Stability of flames in an exothermic-endothermic
system
Peter L. Simon1 , Serafim Kalliadasis2 ,
John H. Merkin3 , Stephen K. Scott1
1
Department of Chemistry, 2 Department of Chemical Engineering,
3
Department of Applied Mathematics,
The University of Leeds, Leeds, LS2 9JT, UK
E-mail: [email protected]
July 17, 2003
Abstract
The propagation of a premixed laminar flame supported by an exothermic chemical reaction under adiabatic conditions but subject to inhibition through a parallel
endothermic chemical process is considered. The temporal stability to longitudinal
perturbations of any resulting flames is investigated. The heat loss through the
endothermic reaction, represented by the dimensionless parameter α, has a strong
quenching effect on wave propagation. The wave speed–cooling parameter (α, c)
curves are determined for a range of values of the other parameters. These curves
can be monotone decreasing or S-shaped, depending on the values of the parameters
β, representing the rate at which inhibitor is consumed relative to the consumption
of fuel, µ, the ratio of the activation energies of the reactants and the Lewis numbers. This gives the possibility of having either one, two or three different flame
velocities for the same value of the cooling parameter α. For Lewis numbers close
to unity, when there are three solutions, two of them are stable and one is unstable,
with two saddle-node bifurcation points on the (α, c) curve. For larger values of the
Lewis numbers there is a Hopf bifurcation point on the curve, dividing it into a stable and an unstable branch. The saddle-node and Hopf bifurcation curves are also
determined. The two curves have a common, Takens-Bogdanov bifurcation point.
Keywords: quenching, travelling wave, multiple solutions, Evans function
AMS classification: 80A25, 35K57, 35B32.
1
1
Introduction
We consider the propagation of a flame supported by an exothermic chemical reaction
under adiabatic conditions but subject to possible inhibition through a parallel endothermic chemical process. In particular, we are concerned with determining the conditions
for the existence of constant-velocity constant-form flame structures, the dependence of
the flame speed on the reaction parameters and the temporal stability to longitudinal
perturbations of any resulting flames. Our model involves the two chemical steps
A → P1 + heat
rate = k1 a e−E1 /RT
W → P2 − heat
rate = k2 w e−E2 /RT
The first step has a positive exothermicity (negative reaction enthalpy) Q1 > 0, the second step has a negative exothermicity, Q2 < 0.
The equations governing our model, written in a reference frame moving with the
flame, are [7, 27, 30, 31]
mA0 (z) = ρDA A00 (z) − k1 e−E1 /RT ρA(z)
(1)
mW 0 (z) = ρDW W 00 (z) − k2 e−E2 /RT ρW (z)
(2)
Cp mT 0 (z) = λT 00 (z) + Q1 k1 e−E1 /RT ρA(z) + Q2 k2 e−E2 /RT ρW (z)
(3)
For notation see the list of symbols. (Primes denote differentiation with respect to the
travelling wave co-ordinate z.) Equations (1)-(3) are subject to the boundary conditions
A → A0 , W → W0 , T → Ta
A0 → 0, W 0 → 0, T 0 → 0
as
as
z → −∞,
z → +∞.
(4)
(5)
Here A0 and W0 denote the initial mass fractions of reactants A and W and Ta is the
ambient temperature.
Previous work on flame quenching [6, 12, 18, 22, 29, 30] has been concerned with
heat removal through physical processes such as conduction or radiation. In the present
case, the ‘heat loss’ process is controlled by the endothermic species W , which is consumed
during the process at a rate which depends on the temperature and its local concentration.
We have investigated the system in the case of unit Lewis numbers [24]. From this
previous work, we expect the important parameters to be represented by the dimensionless
groups
k2
Q2 W0 k2 (E1 −E2 )/RTb
e
,
β = e(E1 −E2 )/RTb
α=−
Q1 A0 k1
k1
which respectively represent a characteristic rate at which heat is lost by the endothermic
reaction relative to the rate at which it is produced by the exothermic reaction and the
rate at which the inhibitor W is consumed relative to the fuel A. In [23] we considered the
case when the temperature dependence of the reaction rates was given by step functions,
2
being zero below specified ‘ignition temperatures’ and a non-zero constant, independent
of temperature, above the ignition temperatures. In this special case, system (1)-(5) can
be solved analytically, allowing some qualitative features of the model to be readily revealed. A modification of the system (1)-(3) has been studied previously [14, 19] in which
a second-order reaction was taken for the fuel. This model was investigated numerically
in [14] and with high activation energy asymptotics in [19].
The aim of the present work is to determine how the dimensionless flame velocity c
depends on the ‘heat loss’ parameter α for general values of the Lewis numbers and to
determine the stability of the corresponding temperature and concentration profiles. In
particular, we wish to determine whether there are critical values of α at which flame
extinction occurs and values of α at which the system changes stability through a Hopf
bifurcation and time dependent structures occur.
In Section 3 we present our results concerning the shape of the bifurcation diagrams
for c as a function of α, which will be referred to as (α, c) curves. We first derive some
qualitative properties of the solutions analytically. We show that we can have either front
waves or pulse waves in temperature depending on the relative sizes of α and β. We
establish that there are two possibilities for large heat loss (Q2 large) depending on the
value of µ = E2 /E1 , the ratio of the activation energies for reactions. For µ > 1, i.e
E2 > E1 , flame structures exist for all α, whereas for µ ≤ 1 there is a cut-off value α0 of
α at which the flame speed becomes zero with no flame solutions for α > α0 . We then
present our numerical results. To do so we take a fixed value for the activation energy
parameter ε of ε = 0.1 and determine the (α, c) curve with pseudoarclength continuation
for different values of µ and β. Our numerical investigation reveals that the curve can
have three different shapes depending on the values of µ, β and the Lewis numbers LA ,
LW . For µ > 1 the curve is monotone decreasing and unbounded in the α direction, i.e.
there is no quenching. For µ ≤ 1 the curve is bounded in the α direction, as established
theoretically, though it can have two different shapes. For larger values of β the curve
is monotone decreasing whereas for smaller values of β it is S-shaped. Thus there is a
range of α where there are three different wave velocities as well as a quenching value of
α. These results suggest that for µ < 1 a hysteresis bifurcation occurs at certain values
of β.
In Section 4 we deal with the stability of the flame solution. The structure of the
spectrum of the linearised system is investigated theoretically. We present two qualitative
results using the Evans function, the zeros of which are the isolated eigenvalues of the
linearised system. We show, without the exact determination of the eigenvalues, that
Hopf bifurcations and saddle-node bifurcations can occur at certain values of the parameters. We then determine numerically the first pair of isolated eigenvalues along the (α, c)
bifurcation curves by discretising the linearised problem with finite differences and the
computing the eigenvalues of the matrix obtained from the discretisation. Concerning
the stability of the flame, we show that, for Lewis numbers close to unity, when there
are three solutions, two of them are stable and one is unstable, i.e. there are saddle-node
bifurcations at the turning points. For larger values of the Lewis number there is a Hopf
bifurcation point on the (α, c) curve, dividing it into a stable and an unstable branch.
The saddle-node and Hopf bifurcation curves are also determined in the (α, LA ) param3
eter plane, where LA is the Lewis number of the fuel. The two curves have a common,
Takens-Bogdanov bifurcation point, where the linearised system has a double zero eigenvalue. These methods have been widely applied for different models. In [2] the stability of
travelling wave solutions of an autocatalytic system was studied. The stability of solitary
wave solutions of a wide range of nonlinear evolution equations, including the generalised
KdV, Boussinesq and KdV-Burgers equations, was studied in [21]. The spectral stability
of a large family of viscous shock waves including compressive and under compressive
travelling waves in thin film models was investigated in [11, 16].
Finally, we perform some numerical simulations of an initial-value problem, from which
our dimensionless versions of equations (1) - (3) can be derived, in cases when the travelling
waves are unstable. We find only periodic behaviour (single and periodic 2 oscillations).
