We consider the problem of thermal explosion of a gas mixture in the case of an autocatalytic combustion reaction in a homogeneous medium.   We determine the maximal temperature on the trajectories located in the transition region between the slow combustion regime and the explosive one

Schneider, Klaus R.

Shchepakina, Elena A.

English

Publications Server of the Weierstrass Institute for Applied Analysis and Stochastics

Weierstra-Institutfur Angewandte Analysis und Stochastikim Forschungsverbund Berlin e.V.Preprint ISSN 0946 { 8633Maximal temperature of safe combustion in caseof an autocatalytic reactionKlaus R. Schneider1, Elena A. Shchepakina2submitted: 19th November 20031 Weierstrass Institute for AppliedAnalysis and Stochastics,Mohrenstrae 39,10117 BerlinGermanyE-Mail: schneider@wias-berlin.de2 Department of Dierential Equationsand Control TheorySamara State UniversityP.O.B. 10902Samara (443099)RussiaE-Mail: shchepakina@yahoo.comNo. 890Berlin 2003W I A S2000 Mathematics Subject Classication. 34E15, 80A25.Key words and phrases. singular perturbations, canards, combustion.Edited byWeierstra-Institut fur Angewandte Analysis und Stochastik (WIAS)Mohrenstrae 3910117 BerlinGermanyFax: + 49 30 2044975E-Mail: preprint@wias-berlin.deWorld Wide Web: http://www.wias-berlin.de/AbstractWe consider the problem of thermal explosion of a gas mixture in the caseof an autocatalytic combustion reaction in a homogeneous medium. We de-termine the maximal temperature on the trajectories located in the transitionregion between the slow combustion regime and the explosive one.1 IntroductionWe consider the problem of thermal explosion of a gas mixture in case of an auto-catalytic combustion reaction in a homogeneous medium. As a mathematical modelwe use the following dierential system proposed in [1, 2]"ddt= (1  )e   ;ddt= (1  )e:(1.1)Here,  denotes the temperature,  is the depth of conversion of the gas mixture,  describes the volumetric heat loss, and " is a positive parameter which is smallin case of a highly exothermic reaction.The chemically relevant phase space P of system (1.1) is dened by P := f(; ) 2R2 :   0; 0    1g. As " is small, (1.1) represents a singularly perturbed systemof autonomous dierential equations. A qualitative investigation of this system canbe found in [1, 2, 3, 4]. In what follows we recall the main results of these studies.Proposition 1.1 Consider system (1.1) in P for   0; " > 0. Then(i) The region  := f(; ) 2 R2 : 0    1; 0    ="g is positively invariant.(ii)  contains exactly two equilibria: the point P = (0; 1) is a stable node, theorigin O is a saddle, where one separatrix tanges the -axis at the origin, whilethe other one tanges the straight line  = ( + ").(iii) A trajectory of (1.1) starting at any point in  dierent from the origin tendsfor increasing t to the equilibrium P .1The degenerate equation to (1.1) reads0 = (1  )  e : (1.2)Its solution set is called the slow manifold S of (1.1). It consists of equilibria of theassociated dierential equationdd= (1  )  e : (1.3)The following gures represent S for dierent values of .1P PPO1O O1 θθθηηηSsαSuαSs,2αSu,2αSs,1α Su,1αSs,2αSu,2αSu,1αSs,1αFigure 1:  > e=4 Figure 2:  = e=4 Figure 3:  < e=4Figure 1 shows S for some  > e=4 in P . In that case, S consists exactly oftwo dierent curves Ss and Su which represent asymptotically stable and unstableequilibria of the associated equation, respectively. For " suÆciently small, system(1.1) has an attracting invariant manifold Ss;" near Ss and a repelling invariantmanifold Su;" near Su, respectively.To study the behavior of (1.1) for small ", we consider to (1.1) the initial valueproblem(0; ") = 0; (0; ") = 0; 0 < 0 < 1=2: (1.4)Since the initial point (0; 0) belongs to the basin of attraction of the set Ss, aftersome short transition period, the solution of (1.1), (1.4) follows the attracting slowinvariant manifold Ss;" and tends to the equilibrium P = (0; 1) as t!1. We callthis behavior the slow combustion regime.Figure 3 shows S for some  < e=4 in P. As in the case  > e=4, S consistsexactly of two dierent curves S1 and S2. But dierent from the case consideredbefore, each of these curves contains stable equilibria of the associated equation(1.3) (namely, for  < 1, denoted by Ss;1 and Ss;2 ) and unstable equilibria of (1.3)2(for  > 1, denoted by Su;1 and Su;2 ). For  < e=4 and " suÆciently small, system(1.1) has local attracting invariant manifolds Ss;1;" near Ss;1 and Ss;2;" near Ss;2 andrepelling local invariant manifolds Su;1;" near Su;1 and Su;2;" near Su;2 , respectively. Inthat case, we have the following behavior of the solution of the initial value problem(1.1), (1.4): after some short transition period, the solution follows the component ofSs;1;" until it reaches the value  = 1. After this moment, (t) will increase very fast,but the solution remains inside  and ultimately tends to the stable equilibrium P .This behavior characterizes the explosive regime.Figure 2 depictures the special case  = e=4. Here, the slow manifold S can beconsidered to consist of four branches Su;ie=4; Ss;ie=4; i = 1; 2: The branch Ss;1e=4 has therepresentationSs;1e=4 := f(; ) 2 P :  = s1() := 0:5(1 q1  e1 ); 0    1g: (1.5)If we denote by  s1() the inverse function of s1(); then Ss;1e=4 can be represented alsoin the formSs;1e=4 := f(; ) 2 P :  =  s1(); 0    0:5g: (1.6)Analogously, the branch Su;2e=4 can be represented asSu;2e=4 := f(; ) 2 P :  = u2 () := 0:5(1 +q1  e1 );   1g == f(; ) 2 P :  =  u2 ();12  < 1g; (1.7)where  u2 () is the inverse function of u2 ().For  = e=4 and " suÆciently small, system (1.1) possesses the local attractinginvariant manifolds Ss;1;" and Ss;2;" near Ss;1 and Ss;2 respectively and the repellinglocal invariant manifolds Su;1;" and Su;2;" near Su;1 and Su;2 , respectively.If we study the initial value problem (1.1), (1.4), then to given small " and for exponentially near e=4 but less than e=4 we can observe the existence of canardsolutions which describe the transition between the slow combustion regime andthe explosive regime and which exhibits the phenomenon of delayed exchange ofstabilities. That means, to given small ", there is an exponentially small -interval(e("); c(")) containing ("), where(") = 0 + 1"+ 2"2 +O("3); 0 = e=4; 1 =  e=p2; 2 = 49e=36; (1.8)such that for  > c(") ( < e(")) the solution of (1.4) belongs to the slow regime(explosive regime). The interval (e("); c(")) characterizes the critical regime, thatis, for  2 (e("); c(")), after some short transition time, the solution of (1.1),(1.4) follows the attracting invariant manifold Ss;1;" until it reaches the value  = 1.After this moment, it stays near the unstable component Su;2;" which is located inthe region  > 12, up to some point J from which the solution "jumps" towards theattracting manifold Ss;2;" and follows this manifold approaching P as t ! 1 (seeFigure 4).30.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50.000.100.200.300.400.500.600.700.800.901.00Figure 4: Canard trajectories of system (1.1) for " = 0:05, 0 = 0:659941603;00 = 0:659941646; 000 = 0:659952218In the sequel, we are interested in the determination of the jump-o point J , andtherefore the maximal temperature on a canard trajectory as a function of the initialpoint (0; 0). For this purpose, in the following section we present an estimate ofthe delay of exchange of stabilities in scalar singularly perturbed equations [6].2 Delayed exchange of stabilities in scalarnon-autonomous dierential equationsWe consider the scalar singularly perturbed dierential equation"dud= g(u; ; ") (2.1)and study the initial value problemu(0; ") = u0;  2 I := f 2 R : 0 <  < 1g (2.2)for suÆciently small ".If we set " = 0 in (2.1) we get the degenerate equationg(u; ; 0) = 0: (2.3)If this equation has a simple isolated root u =  () which is a stable equilibrium ofthe associated equation to (2.1)dud= g(u; ; 0); (2.4)4and if u0 belongs to region of attraction of  (0), then the asymptotic behaviorof the solution of the initial value problem (2.1), (2.2) is uniquely determined bythe standard theory of singularly perturbed systems (see, e.g., [7]). In what followswe consider this initial value problem in case of exchange of stabilities, that is, weassume that the degenerate equation has two intersecting solutions. Here, we haveto distinguish two dierent situations: immediate exchange of stabilities ([5]) anddelayed exchange of stabilities ([6]). The last case is related to the existence ofa canard trajectory. We recall now a result concerning the delayed exchange ofstabilities in case of transversal bifurcation.Let U be an open bounded interval containing the origin, I"0 is the open intervaldened by I"0 := f" 2 R : 0 < " < "0g, "0 > 0:We consider the initial value problem (2.1), (2.2) under the following assumptions:(A1): g : U  I  I"0 ! R is continuous and twice continuously dierentiable withrespect to u and ".