The stationary upward propagation of a very lean methane/air ﬂame in a long vertical tube open at the bottom and closed at the top is simulated numerically using a single overall chemical reaction to model combustion and assuming an optically thin gas and a transparent or non-reﬂecting tube wall to approx-imately account for radiation losses from CO 2 and H2O. Buoyancy plays a dominant role in the propaga-tion of these ﬂames and causes a large region of low velocity of the burnt gas relative to the ﬂame to appear below the ﬂame front when the equivalence ratio is decreased. The size of this region scales with the radius of the tube, and its presence enhances the effect of radiation losses, which would be otherwise negligible for a standard ﬂammability tube, given the small concentration of radiating species. Heat con-duction is found to be important in the low velocity region and to lead to a conduction ﬂux from the ﬂame to the burnt gas that causes extinction at the ﬂame tip for a value of the equivalence ratio near the ﬂam-mability limit experimentally measured in the standard tube. The effect of radiation losses decreases with the radius of the tube. Numerical results and order-of- magnitude estimates show that, in the absence of radiation, a very lean ﬂame front fails to propagate only after recirculation of the burnt gas extends to its reaction region and drastically changes its structure. This condition is not realized for the standard ﬂammability tube, but it seems to account for the ﬂammability limit measured in a tube of about half the radius of the standard tube. 1. Introduction Upward propagation of ﬂame fronts in tubes ﬁlled with very lean reactive mixtures is a problem of both fundamental and ap- plied interest which has been extensively studied. On the basis of a comprehensive analysis of previous work, Coward and Jones [1] proposed that a standard ﬂammability tube can be used to charac-terize the ﬂammability of gas mixtures. This is a vertical tube 51 mm in diameter and 1.8 m long with an open lower end and a closed upper end, which is ﬁlled with the mixture to be tested. Thismixture is ignited near the lower end of the tube, and it is said to be ﬂammable if a ﬂame ensues and propagates all the way to the upper end. Levy [2]noted that buoyancy plays a dominant role in the propagation of the ﬂame front near the ﬂammability limit.This author observed that, when the Lewis number of the fuel is not far from unity, as in mixtures of methane/air or propane/air, the ﬂame front consists of a spherical cap followed by a long skirt,and that its velocity is close to the velocity of a bubble rising in the tube [3]. The dependence of the ﬂammability limit on the diameter of the tube was studied by Babkin et al. [4]. In agreement with some previous results but contrary to the results of Coward and Jones, these authors found that the minimum fuel concentration for which a ﬂame front can propagate increases with the diameter of the tube. Lewis and von Elbe [5], Jarosinski et al. [6]and Hertz- berg [7]pointed out that the stretch of the ﬂame due to its curva- ture and to the strain rate of the ﬂow it induces in the tube may cause the extinction observed at the ﬂammability limit. Von Lavan-te and Strehlow [8]approximately computed the ﬂow of the fresh gas above the ﬂame front. They found that the stretch is duemainly to the strain rate of the ﬂow; that it is of the order of the inverse of the residence time of the gas across a planar ﬂame prop- agating in the mixture; and that it is maximum at the tip of the ﬂame front, where extinction begins at the ﬂammability limit. The ﬂow on both sides of an axisymmetric steadily rising ﬂame front was further investigated by Higuera [9], with emphasis on the structure of the vortical ﬂow downstream of the ﬂame frontand the generation