These cycles can include periods of rapid propagation and high temperatures (above the
burnt gas temperature) and periods when the flame is virtually stationary.
2
Model
In order to make equations (1)-(3) dimensionless, we introduce the following variables
y = ξz,
a(y) =
A(z)
,
A0
w(y) =
W (z)
,
W0
b(y) =
T (z) − Ta
Tb − Ta
(6)
where
Q1 A0
Cp ρk1 −E1 /RTb
.
(burnt gas temperature), ξ 2 =
e
Cp
λ
Substituting (6) into equations (1)-(3) and introducing the notation
s
RTb
Cp 1/ε
λ
λ
ε=
, LA =
, LW =
, c=m
e
E1
DA ρCp
DW ρCp
ρλk1
Tb = Ta +
α=−
Q2 W0 k2 (E1 −E2 )/RTb
e
,
Q1 A0 k1
β=
k2 (E1 −E2 )/RTb
e
,
k1
f1 (b) = e(b−1)/εb ,
(7)
E2
,
E1
(8)
f2 (b) = eµ(b−1)/εb ,
(9)
µ=
we obtain
00
0
L−1
A a − c a − af1 (b) = 0
(10)
00
0
L−1
W w − c w − βwf2 (b) = 0
(11)
b00 − cb0 + af1 (b) − αwf2 (b) = 0
(12)
on −∞ < y < ∞, where primes now denote differentiation with respect to y and c is the
dimensionless flame velocity. Here we assume that Ta = 0 in order to avoid the mathematical difficulties associated with a lack of a ’cold boundary’ [30, 31]. The boundary
conditions are
a → 1, w → 1, b → 0
a → 0, w0 → 0, b0 → 0
0
4
as
as
y → −∞,
y → +∞.
(13)
(14)
We also assume that
b(y) > 0, a(y) ≥ 0, w(y) ≥ 0, c > 0,
−∞ < y < ∞.
(15)
This assumption excludes the trivial solution b ≡ 0, a ≡ 1, w ≡ 1. Here the Lewis numbers LA and LW and the parameters α, β, µ, ε are non-negative.
3
The shape of the (α, c) curve
3.1
Theoretical results
We use the following proposition, which was established in [24].
Proposition 1
1. The functions a and w are decreasing on −∞ < y < ∞.
2. The limits
a+ = lim a,
y→∞
b+ = lim b,
y→∞
exist and
a+ + b+ −
w+ = lim w
y→∞
α
α
w+ = 1 − .
β
β
(16)
We shall refer to temperature profiles with b+ > 0 as fronts, and when b+ = 0 as
pulses. It is clear from the differential equations (10) and (11) that, in the case of fronts
(b+ > 0) we have a+ = 0 and w+ = 0, since in this case f1 (b+ ) > 0 and f2 (b+ ) > 0. Hence
from (16) we must have that α < β for fronts.
We will refer to the case α = 0 as the adiabatic case, since there is no heat loss in
the system. In this case system (10)-(12) reduces to a system of two equations. In [20]
it was shown that this system has a unique solution, when LA ≥ 1, i.e. there exists a
unique value of c for which this problem has a solution. This value of c will be denoted
by cad . (In the case LA < 1 uniqueness is not known. In [4] it is shown that if f1 is a
suitably chosen step function, then the system can have three solutions. Nevertheless, in
the numerics the adiabatic velocity cad appears to be unique for this case as well.)
In the case of unit Lewis numbers LA = LW = 1 our system can be reduced to a
system of two equations. This case was investigated in [24] in detail, where it was shown
that, for any point on the (α, c) curve, c < cad and that, for µ ≤ 1, we have α ≤ β + 1.
The first estimate is confirmed by the numerics in this paper for other values of the Lewis
numbers. The second estimate is established for general Lewis numbers in Proposition 2.
Our numerical results suggest that the estimate for the ‘cut-off’ value for α when µ ≤ 1
is somewhat of an overestimate. However we will prove that at the turning point α < β,
which means that the extinction value of α is less than β, see Proposition 5 below.
Proposition 2 If µ ≤ 1 and system (10)-(14) has a solution, then α ≤ β + 1.
5
PROOF. The proof is based on an estimate for the nonlinear terms in equation (12). In
order to get the necessary estimates we need the following inequalities.
β
w(y) ≥ 1 − , a(y) ≤ 1
α
µ
¶
(b(y) − 1)
exp (µ − 1)
≥1
εb(y)
for all y ∈ R,
(17)
for all y ∈ R.
(18)
The inequalities in (17) follow from Proposition 1. To prove (18) we only have to show
that b(y) < 1. From (16) we obtain b+ = 1 − a+ − (1 − w+ )α/β < 1. Hence, if there
exists y for which b(y) > 1, then, for the maximum point y ∗ of b, we have b(y ∗ ) > 1 and
b0 (y ∗ ) = 0. Adding (10) and (12) and integrating on (−∞, y) we obtain
0
0
L−1
A a + b − ca − cb + c ≥ 0.
(19)
Applying this inequality at y = y ∗ we get
0 ∗
∗
∗
L−1
A a (y ) ≥ c a(y ) + c(b(y ) − 1) > 0,
which contradicts to the first statement of Proposition 1. Thus we have shown that (18)
holds.
Using (17) and (18) we obtain
µ
¶
b−1
af1 (b) − αwf2 (b) = f1 (b) a − αw exp((µ − 1)
) < f1 (b)(1 + β − α).
εb
(20)
Now assume that system (10)-(14) has a solution for α > β + 1. Then according to (20)
the nonlinear term in (12) is negative. On multiplying (12) by e−cy and integrating over
(y, ∞) we get
Z∞
0
b (y) = e
e−cs (a(s)f1 (b(s)) − αw(s)f2 (b(s)))ds < 0.
cy
y
Thus b would be a decreasing function, hence from boundary condition (13) we would get
b(y) < 0, which, with (15), is a contradiction.
3.2
Numerical results
Here we present our numerical results obtained by the finite-difference discretisation of
our boundary-value problem. The (α, c) curves are obtained with a pseudoarclength continuation method [13]. Our numerical method was described fully in [24]. There we
considered the case of unit Lewis numbers LA = LW = 1, in which case the system can
be reduced to two equations. We now present results for the full three variable system
for representative values of LA and LW .
As in the case of unit Lewis numbers we introduced the new independent variable
y = c y and then approximated problem (10)-(14) with a boundary-value problem on
a bounded interval [0, L]. The approximation is based on the fact that the nonlinear
6
(reaction) terms tend to zero as y → ±∞. Neglecting the nonlinear terms in (−∞, 0)
and in (L, ∞) the differential equations can be solved analytically in these regions, hence
we get boundary conditions for all three functions at y = 0 and at y = L ensuring the
smoothness of the solution. This gives six boundary conditions. Using the translational
invariance of the differential equations we get an extra boundary condition, namely we can
assume that b(0) = bI , where bI is a suitably chosen small number, corresponding to an
ignition temperature. Thus we have seven undetermined boundary conditions. The extra
boundary condition is used as the final equation of the discretised system, and enables us
to determine the value of the velocity c.
Introducing N grid points in the interval [0, L] we discretise the differential equations
with finite differences. Assuming that the differential equations are satisfied at the N − 2
internal grid points, we get 3(N − 2) equations. Hence, together with the seven boundary
conditions we have 3N + 1 equations for the same number of unknowns. This system can
be solved using Newton-Raphson iteration if a good initial guess is known. The solution
for the adiabatic case α = 0 can be obtained easily starting from linear profiles as an
initial guess. Then increasing the value of α the solution can be obtained by continuation. We applied a standard pseudoarclength continuation method in the space R3N +2
[13]. We chose the values of bI and L to ensure that problem given in [0, L] is a good
approximation of (10)-(14). It turned out that L = 10 and bI = 0.1 was suitable for most
of the parameter range considered. The choice bI = 0.1 is unsatisfactory only when c is
very small (i.e. α is large). In that case the maximum temperature can be less than 0.1,
which clearly means that b(0) = 0.1 cannot be satisfied. For the values of c shown in the
Figures this is not the case. To have values of b(0) < 0.1 requires c . 10−15 where the
numerical method is probably unreliable.
