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ResearchCite this article: Forbes LK. 2013 Thermalsolitons: travelling waves in combustion. ProcR Soc A 469: 20120587.http://dx.doi.org/10.1098/rspa.2012.0587

Received: 8 October 2012Accepted: 21 November 2012

Subject Areas:applied mathematics

Keywords:asymptotic analysis, combustion, solitons,stability, strained coordinates, travelling waves

Author for correspondence:Lawrence K. Forbese-mail: larry.forbes@utas.edu.au

Thermal solitons: travellingwaves in combustionLawrence K. Forbes

School of Mathematics and Physics, University of Tasmania,PO Box 37, Hobart, Tasmania 7001, Australia

A competitive reaction system is considered, underwhich some chemical reagent decays by means of twosimultaneous chemical reactions to form two separateinert products. One reaction is exothermic, and theother is endothermic. The governing equations forthe model are presented, and a weakly nonlineartheory is then generated using the method of strainedcoordinates. Travelling-wave solutions are possiblein the model, and the temperature is found tohave a classical sech-squared profile. The stabilityof these moderate-amplitude temperature solitons isconfirmed both analytically and numerically.

1. IntroductionNonlinear travelling waves of permanent form in achannel of water have been the subject of intense interestfor a long time. When the waves are of moderateamplitude and are moving at close to the criticalspeed, it is known that their shape may be describedapproximately by the Korteweg–de Vries equation. Thisfamous partial differential equation is discussed indetail by Whitham [1], for example, and in numerousresearch monographs such as the recent work of Vanden-Broeck [2, §5.2]. There is now a vast literature on thisequation and variants of it, and the recent overviewarticle by Grimshaw [3] surveys some of this work.There are also other related weakly nonlinear modelequations for describing water waves, and some of thesegeneralizations are presented by Johnson [4].

There is similarly an extensive literature on travellingwaves in a variety of reaction–diffusion systems.Nonlinear reaction fronts occur throughout populationand disease-spread models in biology [5] and also inauto-catalytic chemical reactions, as discussed by Gray &Scott [6]. Thermokinetic models of combustion in eithersolid or gaseous fuel beds also predict the existence oftravelling fronts of temperature, and have been analysedby Matkowsky and Sivashinsky [7] and Weber et al. [8],

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for example. More recently, there has been some interest in competitive type combustionreactions, in which a compound decays to form products through either of two possible differentreaction pathways. One of these is exothermic (releasing heat), and the other is endothermic(requiring heat input), and such systems have been analysed fairly comprehensively by Ball et al.[9], Hmaidi et al. [10] and Sharples et al. [11]. Recently, a weakly nonlinear asymptotic solutionwas presented by Forbes [12], and it permitted a solution in a phase plane.

In many combustion systems, the chemical reactions cannot be considered in isolation, butinstead are coupled to fluid-flow behaviour. The text by Zel’dovich et al. [13] discusses the basicfluid-combustion problem, and in particular describes how the rate of change in the concentrationof a chemical species is due not only to reaction dynamics but also to the convective movement ofthe species in the fluid. Forbes [14] used such ideas in a model of bushfire spread, and an analysisof one-dimensional fluid-combustion behaviour was presented by Forbes & Derrick [15]. Theydemonstrated that small conflagrations can occur as soliton-type phenomena in the temperature,but that disturbances of larger amplitude could result in shock waves, even to the extent of thecomplete exhaustion of the fuel.

In this paper, a competitive reaction scheme is considered, in which the reagents arecomponents of a flowing gas. The scheme is somewhat similar to that considered by Forbes &Derrick [15] and its governing equations are outlined in §2. Here, however, a weakly nonlinearspatio-temporal approximation is developed in §3 and makes use of the method of strainedcoordinates. Travelling waves in this system are discussed explicitly in §4, and the stability ofthese solutions is analysed in §5. Some concluding observations and suggestions for furtherinvestigation are given in §6.

