Experts of fluid dynamics agree that turbulence is nonlinear and nonlocal. Because of a direct correspondence, nonlocality also implies fractionality. Fractional dynamics is the physics related to fractal (geometrical) systems and is described by fractional calculus. Up-to-present, numerous criticisms of linear and local theories of turbulence have been published. Nonlinearity has established itself quite well, but so far only a very small number of general nonlocal concepts and no concrete nonlocal turbulent flow solutions were available. This book presents the first analytical and numerical solutions of elementary turbulent flow problems, mainly based on a nonlocal closure. Considerations involve anomalous diffusion (Levy flights), fractal geometry (fractal- , bi-fractal and multi-fractal model) and fractional dynamics. Examples include a new 'law of the wall' and a generalization of Kraichnan's energy-enstrophy spectrum that is in harmony with non-extensive and non-equilibrium thermodynamics (Tsallis thermodynamics) and experiments. Furthermore, the presented theories of turbulence reveal critical and cooperative phenomena in analogy with phase transitions in other physical systems, e.g., binary fluids, para-ferromagnetic materials, etc.; the two phases of turbulence identifying the laminar streaks and coherent vorticity-rich structures. This book is intended, apart from fluids specialists, for researchers in physics, as well as applied and numerical mathematics, who would like to acquire knowledge about alternative approaches involved in the analytical and numerical treatment of turbulence.

This third volume describes continuous bodies treated as classical (Boltzmann) and spin (Cosserat) continua or fluid mixtures of such bodies. It discusses systems such as Boltzmann continua (with trivial angular momentum) and Cosserat continua (with nontrivial spin balance) and formulates the balance law and deformation measures for these including multiphase complexities. Thermodynamics is treated in the spirit of Muller-Liu: it is applied to Boltzmann-type fluids in three dimensions that interact with neighboring fluids on two-dimensional contact surfaces and/or one-dimensional contact lines. For all these situations it formulates the balance laws for mass, momenta, energy, and entropy. Further, it introduces constitutive modeling for 3-, 2-, 3-d body parts for general processes and materially objective variable sets and their reduction to equilibrium and non-equilibrium forms. Typical (reduced) fluid spin continua are liquid crystals. Prominent nematic examples of these include the Ericksen-Leslie-Parodi (ELP) formulation, in which material particles are equipped with material unit vectors (directors). Nematic liquid crystals with tensorial order parameters of rank 1 to n model substructure behavior better, and for both classes of these, the book analyzes the thermodynamic conditions of consistency. Granular solid-fluid mixtures are generally modeled by complementing the Boltzmann laws with a balance of fluctuation (kinetic) energy of the particles. The book closes by presenting a full Reynolds averaging procedure that accounts for higher correlation terms e.g. a k-epsilon formulation in classical turbulence. However, because the volume fraction is an additional variable, the theory also incorporates 'k-epsilon equations' for the volume fraction.

This third volume describes continuous bodies treated as classical (Boltzmann) and spin (Cosserat) continua or fluid mixtures of such bodies. It discusses systems such as Boltzmann continua (with trivial angular momentum) and Cosserat continua (with nontrivial spin balance) and formulates the balance law and deformation measures for these including multiphase complexities. Thermodynamics is treated in the spirit of Muller-Liu: it is applied to Boltzmann-type fluids in three dimensions that interact with neighboring fluids on two-dimensional contact surfaces and/or one-dimensional contact lines. For all these situations it formulates the balance laws for mass, momenta, energy, and entropy. Further, it introduces constitutive modeling for 3-, 2-, 3-d body parts for general processes and materially objective variable sets and their reduction to equilibrium and non-equilibrium forms. Typical (reduced) fluid spin continua are liquid crystals. Prominent nematic examples of these include the Ericksen-Leslie-Parodi (ELP) formulation, in which material particles are equipped with material unit vectors (directors). Nematic liquid crystals with tensorial order parameters of rank 1 to n model substructure behavior better, and for both classes of these, the book analyzes the thermodynamic conditions of consistency. Granular solid-fluid mixtures are generally modeled by complementing the Boltzmann laws with a balance of fluctuation (kinetic) energy of the particles. The book closes by presenting a full Reynolds averaging procedure that accounts for higher correlation terms e.g. a k-epsilon formulation in classical turbulence. However, because the volume fraction is an additional variable, the theory also incorporates 'k-epsilon equations' for the volume fraction.