In our numerical studies we fixed the value of ε at ε = 0.1 and determined the (α, c)
curves for different values of LA , LW , µ and β. We found, similar to the case of unit
Lewis numbers, that an (α, c) curve can have three different shapes depending on the
value of µ and β. For µ > 1 the curve is monotone decreasing with increasing α and is
unbounded in the α direction, i.e. there is no quenching. For µ ≤ 1 the curve is bounded
in the α direction, as was proved in Proposition 2, but can have two different shapes.
For larger values of β the curve is monotone decreasing and for smaller values of β it is
S-shaped. Thus there is a quenching value of α and there is a range of α for which there
are 3 different wave velocities. These results suggest that for µ < 1 hysteresis bifurcations
occur at certain values of β. The hysteresis curve will be dealt with below (see Figure 3).
In order to reveal the possible shapes of the (α, c) curves we fixed the value of β and µ
and determined the curves for 16 different pairs of (LA , LW ), namely for LA = 0.5, 1, 2, 5
and LW = 0.5, 1, 2, 5. This has been done for different values of β and µ. We found that
the (α, c) curves can have the three shapes mentioned above. The results corresponding
to the case β = 0.01 are shown in Figure 1. (For other values of β we got qualitatively
similar curves.) In Figure 1 (a), µ = 0.1 is fixed and LA , LW are changed, the values
used are shown in the Figure. We can see that this value of (β, µ) is above the hysteresis
curve (for the LA values chosen), because all the (α, c) curves are monotone decreasing.
In Figure 1 (b), µ = 0.8 and LA , LW take the same values as in Figure 1 (a). Now
the (µ, β) parameter pair is below the hysteresis curve for LA = 1, 2, 5, because their
7
(α, c) curves are S-shaped, and is above the hysteresis curve for LA = 0.5, because its
(α, c) curve is monotone decreasing. (For these values of the parameters the lower turning
point cannot be really seen. The S-shaped curves belonging to other parameter values
are shown in Figure 8.) In Figure 1 (c), µ = 1 and LA , LW are as before. In this case
the (α, c) curves are monotone decreasing for all values of LA and LW , and are bounded
in the α direction, as was shown in Proposition 2. We can see that the dependence of the
curves on LW is less significant then in the previous cases. For µ > 1 this dependence
became unnoticeable as is shown in Figure 1 (d). In this figure µ = 2 and the curves for
the different values of LW are indistinguishable from each other.
Summarising, we can say that increasing the value of LW the (α, c) curves move to
the left, but their qualitative shapes do not change. However, varying the value of LA
not only the position but also the shape of the curve can change, increasing the value
of LA the first section (close to the c axis) of the (α, c) curve moves upwards, because
the adiabatic flame velocity increases. We determined the dependence of the adiabatic
flame velocity on LA . This is shown in Figure 2, where the upper and lower estimates
for the flame velocity for LA → ∞ are also shown. In this limit (solid combustion) the
adiabatic system can be reduced to a single equation, hence upper and lower bounds for
the adiabatic flame velocity can be derived analytically (for any value of the activation
energy, not only for high activation
R 1energies). According to [28] (pp. 375) the bounds are
2
σ < cad < σ/(1 − σ), where σ = 0 f1 (s)/sds. These values are also shown in the figure
(by the broken lines).
Concerning the shape of the (α, c) curves, we have seen that, for fixed value of µ and
β, there is a hysteresis bifurcation value of LA , for which the curve has an inflexion. For
smaller values of LA the curve is monotone decreasing, and for larger values the (α, c)
curve is S-shaped. In the latter case the curve has two turning points, i.e. there are two
values of α where saddle-node bifurcations occur. The loci of the saddle-node points form
a cusp curve in the (α, LA ) parameter plane, as is shown in Figure 9. In this Figure we put
β = 0.001, µ = 0.1, LW = LA and using a continuation algorithm we changed the value of
LA and determined the value of α for which the curve has a turning point. The left branch
of the cusp curve belongs to the lower turning points and the right belongs to the upper
turning points. For these values of ε, β, µ we found that the cusp point is at LA = 0.4124.
Figure 1a shows that, for µ = 0.1, the hysteresis value of LA is above LA = 5, because
for LA = 5 the curve is monotone. Figure 1b shows that, for µ = 0.8, the hysteresis value
of LA is below LA = 1, because for LA = 1 the curve has a turning point. Therefore we
can see from Figure 1 that increasing the value of µ the hysteresis value of LA decreases.
Hence we can expect that increasing the value of LA the hysteresis bifurcation curve in
the (µ, β) parameter plane moves to the left. In Figure 3 the hysteresis bifurcation curve
is shown for LA = 1 and LA = 2. (In both cases LW = LA .) We recall that this curve
consists of those (µ, β) parameter pairs for which the (α, c) curve has an inflexion. If the
(µ, β) parameter pair is above the curve, then the (α, c) curve is decreasing. If it is below
the curve, then the (α, c) curve is S-shaped. The hysteresis bifurcation curve appears
to tend to the vertical line µ = 1, but our numerical method does not reveal whether it
is bounded or unbounded in the β direction, i.e. we cannot determine whether the line
µ = 1 is a tangent or an asymptote of the curve.
8
4
Stability of the flame
In this Section we determine the (linear) stability of the propagating flames, regarded
as permanent-form travelling wave solutions of the time dependent system, based on the
thermal/diffusive model,
2
∂τ a = L−1
A ∂x a − af1 (b)
2
∂τ w = L−1
W ∂x w − βwf2 (b)
∂τ b = ∂x2 b + af1 (b) − αwf2 (b)
(21)
(Here τ and x are dimensionless time and space variables.) The travelling wave solutions
of (10)-(14) are then functions only of the travelling co-ordinate x + c τ and are linearly
stable if the spectrum of the operator L (except the zero eigenvalue) is in the left half of
the complex plane, where
 −1 00

LA V1 − cV10 − f1 (b)V1 − af10 (b)V3
00
0
0

LV =  L−1
(22)
W V2 − cV2 − βf2 (b)V2 − βwf2 (b)V3
00
0
0
0
V3 − cV3 + f1 (b)V1 − αf2 (b)V2 + (af1 (b) − αwf2 (b))V3
We consider L as an operator defined for the C 2 functions in the space
C0 = {V : R → C3 | V is continuous, lim V = 0}
±∞
endowed with the supremum norm. We note that we could work alternatively with the
function space L2 . The complex number λ is a regular value of L if the operator L−λI has
a bounded inverse. The spectrum σ(L) of the operator L consists of the non-regular values.
Following [15] the essential spectrum σess (L) is the set of those points in the spectrum that
are not isolated eigenvalues of L. The location of the essential spectrum can be estimated
and, for our system, it can be shown that it lies in the left half plane ([15, 28]). In the next
section we will investigate the essential spectrum in detail, obtaining more information
about its location. For example, we will see that, if the imaginary axis of the complex λ
plane is a tangent of the essential spectrum, then the tangent point is at the origin, and we
can determine the domain of the Evans function. The isolated eigenvalues of L are then
investigated. First, we present two qualitative results using the Evans function, the zeros
of which are the isolated eigenvalues. We show, without the exact determination of the
eigenvalues, that Hopf bifurcations and saddle-node bifurcations occur at certain values
of the parameters. We then determine numerically the first pair of isolated eigenvalues
along the (α, c) bifurcation curves. These eigenvalues are obtained by discretising the
linear eigenvalue problem
LV = λV,
lim V = 0
(23)
±∞
with finite differences and to compute the eigenvalues of the matrix obtained from the
discretisation.
