Traveling wave solutions to combustion models and their singular limits.

*(English)*Zbl 0596.76096The authors investigate the problem of a low-speed travelling deflagration wave moving with constant velocity V in a non-isothermal compressible reacting gas consisting of two chemical species with a single-step chemical reaction. The locus of the wave is determined by the ignition temperature. According to classical approximations in the theory of combustion, the system of equations of balance reduces to the energy equation for the temperature T and the species continuity equation for the mass fraction Y of the reactant.

In a frame moving with V, two coupled ordinary differential equations of the second order each are obtained. These equations are nonlinear due to the coefficient \(\lambda\) of thermal conductivity depending on T, the diffusion coefficient \(\mu\) depending on T and Y, and the Arrhenius reaction function depending on T. The specific heat \(c_ p\) is assumed constant. The system degenerates to one differential equation of the second order if \(Le=1\) where \(Le:=\lambda /\mu c_ p\) is the Lewis- number. Boundary conditions of the first kind for T and Y are prescribed as \(x\to \pm \infty.\)

The authors also consider the highly activated energy limit as a parameter \(\epsilon\) goes to zero. In this limit, the Arrhenius function approaches a Dirac delta function centered at the locus of the wave. For both the cases of \(Le=1\) and Le\(\neq 1\), the nonlinear boundary value problems depend on a free parameter c representing the mass flux of reactant and product. For \(Le=1\), existence and uniqueness are shown for the eigenvalue of c and the corresponding solution of the differential equation; for this purpose, shooting is used.

Via energy estimates, an asymptotic analysis is executed for \(\epsilon\) \(\to 0\) which yields a free boundary value problem in the limit. An approximation by use of a truncated finite domain [-a,a] is discussed. For Le\(\neq 1\), and a confinement to [-a,a], existence is shown by means of energy estimates, a fixed point representation, and the Leray-Schauder theory. Existence then is shown in the limit as \(a\to \infty\). By use of the construction of bounds, the limit as \(\epsilon\) \(\to 0\) is investigated, which yields a free boundary value problem. This analysis of the case of Le\(\neq 1\) is extended to a discussion of single-step nth order reactions.

In a frame moving with V, two coupled ordinary differential equations of the second order each are obtained. These equations are nonlinear due to the coefficient \(\lambda\) of thermal conductivity depending on T, the diffusion coefficient \(\mu\) depending on T and Y, and the Arrhenius reaction function depending on T. The specific heat \(c_ p\) is assumed constant. The system degenerates to one differential equation of the second order if \(Le=1\) where \(Le:=\lambda /\mu c_ p\) is the Lewis- number. Boundary conditions of the first kind for T and Y are prescribed as \(x\to \pm \infty.\)

The authors also consider the highly activated energy limit as a parameter \(\epsilon\) goes to zero. In this limit, the Arrhenius function approaches a Dirac delta function centered at the locus of the wave. For both the cases of \(Le=1\) and Le\(\neq 1\), the nonlinear boundary value problems depend on a free parameter c representing the mass flux of reactant and product. For \(Le=1\), existence and uniqueness are shown for the eigenvalue of c and the corresponding solution of the differential equation; for this purpose, shooting is used.

Via energy estimates, an asymptotic analysis is executed for \(\epsilon\) \(\to 0\) which yields a free boundary value problem in the limit. An approximation by use of a truncated finite domain [-a,a] is discussed. For Le\(\neq 1\), and a confinement to [-a,a], existence is shown by means of energy estimates, a fixed point representation, and the Leray-Schauder theory. Existence then is shown in the limit as \(a\to \infty\). By use of the construction of bounds, the limit as \(\epsilon\) \(\to 0\) is investigated, which yields a free boundary value problem. This analysis of the case of Le\(\neq 1\) is extended to a discussion of single-step nth order reactions.

Reviewer: E.Adams

##### MSC:

76R99 | Diffusion and convection |

80A99 | Thermodynamics and heat transfer |

34B15 | Nonlinear boundary value problems for ordinary differential equations |