of vorticity at the ﬂame. The results accurately determine the shape and stretch of the ﬂame and point out a ⇑Corresponding author. E-mail address: fhiguera@aero.upm.es (F.J. Higuera). possible instability of the vortical ﬂow. Shoshin et al. [10] recently investigated the effect of preferential diffusion, which can increase the ﬁnal combustion temperature of stretched ﬂames when the Le- wis number of the deﬁcient reactant is smaller than unity. These authors, however, conclude that preferential diffusion cannot al- ways explain the observed behavior of methane/air and propane/ air ﬂame fronts at their ﬂammability limits. Shoshin and Jarosinski [11] and Shoshin et al. [12] proposed that heat losses from the ﬂame due to the effect of radiation in a low velocity region that ap-pears below very lean ﬂames may explain the observed extinction. Radiation losses in near limit ﬂames propagating upward in a tube are further investigated in this paper. Numerical computa- tions and order-of-magnitude estimations are used to show thatthe effect of these losses is very much enhanced by the conditions of the ﬂow below the ﬂame front. For very lean mixtures, this ﬂowfeatures a region of low velocity relative to the ﬂame, whose radial extent scales with the diameter of the tube and whose length may be even larger. Heat conduction is important in this region, despite its large size, leading to a conduction heat ﬂux from the reaction region of the ﬂame to the burnt gas that is sufﬁciently strong to cause extinction of the ﬂame in conditions typical of the standard ﬂammability tube. This singular feature of the ﬂow disappears, and radiation losses cease to play a role, when the equivalence ratio is slightly increased. The effect of the radiation losses also decreases with the radius of the tube. The reaction region at the tip of a very lean ﬂame in a narrow tube is thick and tends to be located in a re- gion of reverse (upward) ﬂow, which may cause ﬂame failure with- out radiation losses. Kinetic effects are important at the ﬂammability limit. However, since analysis of these effects is be- yond the scope of the present work, the combustion is modeled by means of a global irreversible Arrhenius reaction. This drastic simpliﬁcation prevents any accurate analysis of ﬂame extinction but, by keeping the computations affordable, it allows to clarify the mechanism by which low gas velocities and radiation losses act upon the ﬂame. 2. Formulation Consider an axisymmetric ﬂame front rising at constant speed ina very long vertical tube of radius R/C3whose wall is kept at a con- stant temperature Tu, as in the sketch of Fig. 1 . The tube is open to the atmospheric pressure at its lower end and closed at its upper end, and it is ﬁlled with a very lean fuel-air mixture of densityqu, temperature Tu, and fuel mass fraction Yu. A perfect gas with constant speciﬁc heats and mean molecular mass is assumed.Compress ibility effects are left out by setting the pressure equal to a constant in the equation of state. The gas viscosity and thermalconductivity, land k, and the diffusivities of the fuel and the reac- tion products, Diwith i¼CH 4;CO2;H2O in the case of a methane ﬂame, are taken to be powers of the temperature, of the form l=lu¼k=ku¼qDi=quDiu¼ðT=TuÞj, so that the Prandtl and Lewis numbers, Pr¼lcp=kand Lei¼k=qcpDirespectively, are constant. Hereafter a subscript udenotes conditions in the fresh mixture. In what follows, j¼0:75 and Pr¼Lei¼1 unless otherwise is noted. While more accurate molecular transport models are avail-able for lean methane/air mixtures, such as the model developed by Smooke and Giovangigli [13], the simple power law model will sufﬁce for the purposes of this work. Combustion in the ﬂame is modeled using a single irreversible Arrhenius reaction, CH 4þ2O2!CO2þ2H2O for methane/air, with