presents the classification of all possible wave sequences in Riemann solution. Numerical simulations are presented as examples of the theoretical results of the present paper. In this section we also formulate the main result about the existence of traveling waves which is proved in Sect. 5.

Model

The system we consider is

$$\begin{aligned} \displaystyle&\partial _t \theta + a \partial _x \theta = \partial _{xx} \theta + \rho Y\Phi , \end{aligned}$$
(1)
$$\begin{aligned} \displaystyle&\partial _t \rho = - \rho Y\Phi , \end{aligned}$$
(2)
$$\begin{aligned} \displaystyle&\partial _t Y + a \partial _x Y = -\rho Y\Phi , \end{aligned}$$
(3)
$$\begin{aligned}&\Phi = \left\{ \begin{array}{ll} \displaystyle \exp \left( {-1}/{\theta }\right) , &{}\quad \theta >0\\ 0, &{}\quad \theta \le 0. \end{array} \right. \end{aligned}$$
(4)

There are three dimensionless dependent scaled variables: temperature \(\theta \), solid fuel concentration \(\rho \), and oxygen concentration Y. The oxygen is a component of gas that is moving with Darcy velocity \(a>0\). Both oxygen and heat are assumed to be transported with this velocity. An exothermic chemical reaction involving oxygen and solid fuel can occur only when the temperature is above a threshold temperature, which we have normalized to be \(\theta =0\). Because of this convention, the scaled temperature is allowed to be negative. The reaction rate is given as Arrhenius law in (4) by \(\Phi (\theta )\). Equation (1) represents transport and diffusion of temperature, as well as generation of thermal energy by the chemical reaction. Equation (2) represents consumption of the solid fuel, which does not diffuse and is not transported by gas. Equation (3) represents transport and consumption of oxygen in the chemical reaction. Diffusion of oxygen is ignored. A derivation of the model, and discussion of its range of validity, can be found in Chapiro et al. (2014).

We are interested in solutions with \(\rho \ge 0\) and \(Y\ge 0\) everywhere. The injection problem is, therefore, equivalent to considering (1)–(2) on \(-\infty<x<\infty \), \(t\ge 0\), with constant boundary conditions

$$\begin{aligned} (\theta ,\rho ,Y)(-\infty )=(\theta ^L,\rho ^L,Y^L),\qquad (\theta ,\rho ,Y)(\infty ) =(\theta ^R,\rho ^R,Y^R). \end{aligned}$$
(5)

As usual, we assume that the reaction does not occur at the boundaries, i.e., the reaction terms in (1)–(3) vanish at the boundary. There are three reasons for the reaction terms to vanish, where two or all of these conditions can occur simultaneously:

  1. 1.

    Temperature control (TC): the reaction ceases due to low temperature \(\theta \le 0\);

  2. 2.

    Fuel control (FC): the reaction ceases due to lack of fuel \(\rho =0\);

  3. 3.

    Oxygen control (OC): the reaction ceases due to lack of oxygen \(Y=0\).

Previous results

In this section, we review nomenclature and some results obtained in Chapiro et al. (2014). We denote by \((\theta ^{-},\rho ^{-},Y^{-}) \xrightarrow {c} (\theta ^{+},\rho ^{+},Y^{+})\) a wave of velocity c that connects \((\theta ^{-},\rho ^{-},Y^{-})\) at the left to \((\theta ^{+},\rho ^{+},Y^{+})\) at the right. At the end states of the wave, the reaction terms in (1)–(3) vanish.

States at which the reaction terms vanish were classified as TC, FC, OC, \(TC \cap FC\), \(TC\cap OC\), \(FC\cap OC\), or \(TC\cap FC\cap OC\). The type of the state indicates exactly which conditions hold at that state; for example, a \(TC \cap FC\) state has \(\theta \le 0\), \(\rho =0\), and \(Y>0\). Following Chapiro et al. (2014) we assume that the boundary conditions can be FC, OC or TC with no intersections. However, they cannot be ignored as possible intermediate states.

By a “combustion wave” we shall mean a continuous nontrivial traveling wave with velocity \(c\ne 0\), \(c\ne a\). For other approaches see Akkutlu and Yortsos (2003), Aldushin et al. (1999), Chapiro et al. (2012), Ghazaryan et al. (2010), Schult et al. (1996), Zeldovich et al. (1985).

In Chapiro et al. (2014), it was established that there are exactly four types of combustion waves with positive velocity that approach both end states exponentially, two fast (wave velocity \(c_f>a>0\)) and two slow (positive wave velocity \(0<c_s<a\)): \(FC\xrightarrow {c_f}TC\), \(OC\xrightarrow {c_f}TC\), \(FC\xrightarrow {c_s}OC\), \(TC\xrightarrow {c_s}OC\).

