Review articles on the study of combustion. Combustion Theory and Modelling. Volumes Part 2 1-13

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Oscillations in a combustible gas bubble
V. Gol'dshtein a; I. Goldfarb a; I. Shreiber b; A. Zinoviev a
a Department of Mathematics and Computer Sciences, Ben Gurion University of the Negev, POB 653,
84105, Beer Sheva, Israel. b Institute of Industrial Mathematics, 22 Ha-histadrut str., 84213, Beer Sheva,
Israel.
To cite this Article Gol'dshtein, V., Goldfarb, I., Shreiber, I. and Zinoviev, A.(1998) 'Oscillations in a combustible gas
bubble', Combustion Theory and Modelling, 2: 1, 1 — 17
To link to this Article: DOI: 10.1080/713665366
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Combust. TheoryModelling2(1998)1–17. PrintedintheUK PII:S1364-7830(98)83244-0
Oscillations in a combustible gas bubble
V Gol’dshteiny, I Goldfarby, I Shreiberz and A Zinovievy
y Department of Mathematics and Computer Sciences, Ben Gurion University of the Negev,
POB653,84105,BeerSheva,Israel
zInstituteofIndustrialMathematics,22Ha-histadrutstr.,84213,BeerSheva,Israel
Received9April1997,infinalform24November1997
Abstract. The dynamical behaviour of an isolated combustible gas bubble surrounded by
unlimited inviscid liquid is analysed in the case of large activation energy and using spatially
uniform assumptions. The pressure effect is crucial in this problem because of the limited
gasvolume. Themathematicalmodelusedisasystemofthreenonlinearordinarydifferential
equationsincludingtheenergyequation,theconcentrationequationandtheRayleighequation.
Thethermalbehaviourisclassifiedintoslowandexplosiveregimes,andthethermalexplosion
criterionisobtainedanalytically,alongthelinesoftheclassicalSemenovtheory. Thesystemis
showntorevealtemperatureandvolumetricoscillations,theamplitudeandfrequencyofwhich
dependstronglyontheintensityofthethermalprocess. Inparticular,theamplitudeofslowand
10 explosiveregimesdiffersbyatleastanorderofmagnitude.
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1. Introduction
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2

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9: This paper is motivated by a number of industrial and scientific problems concerning
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: flame propagation in multiphase media of complex structures. In particular, a series of
t
A
experimentsonpressurewavepropagationinliquidcontainingcombustiblegasbubbleshas
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e
ad been conducted (Meier and Thompson 1989). There is also the unsolved problem of the
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nl structure and velocity of combustion wave propagation in a gas–liquid foam containing
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D a combustible gas mixture within foam bubbles (Zamashchikov and Kakutkina 1994).
The phenomenon of spontaneous insulation fires represents another example of problems
which are connected with the self-ignition event taking place in a combustible gas bubble
surrounded by liquid media (McIntosh etal 1994).
One of the hypothetical mechanisms of combustion wave propagation in such media is
the following. At the early stages of the process, the self-ignition occurs in an arbitrary
single bubble. The heat produced by the exothermic chemical reaction leads to pressure
changes in the bubble which, in turn, cause changes of the bubble volume (radius). This
generates a spherical pressure wave propagating from the bubble to the surrounding liquid.
The pressure disturbance reaches another bubble (containing the same gas mixture) and
causes compression of this bubble and, consequently, an increase of the gas temperature.
This may produce an intensive self-heating and, as a result, a self-ignition event in this
secondary bubble, since the rate of the exothermic chemical reaction is extremely sensitive
to temperature changes. This secondary bubble repeats the dynamical behaviour of the
originalbubbleandbecomesthesourceofasecondarypressurewavewhichalsopropagates
in all directions. Thus the original bubble generates a chain of self-ignition events in the
neighbouring bubbles, and this mechanism can represent one of the factors determining
1364-7830/98/010001+17$19.50 (cid:13)c 1998IOPPublishingLtd 1
2 VGol’dshteinetal
the structure and velocity of combustion wave propagation in such media. The case of
spontaneous insulation fires is more complicated because of a solid skeleton (which may
conduct heat much better than the liquid) and the evaporation effect from the gas bubble
surface (the porous insulation material is impregnated with a flammable fluid).
