Bayliss, Alvin; Lennon, Erin M.; Tanzy, Matthew C.; Volpert, Vladimir A. Solution of adiabatic and nonadiabatic combustion problems using step-function reaction models. (English) Zbl 1294.76274 J. Eng. Math. 79, 101-124 (2013). Summary: We consider the use of step-functions to model Arrhenius reaction terms for traveling wave solutions to combustion problems. We develop a methodology by which the Arrhenius reaction rate term is replaced by a suitably normalized step-function. The resulting model introduces interior interfaces and allows the conservation equations for energy and species to be solved explicitly within the subdomains bounded by the interfaces. The problem can then be reduced to a small number of nonlinear algebraic equations governing appropriate interface conditions. We apply this methodology to a variety of single-reaction problems and show that the resulting solutions agree with those obtained by the well-known front \(\delta\)-function) approximations for large Zeldovich numbers. We then consider multiple reaction problems, specifically problems involving two independent reactions and problems involving sequential reactions. For these problems we compare the results with simpler front models as well as with Arrhenius kinetics. We show that the step-function models are generally superior to the front models where available and agree, both qualitatively and with reasonable quantitative accuracy, with solutions obtained via full Arrhenius kinetics. Cited in 3 Documents MSC: 76V05 Reaction effects in flows 80A25 Combustion 80A32 Chemically reacting flows Keywords:combustion; traveling waves; multiple reactions PDFBibTeX XMLCite \textit{A. Bayliss} et al., J. Eng. Math. 79, 101--124 (2013; Zbl 1294.76274) Full Text: DOI References: [1] Buckmaster J, Ludford G (1982) Theory of laminar flames. Cambridge University Press, Cambridge · Zbl 0557.76001 [2] Buckmaster J, Ludford G (1983) Lectures on mathematical combustion. Society for Industrial Mathematics · Zbl 0574.76001 [3] Williams FA (1985) Combustion theory. Benjamin Cummings, Menlo Park [4] Zeldovich YB, Barenblatt GI, Librovich VB, Makhviladze GM (1985) The mathematical theory of combustion and explosion. Consultants Bureau, New York [5] Merzhanov AG, Khaikin BI (1988) Theory of combustion waves in homogeneous media. Prog Energy Combust Sci 14(1): 1–98 [6] Zeldovich Y, Frank-Kamenetsky D (1938) A theory of thermal propagation of flame. Acta Physicochim USSR 9: 341–350 [7] Matkowsky BJ, Sivashinsky GI (1978) Propagation of a pulsating reaction front in solid fuel combustion. SIAM J Appl Math 35(3): 465–478 · Zbl 0404.76074 [8] Ferziger JH, Echekki T (1993) A simplified reaction rate model and its application to the analysis of premixed flames. Combust Sci Technol 89(5): 293–315 [9] Goldfeder PM, Volpert VA, Ilyashenko VM, Khan A, Pojman JA, Solovyov SE (1997) Mathematical modeling of free radical polymerization fronts. J Phys Chem 101(18): 3474–3482 [10] Goldfeder PM, Volpert VA (1998) Nonadiabatic frontal polymerization. J Eng Math 34(3): 301–318 · Zbl 0952.76091 [11] Goldfeder PM, Volpert VA (1998) A model of frontal polymerization including the gel effect. Math Probl Eng 4(5): 377–391 · Zbl 0982.82505 [12] Goldfeder PM, Volpert VA (1999) A model of frontal polymerization using complex initiation. Math Probl Eng 5(2): 139–160 · Zbl 0978.80002 [13] Golovaty D (2006) On step-function reaction kinetics model in the absence of material diffusion. SIAM J Appl Math 67(3): 792–809 · Zbl 1117.35316 [14] Perry MF, Volpert VA (2004) Self-propagating free-radical binary frontal polymerization. J Eng Math 49(4): 359–372 · Zbl 1155.80302 [15] Merzhanov A (2003) Combustion and explosion processes in physical chemistry and technology of inorganic materials. Russian Chem Rev 72: 289–310 [16] Abramowitz M, Stegun I (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables. 10th edn. U.S. Government Printing Office. U.S. Government Printing Office, Washington, DC · Zbl 0543.33001 [17] Volpert VA (1997) Combustion waves with wide reaction zones. Appl Math Lett 10: 59–64 · Zbl 0890.34019 [18] Volpert VA, Volpert VA (1991) Propagation velocity estimation for condensed phase combustion. SIAM J Appl Math 51(4): 1074–1089 · Zbl 0791.34022 [19] Zeldovich Y (1941) Theory of limit of slow flame propagation. Zh. Eksperimentalnoy i Teoreticheskoi Fiziki 11: 159 [20] Khaikin BI, Filonenko AK, Khudyaev SI (1968) Flame propagation in presence of two successive gas-phase reactions. Combust Explos Shock Waves 4(4): 591–599 [21] Khaikin BI, Filonenko AK, Khudyaev SI, Martemyanova TM (1973) Stagewise combustion of nonvolatile easily dispersed substances. Combust Explos Shock Waves 9(2): 169–185 [22] Nekrasov E, Timokhin A (1986) Theory of thermal wave propagation of multistage reactions described by simple empirical schemes. Combust Explos Shock Waves 22(4): 48–55 [23] Volpert VA, Khaikin BI, Khudyaev SI (1981) Combustion waves with independent reactions. In: Problems of Technological Combustion. Institute of Chemical Physics, Chernogolovka, pp. 110–113 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.