The papers included in this proceedings volume are mostly original research papers, dealing with life-span of waves, nonlinear interaction of waves, and various applications to fluid mechanics.

From the preface: Fluid dynamics is an excellent example of how recent advances in computational tools and techniques permit the rapid advance of basic and applied science. The development of computational fluid dynamics (CFD) has opened new areas of research and has significantly supplemented information available from experimental measurements. Scientific computing is directly responsible for such recent developments as the secondary instability theory of transition to turbulence, dynamical systems analyses of routes to chaos, ideas on the geometry of turbulence, direct simulations of turbulence, three-dimensional full-aircraft flow analyses, and so on. We believe that CFD has already achieved a status in the tool-kit of fluid mechanicians equal to that of the classical scientific techniques of mathematical analysis and laboratory experiment.

"I do not think at all that I am able to present here any procedure of investiga tion that was not perceived long ago by all men of talent; and I do not promise at all that you can find here anything_ quite new of this kind. But I shall take pains to state in clear words the pules and ways of investigation which are followed by ahle men, who in most cases are not even conscious of foZlow ing them. Although I am free from illusion that I shall fully succeed even in doing this, I still hope that the little that is present here may please some people and have some application afterwards. " Bernard Bolzano (Wissenschaftslehre, 1929) The following book results from aseries of lectures on the mathematical theory of turbulence delivered by the author at the Purdue University School of Aeronautics and Astronautics during the past several years, and represents, in fact, a comprehensive account of the author's work with his graduate students in this field. It was my aim in writing this book to give to engineers and scientists a mathematical feeling for a subject, which because of its nonlinear character has resisted mathematical analysis for many years. On account vii i of its refractory nature this subject was categorized as one of seven "elementary catastrophes." The material presented here is designed for a first graduate course in turbulence. The complete course has been taught in one semester."

With the advent of super computers during the last ten years, the numerical simulation of viscous fluid flows modeled by the Navier-Stokes equations is becoming a most useful tool in Aircraft and Engine Design. In fact, compressible Navier-Stokes solvers tend to constitute the basic tools for many industrial applications occuring in the simulation of very complex turbulent and combustion phenomena. In Aerospace Engineering, as an exemple, their mathematical modelization requires reliable and robust methods for solving very stiff non linear partial differential equations. For the above reasons, it was clear that a workshop on this topic would be of interest for the CFD community in order to compare accuracy and efficiency of Navier-Stokes solvers on selected external and internal flow problems using different numerical approaches. The workshop was held on 4-6 December 1985 at Nice, France and organized by INRIA with the sponsorship of the GAMM Committee on Numerical Methods in Fluid Mechanics.

Turbulent reactive flows are of common occurrance in combustion engineering, chemical reactor technology and various types of engines producing power and thrust utilizing chemical and nuclear fuels. Pollutant formation and dispersion in the atmospheric environment and in rivers, lakes and ocean also involve interactions between turbulence, chemical reactivity and heat and mass transfer processes. Considerable advances have occurred over the past twenty years in the understanding, analysis, measurement, prediction and control of turbulent reactive flows. Two main contributors to such advances are improvements in instrumentation and spectacular growth in computation: hardware, sciences and skills and data processing software, each leading to developments in others. Turbulence presents several features that are situation-specific. Both for that reason and a number of others, it is yet difficult to visualize a so-called solution of the turbulence problem or even a generalized approach to the problem. It appears that recognition of patterns and structures in turbulent flow and their study based on considerations of stability, interactions, chaos and fractal character may be opening up an avenue of research that may be leading to a generalized approach to classification and analysis and, possibly, prediction of specific processes in the flowfield. Predictions for engineering use, on the other hand, can be foreseen for sometime to come to depend upon modeling of selected features of turbulence at various levels of sophistication dictated by perceived need and available capability.