Fast combustion wave

In a fast combustion wave, the burning front moves toward the low-temperature region containing both solid fuel and oxygen; this is often called “premixed combustion.” The heat produced remains behind the combustion front because the moving gas that could transport it has a lower velocity. These fronts were studied in Chapiro et al. (2014), where the following result was proved:

Theorem 1

(Fast combustion waves) Fix \(a>0\). Let \((\theta ^{+},\rho ^{+},Y^{+})\) be a state of type TC, i.e., \(\theta ^{+}\le 0\), \(\rho ^{+}>0\), \(Y^{+}>0\). Assume in addition that \(\theta ^{+}+Y^+>0\). Then there exists a state \((\theta ^{-},\rho ^{-},Y^{-})\) and a velocity \(c_f>a\) such that there is a combustion wave \((\theta ^{-},\rho ^{-},Y^{-}) \xrightarrow {c_f} (\theta ^{+},\rho ^{+},Y^{+})\) that approaches its right state exponentially. It has \(\theta ^{-}>0\), and \(\rho ^{-}\) or \(Y^{-}\) or both equal to 0. Moreover, \(\theta ^+ + Y^+ = \theta ^-+ Y^-\) and

$$\begin{aligned} c_f=\frac{aY^+-aY^-}{Y^+-Y^-+\rho ^--\rho ^+}. \end{aligned}$$
(6)

There are no combustion waves with \(c>a\) and \(\theta ^{+}+Y^+\le 0\).

In Chapiro et al. (2014) numerical evidence for the uniqueness was presented.

Slow combustion wave

In a slow combustion wave, gas bringing oxygen flows into a region in which solid fuel is present but oxygen is not. Combustion occurs behind the incoming gas; it cannot occur in front since ahead of the gas there is no oxygen. Thus, the speed c of the combustion front cannot be greater than a. In fact \(c<a\), so heat produced by combustion, which also moves with speed a, is swept head of the combustion front. Hence, the high-temperature region is ahead of the front. The oxygen is entirely consumed in the reaction. These fronts have been called “reaction-trailing smolder waves” (Schult et al. 1996) and “coflow (or forward) filtration combustion waves” (Aldushin et al. 1999). They were studied in Chapiro et al. (2014), where the following result was established.

Theorem 2

(Slow combustion waves)

  1. 1.

    Consider \(FC\xrightarrow {c_s}OC\) and \(FC\cap TC\xrightarrow {c_s}OC\) waves. Fix \(a>0\). Let \((\theta ^{-},0,Y^{-})\) have \(\theta ^{-}\ge 0\) and \(Y^{-}>0\). Then for each \(\rho ^{+}>0\), there are unique numbers \(\theta ^{+}>0\) and \(c_s\), \(0<c_s<a\), such that there exists a combustion wave of velocity \(c_s\) from \((\theta ^{-},0,Y^{-})\) to \((\theta ^{+},\rho ^{+},0)\). In fact,

    $$\begin{aligned} \theta ^{+}=\theta ^{-}+Y^{-}, \quad c_s=\frac{Y^{-}}{\rho ^{+}+Y^{-}}a. \end{aligned}$$
    (7)
  2. 2.

    \(TC\xrightarrow {c_s}OC\) waves. Fix \(a>0\). Let \(\theta ^{-}<0\), \(Y^{-}\) with \(\theta ^{-}+Y^{-}>0\), and \(\rho ^{+}>0\) be given. Then there are numbers \(\rho ^{-}>0\), \(\theta ^{+}>0\), and \(c_s\), \(0<c_s<a\), such that there exists a combustion wave of velocity \(c_s\) from \((\theta ^{-},\rho ^{-},Y^{-})\) to \((\theta ^{+},\rho ^{+},0)\). Moreover \( \theta ^{+}=\theta ^{-}+Y^{-}\), and the quantities \(c_s\) and \(\rho ^-\) are related by the formula

    $$\begin{aligned} c_s = \frac{aY^{-}}{Y^{-}-\rho ^{-}+\rho ^{+}}. \end{aligned}$$
    (8)
  3. 3.

    There are no other combustion waves with velocity satisfying \(0< c< a\). In particular, there are no slow combustion waves with \(\theta ^-+Y^-\le 0\).

In Chapiro et al. (2014) numerical evidence that the triple \((\rho ^{-},\theta ^{+},c_s)\) is unique was presented.

Contact waves

In the absence of reaction and diffusion terms, the characteristic velocities of (1)–(3) are 0 for the solid fuel, a for temperature and oxygen; as they are constant they correspond to contact discontinuity waves. The solid fuel concentration, temperature and oxygen concentration can change across contact discontinuities. Contact discontinuities must separate spatial intervals in which the reaction does not occur (since \((\theta ,\rho ,Y)\) is constant). The waves in a wave train must occur in order of increasing velocity from left to right.

Remark

Because some of these waves widen with time, they are analogous to, but not exactly, classical contact discontinuities. For more details see Chapiro et al. (2014). A better name for a wave with constant velocity a in variable \(\theta \) would be diffusive contact. For simplicity, we abuse notation and use the term “contact wave” to describe it.

In Chapiro et al. (2014) 18 possible wave sequences were classified and numerically illustrated. As our interest in the present paper is investigating combustion traveling waves with negative velocity, we complete the sequences obtained in Chapiro et al. (2014) with the waves of this type as explained below.

Wave sequences

In this section, the counterflow combustion wave speed \(c_c\) is considered negative.

Proposition 1

There are no possible combustion traveling waves of the system (1)–(3) with negative velocity except \(TC\xrightarrow {c_c}OC\) and \(TC\xrightarrow {c_c}FC\).

