This IMA Volume in Mathematics and its Applications DYNAMICAL ISSUES IN COMBUSTION THEORY is based on the proceedings of a workshop which was an integral part of the 1989-90 IMA program on "Dynamical Systems and their Applications." The aim of this workshop was to cross-fertilize research groups working in topics of current interest in combustion dynamics and mathematical methods applicable thereto. We thank Shui-Nee Chow, Martin Golubitsky, Richard McGehee, George R. Sell, Paul Fife, Amable Liiian and Foreman Williams for organizing the meeting. We especially thank Paul Fife, Amable Liiilin and Foreman Williams for editing the proceedings. We also take this opportunity to thank those agencies whose financial support made the workshop possible: the Army Research Office, the National Science Foundation and the Office of Naval Research. Avner Friedman Willard Miller, Jr. ix PREFACE The world ofcombustion phenomena is rich in problems intriguing to the math ematical scientist. They offer challenges on several fronts: (1) modeling, which involves the elucidation of the essential features of a given phenomenon through physical insight and knowledge of experimental results, (2) devising appropriate asymptotic and computational methods, and (3) developing sound mathematical theories. Papers in the present volume, which are based on talks given at the Workshop on Dynamical Issues in Combustion Theory in November, 1989, describe how all of these challenges have been met for particular examples within a number of common combustion scenarios: reactiveshocks, low Mach number premixed reactive flow, nonpremixed phenomena, and solid propellants."

The aim of this book is to comment on, and clarify, the mathematical aspects of the theory of thermodynamics. The standard presentations of the subject are often beset by a number of obscurities associated with the words "state", "reversible", "irreversible", and "quasi-static". This book is written in the belief that such obscurities are best removed not by the formal axiomatization of thermodynamics, but by setting the theory in the wider context of a genuine field theory which incorporates the effects of heat conduction and intertia, and proving appropriate results about the governing differential equations of this field theory. Even in the simplest one-dimensional case it is a nontrivial task to carry through the details of this program, and many challenging problems remain open.

Physicists, when modelling physical systems with a large number of degrees of freedom, and statisticians, when performing data analysis, have developed their own concepts and methods for making the `best' inference. But are these methods equivalent, or not? What is the state of the art in making inferences? The physicists want answers. More: neural computation demands a clearer understanding of how neural systems make inferences; the theory of chaotic nonlinear systems as applied to time series analysis could profit from the experience already booked by the statisticians; and finally, there is a long-standing conjecture that some of the puzzles of quantum mechanics are due to our incomplete understanding of how we make inferences. Matter enough to stimulate the writing of such a book as the present one. But other considerations also arise, such as the maximum entropy method and Bayesian inference, information theory and the minimum description length. Finally, it is pointed out that an understanding of human inference may require input from psychologists. This lively debate, which is of acute current interest, is well summarized in the present work.

This volume contains twenty contributions of work, conducted since 1996 in the French- German Research Programme "Numerical Flow Simulation" of the Centre National de la Recherche Scientifique (CNRS) and the Deutsche Forschungsgemeinschaft (DFG). The main purpose of this publication is to give an overview over the work conducted in this programme, and to make the results obtained available to the pUblic. The reports are grouped under the four headings "Development of Solution Techniques", "Crystal Growth and Melts", "Flows of Reacting Gases" and "Turbulent Flows". AIl contributions to this publica- tion were reviewed by a board consisting of T. Alziary de Roquefort (Poitiers, France), P. Bontoux (Marseille, France), JA Desideri (Sophia-Antipolis, France), W. Kordulla (G6t- tingen, Germany), R. Peyret (Nice, France), R. Rannacher (Heidelberg, Germany), G. War- necke (Magdeburg, ,Germany), and the editor. The responsibility for the contents of the reports nevertheless lies with the authors. E. H. Hirschel Editor Preface The Colloquium on "Numerical Simulation of Flows", Marseille, November 21 and 22, th 1997, was the 6 Joint CNRS-DFG Colloquium organized in the frame of the French- German Research Collaboration on Computational Fluid Dynamics. This Collaborative Program was elaborated progressively since 1991, when the two major research groups were brought together: the Priority Program "Flow Simulation with Super Computers" from the DFG in Germany and the Groupement de Recherche de "Mecanique des Fluides NumCrique" (GDR MFN) from the CNRS in France.

