[...]... completely consumed along with a portion of the other (the surplus reactant) If the mass fraction of the deficient component is F, the reaction can be represented by at a rate The simple models represented by (31) and (33) play a central role in much of the mathematical treatment of combustion Whether they are appropriate or not depends, to a large extent, on the aim of the analysis We shall return to this... consists of the equation of mass conservation, that of momentum, where 2, the stress tensor, is given by 1 Acknowledgment The author was supported by the National Science Foundation and the Army Research Office during the time in which this chapter was written, and the editing chores for the volume were undertaken 3 4 THE MATHEMATICS OF COMBUSTION for a Newtonian fluid with no bulk viscosity; and the equation... designed with the limit 0 —» °c in mind The early development, analogous to (51) for the thermal model, is described by 16 THE MATHEMATICS OF COMBUSTION The nature of the solution depends on the value of D If D is small, termination dominates; if D is large, branching dominates and ignition occurs The critical value of D is the smallest root of and for D greater than the critical TI, YAl and YBl all become... ticipation of the mathematical and scientific community in selection of topics for future volumes The series is being launched with several volumes in "energy mathematics" and the editorial board gratefully acknowledges the faculty and administration of the University of Wyoming for their cooperation in connection with the Special Year on Energy-Related Mathematics held in Laramie during the 1982-83... -=c, w/is the speed of the wave relative to the fresh mixture This is the flame speed, to be calculated as part of the solution For this steady one-dimensional problem mass conservation implies With one-step kinetics (N = 2) the energy and species equations then have the form where pi is a function of p, Y\ and T (the density is related to the temperature by the equation of state (4) where p is the constant... velocity Equation (21) can therefore be replaced by Moreover, from the momentum equation, spatial changes in /? satisfy the estimate 10 THE MATHEMATICS OF COMBUSTION whence Thus p has the representation and it is p0 that appears in the equation of state, the simplified energy equation and the reaction terms, p\ in the momentum equation (the small pressure gradient drives the small velocities) With... each side of a front across which there are connection conditions representing the conservation of mass and momentum If V is the speed of the front back along its normal (Fig 2), these connection conditions are Tb is the adiabatic flame temperature and W = vnf + V is the adiabatic flame speed, the speed of the front relative to the fresh gas Both Tb and W are specified quantities In the case of a stationary... curvature, like the vorticity, is a perturbation for the tube problem FIG 5 Propagation of a flame down a tube 22 THE MATHEMATICS OF COMBUSTION FIG 6 Resolution of the kinematic incompatibility: (a) flame in a tube; (b) flame tip with dead water; (c) flame tip with hydrodynamic singularity The nature of these perturbations has not been investigated and it might be useful to do so since the nature of the flow... for their talents Asymptotic methods and bifurcation theory have made a significant impact in the last decade And the stunning success of constant density, small heat release models in unravelling the mysteries of deflagration instabilities, together with other imporxi Xll PREFACE tant contributions, is convincing an ever widening circle of combustion experts that mathematics can play a vital role Mathematics. .. equations, simpler than the exact ones, is useful in unravelling these difficulties, and such an approach forms the heart of Chapter IV Chapter V discusses the behavior of finite amplitude waves in combustible gases The material in these two chapters is relevant to the important and poorly understood subject of deflagration-to-detonation transition JOHN D BUCKMASTER THE MATHEMATICS OF COMBUSTION This page . already of great interest to a broad spectrum of the scientific community. The Mathematics of Combustion is the second volume in this series. The first is The Mathematics of Reservoir. ex- perts that mathematics can play a vital role. Mathematics is having a new impact on combustion, and it is this, rather than the development of new mathematics, that makes the subject . Foundation and the Army Research Office during the time in which this chapter was written, and the editing chores for the volume were undertaken. 3 4 THE MATHEMATICS OF COMBUSTION for