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 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.

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."

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.

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.

This investigation is an outgrowth of my doctoral dissertation at Princeton University. I am particularly grateful to Professors George F. Pinder and William G. Gray of Princeton for their advice during both my research and my writing. I believe that finite-element collocation holds promise as a numer ical scheme for modeling complicated flows in porous media. However, there seems to be a "conventional wisdom" maintaining that collocation is hopelessly beset by oscillations and is, in some way, fundamentally inappropriate for multiphase flows. I hope to dispel these objections, realizing that others will remain for further work. The U. S. National Science Foundation funded much of this study through grant number NSF-CEE-8111240. TABLE OF CONTENTS ABSTRACT ;; FOREWORD ;; ; CHAPTER ONE. THE PHYSICAL SYSTEM. 1.1 Introduction. 1 1.2 The reservoir and its contents. 5 1.3 Reservoir mechanics. 9 1.4 Supplementary constraints. 18 1.5 Governing equations. 26 CHAPTER TWO. REPRESENTING FLUID-PHASE BEHAVIOR. 39 2.1 Thermodynamics of the fluid system. 40 2.2 Standard equation-of-state methods. 45 2.3 Maxwell-set interpolation.

The scope of the present book is to offer the most efficient tools for the vectorization of serial computer programs. Here, by vectorization we understand the adaptation of computer programs to the special architecture of modern available vector computers to exploit fully their potential, which will often result in remarkable performance improvements. The book is written primarily for users working in the various fields of computational physics, for scientists as well as for programmers running their jobs on a vector computer. The text may, however, also be of value to those who are interested in numerical algorithms. Although the examples discussed in chapter 9 have been taken from Computational Fluid Dynamics, the numerical methods are well-known, and are applied in many fields of Computational Physics. The book is divided into four parts. After a short introduction which outlines the limits of conventional serial computers in contrast to the possibilities offered by the new vector machines, the second part is addressed to the discussion of some main features of existing computer architectures. We restrict ourselves to the vector computers CRAY-1S and CDC-CYBER 205, although, in the meantime, many vector and parallel computers and array processors are available such as DENELCOR's Heterogeneous Element Processor (HEP), ICL's Distributed Array Processor (DAP), SPERRY UNIVAC's Array Processing System (APS), STAR TECHNOLOGIES ST-l00, FLOATING POINT SYSTEMS' Array Processor (FPS), FUJITSU's FACOM VP-l00 and VP-200, HITACHI's Integrated Array Processor (lAP), HITACHI's S 810/10 and S 810/20 and others.

Applied Mathematics is the art of constructing mathematical models of observed phenomena so that both qualitative and quantitative results can be predicted by the use of analytical and numerical methods. Theoretical Mechanics is concerned with the study of those phenomena which can be ob served in everyday life in the physical world around us. It is often characterised by the macroscopic approach which allows the concept of an element or particle of material, small compared to the dimensions of the phenomena being modelled, yet large compared to the molecular size of the material. Then atomic and molecular phenomena appear only as quantities averaged over many molecules. It is therefore natural that the mathemati cal models derived are in terms of functions which are continuous and well behaved, and that the analytical and numerical methods required for their development are strongly dependent on the theory of partial and ordinary differential equations. Much pure research in Mathematics has been stimu lated by the need to develop models of real situations, and experimental observations have often led to important conjectures and theorems in Analysis. It is therefore important to present a careful account of both the physical or experimental observations and the mathematical analysis used. The authors believe that Fluid Mechanics offers a rich field for il lustrating the art of mathematical modelling, the power of mathematical analysis and the stimulus of applications to readily observed phenomena."

The concept of vorticity is of central importance in fluid mechanics and the change and variability of atmospheric flow is dominated by transient vortices of different time- and space scales. Of particular importance are the most in- tense vortices such as hurricanes, typhoons and tornadoes which are associated with extreme and hazardous weather events of great concern to society. In recent years the un- derstanding of these phenomena has grown due to increased and improved surveillance by satellites and aircraft as well as by numerical modelling and simulation, theoretical studies and laboratory experiments. The symposium on "Intense Atmospheric Vortices" was held at the European Centre for Medium Range Weather Forecasts (ECMWF), Reading, England, July 14-17, 1981. The subject area of the Symposium was concerned with observational work, experimental models, theoretical and numerical studies in- volving hurricanes, typhoons, tornadoes and related pheno- mena. The aim was to bring together experts on these meteo- rological processes and on the fundamental fluid-dynamic mechanisms for vorticity intensification from all parts of the world. Thirtyfour scientists participated in the Sympo- sium, including more than half of those leading world ex- perts in the field whom the organizers had invited.

A cell, whose spatial extent is small compared with a surrounding flow, can develop inside a vortex. Such cells, often referred to as vortex breakdown bubbles, provide stable and clean flame in combustion chambers; they also reduce the lift force of delta wings. This book analyzes cells in slow and fast, one- and two-fluid flows and describes the mechanisms of cell generation: (a) minimal energy dissipation, (b) competing forces, (c) jet entrainment, and (d) swirl decay. The book explains the vortex breakdown appearance, discusses its features, and indicates means of its control. Written in acceptable, non-math-heavy format, it stands to be a useful learning tool for engineers working with combustion chambers, chemical and biological reactors, and delta-wing designs.

The content of this book is based, largely, on the core curriculum in geophys ical fluid dynamics which land my colleagues in the Department of Geophysical Sciences at The University of Chicago have taught for the past decade. Our purpose in developing a core curriculum was to provide to advanced undergraduates and entering graduate students a coherent and systematic introduction to the theory of geophysical fluid dynamics. The curriculum and the outline of this book were devised to form a sequence of courses of roughly one and a half academic years (five academic quarters) in length. The goal of the sequence is to help the student rapidly advance to the point where independent study and research are practical expectations. It quickly became apparent that several topics (e. g., some aspects of potential theory) usually thought of as forming the foundations of a fluid-dynamics curriculum were merely classical rather than essential and could be, however sadly, dispensed with for our purposes. At the same time, the diversity of interests of our students is so great that no curriculum can truly be exhaust ive in such a curriculum period. It seems to me that the best that can be achieved as a compromise is a systematic introduction to some important segment of the total scope of geophysical fluid dynamics which is illustrative of its most fruitful methods."

Introduction to Critical Phenomena in Fluids encompasses the fundamentals of this relatively young field, as well as applications in the fields of chemical engineering, analytical chemistry, and environmental remediation processing. The exercises in the text have been developed in a way that makes the book suitable for graduate courses in chemical engineering thermodynamics and physical chemistry.