frequency factor Band activation temperature Ta, so that the mass of fuel consumed per unit volume of gas and unit time isw¼qBYFexpð/C0Ta=TÞ, where qand Tare the local gas density and temperature, and YFis the fuel mass fraction. Admittedly, a single-step chemistry is a poor model of the real kinetics. It cannot account for kinetic effects on ﬂame extinction and does not allow to accurately determine the ﬂammability limit. However, it is still useful to analyze the effects of gravity and thermal expansion, as well as the effects of preferential diffusion and radiation losses, which are the subject of this paper and seem to play an important role for near limit ﬂames in the standard ﬂammability tube. The adiabatic ﬂame temperature is Tb¼TuþqYu=cp, where qis the heat released per unit mass of fuel consumed. The dimension- less parameter c¼Tb/C0Tu ðÞ =Tumeasures the thermal expansion of the gas in the ﬂame, and the Zeldovich number b¼TaTb/C0Tu ðÞ =T2 b measures the temperature sensitivity of the chemical reaction. The equivalence ratio is /¼17:39Yu=ð1/C0YuÞfor methane/air mix- tures. Well-known asymptotic analysis [14] shows that, in the lim- itb!1 , the velocity and thickness of a planar adiabatic ﬂame are UL¼2DbBðÞ1=2LeF bcþ1ðÞexp /C0b 2cþ1 c/C18/C19 and dL¼ku quULcp; ð1Þ where Db¼DFðTbÞand LeFare the diffusivity and the Lewis number of the fuel. The values of the frequency factor and the activation tempera- ture are chosen for the single-step chemistry to give planar ﬂame velocities in agreement with experimental results in the range of equivalence ratios of interest. For this purpose, a reference case is considered (magnitudes denoted with a subscript r) where Tu¼300 K and Yur¼0:03, so that /r¼0:538 ;Tbr¼1500 K and cr¼4 (using cp¼1287 J/kg K), and the planar ﬂame velocity is ULr¼4:63 cm/s according to Ref. [15]. For each value of the activa- tion temperature Ta, the frequency factor Bis chosen for the asymptotic formula (1)to reproduce this velocity in the reference case. The value Ta¼18;750 K, for which br¼10, is then deter- mined for the ﬂame velocity computed numerically with the sin- gle-step chemistry to ﬁt the experimental results of Yamaoka and Tsuji [16] for very lean mixtures, corrected to zero stretch by Wang et al. [17]. This value of the activation temperature is in line with the results of Westbrook and Dryer [18].Figure 2 shows the computed ﬂame velocity (solid), the velocity given by the asymp-totic formula (1)(dashed) and the experimental velocity (triangles) as functions of the equivalence ratio. Also shown in this ﬁgure arethe planar ﬂame velocities determined by Wang et al. [17] from their microgravity experiments (circles) and the ﬁts obtained withthe single-step chemistry for T a¼37;500 K and 52,500 K (lower and upper dotted curves), though these values will not be usedin what follows. Fig. 1. Deﬁnition sketch. R/C3¼25:5 mm for the standard ﬂammability tube. Radiation from the reaction products CO 2and H 2O is taken into account assuming that the gas is optically thin and the wall of the tube is transparent or non-reﬂecting. The energy lost by radiation per unit volume of gas and unit time is then L¼4rKPT4/C0A, where ris the Stefan-Boltzmann constant, KPis the Planck mean absorp- tion coefﬁcient, and Ais the energy absorbed by the gas from the surrounding medium or the wall at temperature Tu; see, e.g., Ref. [19]. The coarse approximation A¼4rKPT4 u, leading to L¼4rKP T4/C0T4u/C16/C17 , will be used, which is admissible when Tuis small com- pared to the temperature of the radiating gas. The Planck mean absorption coefﬁcient is written in the form KP¼PCO2KCO2þ PH2OKH2O, where PCO2and PH2Oare the partial pressures of CO 2 and H 2O, and KCO2and KH2Oare functions of the gas temperature computed and tabulated by Ju et al. [20] using the