In this book fluid mechanics and thermodynamics (F&T) are approached as interwoven, not disjoint fields. The book starts by analyzing the creeping motion around spheres at rest: Stokes flows, the Oseen correction and the Lagerstrom-Kaplun expansion theories are presented, as is the homotopy analysis. 3D creeping flows and rapid granular avalanches are treated in the context of the shallow flow approximation, and it is demonstrated that uniqueness and stability deliver a natural transition to turbulence modeling at the zero, first order closure level. The difference-quotient turbulence model (DQTM) closure scheme reveals the importance of the turbulent closure schemes' non-locality effects. Thermodynamics is presented in the form of the first and second laws, and irreversibility is expressed in terms of an entropy balance. Explicit expressions for constitutive postulates are in conformity with the dissipation inequality. Gas dynamics offer a first application of combined F&T. The book is rounded out by a chapter on dimensional analysis, similitude, and physical experiments.

This first volume discusses fluid mechanical concepts and their applications to ideal and viscous processes. It describes the fundamental hydrostatics and hydrodynamics, and includes an almanac of flow problems for ideal fluids. The book presents numerous exact solutions of flows in simple configurations, each of which is constructed and graphically supported. It addresses ideal, potential, Newtonian and non-Newtonian fluids. Simple, yet precise solutions to special flows are also constructed, namely Blasius boundary layer flows, matched asymptotics of the Navier-Stokes equations, global laws of steady and unsteady boundary layer flows and laminar and turbulent pipe flows. Moreover, the well-established logarithmic velocity profile is criticised.

The ongoing thread in this volume of Physics of Lakes is the presentation of different methods of investigation for processes taking place in real lakes with a view to understanding lakes as components of the geophysical environment. It is divided into three parts. Part I is devoted to numerical modeling techniques and demonstrates that (i) wind-induced currents in depth-integrated models can only adequately predict current fields for extremely shallow lakes, and (ii) that classical multi-layered simulation models can only adequately reproduce current and temperature distributions when the lake is directly subjected to wind, but not the post-wind oscillating response. This makes shock capturing discretization techniques and Mellor-Yamada turbulence closure schemes necessary, as well as extremely high grid resolution to reduce the excessive numerical diffusion. Part II is devoted to the presentation of principles of observation and laboratory experimental procedures. It details the principles of operation for current, temperature, conductivity and other sensors applied in the field. It also discusses the advantages and limitations of common measuring methods like registration from stationary or drifting buoys, sounding and profiling from a boat, etc. Questions of data accuracy, quality, and reliability are also addressed. The use of laboratory experiments on a rotating platform is based on an exposition of dimensional analysis and model theory and illustrated using Lake Constance as an example. Part III gives an account of the dynamics of lake water as a particle-laden fluid, which, coupled with the transport of the bottom sediments, leads to morphodynamic changes of the bathymetry in estuarine and possibly whole lake regions. An elegant spatially one-dimensional theory makes it possible to derive analytic solutions of deltaic formations which are corroborated by laboratory experiments. A full three-dimensional description of the evolution of the alluvial bathymetry under prescribed tributary sediment input indicates a potential subject for future research.

In this book fluid mechanics and thermodynamics (F&T) are approached as interwoven, not disjoint fields. The book starts by analyzing the creeping motion around spheres at rest: Stokes flows, the Oseen correction and the Lagerstrom-Kaplun expansion theories are presented, as is the homotopy analysis. 3D creeping flows and rapid granular avalanches are treated in the context of the shallow flow approximation, and it is demonstrated that uniqueness and stability deliver a natural transition to turbulence modeling at the zero, first order closure level. The difference-quotient turbulence model (DQTM) closure scheme reveals the importance of the turbulent closure schemes' non-locality effects. Thermodynamics is presented in the form of the first and second laws, and irreversibility is expressed in terms of an entropy balance. Explicit expressions for constitutive postulates are in conformity with the dissipation inequality. Gas dynamics offer a first application of combined F&T. The book is rounded out by a chapter on dimensional analysis, similitude, and physical experiments.