9
4.1
Characterisation of the essential spectrum
The complex number λ is a regular value of L if for any function W ∈ C0 there exists a
unique solution of the equation
LV − λV = W
(24)
and there exists K > 0, such that kV k ≤ KkW k. We introduce the first-order system
corresponding to equation (24), putting x1 = V1 , x2 = V2 , x3 = V3 , x4 = V10 , x5 = V20 ,
x6 = V30 . The corresponding first-order system is then
x0 (t) = Aλ (t)x(t) + y(t),
(25)
¶
µ
0
I
,
(26)
Aλ (t) =
−1
D (λI − Q(t)) cD−1


LA 0 0
D−1 =  0 LW 0 
0
0 1
is the inverse of the diagonal matrix formed by the scaled diffusion coefficients, and


−f1 (b(t))
0
−a(t)f10 (b(t))

0
−βf2 (b(t))
−βw(t)f20 (b(t))
Q(t) = 
0
0
f1 (b(t))
αf2 (b(t)) a(t)f1 (b(t)) − αw(t)f2 (b(t))
is the Jacobian of the travelling wave equations. Now t plays the role of the travelling
co-ordinate x + c τ . A complex number λ is a regular value of Aλ if, for any y ∈ C0 (R, C6 ),
there exists a unique solution x ∈ C0 (R, C6 ) of (25), and there exists K > 0, such that
kxk ≤ Kkyk. The complex number λ is called an eigenvalue of Aλ if for y = 0 there exists
a nonzero solution x ∈ C0 (R, C6 ) of (25). It can be shown that σ(L) = σ(Aλ ) and the
eigenvalues of L and those of Aλ coincide. Since the functions a, w, b tend to their limit
exponentially at ±∞, the limits
A±
λ = lim Aλ (t)
t→±∞
−ω|t|
exist and there exist positive numbers K, ω, such that kAλ (t)−A±
for |t| large.
λ k ≤ Ke
In what follows we consider a general system of the form (25) in Cn , and assume that
the limits above exist at ±∞ with exponential decay. We use the notation C0 = C0 (R, Cn ),
and denote the fundamental system of equation (25) by Ψλ (t), i.e. the n columns of the
matrix Ψλ (t) are n independent solutions of the homogeneous system ẋ(t) = Aλ (t)x(t).
The dimension of the stable, unstable and centre subspaces of the matrices A±
λ play
an important role in the construction of the Evans function and in the determination
of the essential spectrum. Denote the number of eigenvalues (with multiplicity) of A+
λ
+
+
with positive, negative, zero real part by n+
u (λ), ns (λ), nc (λ), respectively. We define
−
−
−
n−
u (λ), ns (λ), nc (λ) similarly using Aλ . The following statement can be proved using
exponential dichotomies and perturbation theorems [8, 9, 26].
Lemma 1
n
+
1. There exists an n+
s (λ) dimensional subspace Es (λ) ⊂ C , such that
lim Ψ(t)x0 = 0
t→+∞
and for x0 ∈
/ Es+ (λ) the limit is non-zero.
10
for all x0 ∈ Es+ (λ)
−
n
2. There exist an n−
u (λ) dimensional subspace Eu (λ) ⊂ C , such that
lim Ψ(t)x0 = 0
t→−∞
for all x0 ∈ Eu− (λ)
and for x0 ∈
/ Eu− (λ) the limit is non-zero.
−
n
3. If n+
c (λ) ≥ 1 or nc (λ) ≥ 1, there exists x0 ∈ C , such that the function Ψ(t)x0 is
bounded in R+ or in R− , but does not tend to zero at +∞, or −∞.
+
n
This lemma shows that there exists an n+
s (λ) dimensional subspace Es (λ) ⊂ C of
initial conditions from which the solution of the homogeneous system tends to zero at ∞,
−
n
and there exists an n−
u (λ) dimensional subspace Eu (λ) ⊂ C of initial conditions from
which the solution of the homogeneous system tends to zero at −∞. The Lemma implies
the following, see [15].
−
Proposition 3 If n+
c (λ) ≥ 1 or nc (λ) ≥ 1, then λ ∈ σ(Aλ ).
−
Hence we have to consider only the case when n+
c (λ) = nc (λ) = 0. For this case the
following proposition of primary importance for our discussion.
−
Proposition 4 Assume n+
c (λ) = nc (λ) = 0.
1. If dim Es+ (λ) + dim Eu− (λ) > n, then λ is a non-isolated eigenvalue.
2. If dim Es+ (λ) + dim Eu− (λ) < n, then λ ∈ σ(Aλ ).
3. If dim Es+ (λ) + dim Eu− (λ) = n and dim(Es+ (λ) ∩ Eu− (λ)) = 0 , then λ is a regular
value.
4. If dim Es+ (λ) + dim Eu− (λ) = n and dim(Es+ (λ) ∩ Eu− (λ)) > 0, then λ is an isolated
eigenvalue.
See [25] for a proof. The dimensions of Es+ (λ) and Eu− (λ) can be explicitly calculated since
only constant coefficient linear systems have to be solved to obtain their values. Hence
the values of λ belonging to 1. and 2. in Proposition 4 can be explicitly determined, i.e.
the essential spectrum can be given analytically.
Corollary 1 For the essential spectrum of Aλ we have.
−
+
−
σess (Aλ ) = {λ ∈ C : n+
c (λ) ≥ 1 or nc (λ) ≥ 1 or ns (λ) + nu (λ) 6= n}.
However, we cannot calculate the dimension of Es+ (λ)∩Eu− (λ) directly since the subspaces
can be determined only via solving the differential equation ẋ(t) = Aλ (t)x(t). Therefore
to make the distinction between 3. and 4. (in Proposition 4) we need a numerical solution,
i.e. the isolated eigenvalues can be determined only numerically. This can be done using
the Evans function which we now consider.
11
Let
−
+
−
+
−
Ω = {λ ∈ C : n+
c (λ) = 0 = nc (λ), ns (λ) 6= 0 6= nu (λ), ns (λ) + nu (λ) = n}.
(Hence Ω contains the isolated eigenvalues, because according to Proposition 4 for the
−
+
isolated eigenvalues n+
s (λ) + nu (λ) = n.) For λ ∈ Ω denote a basis of the subspace Es (λ)
by v1+ , . . . , vn++ , and a basis of the subspace Eu− (λ) by v1− , . . . , vn−− . The assumption that
s
u
dim(Es+ (λ) ∩ Eu− (λ)) > 0 means that the two bases together give a linearly dependent
system of vectors. That is the determinant formed by these n vectors is zero. The Evans
function is defined as this determinant.
Definition 1 The Evans function for the matrix Aλ is D : Ω → C
³
´
+ −
−
+
D(λ) = det v1 . . . vn+ v1 . . . vn−
s
u
We have seen that the isolated eigenvalues are the zeros of the Evans function. It can also
be shown that the multiplicity of an eigenvalue is equal to the multiplicity of the zero of
the Evans function, and that the Evans function is an analytic function on the domain Ω
[1].