First published in 1927 as the second edition of a 1925 original, this book was intended as a general introduction to the principles of thermodynamics for university students involved in the physical sciences. Birtwistle gives a brief overview of the history of science relating to heat and the conservation of energy as well as practical examples to back up salient points in his explanation of key ideas. This book will be of value to anyone with an interest in the history of science.

Written for researchers and advanced students the book exhibits a combination of various methods and tools required to describe the complexity of the chemical and physical behaviour of fluid surfaces. The common denominator for all the contributions presented here is the simultaneous use of concepts from surface chemistry and physics and from hydrodynamics where external force fields can be introduced. Theoretical and experimental work is equally represented. Most of the basic problems in the area of nonequilibrium multiphase systems have not yet received extensive treatment. This volume should be a reference for physicists, physico-chemists, and chemical engineers and will serve as a jumping-off point for new directions and new points of view.

The review articles in this book treat the overall nonlinear and complex behavior of nature from the viewpoint of such diverse research fields as fluid mechanics, condensed matter physics, biophysics, biochemistry, biology, and applied mathematics. Attention is focussed on a broad and comprehensive overview of recent developments and perspectives. Particular attention is given to the so-far unsolved problem of how to capture the mutual interplay between the microscopic and macroscopic dynamics that extend over various length and time scales. The book addresses researchers as well as graduate students.

This book covers a wide area of topics, from fundamental theories to industrial applications. It serves as a useful reference for everyone interested in computational modeling of partial differential equations pertinent primarily to aeronautical applications. The reader will find three survey articles on the present state of the art in numerical simulation of the transition to turbulence, in design optimization of aircraft configurations, and in turbulence modeling. These are followed by carefully selected and refereed articles on algorithms and their applications, on design methods, on grid adaption techniques, on direct numerical simulations, and on parallel computing, and much more.

Modern astronomy has been characterized by an enormous growth in data acquisition - from new technologies in telescopes, detectors, and computation. One can now compile catalogs of tens or hundreds of millions of stars or galaxies and databases from satellite-based observations are reaching terabit proportions. This wealth of data gives rise to statistical challenges not previously encountered in astronomy. This book is the result of a workshop held at Pennsylvania State University in August 1991 that brought together leading astronomers and statisticians to consider statistical challenges encountered in modern astronomical research. The chapters have all been thoroughly revised in the light of the discussions at the conference, and some of the lively discussion is recorded here as well.

Among various branches of polymer physics an important position is occupied by that vast area, which deals with the thermal behav ior and thermal properties of polymers and which is normally called the thermal physics of polymers. Historically it began when the un usual thermo-mechanical behavior of natural rubber under stretch ing, which had been discovered by Gough at the very beginning of the last century, was studied 50 years later experimentally by Joule and theoretically by Lord Kelvin. This made it possible even at that time to distinguish polymers from other subjects of physical investigations. These investigation laid down the basic principles of solving the key problem of polymer physics - rubberlike elasticity - which was solved in the middle of our century by means of the statistical thermodynamics applied to chain molecules. At approx imately the same time it was demonstrated, by using the methods of solid state physics, that the low temperature dependence of heat capacity and thermal expansivity of linear polymers should fol low dependencies different from that characteristic of nonpolymeric solids. Finally, new ideas about the structure and morphology of polymers arised at the end of the 1950s stimulated the development of new thermal methods (differential scanning calorimetry, defor mation calorimetry), which have become very powerful instruments for studying the nature of various states of polymers and the struc tural heterogeneity."