This first volume discusses fluid mechanical concepts and their applications to ideal and viscous processes. It describes the fundamental hydrostatics and hydrodynamics, and includes an almanac of flow problems for ideal fluids. The book presents numerous exact solutions of flows in simple configurations, each of which is constructed and graphically supported. It addresses ideal, potential, Newtonian and non-Newtonian fluids. Simple, yet precise solutions to special flows are also constructed, namely Blasius boundary layer flows, matched asymptotics of the Navier-Stokes equations, global laws of steady and unsteady boundary layer flows and laminar and turbulent pipe flows. Moreover, the well-established logarithmic velocity profile is criticised.

The overwhelming focus of this 2nd volume of "Physics of Lakes" is adequately expressed by its subtitle "Lakes as Oscillators". It deals with barotropic and baroclinic waves in homogeneous and stratified lakes on the rotating Earth and comprises 12 chapters, starting with rotating shallow-water waves, demonstrating their classification into gravity and Rossby waves for homogeneous and stratified water bodies. This leads to gravity waves in bounded domains of constant depth, Kelvin, Poincare and Sverdrup waves, reflection of such waves in gulfs and rectangles and their description in sealed basins as barotropic 'inertial waves proper'. The particular application to gravity waves in circular and elliptical basins of constant depth leads to the description of Kelvin-type and Poincare-type waves and their balanced description in basins of arbitrary geometry on the rotating Earth. Consideration of two-, three- and n-layer fluids with sharp interfaces give rise to the description of gravity waves of higher order baroclinicity with experimental corroboration in a laboratory flume and e.g. in Lake of Lugano, Lake Banyoles and Lake Biwa. Barotropic wave modes in Lake Onega with complex geometry show that data and computational output require careful interpretation. Moreover, a summer field campaign in Lake of Lugano and its two-layer modal analysis show that careful statistical analyses of the data are requested to match data with computational results. Three chapters are devoted to topographic Rossby waves. Conditions are outlined for which these waves are negligibly affected by baroclinicity. Three classes of these large period modes are identified: channel modes, so-called Ball modes and bay modes, often with periods which lie very close together. The last chapter deals with an entire class of Chrystal-type equations for barotropic waves in elongated basins which incorporate the effects of the rotation of the Earth.

The ongoing thread in this volume of Physics of Lakes is the presentation of different methods of investigation for processes taking place in real lakes with a view to understanding lakes as components of the geophysical environment. It is divided into three parts. Part I is devoted to numerical modeling techniques and demonstrates that (i) wind-induced currents in depth-integrated models can only adequately predict current fields for extremely shallow lakes, and (ii) that classical multi-layered simulation models can only adequately reproduce current and temperature distributions when the lake is directly subjected to wind, but not the post-wind oscillating response. This makes shock capturing discretization techniques and Mellor-Yamada turbulence closure schemes necessary, as well as extremely high grid resolution to reduce the excessive numerical diffusion. Part II is devoted to the presentation of principles of observation and laboratory experimental procedures. It details the principles of operation for current, temperature, conductivity and other sensors applied in the field. It also discusses the advantages and limitations of common measuring methods like registration from stationary or drifting buoys, sounding and profiling from a boat, etc. Questions of data accuracy, quality, and reliability are also addressed. The use of laboratory experiments on a rotating platform is based on an exposition of dimensional analysis and model theory and illustrated using Lake Constance as an example. Part III gives an account of the dynamics of lake water as a particle-laden fluid, which, coupled with the transport of the bottom sediments, leads to morphodynamic changes of the bathymetry in estuarine and possibly whole lake regions. An elegant spatially one-dimensional theory makes it possible to derive analytic solutions of deltaic formations which are corroborated by laboratory experiments. A full three-dimensional description of the evolution of the alluvial bathymetry under prescribed tributary sediment input indicates a potential subject for future research.