2. The combustion modelConsider some chemical species X, which decays by one of two possible mechanisms. The first isan endothermic reaction, at some temperature-sensitive rate k1, to give an inert product A. Thisis in competition with a second reaction, at rate k2 and occurring exothermically, under which Xforms an alternative inert product B. This reaction scheme may be represented in the form

X↗k1(T) A endothermic↘k2(T) B exothermic.

(2.1)

A scheme of the form (2.1) was considered by Hmaidi et al. [10], and travelling waves in thesecompetitive reactions have been studied by Ball et al. [9] and Sharples et al. [11]. Recently, a weaklynonlinear asymptotic investigation of these travelling waves was presented by Forbes [12] in ananalysis that permitted a complete solution in a phase plane. All these studies involved the purechemical system (2.1) and its associated thermodynamics alone; in this present study, this is nowgeneralized to include the effects of the reaction in a compressible gas.

The competitive reaction (2.1) occurs in a gas of total density ρ. For simplicity, it is assumedthat the flow is essentially one-dimensional, with local gas speed u in the direction of the x-axis.The rate equation for the chemical species X in equation (2.1) then becomes

∂[X]∂t

+ ∂

∂x(u[X]) = −k1(T)[X] − k2(T)[X], (2.2)

in which [X] denotes the molar concentration of species X. For the present, it will be assumedthat the temperature-sensitive reaction rates follow Arrhenius kinetics (see [6, p. 84]) and havethe forms

k1(T) = Z1 exp(

− E1

RT

)

and k2(T) = Z2 exp(

− E2

RT

).

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (2.3)

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Here, E1 and E2 are activation energies for the two reactions, and the constants Z1 and Z2 give therates of reactions at some standard temperature. The quantity R is the universal gas constant.

Conservation of mass within the gas requires that the continuity equation

∂ρ

∂t+ ∂

∂x(ρu) = 0 (2.4)

be satisfied. In addition, the Euler equation, expressing the conservation of linear momentum inthe absence of viscous diffusion, takes the form

∂

∂t(ρu) + ∂

∂x(p + ρu2) = 0. (2.5)

Here, p is the pressure and it is related to the temperature T by the gas law

p = ρRT. (2.6)

Finally, there is an equation expressing the conservation of energy. This may be derived from thefirst law of thermodynamics, as described by Liepmann & Roshko [16, p. 190]. It follows that

ρ∂

∂t

[γ RTγ − 1

+ 12

u2]

+ ρu∂

∂x

[γ RTγ − 1

+ 12

u2]

− ∂p∂t

= −N(T − Ta) − Q1k1(T)[X] + Q2k2(T)[X], (2.7)

where k1 and k2 are the temperature-sensitive rates (2.3), and Q1 and Q2 are the enthalpies ofexothermic and endothermic reactions per mole, respectively. The constant γ represents the ratioof specific heats, and for a diatomic gas its value is approximately 1.4. The first term on the right-hand side of (2.7) represents Newtonian cooling to ambient temperature Ta. This form (2.7) isconsistent with that given by Anderson [17, p. 196] and Zel’dovich et al. [13, p. 238], and the firsttwo terms in parentheses on the left-hand side correspond to the total enthalpy in the system.

Non-dimensional variables are now introduced, based on the time and energy scalesappropriate to the exothermic reaction in the scheme (2.1). Thus, the unit of time is taken tobe 1/Z2 and lengths are scaled relative to

√E2/Z2 with these two parameters taken from the

Arrhenius rate formula for k2(T) in equation (2.3). The temperature is scaled using the quantityE2/R and speeds are referenced to the term

√E2. Far ahead of the reaction front, the gas density

has some value ρ0 and this is taken as the reference for density. The pressure is scaled relative toρ0E2, and concentrations are made dimensionless by reference to ρ0E2/Q2. There are thus five keydimensionless parameters in this problem, and these are

α = Z1

Z2; ε = E1

E2; γQ = Q1

Q2

and θa = RTa

E2; β = N

Z2ρ0R.

