Bernard J. Matkowsky

Bio: Bernard J. Matkowsky is an academic researcher from Northwestern University. The author has contributed to research in topics: Combustion & Premixed flame. The author has an hindex of 41, co-authored 228 publications receiving 6418 citations. Previous affiliations of Bernard J. Matkowsky include Siena College & New York University.

Flames as gasdynamic discontinuities

Abstract: 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

Abstract: 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 oscillatory front is less than the velocity of the uniformly propagating plane front. diffusion equations for temperature and concentration, with the diffusion coefficient of the combustible component which limits the chemical reaction taken to be zero. In an interesting recent paper, Merzhanov, Filonenko and Borovinskaya (2) report on various new phenomena in the combustion of condensed systems. Among these phenomena they describe what they refer to as autooscillatory combustion for a mixture of 3Nb + 2B. They find that the velocity of propagation of the reaction front exhibits periodic pulsations. They also noted that the burned samples have a layered structure, normal to the front, with the number of layers equal to the number of pulsations. This work is described in the recent monographs of Novozhilov (3), and Zeldo- vich, Leypunsky and Librovich (4) in which they attribute the cause of these pulsations to the absence of diffusion. The existence of pulsating combustion fronts was first described theoretically by Shkadinsky, Khaikin and Merzhanov (5) who obtained

The Exit Problem for Randomly Perturbed Dynamical Systems

Abstract: 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 h...

A direct approach to the exit problem

Abstract: This paper considers the problem of exit for a dynamical system driven by small white noise, from the domain of attraction of a stable state. A direct singular perturbation analysis of the forward equation is presented, based on Kramers’ approach, in which the solution to the stationary Fokker–Planck equation is constructed, for a process with absorption at the boundary and a source at the attractor. In this formulation the boundary and matching conditions fully determine the uniform expansion of the solution, without resorting to “external” selection criteria for the expansion coefficients, such as variational principles or the Lagrange identity, as in our previous theory. The exit density and the mean first passage time to the boundary are calculated from the solution of the stationary Fokker–Planck equation as the probability current density and as the inverse of the total flux on the boundary, respectively. As an application, a uniform expansion is constructed for the escape rate in Kramers’ problem o...

Reaction-rate theory: fifty years after Kramers

Abstract: The calculation of rate coefficients is a discipline of nonlinear science of importance to much of physics, chemistry, engineering, and biology. Fifty years after Kramers' seminal paper on thermally activated barrier crossing, the authors report, extend, and interpret much of our current understanding relating to theories of noise-activated escape, for which many of the notable contributions are originating from the communities both of physics and of physical chemistry. Theoretical as well as numerical approaches are discussed for single- and many-dimensional metastable systems (including fields) in gases and condensed phases. The role of many-dimensional transition-state theory is contrasted with Kramers' reaction-rate theory for moderate-to-strong friction; the authors emphasize the physical situation and the close connection between unimolecular rate theory and Kramers' work for weakly damped systems. The rate theory accounting for memory friction is presented, together with a unifying theoretical approach which covers the whole regime of weak-to-moderate-to-strong friction on the same basis (turnover theory). The peculiarities of noise-activated escape in a variety of physically different metastable potential configurations is elucidated in terms of the mean-first-passage-time technique. Moreover, the role and the complexity of escape in driven systems exhibiting possibly multiple, metastable stationary nonequilibrium states is identified. At lower temperatures, quantum tunneling effects start to dominate the rate mechanism. The early quantum approaches as well as the latest quantum versions of Kramers' theory are discussed, thereby providing a description of dissipative escape events at all temperatures. In addition, an attempt is made to discuss prominent experimental work as it relates to Kramers' reaction-rate theory and to indicate the most important areas for future research in theory and experiment.

Fundamentals of Queueing Theory

Abstract: Praise for the Third Edition: "This is one of the best books available. Its excellent organizational structure allows quick reference to specific models and its clear presentation . . . solidifies the understanding of the concepts being presented."IIE Transactions on Operations EngineeringThoroughly revised and expanded to reflect the latest developments in the field, Fundamentals of Queueing Theory, Fourth Edition continues to present the basic statistical principles that are necessary to analyze the probabilistic nature of queues. Rather than presenting a narrow focus on the subject, this update illustrates the wide-reaching, fundamental concepts in queueing theory and its applications to diverse areas such as computer science, engineering, business, and operations research.This update takes a numerical approach to understanding and making probable estimations relating to queues, with a comprehensive outline of simple and more advanced queueing models. Newly featured topics of the Fourth Edition include:Retrial queuesApproximations for queueing networksNumerical inversion of transformsDetermining the appropriate number of servers to balance quality and cost of serviceEach chapter provides a self-contained presentation of key concepts and formulae, allowing readers to work with each section independently, while a summary table at the end of the book outlines the types of queues that have been discussed and their results. In addition, two new appendices have been added, discussing transforms and generating functions as well as the fundamentals of differential and difference equations. New examples are now included along with problems that incorporate QtsPlus software, which is freely available via the book's related Web site.With its accessible style and wealth of real-world examples, Fundamentals of Queueing Theory, Fourth Edition is an ideal book for courses on queueing theory at the upper-undergraduate and graduate levels. It is also a valuable resource for researchers and practitioners who analyze congestion in the fields of telecommunications, transportation, aviation, and management science.

TRANSITION PATH SAMPLING: Throwing Ropes Over Rough Mountain Passes, in the Dark

Abstract: This article reviews the concepts and methods of transition path sampling. These methods allow computational studies of rare events without requiring prior knowledge of mechanisms, reaction coordinates, and transition states. Based upon a statistical mechanics of trajectory space, they provide a perspective with which time dependent phenomena, even for systems driven far from equilibrium, can be examined with the same types of importance sampling tools that in the past have been applied so successfully to static equilibrium properties.

Noise-induced transitions

Abstract: In this paper I will deal with the effect of external random perturbations, “noise”, on chemical systems and other open nonlinear systems. As a concrete example let us consider a CSTR. This is an open system and as such subject to external constraints, namely the concentrations of the chemical species in the feed streams, the flow rate, the stirring rate, the temperature, and the incident light intensity in the case of a photochemical reaction. These external constraints characterize the state of the environment of the open system and will, in general, fluctuate more or less strongly. Such environmental fluctuations are particularly important for natural systems; here random fluctuations are always present and their amplitude is not necessarily small as in laboratory systems. In the latter systems the experimenter will of course try to minimize the effect of random perturbations, though it is impossible to eliminate noise completely. Clearly, random external noise is ubiquitous in open systems, but this fact by itself would hardly warrant a systematic study of the effects of external fluctuations. The question is whether noise is more than a mere nuisance we have to live with. Is there any hope of finding interesting physics? The intuitive, and wrong, answer would be negative: The system averages out rapid fluctuations and the only trace of external noise would be a certain fuzziness in the state of the system. Of course, if the state of the system becomes unstable, the fluctuations initiate the departure from the unstable state. Then the dynamics of the system take over and the system evolves to a new stable state.

Stochastic Differential Equations and Applications

Abstract: A stochastic process x(t), tϵI is a family of random variables x(t) defined in a measure space (Ω,ℱ) or in a probability space (Ω,ℱ P); here x(t) is either real valued or n-vector valued and I is an interval, usually [0,∞). Notice that x(t) is a function x(t,ω)), ωϵΩ.