This valuable volume provides a broad understanding of the main computational techniques used for processing reclamation of fluid and solid mechanics. The aim of these computational techniques is to reduce and eliminate the risks of mechanical systems failure in hydraulic machines. Using many computational methods for mechanical engineering problems, the book presents not only a platform for solving problems but also provides a wealth of information to address various technical aspects of troubleshooting of mechanical system failure. The focus of the book is on practical and realistic fluids engineering experiences. Many photographs and figures are included, especially to illustrate new design applications and new instruments.

The Dutch Association for Numerical Fluid Mechanics (Kontaktgroep Numerieke Stromingsleer, KNSL) was founded in The Netherlands in November 1974. Since then, the Association has organized meetings twice a year. The present volume contains the proceedings of the 25th meeting, held on October 20, 1986, at Delft University of Technology. The purpose of the KNSL is to provide an opportunity for researchers in numerical fluid mechanics to meet regularly and to inform each other about their research in an informal atmosphere. Presentations preferably describe work in progress, and discussion of unsolved problems and unresolved difficulties is encouraged. The working language is Dutch. Nevertheless, science and technology are worldwide activities, and therefore it was decided to publish the proceedings of the 25th meeting in English. The nine contributions to the 25th meeting were selected by profs. A.I. van de Vooren, C.B. Vreugdenhil and the editor. These works are far from covering completely all activity in this field in this country, but they are typical of what is going on. A wide range of subjects is discussed, including fundamental aspects of spectral methods, solution methods for the Euler equations and aeronautical applications, viscous ship hydrodynamics, shallow water equations, viscous flows with capillary and non-Newtonian effects, and turbulent heat transfer with industrial applications. The 25th meeting of the KNSL was supported financially by ECN (Netherlands Energy Research Foundation), MARIN (Maritime Research Institute Netherlands), NLR (National Aerospace Laboratory), WL (Delft Hydraulics Laboratory), VEG Gasinstituut, Delft University of Technology and University of Twente.

This textbook presents numerical solution techniques for incompressible turbulent flows that occur in a variety of scientific and engineering settings including aerodynamics of ground-based vehicles and low-speed aircraft, fluid flows in energy systems, atmospheric flows, and biological flows. This book encompasses fluid mechanics, partial differential equations, numerical methods, and turbulence models, and emphasizes the foundation on how the governing partial differential equations for incompressible fluid flow can be solved numerically in an accurate and efficient manner. Extensive discussions on incompressible flow solvers and turbulence modeling are also offered. This text is an ideal instructional resource and reference for students, research scientists, and professional engineers interested in analyzing fluid flows using numerical simulations for fundamental research and industrial applications.

In full multigrid methods for elliptic difference equations one works on a sequence of meshes where a number of pre- and/or postsmoothing steps are performed on each level. As is well known these methods can converge very fast on problems with a smooth solution and a regular mesh, but the rate of convergence can be severely degraded for problems with unisotropy or discontinuous coefficients unless some form of robust smoother is used. Also problems can arise with the increasingly coarser meshes because for some types of discretization methods, coercivity may be lost on coarse meshes and on massively parallel computers the computation cost of transporting information between computer processors devoted to work on various levels of the mesh can dominate the whole computing time. For discussions about some of these problems, see (11). Here we propose a method that uses only two levels of meshes, the fine and the coarse level, respec tively, and where the corrector on the coarse level is equal to a new type of preconditioner which uses an algebraic substructuring of the stiffness matrix. It is based on the block matrix tridiagonal structure one gets when the domain is subdivided into strips. This block-tridiagonal form is used to compute an approximate factorization whereby the Schur complements which arise in the recursive factorization are approximated in an indirect way, i. e."