The purpose of this work is to investigate qualitatively the dynamics of an isolated
combustiblegasbubblesurroundedbyunlimitedinviscidliquid. Weareprimarilyseekingto
studytheinteractionbetweenthethermalprocessestakingplacewithinthebubble(including
theself-ignitionphenomenon)andthehydrodynamicsoftheunlimitedliquidsurroundingthe
bubble. Thesimplifiedphysicalmodelusedheretakesaspatiallyuniformapproachtoboth
the energy and concentration equations. The gas temperature is controlled by the following
main processes: heat release associated with the exothermic oxidation reaction, heat losses
due to the cooling and the pressure changes. The chemical reaction is considered as a
single-step reaction obeying an Arrhenius temperature dependence, and the hydrodynamics
of the liquid surrounding the gas bubble is governed by the Rayleigh equation.
Themathematicalmodelisdevelopedasasystemofthreenonlinearordinarydifferential
equations: theenergyequation, theconcentrationequation, andtheRayleighequation. The
systemcanbeconsideredasamulti-scalesystembecauseofthehighlyexothermicchemical
reaction. The energy equation is fast (with a small parameter before the derivative) and
the other equations are slow. This system hierarchy allows us to apply the geometrical
asymptotic methods developed in earlier works on self-ignition (Babushok and Gol’dshtein
1988, Babushok etal 1990, 1992, Gol’dshtein and Sobolev 1992, Gol’dshtein etal 1996).
Conceptual qualitative information about the system behaviour is obtained analytically and
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1 detailed quantitative characteristics are the object of consequent numerical simulations.
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y Theprincipaldifferenceoftheproposedmodelfromtheconventionalapproachestoself-
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ua ignition theory is that it takes into account the pressure effects on the process development.
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J The system is shown to reveal temperature and volumetric oscillations, for any initial

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2 conditions(withintheframeworkofourmodel). Thedistinctcharacterofthenonlinearities
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start after the exothermic reaction is almost completed. The amplitude and frequency of
:
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A these oscillations depends strongly on the thermal process intensity. The thermal behaviour
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l these two types of regimes is obtained analytically. The oscillation amplitude for slow and
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Do explosive regimes differs by at least an order of magnitude.
2. Problem statement
A single gas bubble is considered which is surrounded by unlimited liquid, and contains
someinitialamountofacombustiblegasmixture(combustiblegas, oxidizerandinertgas).
Themaininterestistheinvestigationofthebubbledynamicsduringtheself-ignitionevent.
The main physical assumptions of the suggested model are the following. We restrict
ourselves to a spatially homogeneous approximation. The liquid flow is assumed to be
inviscid and the liquid to be incompressible. Normally in the open air the velocity of the
pressure wave propagation is much larger than the rate of the thermal effects. Therefore
pressuredisturbancesrunawayfromthereactionzone. Thiscanexplainwhythetraditional
approach to the thermal explosion theory ignores the effect of pressure changes on the
process. Inthepresentproblemthepressurevariationswithinthebubblearecrucialbecause
of the limited gas volume. We assume that the gas is ideal, and that its equation of state
is the Mendeleev–Clapeiron equation and the total amount of the gas within the bubble
(number of moles) is constant, i.e. the stoichiometry of the chemical reaction conserves the
Oscillationsinacombustiblegasbubble 3
number of moles. We also assume that the molecular masses of all the gas components are
equal.
The energy equation describes the temperature changes of the gaseous phase within the
bubble. To simplify the mathematical description of the real complex problem we make
someadditionalassumptions. Weassumethattheliquidandtheinterphaseboundaryremain
at a constant temperature during the process, and that this temperature is low enough for
evaporation effects to be neglected. The heat flux from the gas phase to the liquid phase is
considered to be proportional to the temperature difference between the phases. It is also
assumed that the thermal conductivity of the liquid is much higher than that of the gas
for the heat transfer coefficient to be defined by the thermal properties of the gas phase.