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (2.8)

The constants α and ε are respectively the ratio of rates and activation energies for the twocompeting reactions and γQ is the ratio of their enthalpies. The dimensionless rate of Newtoniancooling is β and θa is the ambient temperature. In addition to these five parameters, the solutionis also dependent upon the ratio γ ≈ 1.4 of specific heats for the gas.

In these new dimensionless variables, the rate law (2.2) takes the form

∂X∂t

+ ∂

∂x(uX) = −αk1(T)X − k2(T)X, (2.9)

in which the Arrhenius relations (2.3) become simply

k1(T) = exp(−ε/T); k2(T) = exp(−1/T). (2.10)

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The continuity equation (2.4) and momentum equation (2.5) retain the same forms in the newvariables, with the gas law (2.6) becoming simply p = ρT. Finally, the energy equation (2.7)now becomes

ρ∂

∂t

[γ T

γ − 1+ 1

2u2

]+ ρu

∂

∂x

[γ T

γ − 1+ 1

2u2

]− ∂p

∂t

= −β(T − θa) − αγQk1(T)X + k2(T)X. (2.11)

The appropriate conditions far ahead of the reaction front are

T → θa; u → 0; X → X0;

and ρ → 1; p → θa as x → ∞

}(2.12)

in these non-dimensional variables, and the problem is considered over the entire real x-axis.

3. Weakly nonlinear approximationIn this section, an approximate set of equations is derived, under the assumption that thevariables differ from the steady-state conditions (2.12) only by some small amount, characterizedby a parameter κ assumed to be small. In addition, the two dimensionless rates (2.10) for thecompeting reactions are approximated with expressions of the form

k1(T) ={

0, T < θa

1 − e−ε(T−θa), T > θa

k2(T) ={

0, T < θa

1 − e−(T−θa), T > θa.

(3.1)

These relations (3.1) mimic the high-temperature behaviour of the Arrhenius rates (2.10), butthey avoid the ‘cold boundary’ problem discussed by Matkowsky & Sivashinsky [7] and Grayet al. [18], in which the reactions would proceed at some finite rate for any temperature aboveabsolute zero. Similar forms to (3.1) were used by Forbes & Derrick [15].

The approximation technique is based on the method of strained coordinates, an example ofwhich is given in the text by Jordan & Smith [19, p. 148]. New time and space coordinates aredefined by means of the relations

t = κμt; x = κx, (3.2)

in which κ is the small parameter and μ is an unknown constant.The dimensionless equations in §2 are now written in terms of the strained coordinates (3.2).

The dependent variables, along with the constant μ in equation (3.2), are represented by meansof expansions in powers of the small parameter κ , according to the expressions

X = X0 + κX1 + κ2X2 + O(κ3),

u = κU1 + κ2U2 + O(κ3),

ρ = 1 + κρ1 + κ2ρ2 + O(κ3),

T = θa + κT1 + κ2T2 + O(κ3),

p = θa + κP1 + κ2P2 + O(κ3)

and μ = μ0 + κμ1 + κ2μ2 + O(κ3).

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭(3.3)

These expansions (3.3) are substituted into the governing equations, and terms at each orderof the parameter κ are collected. Without loss of generality, it is possible to set μ0 = 1, and this isassumed from now on.

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The modified reaction rates (3.1) yield the expressions

k1(T) = κεT1 + κ2(εT2 − 12 ε2T2

1) + O(κ3)

and k2(T) = κT1 + κ2(T2 − 12 T2

1) + O(κ3).