Proof

Indeed, notice that there is no advection term in the fuel balance equation (2), thus there is no possibility for FC to be a left state. As the combustion wave is moving in opposite direction as oxygen flow it is impossible for the left state to be OC. On the other side it is easy to see that there is no combustion wave connecting states of the same type. The only possibilities left are \(TC\xrightarrow {c_c}OC\) and \(TC\xrightarrow {c_c}FC\). \(\square \)

The main result of this paper follows; it is proved in Sect. 5.

Theorem 3

Fix \(a>0\). Let (\(\theta ^{-},\rho ^{-},Y^{-}\)) be a state of type TC (i.e., \(\theta ^{-}\le 0\), \(\rho ^{-}>0\), \(Y^{-}>0\)). Assume in addition that \(\theta ^{-}+Y^->0\). Then there exists a state \((\theta ^{+},\rho ^{+},Y^{+})\) and a velocity \(c_c<0\) such that there is a combustion wave \((\theta ^{-},\rho ^{-},Y^{-}) \xrightarrow {c_c} (\theta ^{+},\rho ^{+},Y^{+})\). It has \(\theta ^{+}>0\), and \(\rho ^{+}\) or \(Y^{+}\) or both equal to 0. Moreover, \(\theta ^+ + Y^+ = \theta ^-+ Y^-\) and

$$\begin{aligned} c_c=\frac{aY^+-aY^-}{Y^+-Y^-+\rho ^--\rho ^+}. \end{aligned}$$
(9)

There are no combustion waves with \(c_c<0\) and \(\theta ^{-}+Y^-\le 0\).

Using Proposition 1 and Theorem 3 we can extend the classification of all possible sequences in Riemann solution obtained in Chapiro et al. (2014) by including the combustion waves with negative speed \(c_c\) as presented in Fig. 1.

Fig. 1
figure1

All possible sequences of the waves with (1) left state of type TC, FC, or OC, (2) increasing wave speed, and (3) sequences extended as far as possible. Red arrow dimension number 0. Black arrow dimension number 1. Green arrow dimension number 2 (color figure online)

Using Theorem 3 the following sequences containing counterflow combustion waves appear in the Riemann problem solution. We show numerical examples to evidence these sequences. We simulate System (1)–(4) using the Crank–Nicolson difference scheme. The simulation used the parameter \(a=0.01\), space step \(\Delta x = 1.3\), variable time step to speed up the simulations and a total grid number 1940. Some simulations present small perturbation in fuel concentration due to initial step adaptation and do not interfere in the final result.

Waves containing \(TC\xrightarrow {c_c}FC\)

This wave corresponds to the parameters belonging to the curve C studied in Sect. 5. The following sequences containing \(TC\xrightarrow {c_c}FC\) are possible. All examples correspond to the boundary conditions TC on the left and on the right.

Fig. 2
figure2

Initial condition is plotted on the left and the simulation result at time 15,000 on the right. Both figures correspond to the sequence \(TC\xrightarrow {c_c}FC\xrightarrow {a}OC\cap FC\xrightarrow {c_f}TC\). Horizontal axis variable x

Fig. 3
figure3

Initial condition is plotted on the left and the simulation result at time 10,000 on the right. Both figures correspond to the sequence \(TC\xrightarrow {c_c}FC\xrightarrow {a}FC\xrightarrow {c_f}TC\). Horizontal axis variable x

Fig. 4
figure4

Initial condition is plotted on the left and the simulation result at time 8000 on the right. Both figures correspond to the sequence \(TC\xrightarrow {c_c} FC \xrightarrow {c_s}OC \xrightarrow {a}OC \xrightarrow {c_f}TC\). Horizontal axis variable x

  • \(\bullet \) Wave sequence \(TC\xrightarrow {c_c} FC \xrightarrow {a}OC\cap FC \xrightarrow {c_f}TC \). The simulation starts with the traveling and contact wave profiles, as plotted on the left side of Fig. 2. The stable combustion front can be observed on the right side of Fig. 2.

  • \(\bullet \) Wave sequence \(TC\xrightarrow {c_c} FC \xrightarrow {a}FC \xrightarrow {c_f}TC\). The simulation starts with the traveling and contact wave profiles, as plotted on the left side of Fig. 3. The stable combustion front can be observed on the right side of Fig. 3.

  • \(\bullet \) Wave sequence \(TC\xrightarrow {c_c} FC \xrightarrow {c_s}OC \xrightarrow {a}OC \xrightarrow {c_f}TC\). The simulation starts with the traveling and contact wave profiles, as plotted on the left side of Fig. 4. The stable combustion front can be observed on the right side of Fig. 4. This simulation used \(\delta x=0.5\).

  • \(\bullet \) Wave sequence \(TC\xrightarrow {c_c} FC \xrightarrow {c_s}OC \xrightarrow {a}TC\). It is similar to the previous case, see the wave sequence \(TC\xrightarrow {c_c} FC \xrightarrow {c_s}OC \xrightarrow {a}OC \xrightarrow {c_f}TC\).