This work was begun quite some time ago at the University of Oxford during the tenure of an Overseas Scholarship of the Royal Commission for the Exhibition of 1851 and was completed at Banga lore when the author was being supported by a maintenance allowance from the CSIR Pool for unemployed scientists. It is hoped that significant developments taking place as late as the beginning of 1965 have been incorporated. The initial impetus and inspiration for the work came from Dr. K. Mendelssohn. To him and to Drs. R. W. Hill and N. E. Phillips, who went through the whole of the text, the author is obliged in more ways than one. For permission to use figures and other materials, grateful thanks are tendered to the concerned workers and institutions. The author is not so sanguine as to imagine that all technical and literary flaws have been weeded out. If others come across them, they may be charitably brought to the author's notice as proof that physics has become too vast to be comprehended by a single onlooker. E. S. RAJA GoPAL Department of Physics Indian Institute of Science Bangalore 12, India November 1965 v Contents Introduction ................................................................. .

The five contributions describe some key mathematical concepts involved in the study of complex systems and non-perturbative problems. The selection of topics is intended to cross-fertilize the various fields where complex systems theory has made an impact. The book presents specific and detailed results meant for a wide audience of researchers and students. It begins with those contributions which help to set up a general theoretical framework and ends with selected applications to the particular areas of biophysics, statistical physics, astrophysics and cosmology. It also includes an extensive bibliography. This pedagogically written text can be used as an introduction to the fundamental ideas behind complex systems theory.

This critical overview describes fluid flow driven by the thermocapillary effect in models of crystal growth. Simple models are the floating-zone technique and the open-boat technique. Basic equations, boundary layer scaling, stability analysis, pattern formation and additional buoyancy are discussed. Particular emphasis is given to the understanding of the physics of flow. This reference book gives a state-of-the-art report and reviews the literature available.

Over the last few years it has become apparent that fluid turbulence shares many common features with plasma turbulence, such as coherent structures and self-organization phenomena, passive scalar transport and anomalous diffusion. This book gathers very high level, current papers on these subjects. It is intended for scientists and researchers, lecturers and graduate students because of the review style of the papers.

This monograph treats, for the first time, major aspects of gas dynamics of nozzles from a general point of view. Its outstanding feature is the pre- sentation of the modern theory of gas flows and modern analytical and nu- merical methods, together with numerous examples of practical applications. At the same time, quite diverse physico-chemical processes, such as disso- ciation and recombination, relaxation of vibrational degrees of freedom, two-phase flows wi th phase trans formati ons and e 1 ectromagneti c i nf1 uences, are considered. The material is presented in such a way as to help the reader to use numer- ous methods and approaches, not only for the study of gas flows in nozzles, but also for the investigation of a wide variety of problems of physical gas dynamics in different areas of application. The number of applications which may benefit from the use of the methods and results presented in this book is constantly growing. Theoretical, numerical and analytical methods of physical gas dynamics of internal flows may be, and are nowadays, ap- plied to solving the problem of preventing pollution of the air basin with toxic substances. These methods make it possible to describe the formation and transformation of toxic components in the vapour generators of thermal power plants, internal combustion engines and various metallurgical instal- lations. The methods of physical gas dynamics may be used in meteorology and powder metallurgy to create ultradispersed media and predict their proper- ties.

Universality is one of the fascinating features of condensed matter physies: it is the property whereby systems of radieally different composition and structure ex- hibit similar behavior. In the mid-1960s the word entered usage to express the fact that the equations of state of several substances could be mapped onto one another near the critical point: critieal universality. Renormalization group theory in the early 1970s provided both an explanation and a sharper definition of universality. Systems with similar behavior - universality classes - correspond to the same fixed point of a renormalization group transformation. A number of brilliant con- tributions showed how the same concepts could be applied to non-thermodynamie systems, such as the statisties of self-avoiding walks or of connected clusters on a lattice. A few years later it was realized that chaotie dynamieal systems mayaiso exhibit some degree of universality, the paradigmatic example being the period doubling cascade in the iterated maps of the unit interval into itself.

