The fast-time instability of diffusion flames

A diffusion flame burns with varying intensity if the pressure is varied, and the nonlinear steady-state response is typically S-shaped; that is, multiple solutions exist for some range of pressure. The physically relevant branch can only be determined by unsteady analyses, and in this paper we discuss the stability of a large class of diffusion flames when the activation energy characterizing the reaction rate is large. Specifically, we examine the evolution from the stationary solution on a time scale so short that changes are confined to the thin flame sheet where all the reaction occurs. In this region time derivatives are added to the steady state equations, which otherwise describe a balance between diffusion and chemical reaction. The stability problem is then reduced to the determination of the spectrum of Schrödinger's equation, defined on the infinite interval, with a potential that is not of one sign on this interval. In this way certain conclusions about extinction are drawn and certain past misconceptions are clarified.     

Onsager’s Conjecture Almost Everywhere in Time

Communications in Mathematical Physics - In recent works by Isett (Hölder continuous Euler flows in three dimensions with compact support in time, pp 1–173, 2012), and later by...     

Response of a burning heterogeneous propellant to small pressure disturbances

A numerical code for the simulation of heterogeneous propellant burning is used to examine the response to the pressure field of an incident acoustic wave. It is shown that there are two natural response functions, one defined by the perturbation mass flux beyond the combustion field, the other by the perturbation energy flux beyond the combustion field. The first of these plays an essential role in L* instability (a coupled propellant–rocket-chamber phenomenon), and the second in whether the acoustic wave is amplified on reflection (a phenomenon exclusive to the propellant). We show, for several choices of parameter values and propellant morphology, that the two responses are qualitatively similar, albeit they differ in magnitude by modest amounts. We show that in some cases each response displays a single maximum or peak as a function of frequency, in other cases multiple peaks are obtained. A tentative hypothesis is proposed as a predictor of multiple peaks.     

Smolder waves, smolder spots, and the genesis of tribrachial structures in smolder combustion

We examine the evolution of reverse smolder waves with edges, motivated by the existing literature on edge-flames. An advancing smolder edge proves to be of little interest, and a well-defined invariant structure does not emerge following initial transients. A retreating smolder edge also lacks invariant structure, but retreats at a reasonably well-defined speed and can lead to novel structures and novel evolutions. Thus, a transient propagating smolder spot can be generated; and a tribrachial structure evolves from this spot with a tail that has the form of a forward smolder wave, and two leading branches, fuel-lean and fuel-rich reverse smolder waves. High temperatures and high reaction rates accompany these evolutions, and it is noted that this could lead to flaming (gas-phase) combustion.     

Combustion theory and modeling

In honor of the fiftieth anniversary of the Combustion Institute, we are asked to assess accomplishments of theory in combustion over the past fifty years and prospects for the future. The title of our article is chosen to emphasize that development of theory necessarily goes hand-in-hand with specification of a model. Good conceptual models underlie successful mathematical theories. Models and theories are discussed here for deflagrations, detonations, diffusion flames, ignition, propellant combustion, and turbulent combustion. In many of these areas, the genesis of mathematical theories occurred during the past fifty years, and in all of them significant advances are anticipated in the future. Increasing interaction between theory and computation will aid this progress. We hope that, although certainly not complete in topical coverage or reference citation, the presentation may suggest useful directions for future research in combustion theory.     

Dissipative Euler Flows with Onsager‐Critical Spatial Regularity

For any ? > 0 we show the existence of continuous periodic weak solutions v of the Euler equations that do not conserve the kinetic energy and belong to the space ; namely, x ? v (x,t) is ??ε‐Hölder continuous in space at a.e. time t and the integral is finite. A well‐known open conjecture of L. Onsager claims that such solutions exist even in the class .© 2016 Wiley Periodicals, Inc.     

Interactions of flame balls

Flame ball interactions are numerically investigated in a reaction–diffusion system characterized by single-step Arrhenius kinetics and radiative heat losses. It is found that the interactions of two neighbouring flame balls are characterized by two distinct regimes – a repulsion regime and an attraction regime, depending upon the separation distance. The two regimes join at a critical separation distance, which corresponds to an unstable equilibrium state. For supercritical separation distances, the two flame balls repel and drift apart from each other; whereas for sub-critical separation distances, they move towards each other and eventually merge into a single stationary flame ball. In this connection, flame ball interactions are found to exhibit a qualitatively reverse character in comparison with the well-known van der Waals curve which characterizes intermolecular forces.