Waves containing \(TC\xrightarrow {c_c}OC\)

This wave corresponds to the parameters belonging to the Region 1 studied in Sect. 5. The following sequences containing \(TC\xrightarrow {c_c}OC\) are possible.

Fig. 5
figure5

Initial condition is plotted on the left and the simulation result at time 15,000 on the right. Both figures correspond to the sequence \(TC\xrightarrow {c_c}OC\xrightarrow {0}OC\cap FC\xrightarrow {a}FC\xrightarrow {c_f}TC\). Horizontal axis variable x

Fig. 6
figure6

Initial condition is plotted on the left and the simulation result at time 15,000 on the right. Both figures correspond to the sequence \(TC\xrightarrow {c_c} OC \xrightarrow {0} OC \xrightarrow {a} OC \xrightarrow {c_f}TC\). Horizontal axis variable x

  • \(\bullet \) Wave sequence \(TC\xrightarrow {c_c} OC \xrightarrow {0} OC\cap FC \xrightarrow {a} FC \xrightarrow {c_f} TC \). Numerical simulation corresponds to the boundary conditions TC on the left and on the right. The simulation starts with the traveling and contact wave profiles, as plotted on the left side of Fig. 5. The stable combustion front can be observed on the right side of Fig. 5.

  • \(\bullet \) Wave sequence \(TC\xrightarrow {c_c} OC \xrightarrow {0} OC\cap FC \xrightarrow {c_f}TC\). It is similar to the previous case, see wave sequence \(TC\xrightarrow {c_c} OC \xrightarrow {0} OC\cap FC \xrightarrow {a} FC \xrightarrow {c_f} TC \).

  • \(\bullet \) Wave sequence \(TC\xrightarrow {c_c} OC \xrightarrow {0} OC \xrightarrow {a} OC \xrightarrow {c_f}TC\). Numerical simulation corresponds to the boundary conditions TC on the left and on the right. The simulation starts with the traveling and contact wave profiles, as plotted on the left side of Fig. 6. The stable combustion front can be observed on the right side of Fig. 6.

  • \(\bullet \) Wave sequence \(TC\xrightarrow {c_c} OC \xrightarrow {0} OC \xrightarrow {a} TC \). It is similar to the previous case, see the wave sequence \(TC\xrightarrow {c_c} OC \xrightarrow {0} OC \xrightarrow {a} OC \xrightarrow {c_f}TC\).

Waves containing \(TC\xrightarrow {c_c}OC\cap FC\)

This wave corresponds to the parameters belonging to the Region 2 studied in Sect. 5. The following sequences containing \(TC\xrightarrow {c_c}OC\cap FC\) are possible.

Fig. 7
figure7

Initial condition is plotted on the left and the simulation result at time 15,000 on the right. Both figures correspond to the sequence \(TC\xrightarrow {c_c} OC\cap FC \xrightarrow {a} FC \xrightarrow {c_f} TC\). Horizontal axis variable x

  • \(\bullet \) Wave sequence \(TC\xrightarrow {c_c} OC\cap FC \xrightarrow {a} FC \xrightarrow {c_f} TC\). Numerical simulation corresponds to the boundary conditions TC on the left and on the right. The simulation starts with the traveling and contact wave profiles, as plotted on the left side of Fig. 7. The stable combustion front can be observed on the right side of Fig. 7.

    \(\bullet \) Wave sequence \(TC\xrightarrow {c_c} OC\cap FC \xrightarrow {c_f}TC\). It is similar to the previous case.

Traveling wave solution: existence and uniqueness

We use the traveling coordinate \(\xi =x-ct\), where \(c<0\), and set \(v_1=\partial _\xi \theta \). Using dot to denote derivative with respect to \(\xi \) we obtain the system describing the combustion traveling wave

$$\begin{aligned} \dot{\theta }&= v_1, \end{aligned}$$
(10)
$$\begin{aligned} \dot{v}_1&= (a-c)v_1-\rho Y\Phi (\theta ), \end{aligned}$$
(11)
$$\begin{aligned} w_1&=(c-a)\theta +v_1+c\rho , \end{aligned}$$
(12)
$$\begin{aligned} w_2&=(c-a)Y-c\rho , \end{aligned}$$
(13)

where \(w_1\) and \(w_2\) are constants. Substituting the values for \((\theta ,\rho ,Y)\) at the generic boundary states we obtain

$$\begin{aligned} w_1&=(c-a)\theta ^-+c\rho ^- = (c-a)\theta ^+ + c\rho ^+, \end{aligned}$$
(14)
$$\begin{aligned} w_2&=(c-a)Y^- -c\rho ^- = (c-a)Y^+ - c\rho ^+. \end{aligned}$$
(15)

We solve for Y using (12), and we solve for \(v_1\) using (13). Substituting into (10)–(11) and dividing the second equation by c (we recall the assumption that \(c<0\)), we obtain the reduced traveling wave system describing the internal structure of the combustion wave, in which \((w_1,w_2)\) is a vector of parameters,

$$\begin{aligned} \dot{\theta }&= (a-c)\theta -c\rho +w_1, \end{aligned}$$
(16)
$$\begin{aligned} \dot{\rho }&= \frac{c \rho +w_2}{c(c-a)} \rho \Phi (\theta ). \end{aligned}$$
(17)

