This book presents an overview of the computational framework in which calculations of relativistic hydrodynamics have been developed. It summarizes the jargon and methods used in the field, and provides illustrative applications to real physical systems. The authors explain how to break down the complexities of Einstein's equations and fluid dynamics, stressing the viability of the Euler-Lagrange approach to astrophysical problems. The book contains techniques and algorithms enabling one to build computer simulations of relativistic fluid problems for various astrophysical systems in one, two and three dimensions. It also shows the reader how to test relativistic hydrodynamics codes. Suitable for graduate courses on astrophysical hydrodynamics and relativistic astrophysics, this book also provides a valuable reference for researchers already working in the field.

The main aim of this paper is to present some new and general results, ap plicable to the the equations of two phase flow, as formulated in geothermal reservoir engineering. Two phase regions are important in many geothermal reservoirs, especially at depths of order several hundred metres, where ris ing, essentially isothermal single phase liquid first begins to boil. The fluid then continues to rise, with its temperature and pressure closely following the saturation (boiling) curve appropriate to the fluid composition. Perhaps the two most interesting theoretical aspects of the (idealised) two phase flow equations in geothermal reservoir engineering are that firstly, only one component (water) is involved; and secondly, that the densities of the two phases are so different. This has led to the approximation of ignoring capillary pressure. The main aim of this paper is to analyse some of the consequences of this assumption, especially in relation to saturation changes within a uniform porous medium. A general analytic treatment of three dimensional flow is considered. Pre viously, three dimensional modelling in geothermal reservoirs have relied on numerical simulators. In contrast, most of the past analytic work has been restricted to one dimensional examples."

Moving Loads on Ice Plates is a unique study into the effect of vehicles and aircraft travelling across floating ice sheets. It synthesizes in a single volume, with a coherent theme and nomenclature, the diverse literature on the topic, hitherto available only as research journal articles. Chapters on the nature of fresh water ice and sea ice, and on applied continuum mechanics are included, as is a chapter on the subject's venerable history in related areas of engineering and science. The most recent theories and data are discussed in great depth, demonstrating the advanced state of the modelling and experimental field programmes that have taken place. Finally, results are interpreted in the context of engineering questions faced by agencies operating in the polar and subpolar regions. Although the book necessarily contains some graduate level applied mathematics, it is written to allow engineers, physicists and mathematicians to extract the information they need without becoming preoccupied with details. Structural, environmental, civil, and offshore engineers, and groups who support these industries, particularly within the Arctic and Antarctic, will find the book timely and relevant.

The Third Symposium on Numerical and Physical Aspects of Aerodynamic Flows, like its immediate predecessor, was organized with emphasis on the calculation of flows relevant to aircraft, ships, and missiles. Fifty-five papers and 20 brief communications were presented at the Symposium, which was held at the California State University at Long Beach from 21 to 24 January 1985. A panel discussion was chaired by A. M. O. Smith and includeq state ments by T. T. Huang, C. E. lobe, l. Nielsen, and C. K. Forester on priorities for future research. The first lecture in memory of Professor Keith Stewartson was delivered by J. T. Stuart and is reproduced in this volume together with a selection of the papers presented at the Symposium. In Volume II of this series, papers were selected so as to provide a clear indication of the range of procedures available to represent two-dimensional flows, their physical foundation, and their predictive ability. In this volume, the emphasis is on three-dimensional flows with a section of five papers concerned with unsteady flows and a section of seven papers on three dimensional flows: The papers deal mainly with calculation methods and encompass subsonic and transonic, attached and separated flows. The selec tion has been made so as to fulfill the same purpose for three-dimensional flows as did Volume II for two-dimensional flows."

It has become almost a cliche to preface one's remarks about asymptotic tech niques with the statement that only a very few special problems in diffrac tion theory (be it electromagnetic, acoustic, elastic or other phenomena) are possessed of closed form solutions, but as with many cliches, this is because it is true. One only has to scan the literature to see the large amount of effort (both human and computer) expended to solve diffraction problems involving complicated geometries which do not permit such simplifications as separation of variables, It was a desire for techniques more straightforward than frontal numerical assaults, as well as for a theory \ hich \ ould explain the basic physical phenomena involved, which stimulated research into asymptot ic methods. Geometrical optics (GO) and, now, even Keller's geometrical theory of dif fraction (GTD) have been with us for some time, and have become standard tools in the analysis of high-frequency wave phenomena, Of course, it was always recognized that these approaches broke down in certain regions: GO in the shadow region; GTD along shadow boundaries and caustics. One remedy for these defects is to construct an expansion, based upon a more general ansatz than GO or GTD, which is made to be valid in one or more of the areas where GO or GTD break down."