The energy equation used here is similar to that in the work of Gol’dshtein et al (1994)
in which the phenomenon of creeping detonation in filtration combustion was analysed for
the adiabatic and spatially distributed assumptions. The present energy equation has a heat
release term associated with the chemical oxidation reaction, a heat loss term controlled by
the heat transfer from the gas to the liquid, and also a term associated with the pressure
changes within the bubble:
(cid:0) (cid:1)
dT (cid:11) 4(cid:25)R2 dP
C (cid:26) g Dc Q (cid:22) W − (cid:0) b(cid:1).T −T/C g (2.1)
pg g dt f f f 4(cid:25)R3 g l dt
3 b0
where C is the specific heat capacity (Jkg−1 K−1), P the pressure (Pa), Q the combustion
energy (Jkg−1), R the radius (m), T the temperature (K), W the reaction rate (s−1), c the
combustible gas concentration (kmolm−3), (cid:11) the heat transfer coefficient (Wm−2 K−1),
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01 (cid:22) the molecular weight (kgkmol−1), (cid:26) the density (kgm−3); subscripts are related to: b,
2

ry bubble; g, gas; f, combustible gas component of the mixture; l, liquid; p, under constant
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nu pressure; 0, undisturbed state.
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The reaction rate W is given in the Arrhenius form:
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2 (cid:18) (cid:19)

00 E
9: W DAexp − (2.2)
: 0 BTg
t
d A where A is the pre-exponential factor .s−1/, E the activation energy .Jmol−1/ and B the
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ad universal gas constant.
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n The combustible gas content (per unit volume) in the bubble is controlled by the
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D oxidationchemicalreactionandbythebubblevolumetricvariations. Toderivetherelevant
governing equation we first write the continuity equation for the gas in the bubble, using
the spherical coordinates and taking into account the gas consumption due to the chemical
reaction:
(cid:0) (cid:1)
@c 1 @
f C r2u c D−c W (2.3)
@t r2@r g f f
where u is the velocity (m s−1) and r the current coordinate from the centre of the bubble
(m).
Multiplying by r2 and then integrating the obtained equality with respect to r from 0
to R we obtain the relation between the concentration, temperature and velocity of the
b
gas on the bubble boundary r D R (we remind the reader that the concentration and
b
temperature are spatially uniform in our assumptions). In the absence of phase transitions
on the interphase boundary we may put u D dR =dt (Nakoryakov et al 1993). The
g b
concentration equation thus becomes
dc 3 dR
f D−c W −c b: (2.4)
f f
dt R dt
b
4 VGol’dshteinetal
The chemical reaction within the bubble causes the gas temperature to increase, which,
in turn, leads to the bubble radius variations. Those are controlled by the unlimited
liquid surrounding the gas bubble. To take this hydrodynamical mechanism into account,
we suggest using the Rayleigh equation (Nakoryakov et al 1993) which relates pressure
variationsinaliquidtoabubbleradius. RecallhowtheRayleighequationcanbeobtained.
Let us write the equations of hydrodynamics in a spherically symmetrical case:
(i) the Euler equation:
@u @u 1 @P
l Cu l D− l (2.5)
l
@t @r (cid:26) @r
l
(ii) the continuity equation:
(cid:0) (cid:1)
@
r2u D0: (2.6)
l
@r
Integratingequation(2.6)givesr2u DF.t/,whereF.t/isanarbitrarytimefunction. Using
l
the boundary conditions, we obtain F.t/ D R2.dR =dt/: Hence we get the distribution of
b b
u with respect to r:
l
(cid:18) (cid:19)
R2 dR
u D b b: (2.7)
l r2 dt
Under the assumption that the liquid is incompressible and the pressure changes are small
enough (of the same order as the bubble radius), equation (2.5) can be integrated with
respecttor fromthevaryingvalueR toinfinity. Theboundaryconditionsare: Pl DPl1 at
2010 r D1, Pl DPl.Rb/ and ul.r DRb/DdRb=dt at r DRb. Initially, the system is supposed
ry to be at an equilibrium state and therefore the following equality is true: Pg0 D Pl1.
ua Integrating equation (2.5), taking account of the expression for u (2.7), we get the well
n l
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24 (cid:18) (cid:19) (cid:18) (cid:19)
:00 R d2R C 3 dR 2 D Pl.Rb/−Pl1:
09 dt2 2 dt (cid:26)
l
:
t
A TodefineP.R /weneglectbothviscouseffectsontheboundaryandsurfacetensioneffects
l b
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de (the heat transfer effect has already been taken into account in the simplest approximation).