⎫⎬⎭ (3.4)

The rate law (2.9) and the energy equation (2.11) are now expanded in powers of the smallparameter κ , as in equations (3.3). At first order in κ , both these equations require that

T1(x, t) ≡ 0. (3.5)

It then follows from the gas law p = ρT that

P1 = θaρ1; P2 = T2 + θaρ2 (3.6)

in view of the relation (3.5). The mass equation (2.4) and the energy equation (2.11), combinedwith the expressions (3.4) then show that the second-order temperature term is expressed in termsof the first-order speed function by means of the relation

T2(x, t) = −√

θa

Γ1

∂U1

∂ x, (3.7)

in which the constant

Γ1 = β + (αγQε − 1)X0√θa

(3.8)

has been defined for convenience.The momentum equation (2.5), with the mass equation (2.4), and the relations (3.6) give rise to

the condition∂2U1

∂ t2= θa

∂2U1

∂ x2

at first order in the small parameter κ , and in fact, a similar relationship holds for the second-orderterm U2 in the system (3.3). This relation suggests that a travelling-wave solution should exist forthis problem, and this is examined further in §4. After some algebra, it becomes evident that thefirst-order speed function U1 must satisfy a compatibility condition at the second order in κ . Thisrelation has the form

∂3U1

∂ x2∂ t= Γ1√

θa

[∂2

∂ x∂ t(U2

1) + 2μ1∂2U1

∂ t2

], (3.9)

in which Γ1 is the constant defined in (3.8). This equation (3.9) can be integrated with respect toscaled time t, using the upstream boundary conditions (2.12), to give the weakly nonlinear partialdifferential equation

∂2U1

∂ x2 = Γ1√θa

[∂

∂ x(U2

1) + 2μ1∂U1

∂ t

](3.10)

for the perturbation speed U1(x, t). The second-order temperature function T2 can then becalculated from equation (3.7).

4. Travelling wavesThe governing relation (3.10) in the weakly nonlinear approximation is essentially Burgers’equation (see [1, ch. 4]), and it yields solutions in the form of travelling waves. These are obtainedby making use of the moving coordinate

η = x −√

θa t (4.1)

in the strained system. At zeroth order in the small parameter κ , the speed in these coordinates istherefore

√θa.

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In this moving frame (4.1), the partial differential equation (3.10) reduces to the ordinarydifferential equation

d2U1

dη2 = Γ1√θa

[d

dη(U2

1) − 2μ1√

θadU1

dη

].

This expression may immediately be integrated once to give

dU1

dη= Γ1√

θa[U2

1 − 2μ1√

θaU1], (4.2)

using the upstream boundary conditions (2.12). The relation (4.2) now admits a solution in thefinal form

U1(η) = 2μ1√

θa

1 + exp(2μ1Γ1η), (4.3)

with no loss of generality. Essentially, this same form (4.3) was found by Forbes & Derrick [15] ina related combustion problem.

Equation (3.7) for the perturbation to the temperature takes the simpler form

T2(η) = −√

θa

Γ1

∂U1

∂η, (4.4)

and when this is combined with the solution (4.3), it yields the temperature profile

T2(η) = μ21θa sech2(μ1Γ1η). (4.5)

The temperature may therefore be represented as

T(x, t) = θa + κ2μ21θa sech2(μ1Γ1(x −

√θa t)) + O(κ3), (4.6)

and has the form of a classical soliton (see [1, p. 468]). Similar temperature profiles were observedby Forbes & Derrick [15].

It is interesting to re-construct the governing differential equation for the second-ordertemperature perturbation in equation (4.5). This is done in a straightforward manner, usingequations (4.2) and (4.4). After some algebra, the result is

T2d2T2

dη2 −(

dT2

dη

)2+ 2

Γ 21

θaT3

2 = 0. (4.7)

It can be verified by direct differentiation that (4.5) is a solution of this differential equation (4.7).Nevertheless, the properties of this equation (4.7) are very different from those of the famousKorteweg–de Vries equation, even although both admit the soliton-type solution (4.5). Inparticular, the equation (4.7) has only one equilibrium point (T2, T′

2) = (0, 0) in the phaseplane, and it is a degenerate saddle. This may be confirmed by using the solution (4.5) todemonstrate that

d(T′2)

dT2(0, 0) = ±2μ1Γ1

at the origin in the phase plane; thus two separatrices emerge from the equilibrium point, buttheir slope depends on the parameter μ1 that defines the amplitude AT = μ2

1θa of the soliton andis arbitrary. Figure 1 shows the (T2, T′

2) phase plane for solitons of 10 different amplitudes, at arepresentative set of parameter values.