statistical nar- row band model; see also Fiveland [21] and Souﬁani and Taine [22]. For the reference case, KPis of the order of 1 m/C01in the burnt gas, where PCO2/C250:054 P;PH2O/C250:108 P, and P/C25105Pa is the total pressure of the gas. The absorption length K/C01 Pis thus large compared to the radius of the tube (R/C3¼25:5 mm for the standard ﬂammability tube), which justiﬁes the assumption of an optically thin gas. An additional approximation will be used to simplify the com- putation of the radiation losses. Since the diffusivities of CO 2and H2O are not very different from each other, the approximation LeCO2¼LeH2Ois made. This gives PH2O¼2PCO2, because the two spe- cies are produced and transported in the same conditions. Further- more, since the mass fractions of these species are small for the lean mixtures considered, the approximation PCO2=P/C25YCO2=44/C0/C1 = 0:23=32þ0:77=28 ðÞ ¼0:655 YPcan be used, where YPis the mass fraction of CO 2. Thus, summarizing and using the equation of state of the gas, L¼2:62rRgqYPTK CO2þ2KH2O/C0/C1 T4/C0T4 u/C16/C17 , where Rgis the constant of the air. To assess the effect of this approximation, computations have been carried out in which the mass fractions of CO 2and H 2O are separately computed by solving the conserva- tion equations for these species (analogous to Eq. (7)below) with LeCO2¼1:39 and LeH2O¼0:83. Some results are shown by dotted curves in Fig. 9 below. As can be seen, the approximation has only a small effect on the results. The factor TK CO2þ2KH2O/C0/C1 depends only weakly on the temperature and decreases when the temper-ature increases. Since most of the radiation losses occur from the hot burnt gas below the ﬂame, where YPand T4/C0T4 uare largest, this factor is replaced by a constant equal to its value 0.25 K/ Pa m at T¼1300 K. Again, additional computations in which the temperature variation of TK CO2þ2KH2O/C0/C1 is retained show that this approximation has no important effect on the results. In most cases of interest, the wall of the tube is not transparent to the visible and infrared radiation emitted by the burnt gas. The wall absorbs and reﬂects this radiation partially (for Plexiglas, Pyr-ex or quartz tubes) or totally (for black or metallic tubes), and thus plays a role in the radiation exchange. The effect depends on the wall thickness and the optical and thermal properties of the tube material, and its analysis is beyond the scope of this work. The sim- ple model of an optically thin gas enclosed by a transparent or non- reﬂecting wall is intended to give qualitative results only. Its use relies on the following estimations. First, a coarse, conservative estimate of the temperature in- crease of the wall in a region of characteristic size R/C3around the ﬂame tip due to the absorbed radiation, DTwrsay, can be worked out assuming that all the radiation emitted by the burnt gas in this region, of order LR/C33with Levaluated at temperature Tb, is ab- sorbed by the wall, and leaving out heat losses from the outer sur-face of the tube. In a reference frame moving with the ﬂame front, in which the wall is seen to move downward with a velocity U 0 (see Fig. 1 ), an order-of-magnitude energy balance for the heated layer of the wall reads qscsU0R/C3bDTwr/C24LR/C33, where qsand csare the density and speciﬁc heat of the tube material and bis the thick- ness of the heated layer; b/C24minðdw;ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ ksR/C3=qscsU0p Þ, with dw denoting the thickness of the wall and ksits thermal conductivity. ThusDTwr=Tu/C24ðrKPT4 bR/C32Þ=ðqscsU0bTuÞ, which should be small for the cold wall condition used in this work to be realistic. For a stan- dard tube ( R/C3¼25:5 mm) with the thermal properties of Plexiglas, taking U0¼21 cm/s (see Section 3below) and b¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ ksR/C3=qscsU0p ¼ 0:11 mm, the estimate gives DTwr=Tu/C241:5/C210/C02in the condi- tions of the reference case mentioned above. Similarly, DTwr=Tu/C245:6/C210/C03for a quartz tube, and 3 :84/C210/C04for an alu- minum tube (with dwlarger than about 3 mm). The estimate of DTwris for a region of characteristic size R/C3moving with the ﬂame tip. The radiation-induced wall temperature increase may be larger than DTwrfar downstream of the ﬂame tip, but this is not expected to signiﬁcantly affect the propagation of the ﬂame.Radiative heating of the wall far upstream of the ﬂame tip isassumed to be small because the angle subtended by the burnt gas is then small. Second, the reﬂection of gas-emitted radiation in the tube wall increases the intensity of the radiation in the tube and thus leads to an increase of gas absorption ( Aabove) and a decrease of the net radiation losses, L. However, analysis of a hot isothermal gas en- closed by a cold wall [19] suggests that, for the optically thin gas envisaged here, the relative decrease of radiation losses due to multiple reﬂections is of order KPR/C3=ð1/C0qwÞ, where qwis the reﬂectivity of the wall. Since KPR/C3is small, the decrease of the radi- ation losses is signiﬁcant only for a nearly perfectly reﬂecting wall, with qwclose to unity. The effect approximately amounts to reduc- ing the Planck mean absorption coefﬁcient. Dimensionless variables are now introduced. The values of the planar ﬂamevelocityandthicknessgivenbytheasymptoticformulae (1)forthereferencecase,namely ULr¼4:63 cm/sand dLr¼0:43 mm, willbe used as units of velocityand length. When Yuis changedkeep- ingTuconstant, the parameters c;bandULchange as c cr¼Yu Yur;b br¼c crcrþ1 cþ1/C16/C172 ; Uf¼UL ULr¼cþ1 crþ1/C16/C17ðjþ3Þ=2cr cexp/C0b 2cþ1 crþ11/C0cr c/C16/C17 hi ;9 >= >;ð2ÞFig. 2. Velocity of the planar ﬂame as a function of the equivalence ratio. Triangles: experimental results of Yamaoka and Tsuji [16] corrected as in Ref. [17]. Solid: numerical results for single-step chemistry with Ta¼18;750 K. Dashed: asymp- totic ﬂame velocity (1)for this activation temperature. Circles: experimental results determined by Wang et al. [17] from their microgravity experiments. Dotted: numerical results for single step chemistry with Ta¼37;500 K (lower curve) and Ta¼52;500 K (upper curve). which give a cproportional to Yu;abthat decreases when Yuin- creases in the range of values of interest (above c¼1); and a Ufthat increases rapidly with Yu. In addition, the density is scaled with qu, the mass fractions of all the species are scaled with Yu, and the dimensionless temperature h¼T/C0Tu ðÞ =Tb/C0Tu ðÞ is used. In terms of these variables, the governing equations and boundary condi- tions in the reference frame moving with the ﬂame front, in which the solution is stationary, are $/C1qvðÞ ¼ 0; ð3Þ qv/C1$v¼/C0$pþqGiþ$/C1s0; ð4Þ qv/C1$h¼$k$hðÞ þ W/C0L; ð5Þ qv/C1$YF¼1 LeF$k$YFðÞ /C0 W; ð6Þ qv/C1$YP¼1 LeP$k$YPðÞ þ11 4W; ð7Þ W¼b2U2 f 2ðcþ1Þj/C01LeFqYFexp bðcþ1Þh/C01 1þch/C20/C21 ; ð8Þ L¼qRYP1þch ðÞ4/C01hi ; ð9Þ q1þch ðÞ ¼ 1;k¼1þch ðÞj; ð10Þ r¼R:v¼U0i;h¼@YF @r¼@YP @r¼0; ð11Þ x!/C0 1 :v¼U0i;h¼0;YF¼1;YP¼0; ð12Þ x!1:@v @x¼@h @x¼@YF @x¼@YP @x¼0; ð13Þ where dimensionless variables are denoted with the same symbols used before for their dimensional counterparts. Here ris the dimen- sionless distance from the axis of the tube and xis the dimension- less distance along the axis measured downward from an origin which is deﬁned by the condition that h0¼hð0;0Þhave a chosen constant value, typically 0.7 or 0.8; iis a unit vector pointing down- ward; and U0is the dimensionless velocity of the ﬂame front rela- tive to the tube, which is to be found as part of the solution. Eqs.