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nlo Thus Pl.Rb/ D Pg where Pg is the gas pressure in the bubble determined by the equation
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D
(cid:26) m BT
P D gBT (cid:17) g g : (2.8)
g (cid:22) g (cid:22) 4(cid:25)R3
g g3 b
From the above it follows that the system of governing equations is given by:
(i) the energy equation for the reacting gas:
dT 3(cid:11)R2 dP
C (cid:26) g Dc Q (cid:22) W − b.T −T/C g (2.9)
pg g dt f f f R3 g l dt
b0
(ii) the concentration equation for the reacting gas:
dc 3 dR
f D−c W −c b (2.10)
f f
dt R dt
b
(iii) the Rayleigh equation:
(cid:18) (cid:19) (cid:18) (cid:19)
d2R 3 dR 2 .P −P /
R b C b D g g0 : (2.11)
b dt2 2 dt (cid:26)
l
Here W and P are defined by (2.2) and (2.8), respectively.
g
Oscillationsinacombustiblegasbubble 5
The initial conditions for equations (2.9)–(2.11) are given by
t D0 T DT (cid:17)T c Dc R DR P DP : (2.12)
g g0 l f f0 b b0 g g0
We will rewrite the system (2.9)–(2.11) in the dimensionless form using the conventional
dimensionlesstemperature(cid:18) andconcentration(cid:17)ofthecombustiblegas(Frank-Kamenetskii
1969), and the natural dimensionless radius of the bubble r:
T −T E c R
(cid:18) D g g0 (cid:17)D f r D b : (2.13)
T BT c R
g0 g0 f0 b0
As in the previous works (e.g. Babushok et al 1990, Gol’dshtein and Sobolev 1992,
Gol’dshtein et al 1996), we choose the characteristic reaction time as the basis for the
time scale, i.e.
(cid:18) (cid:19)
(cid:14)
t E
(cid:28) D t Dexp A: (2.14)
r
t BT
r g0
The reaction time t is the time required for the reactant concentration to fall e-times from
r
its initial value under the isothermal condition T D T . In the dimensionless form, the
g g0
system of governing equations reads
(cid:18) (cid:19)
γ(cid:18)P Dγ (cid:17)r3exp (cid:18) −(cid:15) r2(cid:18) −(cid:15) dr 1C(cid:12)(cid:18) (2.15)
c 1C(cid:12)(cid:18) 1 2d(cid:28) r
(cid:18) (cid:19)
(cid:17) dr (cid:18)
(cid:17)P D−3 −(cid:17)exp (2.16)
r d(cid:28) 1C(cid:12)(cid:18)
0 (cid:18) (cid:19)
1
20 3 1C(cid:12)(cid:18)
ary rrRC 2rP2 D(cid:15)3 r3 −1 : (2.17)
u
n
Ja The initial conditions are (cid:28) D 0; (cid:18) D 0; (cid:17) D r D 1; rP D 0, where a dot denotes a

0 24 derivative with respect to the dimensionless time (cid:28). The parameter γc Dcp=cv is the ratio
:0 of the specific heat capacities. The system dynamics is defined by the five dimensionless
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0 parameters(cid:15) ;(cid:15) ;(cid:15) ;(cid:12) andγ. Theparameters(cid:12) andγ arecommonlyusedinthecontextof
: 1 2 3
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ownl (cid:12) D BTg0 γ D Cpg(cid:26)g0Tg0(cid:12): (2.18)
D E c Q (cid:22)
f0 f f
Theparameters(cid:15) ;(cid:15) and(cid:15) arenewintheconsideredresearchfield. Theparameters(cid:15) ;(cid:15)
1 2 3 1 2
describetherelationsbetweenthethermophysicalpropertiesofthegasandtheliquidphases
3(cid:11)γ (cid:12)T 3γ (cid:26) BT
(cid:15) D c g0 exp.1=(cid:12)/ (cid:15) D c g0 g0: (2.19)
1 Ac Q (cid:22) R 2 Q (cid:22)2c
f0 f f b0 f f f0
The parameter (cid:15) correlates the two characteristic times: the characteristic reaction time t
3 r
and the characteristic time of harmonic bubble oscillations t :
osc
(cid:18) (cid:19)
t 24
(cid:15) D r (cid:25)2=γ (2.20)
3 c
t 3
osc
where the characteristic time of harmonic bubble oscillations t is proportional to the
osc
reciprocal of the Minnaert frequency of the bubble under initial values of the size and the
gas pressure (Nakoryakov etal 1993):
(cid:18) (cid:19)
(cid:26)R2 1=2
t D2(cid:25)=! D2(cid:25) l b0 :