The subject of this book is the physics of vortices. A detailed analysis of the dynamics of vortices will be presented. The important topics of vorticity and molecular spin will be dealt with, including the electromagnetic analogy and quantization in superfluids. The effect of molecular spin on the dynamics of molecular nano-confined fluids using the extended Navier-Stokes equations will also be covered -especially important to the theory and applicability of nanofluidics and associated devices. The nanoscale boundary layer and nanoscale vortex core are regions of intense vorticity (molecular spin). It will be shown, based on molecular kinetic theory and thermodynamics, that the macroscopic (solid body) rotation must be accompanied by internal rotation of the molecules. Electric polarization of the internal molecular rotations about the local rotation axis -the Barnett effect - occurs. In such a spin aligned system, major changes in the physical properties of the fluid result.

These two volumes contain the proceedings of the workshop on the Institute for Computer Instability and Transition, sponsored by Applications in Science and Engineering (ICASE) and the Langley Research Center (LaRC), during May 15 to June 9, 1989. The work shop coincided with the initiation of a new, focused research pro gram on instability and transition at LaRC. The objectives of the workshop were to (i) expose the academic community to current technologically important issues of instability and transition in shear flows over the entire speed range, (ii) acquaint the academic com munity with the unique combination of theoretical, computational and experimental capabilities at LaRC and foster interaction with these facilities, (iii) review current state-of-the-art and propose fu ture directions for instability and transition research, (iv) accelerate progress in elucidating basic understanding of transition phenomena and in transferring this knowledge into improved design methodolo gies through improved transition modeling, and (v) establish mech anisms for continued interaction. The objectives (i) to (iii) were of course immediately met. It is still premature to assess whether ob jectives (iv) and (v) are achieved. The workshop program consisted of tutorials, research presenta tions, panel discussions, experimental and computational demonstra tions, and collaborative projects.

This book is drawn from across many active fields of mathematics and physics. It has connections to atmospheric dynamics, spherical codes, graph theory, constrained optimization problems, Markov Chains, and Monte Carlo methods. It addresses how to access interesting, original, and publishable research in statistical modeling of large-scale flows and several related fields. The authors explicitly reach around the major branches of mathematics and physics, showing how the use of a few straightforward approaches can create a cornucopia of intriguing questions and the tools to answer them.

Large Eddy Simulation is a relatively new and still evolving computatio nal strategy for predicting turbulent flows. It is now widely used in research to elucidate fundamental interactions in physics of turbulence, to predict phe nomena which are closely linked to the unsteady features of turbulence and to create data bases against which statistical closure models can be asses sed. However, its applicability to complex industrial flows, to which statisti cal models are applied routinely, has not been established with any degree of confidence. There is, in particular, a question mark against the prospect of LES becoming an economically tenable alternative to Reynolds-averaged N avier-Stokes methods at practically high Reynolds numbers and in complex geometries. Aerospace flows pose particularly challenging problems to LES, because of the high Reynolds numbers involved, the need to resolve accura tely small-scale features in the thin and often transitional boundary layers developing on aerodynamic surfaces. When the flow also contains a separated region - due to high incidence, say - the range and disparity of the influen tial scales to be resolved is enormous, and this substantially aggravates the problems of resolution and cost. It is just this combination of circumstances that has been at the heart of the project LESFOIL to which this book is devoted. The project combined the efforts, resources and expertise of 9 partner organisations, 4 universities, 3 industrial companies and 2 research institu tes."