(3) and (4) are the continuity and momentum equations, where G¼gd Lr=U2 Lris the dimensionless acceleration of gravity, s0¼k$vþ$vðÞThi is the non-spherical part of the viscous stress tensor, and pis the pressure of the gas augmented with the spher- ical part of the viscous stress tensor, both scaled with quU2Lr. Eqs. (5)–(7) are the energy, fuel, and product (CO 2) conservation equa- tions, with the reaction and radiation terms given by Eqs. (8) and (9). Eqs. (10) are the equation of state and the equation giving the power law temperature dependence of the dimensionless viscosity, conductivity and mass diffusivities. The boundary conditions (11)– (13), express that the wall of the tube and the fresh gas are seen to move downward with velocity U0in the reference frame tied to the ﬂame front. Here R¼R/C3=dLris the dimensionless radius of the tube. The dimensionless parameters appearing in (3)–(13) are c¼Tb/C0Tu Tu;b¼TaTb/C0Tu ðÞ T2 b;Uf¼UL ULr; G¼gdLr ULr;R¼R/C3 dLr;LeF;LeP;Pr;j; R¼2:62rRgT3 udLrYur cpcrULrTK CO2þ2KH2O/C0/C1/C2/C3 1300 K:9 >>>>>= >>>>>;ð14Þ The ﬁrst three parameters are given by (2)in terms of cr,brand Yu=Yur. Parameter Ufis artiﬁcial. It appears due to the use of ULr and dLras velocity and length scales instead of the values ULand dLgiven by (1)for the current composition of the fresh gas. The magnitudes ULrand dLrof the reference case are used for conve- nience, to facilitate comparison of results for mixtures with differ- ent fuel mass fractions in the same tube and setup. Since these magnitudes appear in most of the dimensionless parameters, allthese parameters would change when Yuis changed, should UL and dLhave been used in their deﬁnitions. Values of the dimension- less parameters used in the computations of the following sectionare cr¼4;br¼10;LeP¼Pr¼1 and j¼0:75, as mentioned above; LeF¼0:8, 1 and 1.2; G¼2 corresponding to normal gravity; R¼60 corresponding approximately to the standard ﬂammability tube; and R/C252/C210/C05. The numerical method used to solve (3)–(13) is similar to the method used in Ref. [9]. The stream function/vorticity formulation equivalent to (3) and (4) is used together with a pseudo-transient iteration which amounts to adding time derivatives to Eqs. (5)–(7) and the vorticity equation that results from (4). The equations are then rewritten in terms of the new independent variables ðn;rÞ, where n¼xþmR ðÞ =hðrÞþmR ðÞ ,mis an adjustable constant, hð0Þ¼0, and the function hðrÞand the constant U0in(11) and (12) are iteratively chosen to ensure that hðn¼1;rÞ¼h0in a region around the axis of the tube. The equations are discretized using second order ﬁnite differences in a rectangular, nonuniform grid covering the domain 0 ;n1½/C138 /C2 0;R½/C138 and ﬁner around n¼1. Typical values of the numerical parameters are m¼1 and n1¼3—6. Numerical tests show that grid independence is obtained with 360/C2120 grid points. 3. Results and discussion In agreement with experimental results and previous numerical computati ons[9,10] , the ﬂame fronts computed for very lean mix- tures consist of a round cap followed by a long skirt; see, e.g. Figs. 3, 6 and 7 below. The shape of the cap is similar to that of a bubble rising in the tube, and its velocity U0is nearly independent of the equivalence ratio and close to the velocity of a bubble given by the Davies and Taylor formula [3], UDT¼0:467c cþ1GR/C18/C191=2 ð15Þ in our dimensionless variables. Eq. (15) gives UDT¼4:58 (which amounts to 21 cm/s) for c¼4;G¼2 and R¼60. Since UDTis scaled with ULr, this relatively large value implies that the tip of the ﬂame front is conﬁned to a neighborhood of the stagnation point of the equivalent bubble ﬂow, and the rest of the ﬂame front lies near the streamline through this point. Also, insofar as the dimensionless burning velocity of the ﬂame is of the order of Ufin Eq. (2), the length of the skirt scaled with the radius of the tube is large ofthe order of U