osc m
3γ P
c 0
6 VGol’dshteinetal
Let us analyse the physical meaning of parameters (cid:15) and (cid:15) in more detail. Parameter (cid:15)
1 2 1
can be presented as (cid:15) Dγ .t =t /. Here the characteristic time t is given by
1 c ad N ad
C (cid:26) (cid:12)T
t D pg g g0 exp.1=(cid:12)/
ad
Ac Q (cid:22)
f0 f f
andrepresentsthetimeafterwhichthetemperaturewouldbecomeinfiniteintheabsenceof
heatlosses(byheattransferfromthegastothecoolliquidphase)andintheapproximation
(cid:12) D0 (Frank-Kamenetskii 1969). The thermal relaxation time (Newtonian cooling time)
C (cid:26) R
t D pg g b0
N
3(cid:11)
is the characteristic time of gas cooling caused by the heat transfer only. To calculate t
N
we solve the energy equation under the assumptions:
(i) an absence of chemical heat sources .c D0/;
f
(ii) bubble radius R remains at the initial value R during the thermal relaxation
b b0
(such an approach is acceptable during the initial stages of the process). A solution of
the energy equation (2.9) under the above-mentioned conditions involves the connection
.T −T/(cid:24)exp.−t=t /, and thereby t is the time required for the temperature difference
g l N N
.T −T/ to fall e-times from its arbitrary value. Thus the parameter (cid:15) is proportional to
g l 1
theratioofthecharacteristictimesofthecombustionandheattransferprocesses,andhence
it is defined by the competition between the intensities of the combustion and heat transfer
processes.
Parameter (cid:15) can be rewritten as
0 2
1
20 N BT
uary (cid:15)2 D3γcNfgQf(cid:22)g0f
n
4 Ja where Ng D mg=(cid:22)g is the total number of gas moles in the bubble and Nf D cf0Vb0 is
2 the total number of combustible gas moles in the bubble at the initial conditions. We can
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gas in the bubble at the initial moment (i.e. the energy to be released after total burning of
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D 3. Dimensionless model and its phenomenology
The system of governing equations (2.15)–(2.17) can be rewritten as a system of four
ordinary differential equations of the first order, by introducing the new formal variable
v dDefrP:
(cid:18) (cid:19)
γ(cid:18)P Dγ (cid:17)r3exp (cid:18) −(cid:15) r2(cid:18) −(cid:15) v1C(cid:12)(cid:18) dDefF.(cid:18);(cid:17);v;r/ (3.1)
c 1C(cid:12)(cid:18) 1 2 r
(cid:18) (cid:19)
(cid:17)v (cid:18)
(cid:17)P D−3 −(cid:17)exp (3.2)
r 1C(cid:12)(cid:18)
(cid:18) (cid:19)
1C(cid:12)(cid:18) 1 3v2
vP D(cid:15) − − : (3.3)
3 r4 r 2 r
rP Dv: (3.4)
The subsystem (3.1), (3.2) describes thermo-chemical processes and the subsystem (3.3),
(3.4) describes hydrodynamic processes. The parameters γ and (cid:12) are small, due to the
assumption of a highly exothermic chemical reaction. This defines the scaling of the
Oscillationsinacombustiblegasbubble 7
system: (cid:18) is a rapidly varying variable, and (cid:17);v and r are slowly varying variables. Note
that the big parameter (cid:15) ((cid:15) is large because of the small pulsation time) in the right-hand
3 3
s(cid:0)ide of equation (3.3(cid:1)) does not affect the system hierarchy. This is because of the term
.1C(cid:12)(cid:18)/=r4 −1=r which is small for two reasons: first, the term (cid:12)(cid:18) is small because
of the small parameter (cid:12), and secondly variable r changes slowly from the initial value
r D 1, and hence 1=r4 is comparable with 1=r. The same conclusion evidently holds in
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any case when the radius is close to unity that is always true for the considered system, as
will be shown below.