From the reviews of the first edition: "This book is directed to graduate students and research workers interested in the numerical solution of problems of fluid dynamics, primarily those arising in high speed flow. ...The book is well arranged, logically presented and well illustrated. It contains several FORTRAN programms with which students could experiment ... It is a "practical "book, with emphasis on methods and their implementation. It is an excellent text for the fruitful research area it covers, and is highly recommended." "Journal of Fluid Mechanics" #1 From the reviews of the second edition: "The arrangement of chapters in the book remains practically the same as that in the first editon (1977), except for the inclusion of Glimm's method ... This book is higly recommended for both graduate students and researchers." "Applied Mechanics Reviews" #1

This volume collects papers dedicated toWalterNoll on his sixtieth birthday, January 7, 1985. They first appeared in Volumes 86-97 (1984-1987) of the Archive for Rational Mechanics and Analysis. At the request ofthe Editors the list of authors to be invited was drawn up by B.D. Coleman, M. Feinberg, and J. Serrin. WalterNoll's influence upon research into the foundations of mechanics and thermodynamics is plain, everywhere acknowledged. Less obvious is the wide effect his writings have exerted upon those who apply mechanics to special problems, but it is witnessed by the now frequent use of terms, concepts, and styles of argument he introduced, use sometimes by young engineers who have learnt them in some recent textbook and hence take them for granted, oftenwith no idea whence they come. Examples are "objectivity", "material frame- indifference", "constitutive equation", "reduced form" of the last-named, "sim- plematerial", "simplesolid", "simplefluid", "isotropygroup",andtheassociated notations and lines of reasoning.

This volume is the collection of papers presented at the workshop on 'The Stability of Spatially Varying and Time Dependent Flows" sponsored by the Institute for Computer Applications in Science and Engineering (lCASE) and NASA Langley Research Center (LaRC) during August 19- 23, 1985. The purpose of this workshop was to bring together some of the experts in the field for an exchange of ideas to update the current status of knowledge and to help identify trends for future research. Among the invited speakers were D.M. Bushnell, M. Goldstein, P. Hall, Th. Herbert, R.E. Kelly, L. Mack, A.H. Nayfeh, F.T. Smith, and C. von Kerczek. The contributed papers were by A. Bayliss, R. Bodonyi, S. Cowley, C. Grosch, S. Lekoudis, P. Monkewitz, A. Patera, and C. Streett. In the first article, Bushnell provides a historical background on laminar flow control (LFC) research and summarizes the crucial role played by stability theory in LFC system design. He also identifies problem areas in stability theory requiring further research from the view-point of ap plications to LFC design. It is an excellent article for theoreticians looking for some down-to-earth applications of stability theory."

The present lecture notes cover a first course in th most common types of stratified flows encountered in Environ mental Hydraulics. Most of the flows are buoyancy flows, i.e. currents in which gravity acts on small density differences. Part I presents the basic concepts of stagnant, densit- stratified water, and of flowing non-miscible stratified fluids. The similarity to the (presumed) well-known open channel flow, subject to a reduced gravity, is illustrated. Part II treats the miscible density stratified flows. In outlining the governing equations, the strong coupling between the turbulence (the mixing) and the mean flow is emphasized. The presentation and discussions of the basic governing equa tions are followed by illustrative examples. Separate chapters are devoted to Dense Bottom Currents, Free Penetrative Convec tion, Wind-driven Stratified Flow, Horizontal Buoyancy Flow and Vertical jet/plumes. Part III presents some examples of practical problems solved on the basis of knowledge given in the present lecture notes. It is the author's experience that the topics treated in chapter 8 and in the subsequent chapters are especially well suited for self-tuition, followed by a study-circle. ACKNOWLEDGEMENT The author has benefited by the valuable help of his col legues at the Institute of Hydrodynamics and Hydraulic Engin eering, the Technical University of Denmark, especially our librarian Mrs. Kirsten Djcentsrup, our secretary Mrs. Marianne Lewis and our technical draftsman Mrs. Liselotte Norup."

This text focuses on the physics of fluid transport in micro- and nanofabricated liquid-phase systems, with consideration of gas bubbles, solid particles, and macromolecules. This text was designed with the goal of bringing together several areas that are often taught separately namely, fluid mechanics, electrodynamics, and interfacial chemistry and electrochemistry with a focused goal of preparing the modern microfluidics researcher to analyze and model continuum fluid mechanical systems encountered when working with micro- and nanofabricated devices. This text is not a summary of current research in the field, and it omits any discussion of microfabrication techniques or any attempt to summarize the technological state of the art. This text serves as a useful reference for practicing researchers but is designed primarily for classroom instruction. Worked sample problems are inserted throughout to assist the student, and exercises are included at the end of each chapter to facilitate use in classes.