Internal wave dynamics in lakes (and oceans) is an important physical component of geophysical fluid mechanics of 'quiescent' water bodies of the Globe. The formation of internal waves requires seasonal stratification of the water bodies and generation by (primarily) wind forces. Because they propagate in basins of variable depth, a generated wave field often experiences transformation from large basin-wide scales to smaller scales. As long as this fission is hydrodynamically stable, nothing dramatic will happen. However, if vertical density gradients and shearing of the horizontal currents in the metalimnion combine to a Richardson number sufficiently small (< 1/4), the light epilimnion water mixes with the water of the hypolimnion, giving rise to vertical diffusion of substances into lower depths. This meromixis is chiefly responsible for the ventilation of the deeper waters and the homogenization of the water through the lake depth. These processes are mainly formed as a result of the physical conditions, but they play biologically an important role in the trophicational state of the lake.

This first volume in the treatise on the Physics of Lakes deals with the formulation of the mathematical and physical background. A large number of lakes on Earth are described, presenting their morphology as well as the causes of their response to the driving environment. Because the physics of lakes cannot be described without the language used in mathematics, these subjects are introduced first by using the simplest approach and with utmost care, assuming only a limited college knowledge of classical Newtonian physics, and continues with increasing complexity and elegance, starting with the fundamental equations of Lake Hydrodynamics in the form of 'primitive equations' and leading to a detailed treatment of angular momentum and vorticity. Following the presentation of these fundamentals turbulence modeling is introduced with Reynolds, Favre and other non-ergodic filters. The derivation of averaged field equations is presented with different closure schemes, including the k- model for a Boussinesq fluid and early anisotropic closure schemes. This is followed by expositions of surface gravity waves without rotation and an analysis of the role played by the distribution of mass within water bodies on the Earth, leading to a study of internal waves. The vertical structure of wind-induced currents in homogeneous and stratified waters and the Ekman theory and some of its extensions close this first volume of Physics of Lakes. The last chapter collects formulas for the phenomenological coefficients of water.

A "Sonderforschungsbereich" (SFB) is a programme of the "Deutsche For schungsgemeinschaft" to financially support a concentrated research effort of a number of scientists located principally at one University, Research La boratory or a number of these situated in close proximity to one another so that active interaction among individual scientists is easily possible. Such SFB are devoted to a topic, in our case "Deformation and Failure in Metallic and Granular M aterialK', and financing is based on a peer reviewed proposal for three (now four) years with the intention of several prolongations after evaluation of intermediate progress and continuation reports. An SFB is terminated in general by a formal workshop, in which the state of the art of the achieved results is presented in oral or I and poster communications to which also guests are invited with whom the individual project investigators may have collaborated. Moreover, a research report in book form is produced in which a number of articles from these lectures are selected and collected, which present those research results that withstood a rigorous reviewing pro cess (with generally two or three referees). The theme deformation and failure of materials is presented here in two volumes of the Lecture Notes in Applied and Computational Mechanics by Springer Verlag, and the present volume is devoted to granular and porous continua. The complementary volume (Lecture Notes in Applied and Com putational Mechanics, vol. 10, Eds. K. HUTTER & H."

This book is devoted to the deformation and failure in metallic materials, summarizing the results of a research programme financed by the "Deutsche Forschungsgemeinschaft." It presents the recent engineering as well as mathematical key aspects of this field for a broad community. Its main focus is on the constitutive behaviour as well as the damage and fracture of metallic materials, covering their mathematical foundation, modelling and numerics, but also relevant experiments and their verification.