(A2): g(0; ; ")  0 for (; ") 2 I  I"0 (I means the closure of I).From (A1) and (A2) it follows that a solution of (2.1) starting at u = u0 remainspositive (negative) if u0 is positive (negative). In the sequel we restrict ourselves tothe case u0 < 0. We denote by U  the set dened by U  := fu 2 U : u  0g.(A3): The solution set of the degenerate equation g(u; ; 0) = 0 in U   I con-sists of the two curves u  0 and u =   () where    belongs to the classC1([c; 1]; R ) and satises   (c) = 0;   () < 0 for  2 (c; 1] (see Figure5).η1 ηη0 ηcuu = 0u = ψ−(η)Figure 5: Solution set of g(u; ; 0) = 0(A4):gu(0; ; 0)(< 0 for  2 [0; c);> 0 for  2 (c; 1]:5Assumption (A4) implies that u = 0 is an equilibrium point of the associated equa-tion (2.4) which is exponentially stable for  2 [0; c) and unstable for  2 (c; 1]:LetG(; 0; ") :=Z 0gu(0; s; ")ds: (2.5)From assumption (A4) we get that G(; 0; 0) = 0 has at most one root  =  in(0; 1). Therefore, we assume(A5): G(; 0; 0) = 0 has a root  in (0; 1).It is easy to see that  is such that it holds> c; G0(; 0; 0) > 0: (2.6)The following assumption on the function g is fullled if the second derivative of gwith respect to u at u = 0 is positive for all (; ") under consideration.(A6): There are suÆciently small positive numbers c0 and "0, such that [ c0; c0] 2 Uandg(u; ; ")  gu(0; ; ")u for 0    ; " 2 I"0;  c0  u  0:The following result about the delayed exchange of stabilities has been proved in[6].Theorem 2.1 Assume the hypotheses (A1) - (A6) to be valid. Then for suÆcientlysmall " and u0 < 0 there exists a unique solution of (2.1), (2.2) satisfyinglim"!0u(; ") =(0 for  2 [0; );  () for  2 (; 1]:3 Maximal temperature of combustionWe return to the combustion model"ddt= (1  )e   ;ddt= (1  )e:(3.1)As we mentioned in the introduction, for small ", the transition from a slow combus-tion regime to an explosive regime takes place when  is exponentially near (")but less than e=4. It is characterized by the existence of a canard trajectory C;".6From the second equation of (3.1) it follows that we can represent any solution ofsystem (3.1) located in the region 0 <  < 1 in the form  = ~'(; "), where ~'(; ")satises the dierential equation"dd=(1  )e   (1  )e: (3.2)Our goal is for  = (") and suÆciently small " to estimate the maximal temper-ature "max on the canard trajectory C" := f(; ) 2 R2 :  = '(; 0; ")g of system(3.2), which starts at the given initial point ( = 0;  = 0 = 0).It follows from (3.2) that for 0 <  < 1 all extrema of any solution  = ~'(; ") of thisdierential equation are located on the curve S dened in (1.2). By (1.8) it holds(") < e=4 for suÆciently small ". Since any canard solution of (3.2) is bounded,we can conclude that the maximal temperature is determined by the intersection ofthe canard solution with the curve Su;2(") which is located in the region  > 0:5. Atthe same time we get from (3.2) that any canard solution has only one maximumin that region. This follows from the fact that if we calculate d2 ~'d2on S(") we getd2 ~'d2=(1  2)e ~'"(1  )e ~':Hence, in the region  > 0:5 this derivative has negative sign, and we have always amaximum. Therefore, we can conclude that the jumping point J is located on thecurve Su;2("), and that "max uniquely determines its position on that curve.Our idea to estimate the maximal temperature on the canard trajectory C" of thedierential equation"dd=(1  )e   (")(1  )e(3.3)for suÆciently small " is to apply Theorem 2.1. To do this we use the coordinatetransformation = '(; ") + u; (3.4)where '(; ") is the canard trajectory " of (3.3) satisfyinglim!0'(; ") = 0:Since the origin is a saddle point of system (3.1) we can conclude that the canardtrajectory describes the separatrix " of system (3.1) entering the positive orthantfor increasing t.If " tends to zero, then the canard trajectory " tends to a discontinuous curve 0consisting of Ss;1e=4, of the part of Su;2e=4 bounded by the jump{o point J( = ;  = )7and the point ( = 1;  = 12) and of the part of Ss;2e=4 located in the region <   1)(see Fig. 4). In what follows we restrict our investigation on the region 0 <  < ,where 0 has the representation0 =( =  s1() for 0 <   0:5; =  u2 () for 0:5   < :(3.5)Applying the coordinate transformation (3.4) we get from (3.3) the dierential equa-tion"dud=(")(1  )e '(;") '(; ")(eu   1)  ueu! g(u; ; "): (3.6)Now, we want to verify that g(u; ; ") satises all assumptions of Theorem 2.1 foru  0; 0 <   ; " suÆciently small.It is easy to see that g is smooth in the region u 2 R; 0   < ; 0  "  "0and obeys g(0; ; ")  0 for all  and " under consideration. Thus, g satises theconditions (A1), (A2) of Theorem 2.1 for " > 0.To check hypothesis (A3) we note that the degenerate equation g(u; ; 0) = 0 isequivalent to'(; 0)((eu   1)  u) = 0: (3.7)If we represent the solutions of the degenerate equation (1.2) in the form = '1();  = '2()and if we set '(; 0) = '1(), then it can be veried that u = '2()   '1() is asolution of (3.7) intersecting in ( = 1;  = 0:5).OCθ0J1S e/4S e/4S e/4s,1u,2u,1S e/4s,2ηFigure 6: Limit position C0 of the canard trajectory C"8Next we determine the stability of the branch u = 0 of the solution set of g(u; ; 0) =0.From (1.2), (1.5){(1.7) and (3.6) we obtain@g@u ju=0;"=0e( s1()  1)4(1  )e s1()= 1  ( s1) 1() < 0 for 0 <  < 0:5;@g@u ju=0;"=0e( u2 ()  1)4(1  )e u2 ()= 1  ( u2 ) 1() > 0 for 0:5 <  < :(3.8)That means, assumption (A4) is satised.In order to verify the assumption (A5) we consider the functionG(; 0; 0) :=Z 0gu(0; s; 0)ds:From (3.8) we getG(; 0; 0) =Z 0:501  ( s1) 1(s)ds+Z 0:51  ( u2 ) 1(s)ds:We have by (1.5){(1.7)G(; 0; 0) =Z 10(1   1)(s1)0()d +Z 1(1   1)(u2 )0()d := ~G(; 0; 0); (3.9)where 0 is the root of the equation0(1  0)e0  e40 = 0:Since(1   1)(s1)0() =  (1  )2e 14p1  e1 < 0 for 0 <  < 1and(1   1)(s1)0() =(1  )2e 14p1  e1 > 0 for  > 1;there is a number 0, 0 < 0 < 1 such that to 0 2 [0; 1] there is a  = (0) suchthat ~G(; 0; 0) = 0. Thus, hypothesis (A5) is valid.Assumption (A6) is fullled if guu(0; ; ") is positive for 0     and for suÆ-ciently small ". From (3.6) we getguu(0; ; ") =(")(1  )e '(;")(2  '(; "));that is hypothesis (A6) is fullled as long as '(; ") < 2.9θ11.21.41.61.820.6 0.7 0.8 0.9 10∗θFigure 7: Dependence of maximal temperature on 0From (3.9) we get~G(; 0; 0) =121 1(1 +q1  e1 ) 121 10(1 q1  0e1 0)  12"Z 101 p1  e1 2d +Z 11 +p1  e1 2d#= 0:Thus, the equation121 1(1 +q1  e1 ) 1 100 +121 10++12Z 10p1  e1 2d  12Z 1p1  e1 2d = 0determines the point (; ) characterizing the jumping from the slow manifold,where =12+ (1 +q1  e1 ):Theorem 3.1 Consider system (1.1) for  = (") and " suÆciently small. Thenthe maximal temperature "max on the canard trajectory C" starting at the point ( =0;  = 0) with 0 < 0 < 0:5 satiseslim"!0"max = (0):10References[1] B.F. Gray, Critical behaviour in chemical reacting systems: II. An exactlysoluble model. Combust. Flame 21, 319{325 (1973).[2] G. Gorelov, V. Sobolev, Mathematical modeling of critical phenomena inthermal explosion theory, Combust. Flame, 87, 203{210 (1991).[3] G. Gorelov, V.A. Sobolev, Duck-trajectories in a thermal explosion prob-lem, Appl. Math. Lett. 5(6), 3-6 (1992).[4] G. Gorelov, V.A. Sobolev, E.A. Shchepakina, The singularly perturbedmodels of combustion (in Russian). Russian Academy of Natural Sciences,SamWien, Samara, 1999.[5] N.N. Nefedov, K.R. Schneider, Immediate exchange of stabilities in sin-gularly perturbed systems, Di. Int. Equs. 12, 583{599 (1999).[6] N.N. Nefedov, K.R. Schneider,On immediate-delayed exchange of stabil-ities and periodic forced canards, Preprint 872 of the Weierstrass Institute forApplied Mathematics and Stochastics, Berlin 2003.[7] A.B. Vasil'eva, V.F. Butuzov, L.V. Kalachev, The Boundary FunctionMethod for Singular Perturbation Problems, SIAM Studies in Applied Mathe-matics, Philadelphia, 1995.11

Maximal temperature of safe combustion in case of an autocatalytic reaction

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Maximal temperature of safe combustion in case of an autocatalytic reaction

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