We determine the bases of the stable and unstable subspaces numerically in the following way. We calculate the eigenvalues of A+
λ with negative real part, and their corresponding eigenvectors. Denote these eigenvalues by µ1 , . . . , µk , and the eigenvectors by
u1 , . . . , uk (for short we use the notation k = n+
s (λ)). Similarly, denote the eigenvalues of
−
Aλ with positive real part by ν1 , . . . , νl , and the corresponding eigenvectors by v1 , . . . , vl
(where l = n−
u (λ)). Then, choosing a sufficiently large number L, we solve the homogeneous equation ẋ(t) = Aλ (t)x(t) on [0, L] starting from the right end point with initial
condition x(L) = ui eµi L for i = 1, . . . , k. Hence we get k = n+
s (λ) linearly independent
(approximating) solutions of the differential equation, their values at t = 0 giving a base
for Es+ (λ). Similarly, solving the differential equation in [−L, 0] we get a base of Eu− (λ),
and the determinant defining the Evans function can be computed. We note that, if L is
very large and there is a significant difference between the real parts of the eigenvalues
µ1 , . . . , µk , then the solution for the eigenvalue with largest real part will dominate and the
solutions starting from linearly independent initial conditions will be practically linearly
dependent at zero. (A similar case can occur on [−L, 0].) To overcome this difficulty the
problem can be extended to a wedge product space of higher dimension [5].
Now consider our case. From (26) we get
µ
¶
0
I
±
Aλ =
D−1 (λI − Q± ) cD−1
where Q− is a 3 × 3 zero matrix,


0 0 0
Q+ =  0 0 0  for pulses,
0 0 0


−q1
0
0
Q+ =  0 −βq2 0  for fronts,
q1 −αq2 0
and q1 = f1 (1 − α/β), q2 = f2 (1 − α/β). If µ is an eigenvalue of A±
λ with eigenvector
T
3
u = (u1 , u2 ) , (u1 , u2 ∈ C ), then u2 = µu1 and
(Dµ2 − cµI + Q± − λI)u1 = 0,
12
(27)
hence
det(Dµ2 − cµI + Q± − λI) = 0.
(28)
Since Q± are lower triangular matrices, equation (28) is equivalent to
−1 2
2
±
±
2
±
(L−1
A µ − cµ + Q11 − λ)(LW µ − cµ + Q22 − λ)(µ − cµ + Q33 − λ) = 0.
(29)
Therefore the set of those λ values for which n+
c (λ) ≥ 1, that is µ = iω (for some ω ∈ R) is
a solution of equation (29), consists of three parabolas, denoted by P1+ , P2+ , P3+ , see Figure
4 (a) and (b). In Figure 4 (a) the parabolas are shown in the case of a pulse solution,
in Figure 4 (b) the parabolas corresponding to a front solution are shown. We define the
parabolas P1− , P2− , P3− similarly as the loci of those λ values for which n−
c (λ) ≥ 1, they
are shown in Figure 4 (c). The parabolas are given explicitly by
(Imλ)2
},
LA c2
(Imλ)2
= {λ ∈ C : Reλ = −βq2 −
},
LW c2
(Imλ)2
},
P3+ = {λ ∈ C : Reλ = −
c2
P1+ = {λ ∈ C : Reλ = −q1 −
P2+
(Imλ)2
},
LA c2
(Imλ)2
−
P2 = {λ ∈ C : Reλ = −
},
LW c2
(Imλ)2
P3− = {λ ∈ C : Reλ = −
}.
c2
P1− = {λ ∈ C : Reλ = −
According to Corollary 1 these parabolas belong to the essential spectrum. It can easily
2
be seen that if λ is to the left of P1+ , then both solutions for µ of equation L−1
A µ −
+
cµ + Q±
11 − λ = 0 have positive real part, and if λ is to the right of P1 , then one of
the solutions has positive real part, the other one has negative real part. The same is
true for the other parabolas. Therefore we can determine, for any λ, the value of n+
s (λ),
+
−
−
i.e. the dimension of Es (λ), and the value of nu (λ), i.e. the dimension of Eu (λ). The
values of these numbers are shown in Figure 4 in the different domains determined by the
parabolas. As a simple consequence of Proposition 4 we obtain the following properties
of the spectrum.
Proposition 5
values.
1. The domain lying to the left of all parabolas consists of regular
2. The Evans function can be defined in the domain lying to the right of all parabolas.
3. For fronts there are open domains filled with eigenvalues. For pulses this kind of
domain does not exist.
4. The turning points of the (α, c) curves belong to front solutions, hence at the turning
point α < β.
PROOF. We use the dimensions of the spaces Es+ (λ), Eu− (λ) as they are given in Figure
4.
1. If λ is in the domain lying to the left of all parabolas, then dim(Es+ (λ)) = 0,
dim(Eu− (λ)) = 6, hence conditions of Proposition 4 (3) are fulfilled, thus λ is a
regular value.
2. If λ is in the domain lying to the right of all parabolas, then dim(Es+ (λ)) = 3,
dim(Eu− (λ)) = 3, hence λ ∈ Ω, which is the domain of the Evans function.
13
3. For fronts in the domain lying between P1+ and P3+ we have dim(Es+ (λ)) = 2 (see
Figure 4b), dim(Eu− (λ)) = 6, hence conditions of Proposition 4 (1) are fulfilled.
Thus any point λ of this domain is an eigenvalue, i.e. this domain is filled with
eigenvalues. (We can find other domains where conditions of Proposition 4 (1) are
fulfilled.) For pulses Pi+ = Pi− for i = 1, 2, 3, hence from Figure 4a and 4c we can see
that dim(Es+ (λ)) + dim(Eu− (λ)) = 6 holds for any λ ∈ C except on the parabolas.
Hence according to statements (3) and (4) of Proposition 4 any λ which is not on
the parabolas is a regular value or an isolated eigenvalue.
4. The turning points belong to saddle-node bifurcation points. At the saddle-node
bifurcation point a real eigenvalue crosses zero, i.e. in the neighbourhood of the
bifurcation point the linearised system must have positive and negative real eigenvalues. For pulses Figure 4a and 4c show that for λ values in the negative part of
the real axis we have dim(Es+ (λ)) = 0, dim(Eu− (λ)) = 6, hence conditions of Proposition 4 (3) are fulfilled and λ is a regular value. Thus for pulses a saddle-node
bifurcation cannot occur. Therefore the turning points belong to front solutions,
and front solutions can exist only for α < β, see the explanation after Proposition 1
4.2
Bifurcation results obtained with the Evans function
Here we show that Hopf bifurcations and saddle-node bifurcations occur at certain parameter values, without computing the isolated eigenvalues explicitly. The Evans function
is a useful tool for this purpose. As we have seen the zeros of the Evans function are the
isolated eigenvalues, hence solving the equation D(λ) = 0 with Newton-Raphson iteration we can get several eigenvalues. However, the main advantage of the Evans function
method is not to find the explicit values of several eigenvalues but to decide whether a
closed curve in the complex plane encircles some eigenvalues or not. In this Section we
show that, on changing the value of LA , two complex eigenvalues cross the imaginary axis,
i.e. a Hopf bifurcation occurs, and we show that changing the value of α a real eigenvalue
crosses the imaginary axis at zero, i.e. a saddle-node bifurcation occurs.
Let Lmax = max{LA , LW , 1}. If λ is in the domain
(Imλ)2
},
Lmax c2
then dim Es+ (λ) = dim Eu− (λ) = 3, hence the Evans function can be defined in this
domain. The eigenvalues of A+
λ with negative real part are
q
q
√
−1
2
c − c + 4(λ + q1 )LA
c − c2 + 4(λ + βq2 )L−1
W
c − c2 + 4λ
.
µ1 =
, µ2 =
, µ3 =
2
2L−1
2L−1
A
W
Ω∗ = {λ ∈ C : Reλ > −
The eigenvector corresponding to µi is ui = (ui , µi ui )T , where
q1
−αq2
u1 = (1, 0, 2
)T , u2 = (0, 1, 2
)T ,
µ1 − cµ1 − λ
µ2 − cµ2 − λ
u3 = (0, 0, 1)T .
If LA = LW = 1 and q1 = q2 = 0, then let u1 = (1, 0, 0)T and u2 = (0, 1, 0)T . The
eigenvalues of A−
λ with positive real part are
q
q
√
−1
2
c + c + 4λLA
c + c2 + 4λL−1
W
c + c2 + 4λ
ν1 =
.