Stochastic Energetics by now commonly designates the emerging field that bridges the gap between stochastic dynamical processes and thermodynamics. Triggered by the vast improvements in spatio-temporal resolution in nanotechnology, stochastic energetics develops a framework for quantifying individual realizations of a stochastic process on the mesoscopic scale of thermal fluctuations. This is needed to answer such novel questions as: Can one cool a drop of water by agitating an immersed nano-particle? How does heat flow if a Brownian particle pulls a polymer chain? Can one measure the free-energy of a system through a single realization of the associated stochastic process? This book will take the reader gradually from the basics to the applications: Part I provides the necessary background from stochastic dynamics (Langevin, master equation), Part II introduces how stochastic energetics describes such basic notions as heat and work on the mesoscopic scale, Part III details several applications, such as control and detection processes, as well as free-energy transducers. It aims in particular at researchers and graduate students working in the fields of nanoscience and technology.

Many laboratory and astrophysical plasmas show deviations from local ther modynamic equilibrium (LTE). This monograph develops non-LTE plasma spectroscopy as a kinetic theory of particles and photons, considering the radiation field as a photon gas whose distribution function (the radiation in tensity) obeys a kinetic equation (the radiative transfer equation), just as the distribution functions of particles obey kinetic equations. Such a unified ap proach provides clear insight into the physics of non-LTE plasmas. Chapter 1 treats the principle of detailed balance, of central importance for understanding the non-LTE effects in plasmas. Chapters 2, 3 deal with kinetic equations of particles and photons, respectively, followed by a chapter on the fluid description of gases with radiative interactions. Chapter 5 is devoted to the H theorem, and closes the more general first part of the book. The last two chapters deal with more specific topics. After briefly discuss ing optically thin plasmas, Chap. 6 treats non-LTE line transfer by two-level atoms, the line profile coefficients of three-level atoms, and non-Maxwellian electron distribution functions. Chapter 7 discusses topics where momentum exchange between matter and radiation is crucial: the approach to thermal equilibrium through interaction with blackbody radiation, radiative forces, and Compton scattering. A number of appendices have been added to make the book self-contained and to treat more special questions. In particular, Appendix B contains an in troductory discussion of atomic line profile coefficients."

This volume contains the proceedings of the symposium held on Friday 6 July 1990 at the University Pierre et Marie Curie (Paris VI), France, in honor of Professor Henri Cabannes on the occasion of his retirement. There were about one hundred participants from nine countries: Canada, France, Germany, Italy, Japan, Norway, Portugal, the Netherlands, and the USA. Many of his past students or his colleagues were among the participants. The twenty-six papers in this volume are written versions submitted by the authors and cover almost all the fields in which Professor Cabannes has actively worked for more than forty-five years. The papers are presented in four chapters: classical kinetic theory and fluid dynamics, discrete kinetic theory, applied fluid mechanics, and continuum mechanics. The editors would like to take this opportunity to thank the generous spon sors of the symposium: the University Pierre et Marie Curie, Commissariat a l'Energie Atomique (especially Academician R. Dautray and Dr. N. Camarcat) and Direction des Recherches et Etudes Techniques (especially Professor P. Lallemand). Many thanks are also due to all the participants for making the symposium a success. Finally, we thank Professor W. Beiglbock and his team at Springer-Verlag for producing this volume.

The term transport phenomena is used to describe processes in which mass, momentum, energy and entropy move about in matter. Advances in Transport Phenomena provide state-of-the-art expositions of major advances by theoretical, numerical and experimental studies from a molecular, microscopic, mesoscopic, macroscopic or megascopic point of view across the spectrum of transport p- nomena, from scientific enquiries to practical applications. The annual review series intends to fill the information gap between regularly published journals and university-level textbooks by providing in-depth review articles over a broader scope than in journals. The authoritative articles, contributed by international- leading scientists and practitioners, establish the state of the art, disseminate the latest research discoveries, serve as a central source of reference for fundamentals and applications of transport phenomena, and provide potential textbooks to senior undergraduate and graduate students. The series covers mass transfer, fluid mechanics, heat transfer and thermo- namics. The 2009 volume contains the four articles on biomedical, environmental and nanoscale transports. The editorial board expresses its appreciation to the c- tributing authors and reviewers who have maintained the standard associated with Advances in Transport Phenomena. We also would like to acknowledge the efforts of the staff at Springer who have made the professional and attractive pr- entation of the volume. Serial Editorial Board Editor-in-Chief Professor L. Q. Wang The University of Hong Kong, Hong Kong; lqwang@hku. hk Editors Professor A. R. Balakrishnan Indian Institute of Technology Madras, India Professor A.