The matrix of derivatives of (16)–(17) considering \((\theta ,\rho ) \in OC\), FC or TC becomes, respectively,

$$\begin{aligned} J_{1} = \left[ \begin{array}{ll} a-c &{}\quad -c \\ 0 &{}\quad \displaystyle \frac{\rho \Phi (\theta )}{c-a} \end{array} \right] , \; J_{2} = \left[ \begin{array}{ll} a-c &{}\quad -c \\ 0 &{}\quad \displaystyle \frac{w_2\Phi (\theta )}{c(c-a)} \end{array} \right] , \; J_{3} = \left[ \begin{array}{ll} a-c &{}\quad -c \\ 0 &{}\quad 0 \end{array} \right] . \end{aligned}$$
(18)

Using \(c<0\) we notice that \(w_2>0\) in OC, \(w_2<0\) in FC and \(w_2=0\) in \(OC\cap FC\), we can formulate the following result.

Proposition 2

The following statements are valid:

  • An equilibrium in OC is a saddle. One eigenvalue is \(a-c\), with eigenvector pointing along the invariant line \(\rho =-{w_2}/{c}\), which corresponds to \(Y=0\). The other eigenvector is transverse to the invariant line.

  • An equilibrium in FC is a saddle. One eigenvalue is \(a-c\), with eigenvector pointing along the invariant line \(\rho =0\). The other eigenvector is transverse to the invariant line.

  • At an equilibrium in \(FC\cap OC\), one eigenvalue is \(a-c\), with eigenvector pointing along the invariant line \(\rho =0\). The other eigenvalue is 0.

  • At an equilibrium in TC, one eigenvalue is \(a-c\), with eigenvector (1, 0); the other eigenvalue is 0.

Next, we prove Theorem 3 following general lines of Chapiro et al. (2014).

Proof

Notice that \(\theta ^-+Y^- = \theta ^+ + Y^+ \ge 0\). We recall that the left state of a traveling wave in this case is \((\theta ^-,\rho ^-,Y^-)\) with \(\theta ^-<0\), \(\rho ^->0\) and \(Y^->0\).

The invariant line \(Y=0\) of System (16)–(17) corresponds to the line \(\rho =\rho ^--({c-a})Y^-/{c}\), which is not above the line \(\rho =\rho ^-\). Accordingly to the relative positions of the invariant lines \(Y=0\) and \(\rho =0\) the subsets \(C,\;R_1\) and \(R_2\) of are defined as follows, see Fig. 8:

$$\begin{aligned} C= & {} \{(Y^-,c):Y^-<\rho ^-<\infty ,\;\text {and}\;c={aY^-}/(Y^--\rho ^-)\},\end{aligned}$$
(19)
$$\begin{aligned} R_1= & {} \{(Y^-,c):Y^-<\rho ^-<\infty ,\;\text {and}\;c<{aY^-}/(Y^--\rho ^-)\},\end{aligned}$$
(20)
$$\begin{aligned} R_2= & {} \{(Y^-,c):Y^->0,\;\text{ e }\;(Y^-,c)\notin C\cup R_1\}. \end{aligned}$$
(21)

Then the line \(Y=0\) is below the line \(\rho =0\) in \(R_2\), \(Y=0\) is above the line \(\rho =0\) in \(R_1\), and they coincide in C. We will study each case separately. \(\square \)

Fig. 8
figure8

Regions \(R_1\) and \(R_2\) separated by the curve C for the counterflow combustion

Notice that \(Y^--\rho ^-\ne 0\), otherwise, if \(Y^--\rho ^-=0\) then there exists a wave with left state of type \(FC\cap OC\), which is not possible by Proposition 1.

Curve C : Substituting \(c={aY^-}/(Y^--\rho ^-)\) into the System (16)–(17) and rescaling time by multiplying the right side by \(-a\rho ^-(Y^--\rho ^-)>0\) we obtain:

$$\begin{aligned} \begin{array}{l} \dot{\theta }=a^2\rho ^-[\rho ^-(\theta -\theta ^-)+Y^-(\rho -\rho ^-)],\\ \dot{\rho }=-(Y^--\rho ^-)^2\rho ^2\Phi (\theta ). \end{array} \end{aligned}$$
(22)

Notice that \(\dot{\rho }<0\) if \(\theta >0\) and \(\dot{\rho }=0\) if \(\theta \le 0\). Also, the derivative \(\dot{\theta }\) changes from negative to positive when the nullcline \(\rho ^-(\theta -\theta ^-)+Y^-(\rho -\rho ^-)=0\) is crossed from left to right. From Proposition 2 the equilibrium \((\theta ^-,\rho ^-)\) has a 1-dimensional unstable manifold.

As \(\theta ^-+Y^-=\theta ^++Y^+\ge 0\), \((\theta ^-+Y^-,0)\) is an equilibrium in \(FC\cap OC\) since this equilibrium belongs to both invariant lines \(\rho =0\) and \(Y=0\). At the region \(\rho >0\) it has a 1-dimensional center manifold \(W^c(\theta ^-+Y^-,0)\) which is an orbit that tends to \((\theta ^-+Y^-,0)\) in the future. This orbit is tangent to and above the line \(\rho ^-(\theta -\theta ^-)+Y^-(\rho -\rho ^-)=0\), see Fig. 9.