Mixture concepts are nowadays used in a great number of subjects of the - ological, chemical, engineering, natural and physical sciences (to name these alphabetically) and the theory of mixtures has attained in all these dis- plines a high level of expertise and specialisation. The digression in their development has on occasion led to di?erences in the denotation of special formulations as 'multi-phase systems' or 'non-classical mixtures', 'structured mixtures', etc., and their representatives or defenders often emphasise the di?erences of these rather than their common properties. Thismonographisanattempttoviewtheoreticalformulationsofprocesses which take place as interactions among various substances that are spatially intermixedandcanbeviewedtocontinuously?llthespacewhichtheyoccupy as mixtures. Moreover, we shall assume that the processes can be regarded to becharacterisedbyvariableswhichobeyacertaindegreeofcontinuityintheir evolution, so that the relevant processes can be described mathematically by balance laws, in global or local form, eventually leading to di?erential and/or integralequations, towhichtheusualtechniquesoftheoreticalandnumerical analysis can be applied. Mixtures are generally called non-classical, if, apart from the physical laws (e. g. balances of mass, momenta, energy and entropy), also further laws are postulated, whicharelessfundamental, butmaydescribesomefeaturesofthe micro-structure on the macroscopic level. In a mixture of ?uids and solids - thesearesometimescalledparticleladensystems-thefractionofthevolume that is occupied by each constituent is a signi?cant characterisation of the micro-structure that exerts some in?uence on the macro-level at which the equations governing the processes are formulated. For solid-?uid mixtures at high solids fraction where particle contact is essential, friction between the particles gives rise to internal stresses, which turn out to be best described by an internal symmetric tensor valued variable.

Internal wave dynamics in lakes (and oceans) is an important physical component of geophysical fluid mechanics of 'quiescent' water bodies of the Globe. The formation of internal waves requires seasonal stratification of the water bodies and generation by (primarily) wind forces. Because they propagate in basins of variable depth, a generated wave field often experiences transformation from large basin-wide scales to smaller scales. As long as this fission is hydrodynamically stable, nothing dramatic will happen. However, if vertical density gradients and shearing of the horizontal currents in the metalimnion combine to a Richardson number sufficiently small (< 1/4), the light epilimnion water mixes with the water of the hypolimnion, giving rise to vertical diffusion of substances into lower depths. This meromixis is chiefly responsible for the ventilation of the deeper waters and the homogenization of the water through the lake depth. These processes are mainly formed as a result of the physical conditions, but they play biologically an important role in the trophicational state of the lake.

The overwhelming focus of this 2nd volume of "Physics of Lakes" is adequately expressed by its subtitle "Lakes as Oscillators". It deals with barotropic and baroclinic waves in homogeneous and stratified lakes on the rotating Earth and comprises 12 chapters, starting with rotating shallow-water waves, demonstrating their classification into gravity and Rossby waves for homogeneous and stratified water bodies. This leads to gravity waves in bounded domains of constant depth, Kelvin, Poincare and Sverdrup waves, reflection of such waves in gulfs and rectangles and their description in sealed basins as barotropic 'inertial waves proper'. The particular application to gravity waves in circular and elliptical basins of constant depth leads to the description of Kelvin-type and Poincare-type waves and their balanced description in basins of arbitrary geometry on the rotating Earth. Consideration of two-, three- and n-layer fluids with sharp interfaces give rise to the description of gravity waves of higher order baroclinicity with experimental corroboration in a laboratory flume and e.g. in Lake of Lugano, Lake Banyoles and Lake Biwa. Barotropic wave modes in Lake Onega with complex geometry show that data and computational output require careful interpretation. Moreover, a summer field campaign in Lake of Lugano and its two-layer modal analysis show that careful statistical analyses of the data are requested to match data with computational results. Three chapters are devoted to topographic Rossby waves. Conditions are outlined for which these waves are negligibly affected by baroclinicity. Three classes of these large period modes are identified: channel modes, so-called Ball modes and bay modes, often with periods which lie very close together. The last chapter deals with an entire class of Chrystal-type equations for barotropic waves in elongated basins which incorporate the effects of the rotation of the Earth.

The book unifies classical continuum mechanics and turbulence modeling, i.e. the same fundamental concepts are used to derive model equations for material behaviour and turbulence closure and complements these with methods of dimensional analysis. The intention is to equip the reader with the ability to understand the complex nonlinear modeling in material behaviour and turbulence closure as well as to derive or invent his own models. Examples are mostly taken from environmental physics and geophysics.