Con?gurational mechanics has attracted quite a bit of attention from various - search ?elds over the recent years/decades. Having been regarded in its infancy of the early years as a somewhat obscureand almost mystic ?eld of researchthat could only be understood by a happy few of insiders with a pronounced theoretical inc- nation, con?gurational mechanics has developed by now into a versatile tool that can be applied to a variety of problems. Since the seminal works of Eshelby a general notion of con?gurational - chanics has been developed and has successfully been applied to many pr- lems involving various types of defects in continuous media. The most pro- nent application is certainly the use of con?gurational forces in fracture - chanics. However, as con?gurational mechanics is related to arbitrary mat- ial inhomogeneities it has also very successfully been applied to many ma- rials science and engineering problems such as phase transitions and inelastic deformations. Also the modeling of materials with micro-structure evolution is an important ?eld, in which con?gurational mechanics can provide a better understanding of processes going on within the material. Besides these mechanically, physically, and chemically motivated applications, ideas from con?gurational mechanics are now increasingly applied within computational mechanics.

Thermodynamicsandstatisticalphysicsstudythephysicalproperties(mec- nical, thermal, magnetic, optical, electrical, etc.) of the macroscopic system. The tasks and objects of study in thermodynamics and statistical physics are identical. However, the methods of investigationinto macroscopicsystems are di?erent. Thermodynamics is a phenomenological theory. It studies the properties of bodies, without going into the mechanism of phenomena, i.e., not taking into consideration the relation between the internal structure of substance and phenomena, it generalizes experimental results. As a result of such a g- eralization, postulates and laws of thermodynamics made their appearance. These laws make it possible to ?nd general relations between the di?erent properties of macroscopic systems and the physical events occurring in them. Statisticalphysicsisa microscopic theory.Onthebasisoftheknowledgeof the type of particles a system consists of, the nature of their interaction, and thelawsofmotionoftheseparticlesissuingfromtheconstructionofsubstance, it explains the properties being observedon experiment, and predicts the new properties of systems. Using the laws of classical or quantum mechanics, and alsothe theoryofprobability, itestablishesqualitativelynewstatistical app- priatenesses of the physical properties of macroscopic systems, substantiates the laws of thermodynamics, determines the limits of their applicability, gives the statistical interpretation of thermodynamic parameters, and also works out methods of calculations of their means. The Gibbs method is based on statisticalphysics.Thismethodis themostcanonical.Therefore, inthis book, the exposition of the Gibbs method takes an important pla

The present book treats the science of heat conduction to an extent to which it can be taught in the specialized departments of Mechanical, Chemical or Electrical Engineering at a German Engineering University. No special prerequisites are assumed, and the mathematical methods employed draw, essentially, on the content of a normal curriculum in the depart ments mentioned above. The book is intended for adoption in conjunction with a standard lecture course or as an aid to review before examinations. It should also be found helpful to a practicing engineer in solving problems in heat conduction. As far as the readers are concerned, the book should, above all, show that exact and approximate solutions to answer questions which arise in a very large number of important, practical applications are at their disposal. Further, the book will show that in many cases it is possible to perform first estimates in a very elementary manner before engaging in the derivation of complicated analytic solutions. It is in this way that we utilize the past results of the great mathematicians of earlier generations who have bequeathed to us a considerable stock of methods and solutions. The application of such methods is illustrated in this book with the aid of examples drawn from various branches of science and technology. In this manner, the wide field of applicability of heat transfer will be made clear."

Addressing graduate students and researchers in physics and mathematics, this book fills a gap in the literature. It is an introduction into modern constructive physics, field theory and statistical mechanics and a survey on the most recent research in this field. It presents the main technical tools such as cluster expansion and their implementation in the rigorous renormalization group, and studies physical models in some detail. The reader will find a study of the ultraviolet limit of the Gross-Neveu model, of continuous symmetry breaking and of self-avoiding random walks in statistical mechanics, as well as applications to solid-state physics. Mathematicians will find constructive methods useful for studies in partial differential equations.