Fig. 9
figure9

Phase portrait of System (22) under assumption \(\theta ^-+Y^->0\)

The next proposition describe the behavior of System (16)–(17) in C corresponding to the wave of the type \(TC\rightarrow FC\cap OC\).

Proposition 3

Let \(a>0\), \(\theta ^-\le 0\) and \(Y^-<\rho ^-\). Then there is a unique \(\rho ^*\), \(Y^-<\rho ^*<\infty \), such that the unstable manifold of \((\theta ^-,\rho ^*)\) contains a branch of the center manifold of \((\theta ^-+Y^-,0)\).

Proof

Existence. First we consider the limit \(\rho ^-=Y^-\) at which System (22) becomes

$$\begin{aligned} \begin{array}{l} \dot{\theta }=a^2(\rho ^-)^2((\theta -\theta ^-)+(\rho -\rho ^-)),\\ \dot{\rho }=0. \end{array} \end{aligned}$$
(23)

The unstable manifold of \((\theta ^-,\rho ^-)\) is the line \(\rho =\rho ^-\) and the central manifold of \((\theta ^-+Y^-,0)\) is the line \((\theta -\theta ^-)+(\rho -\rho ^-)=0\) with \(\theta ^-<\theta <\theta ^-+Y^-\). Therefore, for \(\rho ^-\) a little larger then \(Y^-\), the former lies above the latter, see Fig. 10. \(\square \)

Fig. 10
figure10

Phase portrait of System (23) for the limit case \(\rho ^-=Y^-\) corresponding to curve C

Now, to study the limit \(\rho ^-=\infty \), fix \(\varepsilon \in (0,\rho ^-)\) and consider the solution of System (22) for \((\theta ,\rho )\in [-\theta ^-+\varepsilon ,-\theta ^-+2\varepsilon ]\times [\rho ^--\varepsilon ,\rho ^-]\). Then

$$\begin{aligned} \frac{\mathrm{d}\theta }{\mathrm{d}\rho }=\frac{-a^2\rho ^-}{(Y^--\rho ^-)^2}\cdot \frac{\rho ^-(\theta -\theta ^-)+Y^-(\rho -\rho ^-)}{\rho ^2\Phi (\theta )} \end{aligned}$$
(24)

and

$$\begin{aligned} \frac{-a^2\rho ^-}{(Y^--\rho ^-)^2}\cdot \frac{\rho ^-(-2\theta ^-+2\varepsilon )}{(\rho ^--\varepsilon )^2\Phi (-\theta ^-+\varepsilon )} \le \frac{\mathrm{d}\theta }{\mathrm{d}\rho } \le \frac{-a^2\rho ^-}{(Y^--\rho ^-)^2}\cdot \frac{\rho ^-(-2\theta ^-+\varepsilon )-Y^-\varepsilon }{(\rho ^-)^2\Phi (-\theta ^-+2\varepsilon )}. \end{aligned}$$
(25)

As \(\rho ^-\rightarrow \infty \), the upper and lower bounds approach 0.

Therefore, for \(\rho ^-\) sufficiently large, the unstable manifold of \((\theta ^-,\rho ^-)\) lies below the center manifold of \((\theta ^-+Y^-,0)\), see Fig. 11.

Fig. 11
figure11

Phase portrait of System (23) for the limit case \(\rho ^-=\infty \) corresponding to curve C

Then for System (22) there exists \(\rho ^*\), \(Y^-<\rho ^*<\infty \), such that the unstable manifold of \((\theta ^-,\rho ^*)\) contains a branch of the center manifold of \((\theta ^-+Y^-,0)\).

Uniqueness. The uniqueness of such \(\rho ^*\) follows from Melnikov’s integral, see Guckenheimer et al. (2013) for details. The splitting of the invariant manifolds for System (22) as \(\rho ^-\) varies is governed by a Melnikov’s integral. Notice that this method can be applied even when the equilibria are degenerate, see Schecter (1987).

Let \(X(\theta ,\rho ) = (X_1(\theta ,\rho ),X_2(\theta ,\rho ))\) be the vector field given by the right side of System (22). Also, let \((\theta ,\rho )(t)\) be the connecting orbit and let \(r(t) =\)div\(X(\theta ,\rho )(t)\). Then the Melnikov integral is given by:

$$\begin{aligned} M= & {} \int ^\infty _{-\infty }\text{ exp }[-\int _0^\tau r(\eta )d\eta ](-\dot{\rho }\;\dot{\theta })\frac{\partial }{\partial \rho ^-}\left( \begin{array}{c}X_1\\ X_2\end{array}\right) d\tau \\= & {} \int ^\infty _{-\infty }\text{ exp }[-\int _0^\tau r(\eta )d\eta ](-\dot{\rho }\;\dot{\theta })\left( \begin{array}{c}2a^2\rho ^-(\theta -\theta ^-)+a^2Y^-(\rho -2\rho ^-)\\ 2(Y^--\rho ^-)\rho ^2\Phi (\theta )\end{array}\right) d\tau \\= & {} \int ^\infty _{-\infty }\text{ exp }[-\int _0^\tau r(\eta )d\eta ]\left[ -\dot{\rho }\left( 2a^2\rho ^-(\theta -\theta ^-)+a^2Y^-(\rho -2\rho ^-)\right) \right. \\&\left. + \dot{\theta }\left( 2(Y^--\rho ^-)\rho ^2\Phi (\theta ) \right) \right] d\tau .\\= & {} \int ^\infty _{-\infty }\text{ exp }[-\int _0^\tau r(\eta )d\eta ]\\&\left[ (Y^--\rho ^-)^2\rho ^2\Phi (\theta )\left( 2a^2\rho ^-(\theta -\theta ^-)+a^2Y^-(\rho -2\rho ^-) \right) \right. \\&+\left. a^2\rho ^-[\rho ^-(\theta -\theta ^-)+Y^-(\rho -\rho ^-)]\left( 2(Y^--\rho ^-)\rho ^2\Phi (\theta ) \right) \right] d\tau .\\= & {} \int ^\infty _{-\infty }\text{ exp }\left[ -\int _0^\tau r(\eta )d\eta \right] \\&a^2\rho ^2\Phi (\theta )Y^-(Y^--\rho ^-)[2\rho ^-(\theta -\theta ^-)+(Y^-+\rho ^-)\rho -2\rho ^-Y^-] d\tau . \end{aligned}$$

Notice that the line \(2\rho ^-(\theta -\theta ^-)+(Y^-+\rho ^-)\rho -2\rho ^-Y^-=0\) stays below the line \(\rho ^-(\theta -\theta ^-)+Y^-(\rho -\rho ^-)=0\), for \(\theta <\theta ^-+Y^-\), which is below the connection orbit. Then we have \(2\rho ^-(\theta -\theta ^-)+(Y^-+\rho ^-)\rho -2\rho ^-Y^->0\) along the connection orbit and, therefore, \(M<0\). This implies that the center manifold of \((\theta ^-+Y^-,0)\) crosses from below to above the unstable manifold of \((\theta ^-,\rho ^-)\) as \(\rho ^-\) increases past \(\rho ^*\). Since this is true for any \(\rho ^*\) where the manifolds meet, it follows that \(\rho ^*\) is unique. \(\square \)

Region 1: In this region, a traveling wave with left state \((\theta ^-,\rho ^-)\) exists if its unstable manifold meets the stable manifold of equilibrium \((\theta ^-+Y^-,\rho ^- + (a-c)Y^- / c)\), corresponding to connection of type \(TC\rightarrow OC\).

Proposition 4

Fix \(a>0\) and \(\theta ^-\le 0\). Let \(Y^-<\rho ^-\) and let \(\rho ^*\) be given by Proposition 3. For every \((Y^-,c)\) in \(R_1\) with \(\rho ^*<\rho ^-\) there exists a velocity \(c^*<0\) such that the unstable manifold of \((\theta ^-,\rho ^-)\) contains part of the stable manifold of \((\theta ^-+Y^-,\rho ^- + (a-c^*)Y^-/c^*)\).

Proof

Proposition 3 gives the relative positions of the unstable manifold of \((\theta ^-,\rho ^-)\) and the center manifold of \((\theta ^-+Y^-,0)\) for \((Y^-,c_0)\in C\). This means that for c sufficiently close to \(c_0=aY^-/(Y^--\rho ^-)\) and smaller than \(c_0\) the unstable manifold of \((\theta ^-,\rho ^-)\) is below the stable manifold of \((\theta ^-+Y^-,\rho ^- + (a-c)Y^- / c)\), see Fig. 12.

Fig. 12
figure12

Phase portrait of System (22) for the case when c is sufficiently close to \(c_0\) corresponding to region \(R_1\)

Now we will study the behavior of these invariant manifolds as \(c\rightarrow -\infty \). For c sufficiently small the equilibrium \((\theta ^-+Y^-,\rho ^- + (a-c)Y^-/c)\) approaches \((\theta ^-+Y^-,\rho ^--Y^-)\). We can consider System (16)–(17), divide its right side of by \(-c\), which is equivalent to a rescaling of time for each c, and let \(c\rightarrow -\infty \), resulting in

$$\begin{aligned} \begin{array}{l} \dot{\theta }=(\theta -\theta ^-)+(\rho -\rho ^-),\\ \dot{\rho }=0. \end{array} \end{aligned}$$
(26)

As in the proof of Proposition 3, we see that for sufficiently small c, the unstable manifold of \((\theta ^-,\rho ^-)\) lies above the stable manifold of \((\theta ^-+Y^-,\rho ^- + {(a-c)}Y^-/{c})\). This shows the existence of a velocity \(c^*<0\) such that the unstable manifold of \((\theta ^-,\rho ^-)\) contains part of the stable manifold of \((\theta ^-+Y^-,\rho ^- + (a-c^*)Y^-/c^*)\). \(\square \)

Region 2: A traveling wave with left state \((\theta ^-,\rho ^-)\) exists if the unstable manifold of equilibrium \((\theta ^-,\rho ^-)\) meets the stable manifold of \((\theta ^- + {c}\rho ^- / (c-a),0)\) corresponding to the connection of type \(TC\rightarrow FC\).