This book contains notes for a one-semester course on viscoelasticity given in the Division of Applied Mathematics at Brown University. The course serves as an introduction to viscoelasticity and as a workout in the use of various standard mathematical methods. The reader will soon find that he needs to do some work on the side to fill in details that are omitted from the text. These are notes, not a completely detailed explanation. Furthermore, much of the content of the course is in the problems assigned for solution by the student. The reader who does not at least try to solve a good many of the problems is likely to miss most of the point. Much that is known about viscoelasticity is not discussed in these notes, and references to original sources are usually not give, so it will be difficult or impossible to use this book as a reference for looking things up. Readers wanting something more like a treatise should see Ferry's Viscoelastic Properties of Polymers, Lodge's Elastic Liquids, the volumes edited by Eirich on Rheology, or any issue of the Transactions of the Society of Rheology. These works emphasize physical aspects of the subject. On the mathematical side, Gurtin and Sternberg's long paper On the Linear Theory of Viscoelasticity (ARMA II, 291 (I962" remains the best reference for proofs of theorems.

Computational fluid flow is not an easy subject. Not only is the mathematical representation of physico-chemical hydrodynamics complex, but the accurate numerical solution of the resulting equations has challenged many numerate scientists and engineers over the past two decades. The modelling of physical phenomena and testing of new numerical schemes has been aided in the last 10 years or so by a number of basic fluid flow programs (MAC, TEACH, 2-E-FIX, GENMIX, etc). However, in 1981 a program (perhaps more precisely, a software product) called PHOENICS was released that was then (and still remains) arguably, the most powerful computational tool in the whole area of endeavour surrounding fluid dynamics. The aim of PHOENICS is to provide a framework for the modelling of complex processes involving fluid flow, heat transfer and chemical reactions. PHOENICS has now been is use for four years by a wide range of users across the world. It was thus perceived as useful to provide a forum for PHOENICS users to share their experiences in trying to address a wide range of problems. So it was that the First International PHOENICS Users Conference was conceived and planned for September 1985. The location, at the Dartford Campus of Thames Polytechnic, in the event, proved to be an ideal site, encouraging substantial interaction between the participants.

This revised edition provides updated fluid mechanics measurement techniques as well as a comprehensive review of flow properties required for research, development, and application. Fluid-mechanics measurements in wind tunnel studies, aeroacoustics, and turbulent mixing layers, the theory of fluid mechanics, the application of the laws of fluid mechanics to measurement techniques, techniques of thermal anemometry, laser velocimetry, volume flow measurement techniques, and fluid mechanics measurement in non-Newtonian fluids, and various other techniques are discussed.

In the past several years, it has become apparent that computing will soon achieve a status within science and engineering to the classical scientific methods of laboratory experiment and theoretical analysis. The foremost tools of state-of-the-art computing applications are supercomputers, which are simply the fastest and biggest computers available at any given time. Supercomputers and supercomputing go hand-in-hand in pacing the development of scientific and engineering applications of computing. Experience has shown that supercomputers improve in speed and capability by roughly a factor 1000 every 20 years. Supercomputers today include the Cray XMP and Cray-2, manufactured by Cray Research, Inc., the Cyber 205, manufactured by Control Data Corporation, the Fujitsu VP, manufactured by Fujitsu, Ltd., the Hitachi SA-810/20, manufactured by Hitachi, Ltd., and the NEC SX, manufactured by NEC, Inc. The fastest of these computers are nearly three orders-of-magnitude faster than the fastest computers available in the mid-1960s, like the Control Data CDC 6600. While the world-wide market for supercomputers today is only about 50 units per year, it is expected to grow rapidly over the next several years to about 200 units per year.