, ν2 =
, ν3 =
2
2L−1
2L−1
A
W
14
The eigenvector corresponding to νi is vi = (v i , νi v i )T , where v 1 = (1, 0, 0)T , v 2 = (0, 1, 0)T
and v 3 = (0, 0, 1)T .
We solved the travelling wave equations (10)-(12) numerically in an interval [0, L],
and neglecting the nonlinear terms we solved the differential equations analytically outside this interval. We will apply the same method for the linearized equations. Hence
the linearized equations become a linear system of differential equations with constant
coefficients in (−∞, 0] that can be solved analytically. The base of the subspace Eu− (λ)
is {v1 , v2 , v3 }. The base of the subspace Es+ (λ) can be obtained as {x1 (0), x2 (0), x3 (0)},
where xi is the solution of the linearised system in the interval [0, L] satisfying the initial
condition xi (L) = ui eµi L for i = 1, 2, 3.
We can decide whether there is an eigenvalue with positive real part by computing
the image of a half circle centred at the origin and lying in the right half plane under the
Evans function D. If the image winds around the origin, then by the argument principle
there is (at least one) zero of D in the half circle. Choosing a sufficiently large half circle
all the eigenvalues with positive real part are inside the half circle, because an estimate
can be derived for the eigenvalues with positive real part. Alternatively, if we can prove
that D(λ) tends to a limit as |λ| → ∞ and Reλ > 0, then we can compute the image of
the imaginary axis, instead of a half circle. We do not prove this here, hence we apply
the half circles.
In Figure 5 we can see the image of a half circle lying in the left half plane under D
for the parameter values ε = 0.1, α = 0, i.e. in the adiabatic case. The centre of the half
circle is at 0.001+0i, in order to avoid the origin, its radius is 0.05 and its straight segment
part is vertical. In part (a) LA = 3, this value is below the Hopf bifurcation value (which
is at LA = 3.2217, see below). Now the image of the half circle does not wind around
the origin, the winding number was computed as 0, as is shown in the Figure. Hence
there is no zero of D in the half circle. We get the same result for half circles with larger
radius. In part (b) of Figure 5 LA = 4, this value is above the Hopf bifurcation value.
Now the image of the half circle winds twice around the origin, the winding number was
computed as 2. Hence there are two zeros of D in the half circle. This shows that the
Hopf bifurcation value of LA is between 3 and 4.
In Figure 6 we can see the image of a half circle lying in the left half of the complex
plane under the Evans function D in the case LA = 1. Now α is below, but close to the
extinction value. In (a) the flame solution belongs to the point on the stable branch of
the (α, c) curve. In (b) the flame solution belongs to the point on the unstable branch.
4.3
Determining the eigenvalues with finite difference discretisation
In this section we solve (23) with finite-difference discretisation. In [3] it is shown that
the eigenvalues of the truncated problem
LV
15
= λV
(30)
V (−l) = V (l) = 0
(31)
tend to those of the original problem (23) as l → ∞. Therefore choosing a sufficiently
large number l we discretise problem (30)-(31) with finite differences on a grid of N points
and solve the 3N dimensional matrix eigenvalue problem
M V = λV .
(32)
It can be shown that the eigenvalues of (32) tends to those of (30)-(31) as N → ∞ [17].
We increased the values of l and N until the required accuracy was achieved.
We now present our numerical results obtained by this method. First we solved the
equations in the adiabatic case (α = 0) for increasing values of LA . We found that a
Hopf-bifurcation occurs at a certain value LH
A of this parameter. For ε = 0.1 this value
H
was found to be LA = 3.2217. Below this value the adiabatic flame is stable, above LH
A it
is unstable with a pair of complex eigenvalues with positive real part.
We then determined the eigenvalues of the linearised system for points on some of the
(α, c) curves. For a given point of the curve, from equation (32), we got 3N eigenvalues.
Since zero must be an eigenvalue (because of the translational invariance), we determined
the eigenvalue with smallest absolute value and omit it from the set of eigenvalues. Then
we determined the two eigenvalues with largest real part. According to the position of
these we can have four cases. The real part of the remaining eigenvalues are negative for
the parameter values used in these computations. We have seen that the value LW has
only a small effect on the shape of the curve, therefore we carried out our computations
for the physically relevant value LW = LA when LA ≥ 1 and we put LW = 1 for LA < 1.
In Figure 7 the stability results are shown for β = 0.01 and for µ = 0.1 in part (a) and
for µ = 0.5 in part (b). For LA ≤ 1 all points of the curve are stable. When 1 < LA < LH
A
there is a Hopf bifurcation point on the curve. If µ and LA are sufficiently large, e.g.
LA = 5 in Figure 7 (b), then there are turning points on the curve. At the upper turning
point one of the real eigenvalues goes to the negative part of the real line and at the lower
turning point it returns to the positive half line.
To see the behaviour of the Hopf bifurcation point we fixed the values β = 0.001 and
µ = 0.1 and computed the (α, c) curve for several values of LA . In this case the curve is
S-shaped, hence the relation of the Hopf and saddle-node points can be investigated. The
results for LA ≥ 1 are shown in Figure 8. For LA = 1 the (α, c) curve consists of three
parts, two of them are stable, one is unstable. From the adiabatic point (α = 0) to the
upper turning point there are two negative real eigenvalues, hence it is a stable branch.
At the turning point one of the eigenvalues crosses zero, a saddle-node bifurcation occurs.
Between the two turning points there is a positive and a negative eigenvalue, therefore it
is the unstable branch. At the lower turning point the positive eigenvalue goes back to
the negative half line, hence the lower branch is stable again. The upper turning point is
an extinction point for the flames developing from the adiabatic case. For higher values
of α the flame solution ‘jumps’ to the lower branch. These flames have a much slower
progagation speed, lower maximum temperatures and thus much lower reaction rates. For
LA = 1.5 the (α, c) curve has further bifurcations. Close to the adiabatic point there are
16
two negative real eigenvalues, they then merge and a complex pair with negative real part
is born. On the upper branch there is a Hopf bifurcation and after the Hopf point there
is a complex pair with positive real part. These merge before the upper turning point
and two positive real eigenvalues are born. At the turning point one of them crosses zero,
there is a saddle-node bifurcation. Between the two turning points there is a positive and
a negative eigenvalue. At the lower turning point the negative eigenvalue goes back to the
positive half line. Then these merge and a complex pair with positive real part appears.
Further along the curve a Hopf bifurcation occurs again, which can be seen in the enlarged
picture, Figure 8 (b). The same scenario can be observed for LA = 2, 2.5, 3, but the first
part with two negative real eigenvalues is missing for LA > 1.5. When LA = 4 (not shown
on the figure) the whole curve is unstable because LH
A < 4 (for ε = 0.1), therefore in the
adiabatic case there is a complex pair of eigenvalues with positive real part.
We can see that there is a Hopf bifurcation curve consisting the Hopf points on the
upper branch. This curve is shown in Figure 9, together with the saddle-node bifurcation
curve in the (α, LA ) parameter plane. In Figure 8 (b) we saw that there is a Hopf
bifurcation on the lower branch for LA = 1.5. Therefore it can be expected that there is
another Hopf bifurcation curve, which is schematically drawn with dotted line in Figure
9. The common point of the Hopf bifurcation and saddle-node curves, denoted by T B, is
a Takens-Bogdanov bifurcation point, where the operator L has a double zero eigenvalue
(besides the zero eigenvalue coming from the translational invariance). The Hopf curves
and the saddle-node curve divide the positive quadrant of the (α, LA ) parameter plane
into 7 parts. The number of stable and unstable solutions is shown in the Figure. (For
example, 2,1) means that in the given region there are two stable and one unstable
solutions.)