The system behaviour can be either slow or explosive, due to the highly exothermic
chemical reaction. These two main types of system behaviour are studied in sections 5 and
6bytheuseofthedifferentasymptotics. Slowregimesareanalysedintheapproximationof
rapidlyandslowlyvaryingvariables(.1(cid:2)3/-approximation: (cid:18) isarapidlyvaryingvariable
and (cid:17);v and r are slowly varying variables). The explosive regimes are investigated by
a combination of the two relevant asymptotics. Namely, the fast explosive temperature
increaseismodelledinanadiabaticapproachsinceheatlossesduringthethermalexplosion
canbeignored,andthefurthertemperaturedecreaseisanalysedinthe.1(cid:2)3/-approximation,
similar to the case of the slow regimes.
Thecriteriawhichseparatetheslowsystembehaviourfromtheexplosiveonearestudied
in the next section.
Remark. The system reveals oscillation behaviour whichever regimes (slow or explosive)
occur. Thecharacteroftheseoscillationsisanalysedqualitativelybyusinga.1(cid:2)3/-system
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0 The thermal explosion problem studies the critical parameter conditions which separate the
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9 slow system behaviour from the explosive one. Remember that the character of the system
0
t: hierarchy is .1(cid:2)3/, i.e. (cid:18) is a rapidly varying variable and (cid:17);v and r are slowly varying
A
d variables. In the zeroth-order approximation of the small parameter γ (γ D0), the slowly
e
ad varying variables are fixed and equal to their initial values (cid:17) D 1; v D 0; r D 1 and the
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own problem is reduced to an anal(cid:18)ysis of th(cid:19)e following equation:
D
d(cid:18) (cid:18)
γ Dγ exp −(cid:15) (cid:18): (4.1)
d(cid:28) c 1C(cid:12)(cid:18) 1
Thefirsttermintheright-handsideofthisequationcorrespondstotheheatproduction,and
the second term corresponds to the heat losses. In this situation we can use the classical
theory of Semenov (1928), according to which the critical parameter condition which
separates slow and explosive regimes is defined as a contact point of the heat production
curve and the heat losses lines (see the Semenov diagram, figure 1). The contact point
condition is given by
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:18) (cid:18) 1
γ exp −(cid:15) (cid:18) D0 γ exp −(cid:15) D0:
c 1C(cid:12)(cid:18) 1 c 1C(cid:12)(cid:18) .1C(cid:12)(cid:18)/2 1
This determines the critical parameter value (cid:15) , and also the maximum subcritical
1cr
temperature (cid:18) (which is the temperature corresponding to the critical parameter value):
m
(cid:15) Dγ e.1−(cid:12)/ (cid:18) D1C2(cid:12): (4.2)
1cr c m
For (cid:15) > (cid:15) the system behaviour is slow with maximum temperatures less than (cid:18) and
1 1cr m
for (cid:15) < (cid:15) the system behaviour is explosive. Figure 2 gives the regions of the slow
1 1cr
8 VGol’dshteinetal
heat
heat release
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II
III
heat losses
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Figure1. TheSemenovdiagram. SituationI,slowregimes;II,criticality;III,ignitionevent.
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o
D Figure2. Parametricdescriptionoftheslowandexplosiveregimes,intheplane.(cid:12);(cid:15)N1/(where
(cid:15)N1isobtainedfrom(cid:15)1givenby(2.19)as(cid:15)N1D3(cid:11)E=ABecf0Qf(cid:22)fRb0). Thetypicalphaseplots
andtimehistoriesaregiveninfigures3–6.
and explosive regimes in the plane .(cid:12);(cid:15)N /, where the parameter (cid:15)N is obtained from the
1 1
parameter (cid:15) (see equation (2.19)) by excluding (cid:12): (cid:15)N D .3(cid:11)E/=.ABec Q (cid:22) R /. This
1 1 f0 f f b0
figure illustrates the effect of the reduced initial temperature (cid:12) (see equation (2.18)) on the
system behaviour. We study the cases of the slow and explosive behaviour using relevant
asymptotics.