Proposition 5

Fix \(a>0\), \(\theta ^-\le 0\) and let \(\rho ^*\) be given by Proposition 3.

  1. 1.

    If \(\rho ^-<Y^-\) then for \(\rho ^*<\rho ^-<\infty \), every point \((Y^-,c)\in R_2\) have \(\theta ^- + {c\rho ^-}/(c-a)>0\). For each such \(\rho ^-\) there exists a velocity \(aY^-/(Y^--\rho ^-)<c^*<0\), such that the stable manifold of the saddle \((\theta ^-+ c^*\rho ^- /(c^*-a),0)\) meets the unstable manifold of the degenerate equilibrium \((\theta ^-,\rho ^-)\).

  2. 2.

    If \(Y^-<\rho ^-\) then for every \(\rho ^*<\rho ^-<\infty \), there exists a velocity \(-\infty<c^*<0\) such that the unstable manifold of \((\theta ^-,\rho ^-)\) contains part of the stable manifold of \((\theta ^-+ c^*\rho ^- /(c^*-a),0)\).

Proof

In the first case, we study the limit \(c\rightarrow c_0=aY^-/(Y^--\rho ^-)\) and limit \(c\rightarrow 0\).

Proposition 3 gives the relative positions of the unstable manifold of \((\theta ^-,\rho ^-)\) and the center manifold of \((\theta ^-+Y^-,0)\) for \((Y^-,c_0)\in C\). For c sufficiently close to \(c_0\), \(c>c_0\), the unstable manifold of \((\theta ^-,\rho ^-)\) is above the central manifold of \((\theta ^-+Y^-,0)\), see Fig. 13.

\(\square \)

Fig. 13
figure13

Phase portrait of System (22) for the case when c is sufficiently close to \(c_0\) corresponding to region \(R_2\)

To study the limit \(c\rightarrow 0\), we can multiply the right side of System (16)–(17) by \(-c\) and let \(c\rightarrow 0\), resulting in

$$\begin{aligned} \begin{array}{l} \dot{\theta }=0,\\ \dot{\rho }=-\rho ^2\Phi (\theta ). \end{array} \end{aligned}$$
(27)

Then, for sufficiently small c we see that the unstable manifold of \((\theta ^-,\rho ^-)\) lies below the central manifold of \((\theta ^-+Y^-,0)\), see Fig. 14.

Fig. 14
figure14

Phase portrait of System (22) for the limit case \(c\rightarrow 0\) corresponding to region \(R_2\)

We can conclude that there exists a velocity \(c_0<c^*<0\), such that the stable manifold of the saddle \((\theta ^-+ c^*\rho ^- /(c^*-a),0)\) meets the unstable manifold of the degenerate equilibrium \((\theta ^-,\rho ^-)\).

In the second case, when \(Y^-<-\theta ^-\), we can use the limits studied previously \(c\rightarrow -\infty \) and \(c\rightarrow 0\).

For c sufficiently small, we can consider System (16)–(17), divide its right side of by \(-c\), which is equivalent to rescaling time by c, and let \(c\rightarrow -\infty \), resulting in

$$\begin{aligned} \begin{array}{l} \dot{\theta }=(\theta -\theta ^-)+(\rho -\rho ^-),\\ \dot{\rho }=0. \end{array} \end{aligned}$$
(28)

We see that for sufficiently small c, the unstable manifold of \((\theta ^-,\rho ^-)\) lies above the central manifold of \((\theta ^-+Y^-,0)\).

For c close to 0, we can multiply the right side of System (16)–(17) by \(-c\) and let \(c\rightarrow 0\), resulting in

$$\begin{aligned} \begin{array}{l} \dot{\theta }=0,\\ \dot{\rho }=-\rho ^2\Phi (\theta ). \end{array} \end{aligned}$$
(29)

We see that the unstable manifold of \((\theta ^-,\rho ^-)\) lies below the central manifold of \((\theta ^-+Y^-,0)\).

Then, we can conclude that there exists a velocity \(-\infty<c^*<0\) such that the unstable manifold of \((\theta ^-,\rho ^-)\) contains part of the stable manifold of \((\theta ^-+ c^*\rho ^- /(c^*-a),0)\).

Considering the results for regions C, \(R_1\) and \(R_2\) the main theorem is proved.

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Acknowledgements

The authors would like to thank Prof. S. Schecter and Prof. D. Marchesin for the preliminary work. We also would like to thank Prof. D. Tadeu, I. Ledoino and D. L. de Albuquerque for developing and improving the numerical code RCD used in this work. We thank anonymous referee for help in improving this text.

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Correspondence to Grigori Chapiro.

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This work was supported in part by the FAPEMIG under Grant APQ 01377/15.

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Chapiro, G., Senos, L. Riemann solutions for counterflow combustion in light porous foam. Comp. Appl. Math. 37, 1721–1736 (2018). https://doi.org/10.1007/s40314-017-0420-6

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Keywords

  • Traveling wave
  • Riemann problem
  • Counterflow combustion
  • Porous medium

Mathematics Subject Classification

  • 80A25
  • 35C07
  • 35K57
  • 34C37