5
Numerical simulations
We solved equations (21) numerically starting with the system in its unreacted state
(a = 1, w = 1, b = 0) and applying a local temperature input to start the reactions. The
numerical scheme is an implicit one based on the Crank-Nicolson method with NewtonRaphson iteration to solve, at each time step, the sets of nonlinear algebraic equations
arising from the discretisation. Further details are given in [14, 19]. Our aim is to check
that sustained time-dependent behaviour can evolve when the steady flame becomes unstable and to gain some insight into the form of this time-dependent behaviour. We limit
attention to two specific cases. A thorough numerical search of the full parameter range
is beyond the scope of the present paper.
We start by looking at the adiabatic (α = 0) case. Here the linear stability analysis
shows that there is a Hopf bifurcation at LA = LH
A = 3.2217 for ε = 0.1 (which is the
value we use throughout) with the system being unstable for LA > LH
A . Our numerical
integrations show that, with LA < LH
,
a
steady
flame
results
as
the
long
time behaviour
A
of the numerical integration, propagating with an average speed corresponding to those
shown in Figure 2. For LA > LH
A periodic behaviour is seen. For values of LA just above
H
LA there are simple periodic oscillations, as can be seen in Figure 10a where we plot the
average wave speed c against τ for LA = 4.0. The figure shows that there is an initial
17
transient period before the oscillations become established, with period τ0 = 159.3. The
numerical integrations also show that the amplitude of the oscillations are smaller as values of LA get closer to LH
A , suggesting that the Hopf bifurcation is supercritical in this case.
As LA is increased a period-doubling bifurcation occurs and period 2 behaviour is
seen, Figure 10b for LA = 5.0. The period of the oscillations increases considerably to
τ0 = 303 in this case. The system undergoes a transition between relatively short periods when it has a high propagation speed to longer periods when it is moving more
slowly, the minimum propagation speed cmin = 0.073 for this value of LA . During the
rapid propagation temperatures above the burnt gas temperature are achieved, i.e. the
waveform has a region where b > 1, with a large gradient at the front of the flame.
In the low speed propagation the waveform is much smoother (smaller gradients in temperature) and the temperature is below the burnt gas temperature, i.e. b < 1, throughout.
This trend becomes more pronounced as LA is increased, see Figure 10c for LA = 15.0.
Here the period of the oscillations has increased to τ0 = 372 and much higher and lower
propagation speeds are reached during the cycle, now cmin = 0.022. This corresponds to
higher temperatures and steeper waveforms during the fast propagation part and smoother
waveforms during the low propagation speed. Note that, during the latter period, the wave
becomes almost stationary. We continued our numerical integrations by increasing the
value of LA , finding only period 2 oscillations for the value of ε = 0.1 used. We did not see
any further bifurcations to more complex behaviour. We illustrate this in Figure 10d for
LA = 150.0. Here the period has increased to τ0 = 470, the maximum wave speed is much
greater and the period of slow propagation becomes longer and slower (cmin = 0.0076).
A system equivalent to (21) has been considered by [29] in the solid combustion limit
(LA → ∞). They presented results for the adiabatic case, though the form of their nondimensional thermal-diffusion equations is different to ours. Their wave speed plots and
temperature profiles are mostly qualitatively similar to the ones we found for finite (and
large) LA . There is one difference in that they found period 3 oscillations for what, in our
terms, would be a value of ε ' 0.13. We modified our numerical scheme to deal with the
limit LA → ∞ and still found only period 2 oscillations for ε = 0.1.
The other case that we considered was to fix values for LA and LW at LA = LW = 4.0
and investigate how the solution changed as α was increased. We took β = 0.01, µ =
0.5, ε = 0.1 (see Figure 7b). When α = 0 the system has simple periodic behaviour, see
Figure 10a. At a small, non-zero value of α a period-doubling bifurcation occurs (before
α = 0.0005) and by α = 0.001 period 2 oscillations are well-established, Figure 11a, with
period τ0 = 751 (and cmin = 0.037). By α = 0.0015, Figure 11b, we still have period 2
oscillations though the secondary peaks seen in Figure 11a have become less pronounced.
The period of the oscillations has increased to τ0 = 1774 with the wave having longer periods when it propagates slowly, cmin = 0.0164. At α = 0.002, Figure 11c, the secondary
(smaller) disappear and the maximum speed is decreased. The system has returned to
period 1 behaviour though the period has increased considerably (τ0 = 5044). In this case
there are long periods of the cycle when the wave is almost stationary (cmin = 0.0066).
We were unable to initiate propagating flames for the larger values of α tried, even
though we increased the temperature input considerably. In these cases there was some
18
initial propagation, with some A and W being consumed, before the temperature started
to fall below the burnt gas temperature (b < 1). This reduced the reaction rates, further
decreasing the temperatures, before finally the reactions stopped and the system returned
slowly to its unreacted state. This propagation failure at α just above 0.002 may be
associated with the form of the eigenvalues for the upper branch solutions changing.
With LA = LW = 4.0, they change from a complex conjugate pair to real (and positive)
at α = 0.00218. The rapid increase in period as propagation failure is approached is
reminiscent of a homoclinic bifurcation.
6
Conclusions
We have considered the conditions for flame propagation and flame inhibition for a firstorder exothermic chemical reaction occurring in the presence of an endothermic reaction.
We have determined the critical conditions in terms of the various system parameters,
with particular attention to a ‘heat loss‘ parameter α which reflects a combination of the
relative thermodynamic and kinetic constants for the endothermic and exothermic processes. Depending on the parameters LA , LW , β and µ, plots of flame speed c against α
may show one of three qualitative forms. Of particular interest are the cases with µ < 1,
for which these bifurcation diagrams show a specific extinction point. Under some conditions, the c − α locus is folded into a characteristic S-shape, showing bistability between
upper and lower branches of steady solutions separated by a branch of saddle point solutions. The critical points correspond to saddle-node turning points in these cases.
The temporal stability to longitudinal perturbations of any resulting flames was studied by investigating the spectrum of the linearised system. We have described the essential
spectrum in detail. The Evans was used to show, without computing the eigenvalues explicitly, that Hopf bifurcations and saddle-node bifurcations can occur in the system.
Then the first couple of isolated eigenvalues were computed along the (α, c) curves by
discretising the linearised eigenvalue problem with finite differences and by computing
the eigenvalues of the matrix obtained from the discretisation. For Lewis numbers close
to unity we found that, if there are three solutions, then two of them are stable and
one is unstable, and there are two saddle-node bifurcation points on the (α, c) parameter
curve. For larger values of the Lewis numbers there is also a Hopf bifurcation point on
the curve, dividing it into a stable and an unstable branch. The saddle-node and Hopf
bifurcation curves were also determined in the (α, LA ) parameter plane. The two curves
have a common point, which is a Takens-Bogdanov bifurcation point. We also solved the
corresponding time dependent system for the values of the Lewis numbers beyond the
Hopf bifurcation. These showed both single period and, away from the Hopf bifurcation
point, period 2 oscillations. These latter oscillatory cycles have periods of rapid propagation when high temperatures (above the burnt gas temperature) are reached and much
longer periods when the flame is virtually stationary and has a lower temperature.
Acknowledgment
This work was supported by the EPSRC (Grant GR/N66537/01).
19
List of symbols
symbol
m
A,W
ρ
T
Ta
E1 , E2
R
k1 , k 2
Q1 , Q 2
DA , D W
Cp
λ
mass flux
mass fractions of the species A and W
total density
temperature
ambient temperature
activation energies
universal gas constant
pre-exponential factors
exothermicities of the reactions
diffusion coefficients of A and W
specific heat for constant pressure
thermal conductivity
units (SI)
kg/m2 s
–
kg/m3
K
K
J/mol
J/K mol
1/s
J/kg
m2 /s
J/kgK
J/msK
References
[1] Alexander, J., Gardner, R., Jones, C., A topological invariant arising in the stability
analysis of travelling waves, J. Reine Angew. Math., 410, 167-212, 1990.