From the astrophysical scale of a swirling spiral galaxy, through the geophysical scale of a hurricane, down to the subatomic scale of elementary particles, vortical motion and vortex dynamics have played a profound role in our understanding of the physical world. Kuchemann referred to vortex dynamics as "the sinews and muscles of fluid motion. " In order to update our understanding of vortex dominated flows, NASA Langley Research Center and the Institute for Computer Applications in Science and Engineering (ICASE) conducted a workshop during July 9-11, 1985. The subject was broadly divided into five overlapping topics vortex dynamics, vortex breakdown, massive separation, vortex shedding from sharp leading edges and conically separated flows. Some of the experts in each of these areas were invited to provide an overview of the subject. This volume is the proceedings of the workshop and contains the latest, theoretical, numerical, and experimental work in the above-mentioned areas. Leibovich, Widnall, Moore and Sirovich discussed topics on the fundamentals of vortex dynamics, while Keller and Hafez treated the problem of vortex break down phenomena; the contributions of Smith, Davis and LeBalleur were in the area of massive separation and inviscid-viscous interactions, while those of Cheng, Hoeijmakers and Munnan dealt with sharp-leading-edge vortex flows; and Fiddes and Marconi represented the category of conical separated flows."

The GAMM Committee for Numerical Methods in Fluid Mechanics organizes workshops which should bring together experts of a narrow field of computational fluid dynamics (CFD) to exchange ideas and experiences in order to speed-up the development in this field. In this sense it was suggested that a workshop should treat the solution of CFD problems on vector computers. Thus we organized a workshop with the title "The efficient use of vector computers with emphasis on computational fluid dynamics." The workshop took place at the Computing Centre of the University of Karlsruhe, March 13-15,1985. The participation had been restricted to 22 people of 7 countries. 18 papers have been presented. In the announcement of the workshop we wrote: "Fluid mechanics has actively stimulated the development of superfast vector computers like the CRAY's or CYBER 205. Now these computers on their turn stimulate the development of new algorithms which result in a high degree of vectorization (sca1ar/vectorized execution-time). But with 3-D problems we quickly reach the limit of present vector computers. If we want e.g. to solve a system of 6 partial differential equations (e.g. for u, v, w, p, k, or for the vectors u, curl u) on a 50x50x50 grid we have 750.000 unknowns and for a 4th order difference method we have circa 60 million nonzero coefficients in the highly sparse matrix. This characterizes the type of problems which we want to discuss in the workshop.""

6. 2 Creeping viscous flow in a semi-infinite channel 140 6. 3 Poiseuille flow in tubes of circular cross-section 144 6. 4 Motion of a Newtonian liquid between two coaxial cylinders 148 151 6. 5 Bodies in liquids 6. 6 liquid flow and intermolecular forces 154 Non-Newtonian liquids 157 6. 7 6. 8 Viscometers 160 Chapter 7 Surface effects 163 7. 1 Introduction 163 7. 2 Excess surface free energy and surface tension of liquids 163 7. 3 The total surface energy of liquids 167 7. 4 Surface tension and intermolecular forces 168 7. 5 Solid surfaces 171 7. 6 Specific surface free energy and the intermolecular potential 172 7. 7 liquid surfaces and the Laplace-Young equation 174 7. 8 liquid spreading 178 7. 9 Young's relation 181 7. 10 Capillary effects 184 7. 11 The sessile drop 187 7. 12 Vapour pressure and liquid-surface curvature 189 7. 13 The measurement of surface free energies 191 Chapter 8 High polymers and liquid crystals 197 8. 1 Introduction 197 8. 2 High polymers 197 8. 3 The mechanisms of polymerisation 198 8. 4 The size and shape of polymer molecules 199 8. 5 The structure of solid polymers 201 8. 6 The glass transition temperature 203 8. 7 Young's modulus of solid polymers 205 Stress-strain curves of polymers 8. 8 206 8. 9 Viscous flow in polymers 209 liquid crystals 8.