This first volume in the treatise on the Physics of Lakes deals with the formulation of the mathematical and physical background. A large number of lakes on Earth are described, presenting their morphology as well as the causes of their response to the driving environment. Because the physics of lakes cannot be described without the language used in mathematics, these subjects are introduced first by using the simplest approach and with utmost care, assuming only a limited college knowledge of classical Newtonian physics, and continues with increasing complexity and elegance, starting with the fundamental equations of Lake Hydrodynamics in the form of 'primitive equations' and leading to a detailed treatment of angular momentum and vorticity. Following the presentation of these fundamentals turbulence modeling is introduced with Reynolds, Favre and other non-ergodic filters. The derivation of averaged field equations is presented with different closure schemes, including the k- model for a Boussinesq fluid and early anisotropic closure schemes. This is followed by expositions of surface gravity waves without rotation and an analysis of the role played by the distribution of mass within water bodies on the Earth, leading to a study of internal waves. The vertical structure of wind-induced currents in homogeneous and stratified waters and the Ekman theory and some of its extensions close this first volume of Physics of Lakes. The last chapter collects formulas for the phenomenological coefficients of water.

This book delivers a thorough derivation of nonrelativistic interaction models of electromagnetic field theories with thermoelastic solids and viscous fluids, the intention being to derive unique representations for the observable field quantities. This volume is intended for and will be useful to students and researchers working on all aspects of electromagneto-mechanical interactions in the materials sciences of complex solids and fluids.

The book unifies classical continuum mechanics and turbulence modeling, i.e. the same fundamental concepts are used to derive model equations for material behaviour and turbulence closure and complements these with methods of dimensional analysis. The intention is to equip the reader with the ability to understand the complex nonlinear modeling in material behaviour and turbulence closure as well as to derive or invent his own models. Examples are mostly taken from environmental physics and geophysics.

 This book is devoted to the deformation and failure in metallic materials, summarizing the results of a research programme financed by the "Deutsche Forschungsgemeinschaft". It presents the recent engineering as well as mathematical key aspects of this field for a broad community. Its main focus is on the constitutive behaviour as well as the damage and fracture of metallic materials, covering their mathematical foundation, modelling and numerics, but also relevant experiments and their verification.

This book summarizes the results of a research programme financed by the "Deutsche Forschungsgemeinschaft" in the field of deformation and failure in granular and porous media. It presents the recent engineering as well as mathematical key aspects of this field for a broad community. Its main focus is on the dynamics response under large and catastrophic deformations, covering the fundamentals and dynamical processes of porous and granular materials as well as their related sub-scale and micro-mechanical effects.Special emphasis is placed on the modeling and theory of avalanches, debris and mud flows.

This book consists of peer-reviewed articles and reviews presented as lectures at the Sixth International Symposium on Thermal Engineering and Sciences for Cold Regions in Darmstadt, Germany. It addresses all relevant aspects of thermal physics and engineering in cold regions, such as the Arctic regions. These environments present many unique freezing and melting phenomena and the relevant heat and mass transfer processes are of basic importance with respect to both the technological applications and the natural context in which they occur. Intended for physicists, engineers, geoscientists, climatologists and cryologists alike, these proceedings cover topics such as: ice formation and decay, heat conduction with phase change, convection with freezing and melting, thermal properties at low temperature, frost heave and permafrost, climate impact in cold regions, thermal design of structures, bio-engineering in cold regions, and many more.

This book is an account on the thermomechanical behaviour of granular and porous materials and deals with experiments, theoretical deduction of macroscale equations by means of averaging from microscale properties, embedding the macroscopic description into a continuum-thermodynamical and statistical context and analysis of solutions of macroscopic models by numerical techniques. It addresses itself to engineers (chemical, civil, mechanical) applied mathematicians and physicists at the advanced student or Ph. D. level at universities, research centres and in industry.