5. Stability of solitonsIt is possible to assess the stability of the travelling waves in §4 directly from the temperatureequation (4.7), but it turns out to be easier to work from the equation (4.2) for the velocityperturbation function U1 and this is presented here.

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0 0.002 0.004 0.006 0.008 0.010−4

−3

−2

−1

0

1

2

3

4×10−3

T2

T¢2

Figure 1. Temperature solitons in a phase-plane representation, for ambient temperature θa = 0.25, upstream materialconcentration X0 = 1 and parameters γQ = 1, ε = 2, α = 1 and β = 0.1. Solutions are shown at 10 different solitonamplitudes AT = 0.001, 0.002, . . . , 0.01. (Online version in colour.)

In the travelling-wave coordinate (4.1), stability of the solution (4.3) is assessed by adding someperturbation U1 to create

U1(η) + U1(η)

and substituting this into the governing equation (4.2), retaining only first-order terms in U1. Thisyields the linearized variational equation

dU1

dη= 2

Γ1√θa

[U1 − μ1√

θa]U1. (5.1)

This stability equation (5.1) admits a general solution in the form

U1(η) = C exp(

2Γ1√θa

∫[U1 − μ1

√θa] dη

), (5.2)

in which C is an arbitrary constant. When the solution (4.3) is used in equation (5.2) and theintegral evaluated, it may be shown that

U1(η)

U1(0)= sech2(μ1Γ1η). (5.3)

Consequently, the ratio in equation (5.3) is always less than 1 and in fact decays to zero asη → ±∞, so that the perturbation U1 decays to zero. Thus, the original profile (4.3) is stable.

Stability of the profile (4.3) in the more general spatio-temporal context has also beenexamined, using the method of lines to solve the partial differential equation (3.10) numerically(see [20, p. 302]). The spatial derivatives are approximated using second-order central differences,and the resulting ordinary differential equations are integrated forward in time, using the

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−50 0 500

10

20

30

40

0

0.1

0.2

0.3

t

x

U1t = 0

Figure2. Theeffect of an initial perturbation to thefirst-order velocity functionU1 at 20 times t = 2, 4, . . . , 40. In this solution,the ambient temperature is θa = 0.25 and upstreammaterial concentration is X0 = 1, with parametersγQ = 1, ε = 2,α = 1andβ = 0.1. The initial soliton amplitude is AT = 0.01. The initial temperature is shown separately. (Online version in colour.)

fourth–fifth-order Runge–Kutta Fehlberg quadrature in the MATLAB package. The system isgiven the initial condition

U1(x, 0) = 2√

AT

1 + exp(2Γ1x√

AT/θa)+ qf (x), (5.4)

for some small parameter q and appropriate perturbation f (x), and integrated forward in time. Ifstable, the effect of the initial perturbation f will diminish in time, leaving simply the travellingwave (4.3).

The results of a sample integration are shown in figure 2. For comparison, the parameters arethe same as those in figure 1. The initial soliton amplitude in the condition (5.4) was AT = 0.01, andthe parameter q was chosen to be q = 0.4 × 2

√AT. The perturbation function used in this figure

was f (x) = exp(−x2), and 501 spatial points were used, over the interval −50 < x < 50. The initialperturbed velocity profile U1(x, 0) is labelled on the diagram, and solutions at the 20 later timest = 2, 4, . . . , 40 are shown on a three-dimensional plot. It is clear that the travelling-wave solution(4.3) is re-established within a very short time, confirming that the velocity profile is stable, inaccordance with the linearized stability analysis presented in equation (5.3).