Chemical reaction systems of practical interest are usually very complex: They consist of a large number of elementary reactions (hundreds or thou sands in a small system), mostly with rate coefficients differing by many orders of magnitude, which leads to serious stiffness, and they are often coupled with surface reaction steps and convective or diffusive processes. Thus, the derivation of a "true" chemical mechanism can be extremely cumbersome. In most cases this is done by setting up "reaction models" which are improved step by step using, for example, perturbation theory, numerical simulation and sensitivity analysis (and - hopefully, in the near future - parameter identification procedures), and by comparison with experimental data on sensitive properties. Because of the complexity of these processes, it was very difficult in the past to convince engineers to apply methods using detailed mecha nisms given in terms of elementary reactions, and even in basic sciences there was scepticism about this ambitious aim. A previous workshop on modelling of chemical reaction systems held in 1980 was an attempt to find a common language of mathematicians, chemists, and engineers working in this interdisciplinary area. Since then considerable progress has been made by the simultaneous development of applied mathematics, an enor mous increase of computer capacity, and the development of experimental techniques in physical chemistry that have made available well-working reaction mechanisms in some fields of reaction kinetics."

This book is dedicated to the memory of a distinguished Russian engineer, Rostislav E. Alexeyev, who was the first in the world to develop the largest ground effect machine - Ekranoplan. One of Alexeyev's design concepts with the aerodynamic configuration of a jlying wing can be seen on the front page. The book presents a description of a mathematical model of flow past a lifting system, performing steady and unsteady motions in close proximity to the underlying solid surface (ground). This case is interesting for practical purposes because both the aerodynamic and the economic efficiency of the system near the ground are most pronounced. Use of the method of matched asymptotic expansions enables closed form solutions for the aerodynamic characteristics of the wings-in-ground effect. These can be used for design, identification, and processing of experimental data in the course of developing ground effect vehicles. The term extreme ground effect, widely used through out the book, is associated with very small relative ground clearances of the order of 10% or less. The theory of a lifting surface, moving in immediate proximity to the ground, represents one of the few limiting cases that can be treated analytically. The author would like to acknowledge that this work has been influenced by the ideas of Professor Sheila E. Widnall, who was the first to apply the matched asymptotics techniques to treat lifting flows with the ground effect. Saint Petersburg, Russia February 2000 Kirill V. Rozhdestvensky Contents 1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ."

The subject of the book is uid dynamics and heat transfer in micro-channels. This problem is important for understanding the complex phenomena associated with single- and two-phase ows in heated micro-channels. The challenge posed by high heat uxes in electronic chips makes thermal management a key factor in the development of these systems. Cooling of mic- electronic components by new cooling technologies, as well as improvement of the existing ones, is becoming a necessity as the power dissipation levels of integrated circuits increases and their sizes decrease. Miniature heat sinks with liquid ows in silicon wafers could signi cantly improve the performance and reliability of se- conductor devices. The improvements are made by increasing the effective thermal conductivity, by reducing the temperature gradient across the wafer, by reducing the maximum wafer temperature, and also by reducing the number and intensity of localized hot spots. A possible way to enhance heat transfer in systems with high power density is to change the phase in the micro-channels embedded in the device. This has motivated a number of theoretical and experimental investigations covering various aspects of heat transfer in micro-channel heat sinks with phase change. The ow and heat transfer in heated micro-channels are accompanied by a n- ber of thermohydrodynamic processes, such as liquid heating and vaporization, bo- ing, formation of two-phase mixtures with a very complicated inner structure, etc., which affect signi cantly the hydrodynamic and thermal characteristics of the co- ing systems.

This book completes the physical foundations and experimental techniques described in volume 1 with an updated review of the accessory equipment indispensable in molecular beam experiments. It extends the subject to cluster beams and beams of hyperthermal and subthermal energies.

It was long ago that group analysis of differential equations became a powerful tool for studying nonlinear equations and boundary value problems. This analysis was especially fruitful in application to the basic equations of mechanics and physics because the invariance principles are already involved in their derivation. It is in no way a coincidence that the equations of hydrodynamics served as the first object for applying the new ideas and methods of group analysis which were developed by 1. V. Ovsyannikov and his school. The authors rank themselves as disciples of the school. The present monograph deals mainly with group-theoretic classification of the equations of hydrodynamics in the presence of planar and rotational symmetry and also with construction of exact solutions and their physical interpretation. It is worth noting that the concept of exact solution to a differential equation is not defined rigorously; different authors understand it in different ways. The concept of exact solution expands along with the progress of mathematics (solu tions in elementary functions, in quadratures, and in special functions; solutions in the form of convergent series with effectively computable terms; solutions whose searching reduces to integrating ordinary differential equations; etc. ). We consider it justifiable to enrich the set of exact solutions with rank one and rank two in variant and partially invariant solutions to the equations of hydrodynamics."