5. Slow regimes ((cid:15) >(cid:15) )
1 1cr
Remember that the system (3.1)–(3.4) has one rapidly varying variable (cid:18) and three slowly
varying variables (cid:17);v and r. The equation
(cid:18) (cid:19)
(cid:18) 1C(cid:12)(cid:18)
γ (cid:17)r3exp −(cid:15) r2(cid:18) −(cid:15) v dDefF.(cid:18);(cid:17);v;r/D0
c 1C(cid:12)(cid:18) 1 2 r
Oscillationsinacombustiblegasbubble 9
defines a three-dimensional surface in the phase space which is termed the slow surface of
the system. On this surface the rapidly varying variable (cid:18) and the slowly varying variables
(cid:17);v and r are comparable, and beyond this surface the slowly varying variables (cid:17);v;r are
assumed to be constant, in the zeroth-order approximation (γ D 0). A trajectory with the
starting point at r D (cid:17) D 1; (cid:18) D v D 0 (this is the initial condition of the process) is
attracted by the stable leaf of the slow surface (where F0.(cid:18);(cid:17);v;r/ < 0). The point at
(cid:18)
which the trajectory hits the slow surface is given by r D(cid:17) D1; v D0 and (cid:18) D(cid:18)(cid:3) where
(cid:18)(cid:3) is a root of the equation
(cid:18) (cid:19)
(cid:18)(cid:3)
F.(cid:18)(cid:3);1;0;1/(cid:17)γ exp −(cid:15) (cid:18)(cid:3) D0 (5.1)
c 1C(cid:12)(cid:18)(cid:3) 1
which satisfies the condition 0<(cid:18)(cid:3) <(cid:18) (cid:25)1 C2(cid:12). The trajectory reaches the maximum
m
temperature(cid:18)(cid:3) andthenpassesalongtheslowsurfacedowntothefinalsteadystate. Letus
analyse the system dynamics on the slow surface. For a fixed value of (cid:18) the system (3.3),
(3.4) can be rewritten as
(cid:18) (cid:19)
1C(cid:11) 1 3
vP D(cid:15) − − v2=r (5.2)
3 r4 r 2
rP Dv (5.3)
where(cid:11) D(cid:12)(cid:18) issmallbecauseofthesmallparameter(cid:12). Thissystemhasauniquesingular
point
v D0 r D.1C(cid:11)/1=3: (5.4)
0
201 The Jacobi mat(cid:18)rix of the system (5.2) a(cid:19)nd (5.3) at this singular point is given by

ary 0 −3(cid:15)3=.1C(cid:11)/2=3
u
n 1 0
a
J

4 and has the purely imaginary eigenvalues
2

0
: 09:0 (cid:21)1;2 D(cid:6).1.3C(cid:15)3(cid:11)/1/=12=3I:
t
A
Hence for the linear system the singular point is a centre, for any set of the parameters. In
d
e
ad this case the character of the singular point of equations (5.2) and (5.3) is not determined
o
nl by the linearized equations, according to the Liapunov theory on the stability of the steady
w
o
D states(see,forexample,Andronovetal 1966). Thispointcancorrespondtoeitheracentre
or a multiple focus, and the character of the singular point can be established by means of
the Liapunov coefficients. More precisely, if at least one Liapunov coefficient is non-zero,
then the steady state is a multiple focus. Otherwise when all the Liapunov coefficients are
zero, the steady state is a centre.
We have shown that the two first Liapunov coefficients are zero, using the formulae
presented in the book by Bautin (1984). For the sake of brevity we do not give these
tedious calculations here (see, for example, Bautin 1984, pp 28, 36). Since the analytical
investigations of the model are approximate, it is reasonable to assume that the steady state
is a centre. Under this assumption the system reveals harmonic radius oscillations with the
amplitude defined as the difference between the radius values at the initial moment and the
steady state:
(cid:0) (cid:1)
amD .1C(cid:11)/1=3−1
and the frequency
.3(cid:15) /1=2
!D 3 :
.1C(cid:11)/1=3