B. J. Matkowsky

Flames as gasdynamic discontinuities

Early treatments of flames as gasdynamic discontinuities in a fluid flow are based on several hypotheses and/or on phenomenological assumptions. The simplest and earliest of such analyses, by Landau and by Darrieus prescribed the flame speed to be constant. Thus, in their analysis they ignored the structure of the flame, i.e. the details of chemical reactions, and transport processes. Employing this model to study the stability of a plane flame, they concluded that plane flames are unconditionally unstable. Yet plane flames are observed in the laboratory. To overcome this difficulty, others have attempted to improve on this model, generally through phenomenological assumptions to replace the assumption of constant velocity. In the present work we take flame structure into account and derive an equation for the propagation of the discontinuity surface for arbitrary flame shapes in general fluid flows. The structure of the flame is considered to consist of a boundary layer in which the chemical reactions occur, located inside another boundary layer in which transport processes dominate. We employ the method of matched asymptotic expansions to obtain an equation for the evolution of the shape and location of the flame front. Matching the boundary-layer solutions to the outer gasdynamic flow, we derive the appropriate jump conditions across the front. We also derive an equation for the vorticity produced in the flame, and briefly discuss the stability of a plane flame, obtaining corrections to the formula of Landau and Darrieus.

PROPAGATION OF A PULSATING REACTION FRONT IN SOLID FUEL COMBUSTION

We consider a system of reaction diffusion equations which describe gasless combustion of condensed systems. To analytically describe recent experimental results, we show that a solution exhibiting a periodically pulsating, propagating reaction front arises as a Hopf bifurcation from a solution describing a uniformly propagating front. The bifurcation parameter is the product of a nondimensional activation energy and a factor which is a measure of the difference between the nondimensionalized temperatures of unburned propellant and the combustion products. We show that the uniformly propagating plant front is stable for parameter values below the critical value. Above the critical value the plane front becomes unstable and perturbations of the system evolve to the bifurcated state, i.e., to the pulsating propagating state. In our nonlinear analysis we calculate the amplitude, frequency and velocity of the propagating pulsating front. In addition we demonstrate analytically that the mean velocity of the os...

The Exit Problem for Randomly Perturbed Dynamical Systems

The cumulative effect on dynamical systems, of even very small random perturbations, may be considerable after sufficiently long times. For example, even if the corresponding deterministic system has an asymptotically stable equilibrium point, random effects can cause the trajectories of the system to leave any bounded domain with probability one. In this paper we consider the effect of small random perturbations of the type referred to as Gaussian white noise, on a (deterministic) dynamical system

Two routes to chaos in condensed phase combustion

\dot x = b(x)

A direct approach to the exit problem

. The vector

Singular Perturbations of Bifurcations

x(t)

Forced forward smolder combustion

then becomes a stochastic process

Fronts, relaxation oscillations, and period doubling in solid fuel combustion

x_\varepsilon (t)

Bifurcation of Pulsating and Spinning Reaction Fronts in Condensed Two-Phase Combustion

which satisfies the stochastic differential equation

Maximal energy accumulation in a superadiabatic filtration combustion wave

dx_\varepsilon = b(x_\varepsilon )dt + \varepsilon \sigma (x_\varepsilon )dw