[2] Balmforth, N.J., Craster, R.V., Malham, S.J.A., Unsteady fronts in an autocatalytic
system, R. Soc. Lond. Proc. Ser. A, 455, 1401-1433, 1999.
[3] Beyn, W.J., Lorenz, J., Stability of traveling waves: dichotomies and eigenvalue
conditions on finite intervals, Numer. Func. Anal. Optim., 20, 201-244, 1999.
[4] Bonnet, A., Non-uniqueness for flame propagation when the Lewis number is less
than 1, European J. Appl. Math., 6, 287-306, 1995.
[5] Brin, L.Q., Numerical testing of the stability of viscous shock waves, Math. Comp.,
70, 1071-1088, 2001.
[6] Buckmaster, J.D., The quenching of deflagration waves, Combustion and Flame, 26,
151-162, 1976.
[7] Buckmaster, J.D., Ludford, G.S.S., Theory of laminar flames, Cambridge University
Press, 1982.
[8] Coppel, W.A., Dichotomies in stability theory, Lect. Notes Math. 629, Springer,
1978.
[9] Eastham, M.S.P., The asymptotic solution of linear differential systems, Clarendon
Press, Oxford, 1989.
[10] Evans, J.W., Nerve axon equations IV: The stable and unstable impulse, Indiana
univ. Math. J., 24, 1169-1190, 1974/75.
20
[11] Gardner, R.A., Zumbrun, K., The gap lemma and geometric criteria for instability
of viscous shock profiles, Comm. Pure Appl. Math., 51, 797-855, 1998.
[12] Giovangigli, V., Nonadiabatic plane laminar flames and their singular limits, SIAM
J. Math. Anal., 21, 1305-1325, 1990.
[13] Govaerts, W.J.F., Numerical methods for bifurcations of dynamical equilibria, SIAM,
2000.
[14] Gray, B.F., Kalliadasis, S., Lazarovici, A., Macaskill, C., Merkin, J.H., Scott, S.K.,
The suppression of an exothermic branched–chain flame through endothermic reaction and radical scavenging. Proc. R. Soc. Lond. A, 458, 2119-2138, 2002.
[15] Henry, D., Geometric theory of semilinear parabolic equations, Springer, 1981.
[16] Hoff, D., Zumbrun, K., Asymptotic behavior of multidimensional scalar viscous shock
fronts, Indiana Univ. Math. J., 49, 427-474, 2000.
[17] Keller, H.B., Numerical methods for two-point boundary-value problems, Blaisdell
Publishing Company, 1968.
[18] Lasseigne, D.G., Jackson, T.L., Jameson, L., Stability of freely propagating flames
revisited, Combust. Theory Modelling, 3, 591-611, 1999.
[19] Lazarovici,A., Kalliadasis, S., Merkin, J.H., Scott, S.K., Flame quenching through
endothermmic reaction, J. Engineering Math., 44, 207-228, 2002.
[20] Marion, M., Qualitative properties of a nonlinear system for laminar flames without
ignition temperature, Nonlinear Anal., 9, 1269–1292, 1985.
[21] Pego, R.L., Weinstein, M.I., Eigenvalues and instabilities of solitary waves, Phil.
Trans. R. Soc. Lond., A 340, 47-94, 1992.
[22] Simon, P.L., Kalliadasis, S., Merkin, J.H., Scott, S.K., Quenching of flame propagation with heat loss, J. Math. Chem., 31, 313–332, 2002.
[23] Simon, P.L., Kalliadasis, S., Merkin, J.H., Scott, S.K., Quenching of flame propagation through endothermic reaction, J. Math. Chem., 32, 73–98, 2002.
[24] Simon, P.L., Kalliadasis, S., Merkin, J.H., Scott, S.K., Inhibition of flame propagation
by an endothermic reaction, Accepted for publication in IMA Jl. Applied Math.
[25] Simon, P.L., Kalliadasis, S., Merkin, J.H., Scott, S.K., On the structure of spectra
of some combustion waves, (submitted for publication).
[26] Sandstede, B., Scheel, A., On the structure of spectra of modulated travelling waves,
Math. Nachr., 232, 39-93, 2001.
[27] Warnatz, J., Maas, U., Dibble, R.W., Combustion. Physical and chemical fundamentals, modeling and simulation, experiments, pollutant formation, Springer, 2001.
[28] Volpert, A.I., Volpert, V.A., Volpert, V.A., Traveling wave solutions of parabolic
systems, AMS, 1994.
21
[29] Weber, R.O., Mercer, G.N., Sidhu, H.S., Gray, B.F., Combustion waves for gases
(Le = 1) and solids (Le → ∞), Proc. Roy. Soc. Lond. A, 453, 1105-1118, 1997.
[30] Williams, F.A., Combustion theory, Addison-Wesley, 1985.
[31] Zeldovich, Ya.B., Barenblatt, G.I., Librovich, V.B., Makhviladze, G.M., The mathematical theory of combustion and explosions, Consultants Bureau, 1985.
22
Captions for Figures
Figure 1. The (α, c) curves for ε = 0.1, β = 0.01 and for four different values of LA and
LW as shown in the figure. The value of µ is changed, (a) µ = 0.1, (b) µ = 0.8, (c) µ = 1,
(d) µ = 2.
Figure 2. A plot of the adiabatic flame velocity cad against LA for ε = 0.1. The values cl
and cu are theoretical lower and upper bounds for the case LA → ∞ (solid combustion)
[21].
Figure 3. The hysteresis bifurcation curve for LA = LW = 1 and for LA = LW = 2 with
ε = 0.1. This curve consists of those (µ, β) parameter pairs, for which the (α, c) curve
has an inflexion. If the (µ, β) parameter pair is above the curve, then the (α, c) curve is
decreasing. If it is below the curve, then the (α, c) curve is S-shaped.
Figure 4. The parabolas determining the essential spectrum of the operator obtained
after linearisation. The dimension of the subspace Es+ (λ) is shown in part (a) and (b).
The dimension of the subspace Eu− (λ) is shown in part (c). (a) corresponds to pulses, (b)
corresponds to fronts.
Figure 5. The image of a half circle lying in the right half of the complex plane under
the Evans function D in the case α = 0 for two different values of LA . (a) LA = 3, the
image does not wind around the origin, the winding number (wn) is zero. (b) LA = 4,
the image winds twice around the origin, the winding number (wn) is two.
Figure 6. The image of a half circle lying in the right half of the complex plane under
the Evans function D in the case LA = 1, α is below, but close to the extinction value for
two different values of c (one in the stable branch, the other in the unstable branch).
Figure 7. The stability of the travelling wave along the (α, c) curves for ε = 0.1, β = 0.01,
for four different values of LA and LW as shown in the figure. The value of µ is changed,
(a) µ = 0.1, (b) µ = 0.5. — correspond to stable solutions, − − to unstable solutions.
Figure 8. The stability of the travelling wave along the (α, c) curves for ε = 0.1, β =
0.001, µ = 0.1 for five different values of LA (LW = LA ) as shown in the figure. (b) The
lower part of the figure is enlarged.
Figure 9. The saddle-node and Hopf bifurcation curves. The dotted line is the schematic
plot of the Hopf curve for the lower branch. The pair of numbers indicate the number of
stable and unstable solutions in the different regions determined by the bifurcation curves.
(The first number is the number of stable travelling waves, the second is the number of
unstable waves.)
Figure 10. Plots of the wave speed c against τ obtained from the numerical integration
of (21) for the adiabatic case (α = 0) with ε = 0.1 for (a) LA = 4.0, (b) LA = 5.0, (c)
LA = 15.0, (d) LA = 150.0.
Figure 11. Plots of the wave speed c against τ obtained from the numerical integration
of (21) with LA = LW = 4.0, ε = 0.1, β = 0.01, µ = 0.5 for (a) α = 0.001, (b) α = 0.0015,
(c) α = 0.002.
23