DT=Uf. 3.1. Results without radiation losses Consider ﬁrst a ﬂame front propagating upward in the absence of radiation losses ( R¼0).Figure 3 shows the reaction region, some isotherms, and some streamlines of the ﬂow relative to the ﬂame for LeF¼1;R¼60, and equivalence ratios /= 0.344, 0.504, 0.538 and 0.581, which correspond to Uf= 0.07, 0.7, 1 and 1.5. Fig- ure 4 (a) summarizes some properties of the ﬂame at its tip. Mea- suring vertical distances from the point where the reaction rateattains its maximum on the symmetry axis, x Wwhere W¼Wmax say, this ﬁgure shows, as functions of /, the distances from this point to the points where W¼0:1Wmaxon either side of the max- imum (solid lines), the distance to the point where h¼0:1 (dashed), and the distance to the point where v¼0 (dotted) when such point, which marks the upper boundary of a recirculation re-gion, exists (in Fig. 3 (a) and (b), for example). Results are shown for R¼60 and Le F= 0.8, 1 and 1.2, increasing as indicated by the ar- rows. Figure 4 (b), to be commented below, shows similar results when radiation losses are taken into account with R¼2/C210/C05. The distance between the two solid curves in Fig. 4 is a measure of the thickness of the reaction region, and the distance between the lower solid curve and the dashed curve is a measure of the thickness of the whole ﬂame at the tip. When LeF¼0:8, the ﬂame transitions from bubble-shaped to tulip-shaped at the right-handside end of the curves, for //C250:573. Similar transitions occur for Le F¼1 and 1.2 at higher values of /, not shown in the ﬁgure. As can be seen in Figs. 3 and 4 (a), the following sequence of events occurs when the ﬂame is weakened by decreasing theequivalence ratio. The downward velocity of the burnt gas relative to the ﬂame front decreases around the axis of the tube (the spac- ing of the streamlines increases); a recirculation region appears be- low the ﬂame front; the reaction region of the ﬂame thickens and approaches the recirculation region; the reaction region enters the recirculation region; the numerical method ceases to converge to a stationary solution. When the Lewis number of the fuel is larger than unity, the strain rate of the ﬂow ahead of the ﬂame weakens it at the tip (see, e.g. Williams [14]) and the sequence of events oc- curs earlier, for larger values of the equivalence ratio than whenLe F¼1. Similarly, the strain rate strengthens the ﬂame and post- pones the sequence of events to smaller values of the equivalenceratio when LeF<1. The structure of the ﬂame changes signiﬁcantly when most of the chemical reaction occurs in the recirculation region. The con- vection, conduction and reaction terms of the energy equation(5)along the axis of the tube are shown in Fig. 5 for/¼0:406 and 0.504. Convection is not important in the reaction region when/¼0:504, in which case the gas moves downward across the ﬂame, but it is of the same order as the conduction and reactionterms when /¼0:406, and even larger than the reaction term when /¼0:344 (not displayed), in which cases the gas ﬂows upward in the reaction region. This ﬂow tends to conﬁne the fuel that enters the recirculation region from above by diffusion across the dividing streamline bounding this region. The increasing relative importance of convection when / decrease s can be explained using simple order-of-magnitude estimates. The Zeldovich number bis fairly large, implying that Fig. 3. Contours of constant reaction rate ( W, solid, red), isotherms (dashed, green) and streamlines of the ﬂow relative to the ﬂame front (solid with arrows, blue) for /¼0:344 ( a), 0.504 ( b), 0.538 ( c) and 0.581 ( d), in the absence of radiation losses ( R¼0). Other parameters have their reference values mentioned below Eq. (14).I n particular, G¼2;R¼60 and LeF¼Pr¼1. (For interpretation of the