This interdisciplinary collection brings together the fundamental research in shock focusing and sonoluminescence. The authors report on their studies on shock focusing and related bubble dynamics, as well as their applications in medical science.

This book springs from the programme Quantized Vortex Dynamics and Sup- ?uid Turbulence held at the Isaac Newton Institute for Mathematical Sciences (University of Cambridge) in August 2000. What motivated the programme was the recognition that two recent developments have moved the study of qu- tized vorticity, traditionally carried out within the low-temperature physics and condensed-matter physics communities, into a new era. The ?rst development is the increasing contact with classical ?uid dynamics and its ideas and methods. For example, some current experiments with - lium II now deal with very classical issues, such as the measurement of velocity spectra and turbulence decay rates. The evidence from these experiments and many others is that super?uid turbulence and classical turbulence share many features. The challenge is now to explain these similarities and explore the time scales and length scales over which they hold true. The observed classical aspects have also attracted attention to the role played by the ?ow of the normal ?uid, which was somewhat neglected in the past because of the lack of direct ?ow visualization. Increased computing power is also making it possible to study the coupled motion of super?uid vortices and normal ?uids. Another contact with classical physics arises through the interest in the study of super?uid vortex - connections. Reconnections have been studied for some time in the contexts of classical ?uid dynamics and magneto-hydrodynamics (MHD), and it is useful to learn from the experience acquired in other ?elds.

This book is a superb tool in virtually all application areas involving the Kinetic Theory of Gases, Rarefied Gas Dynamics, Transport Theory, and Aerosol Mechanics. It has been especially designed to serve a dual function, both as a teaching instrument either in a classroom environment or at home, and as a reference for scientists and engineers working in the fields of Rarefied Gas Dynamics and Aerosol Mechanics.

An extensive critical compilation of the wide range of manufacturing processes that involve the application of spray technology, this book covers design of atomizers as well as the performance of plant and their corresponding spray systems. The needs of practising engineers from different disciplines: project managers, and works, maintenance and design engineers are catered for. Of interest to researchers in the field of liquid sprays, the book includes outlines of the contemporary and possible future research and challenges in the different fields of application and deals with:

sprays and their production;

sprays in industrial production processes;

processes involving vaporisation and cooling or cleaning of gases;

spray-surface impact processes;

fuel sprays for fixed plant;

spraying of hot surfaces for steel making and other metals;

spraying of molten metals.

Guidance is given for the analysis and interpretation of experimental data obtained using different measurement techniques."

Geomaterials consist of a mixture of solid particles and void space that may be ?lled with ?uid and gas. The solid particles may be di?erent in sizes, shapes, and behavior; and the pore liquid may have various physical and chemical properties. Hence, physical, chemical or electrical interaction - tween the solid particles and pore ?uid or gas may take place. Therefore, the geomaterials in general must be considered a mixture or a multiphase material whose state is described by physical quantities in each phase. The stresses carried by the solid skeleton are typically termed "e?ective stress" while the stresses carried by the pore liquid are termed "pore pressure. " The summation of the e?ective stress and pore pressure is termed "total stress" (Terzaghi, 1943). For a free drainage condition or completely undrained c- dition, the pore pressure change is zero or depends only on the initial stress condition; it does not depend on the skeleton response to external forces. Therefore, a single phase description of soil behavior is adequate. For an intermediate condition, however, some ?ow (pore pressure leak) may take place while the force is applied and the skeleton is under deformation. Due to the leak of pore pressure, the pore pressure changes with time, and the e?ective stress changes and the skeleton deforms with time accordingly. The solution of this intermediate condition, therefore, requires a multi-phase c- tinuum formulations that may address the interaction of solid skeleton and pore liquid interaction.

The principal aim of the book is to give a comprehensive account of the variety of approaches to such an important and complex concept as Integrability. Dev- oping mathematical models, physicists often raise the following questions: whether the model obtained is integrable or close in some sense to an integrable one and whether it can be studied in depth analytically. In this book we have tried to c- ate a mathematical framework to address these issues, and we give descriptions of methods and review results. In the Introduction we give a historical account of the birth and development of the theory of integrable equations, focusing on the main issue of the book - the concept of integrability itself. A universal de nition of Integrability is proving to be elusive despite more than 40 years of its development. Often such notions as "- act solvability" or "regular behaviour" of solutions are associated with integrable systems. Unfortunately these notions do not lead to any rigorous mathematical d- inition. A constructive approach could be based upon the study of hidden and rich algebraic or analytic structures associated with integrable equations. The requi- ment of existence of elements of these structures could, in principle, be taken as a de nition for integrability. It is astonishing that the nal result is not sensitive to the choice of the structure taken; eventually we arrive at the same pattern of eq- tions.