The second-order temperature perturbation function T2(x, t) may be computed from thevelocity profiles in figure 2, using straightforward differentiation, following equation (3.7). Thisis achieved here using centred differences in the method of lines solution scheme. Results for thesame case as in figure 2 are displayed in figure 3, at the same times t = 2, 4, . . . , 40, although thetemperature at the initial time is not shown. The first two profiles shown are still influenced bythe effects of the initial disturbance, but it is evident that the solution quickly re-establishes thetravelling soliton form (4.6). Thus later times show a sech-squared profile of permanent form,that moves with speed

√θa = 1

2 for this choice of parameters, and this is confirmed by a carefulinspection of the results.

6. ConclusionsThis paper has presented an analysis of combustion waves of small amplitude, for a competitivereaction system in a flowing compressible fluid. A weakly nonlinear development, based onthe method of strained coordinates, shows that the temperature disturbance may propagate asa classical sech-squared type soliton, although the governing equation for temperature is not theclassical Korteweg–de Vries equation. This soliton has been shown numerically and analyticallyto be stable, at least to the order of approximation for which this theory is valid.

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−50 0 500

20

40

0

0.005

0.010

0.015

x

t

T2

Figure 3. The effect of an initial perturbation to the second-order temperature function T2 at 20 times t = 2, 4, . . . , 40. Inthis solution, the ambient temperature is θa = 0.25 and upstreammaterial concentration is X0 = 1, with parameters γQ = 1,ε = 2,α = 1 andβ = 0.1. The initial soliton amplitude is AT = 0.01. (Online version in colour.)

−80 −60 −40 −20 0 20 40 60 800

10

20

30

40

50

60

0

0.01

0.02

0.03

0.04

t

x

T2

Figure 4. The evolution of an initial condition consisting of two solitons of different amplitudes. The second-order temperatureperturbation T2 is shown at the thirty times t = 2, 4, . . . , 60, for the same parameters as previously. (Online version in colour.)

Nevertheless, as with the classical Korteweg-de Vries equation, the theory is only valid for theconditions under which it has been derived, and too much should not be expected of it otherwise.In particular, Forbes & Derrick [15] showed that, when the amplitude of the disturbance becomestoo large, a more complete theory indicates that the temperature soliton ceases to be stable,and instead develops into a shock wave. Their model did not include the effects either of fluidviscosity or temperature diffusion, as is also the case here, and so it remains an interesting openquestion as to what extent diffusion might render the travelling temperature soliton stable, evenfor larger-amplitude waves.

The second-order temperature perturbation function T2 propagates as a classical sech-squaredsoliton, although it does not satisfy the Korteweg–de Vries equation. It is possible to derive theweakly nonlinear equation for T2 by combining equations (3.10) and (3.7); this is not shown herein the interests of space, although the reduced version of it in travelling-wave coordinates isgiven in equation (4.7) and its phase-plane behaviour is displayed in figure 1. It is therefore aninteresting question as to how solutions to the spatio-temporal equation behave more generally.One such numerical experiment is shown in figure 4, for the same parameter values as inthe previous figures. Here, equation (3.10) has again been solved using the method-of-linestechnique as previously, but now with an initial condition consisting of two disturbances of the

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form (5.4), but centred at x = ±L and added. For illustrative purposes, L = 20 and the amplitudeof the forward disturbance at x = L is taken to be 70 per cent of the rearward disturbance.The temperature perturbation T2 has then been computed from the method-of-lines solution,in this case with 2401 spatial mesh points over the interval −80 < x < 80, and is presented infigure 4. This diagram shows that the larger rearward soliton moves faster than the forwardone and eventually catches up to it, as time progresses. However, unlike the celebrated resultfor the Korteweg–de Vries equation, for which interacting pulses pass through one anotherand regain their identity after the interaction is complete [21], in the present reaction the twotemperature solitons evidently coalesce to form a single larger pulse. A more systematic studyof the interaction between temperature pulses in combustion systems of this type may thereforeprove worthwhile, and is left for future research.

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