Time-Resolved Spectroscopy in Complex Liquids is intended to introduce the experimental researchers to state-of-the-art techniques in the study of the dynamics of complex liquids. The contributors concentrate on time-resolved optical spectroscopy, which recently produced many relevant results and new information about complex liquids. This is an emerging topic of soft-matter science and this book provides the most up-to-date account of new development.

This up-to-date book gives an account of the present state of the art of numerical methods employed in computational fluid dynamics. The underlying numerical principles are treated in some detail, using elementary methods. The author gives many pointers to the current literature, facilitating further study. This book will become the standard reference for CFD for the next 20 years.

An exciting new direction in hydrodynamic stability theory and the transition to turbulence is concerned with the role of disconnected states or finite amplitude solutions in the evolution of disorder in fluid flows. This volume contains refereed papers presented at the IUTAM/LMS sponsored symposium on "Non-Uniqueness of Solutions to the Navier-Stokes equations and their Connection with Laminar-Turbulent Transition" held in Bristol 2004. Theoreticians and experimentalists gathered to discuss developments in understanding both the onset and collapse of disordered motion in shear flows such as those found in pipes and channels.

The central objective of the symposium was to discuss the increasing amount of experimental and numerical evidence for finite amplitude solutions to the Navier-Stokes equations and to set the work into a modern theoretical context. The participants included many of the leading authorities in the subject and this volume captures much of the flavour of the resulting stimulating and lively discussions.

for the fluctuations around the means but rather fluctuations, and appearing in the following incompressible system of equations: on any wall; at initial time, and are assumed known. This contribution arose from discussion with J. P. Guiraud on attempts to push forward our last co-signed paper (1986) and the main idea is to put a stochastic structure on fluctuations and to identify the large eddies with a part of the probability space. The Reynolds stresses are derived from a kind of Monte-Carlo process on equations for fluctuations. Those are themselves modelled against a technique, using the Guiraud and Zeytounian (1986). The scheme consists in a set of like equations, considered as random, because they mimic the large eddy fluctuations. The Reynolds stresses are got from stochastic averaging over a family of their solutions. Asymptotics underlies the scheme, but in a rather loose hidden way. We explain this in relation with homogenizati- localization processes (described within the 3. 4 ofChapter 3). Ofcourse the mathematical well posedness of the scheme is not known and the numerics would be formidable Whether this attempt will inspire researchers in the field of highly complex turbulent flows is not foreseeable and we have hope that the idea will prove useful."

In order to allow the application of the theory from all the three volumes also to processes in combustion engines a systematic set of internally consistent state equations for diesel fuel gas and liquid valid in broad range of changing pressure and temperature are provided also in Volume 3. Erlangen, October 2006 Nikolay Ivanov Kolev Table of contents 1 Some basics of the single-phase boundary layer theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. 1 Flow over plates, velocity profiles, share forces, heat transfer. . . . . . . . . . . . . . . . . . . . 1 1. 1. 1 Laminar flow over the one site of a plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. 1. 2 Turbulent flow parallel to plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. 2 Steady state flow in pipes with circular cross sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1. 2. 1 Hydraulic smooth wall surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1. 2. 2 Transition region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1. 2. 3 Complete rough region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1. 2. 4 Heat transfer to fluid in a pipe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1. 3 Transient flow in pipes with circular cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Nomenclature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 Introduction to turbulence of multi-phase flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2. 1 Basic ideas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2. 2 Isotropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2. 3 Scales, eddy viscosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2. 3. 1 Small scale turbulent motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2. 3. 2 Large scale turbulent motion, Kolmogorov-Pandtl expression. . . . . . . . . 42 2. 4 k-eps framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Nomenclature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3 Sources for fine resolution outside the boundary layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3. 1 Bulk sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3. 1. 1 Deformation of the velocity field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3. 1. 2 Blowing and suction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .