Foundations of Fluid Mechanics with Applications

Fluid mechanics (FM) is a branch of science dealing with the investi­ gation of flows of continua under the action of external forces. The fundamentals of FM were laid in the works of the famous scientists, such as L. Euler, M. V. Lomonosov, D. Bernoulli, J. L. Lagrange, A. Cauchy, L. Navier, S. D. Poisson, and other classics of science. Fluid mechanics underwent a rapid development during the past two centuries, and it now includes, along with the above branches, aerodynamics, hydrodynamics, rarefied gas dynamics, mechanics of multi phase and reactive media, etc. The FM application domains were expanded, and new investigation methods were developed. Certain concepts introduced by the classics of science, however, are still of primary importance and will apparently be of importance in the future. The Lagrangian and Eulerian descriptions of a continuum, tensors of strains and stresses, conservation laws for mass, momentum, moment of momentum, and energy are the examples of such concepts and results. This list should be augmented by the first and second laws of thermodynamics, which determine the character and direction of processes at a given point of a continuum. The availability of the conservation laws is conditioned by the homogeneity and isotrop­ icity properties of the Euclidean space, and the form of these laws is related to the Newton's laws. The laws of thermodynamics have their foundation in the statistical physics.

Contenu

1 Definitions of Continuum Mechanics.- 1.1 Vectors and Tensors.- 1.1.1 Covariant Differentiation.- 1.1.2 The Levi-Civita Tensor.- 1.1.3 Differential Operations.- 1.1.4 Physical Components of Vectors and Tensors.- 1.1.5 Eigenvalues and Eigenvectors of a Symmetric Tensor.- 1.1.6 The Ostrogradsky-Gauss Theorem.- 1.1.7 The Stokes Theorem.- 1.1.8 The Weyl Formula.- 1.2 Eulerian and Lagrangian Description of a Continuum: Strain Tensor.- 1.2.1 Lagrangian and Eulerian Description of a Continuum.- 1.2.2 Strain Tensor.- 1.2.3 A Condition for Compatibility of Deformations.- 1.2.4 Rate-of-Strain Tensor: Cauchy-Helmholtz Theorem.- 1.3 Stress Tensor.- 1.3.1 The Cauchy Stress Tensor in the Accompanying Coordinate System.- 1.3.2 Piola-Kirchhoff Stress Tensors in the Reference Frame and in the Eulerian Coordinates.- 1.3.3 Principal Values and Invariants of the Stress Tensor.- 1.3.4 Differentiation of the Stress Tensor with Respect to Time.- References.- 2 Fundamental Principles and Laws of Continuum Mechanics.- 2.1 Equations of Continuity, Motion, and Energy for a Continuum.- 2.1.1 Continuity Equation.- 2.1.2 Equations of Motion and of Momentum Moment.- 2.1.3 The Energy Conservation Law: The First and Second Laws of Thermodynamics.- 2.1.4 Equation of State (General Relations).- 2.1.5 Equations of an Ideal and Viscous, Heat-Conducting Gas.- 2.2 The Hamilton-Ostrogradsky's Variational Principle in Continuum Mechanics.- 2.2.1 Euler-Lagrange Equations in Lagrangian Coordinates.- 2.2.2 Hamilton's Equations in Lagrangian Coordinates.- 2.2.3 Euler-Lagrange Equations in Eulerian Coordinates and Murnaghan's Formula.- 2.3 Conservation Laws for Energy and Momentum in Continuum Mechanics.- 2.3.1 Conservation Laws in Cartesian Coordinates.- 2.3.2 Conservation Laws in an Arbitrary Coordinate System.- References.- 3 The Features of the Solutions of Continuum Mechanics Problems.- 3.1 Similarity and Dimension Theory in Continuum Mechanics.- 3.2 The Characteristics of Partial Differential Equations..- 3.3 Discontinuity Surfaces in Continuum Mechanics.- References.- 4 Ideal Fluid.- 4.1 Integrals of Motion Equations of Ideal Fluid and Gas.- 4.1.1 Motion Equations in the Gromeka-Lamb Form.- 4.1.2 The Bernoulli Integral.- 4.1.3 The Lagrange Integral.- 4.2 Planar Irrotational Steady Motions of an Ideal Incompressible Fluid.- 4.2.1 The Governing Equations of Planar Flows.- 4.2.2 The Potential Flow past the Cylinder.- 4.2.3 The Method of Conformal Mappings.- 4.2.4 The Problem of the Flow around a Slender Profile.- 4.3 Axisymmetric and Three-Dimensional Potential Ideal Incompressible Fluid Flows.- 4.3.1 Axially Symmetric Flows.- 4.3.2 The Method of Sources and Sinks.- 4.3.3 The Program prog4-5.nb.- 4.3.4 The Transverse Flow around the Body of Revolution: The Program prog4-6.nb.- 4.4 Nonstationary Motion of a Solid in the Fluid.- 4.4.1 Formulation of a Problem on Nonstationary Body Motion in Ideal Fluid.- 4.4.2 The Hydrodynamic Reactions at the Body Motion.- 4.4.3 Equations of Solid Motion in a Fluid under the Action of Given Forces.- 4.5 Vortical Motions of Ideal Fluid.- 4.5.1 The Theorems of Thomson, Lagrange, and Helmholtz.- 4.5.2 Motion Equations in Friedmann's Form.- 4.5.3 The Biot-Savart Formulas and the Straight Vortex Filament.- References.- 5 Viscous Fluid.- 5.1 General Equations of Viscous Incompressible Fluid.- 5.1.1 The Navier-Stokes Equations.- 5.1.2 Formulation of Problems for the System of the Navier-Stokes Equations.- 5.2 Viscous Fluid Flows at Small Reynolds Numbers.- 5.2.1 Exact Solutions of the System of Equations for a Viscous Fluid.- 5.2.2 Viscous Fluid Motion between Two Rotating Coaxial Cylinders.- 5.2.3 The Viscous Incompressible Fluid Flow around a Sphere at Small Reynolds Numbers.- 5.3 Viscous Fluid Flows at Large Reynolds Numbers.- 5.3.1 Prandtl's Theory of Boundary Layers.- 5.3.2 Boundary Layer of a Flat Plate.- 5.4 Turbulent Fluid Flows.- 5.4.1 Basic Properties of Turbulent Flows.- 5.4.2 Laminar Flow Stability and Transition to Turbulence.- 5.4.3 Turbulent Fluid Flow.- References.- 6 Gas Dynamics.- 6.1 One-Dimensional Stationary Gas Flows.- 6.1.1 Governing Equations for Quasi-One-Dimensional Gas Flow.- 6.1.2 Gas Motion in a Variable Section Duct: Elementary Theory of the Laval Nozzle.- 6.1.3 Planar Shock Wave in Ideal Gas.- 6.1.4 Shock Wave Structure in Gas.- 6.2 Nonstationary One-Dimensional Flows of Ideal Gas.- 6.2.1 Planar Isentropic Waves.- 6.2.2 Gradient Catastrophe and Shock Wave Formation.- 6.3 Planar Irrotational Ideal Gas Motion (Linear Approximation).- 6.3.1 Governing Equations and Their Linearization.- 6.3.2 The Problem of the Flow around a Slender Profile.- 6.4 Planar Irrotational Stationary Ideal Gas Flow (General Case).- 6.4.1 Characteristics of Stationary Irrotational Flows of Ideal Gas, Simple Wave: The Prandtl-Meyer Flow.- 6.4.2 Chaplygin's Equations and Method.- 6.4.3 Oblique Shock Waves.- 6.4.4 Interference of Stationary Shock Waves.- 6.5 The Fundamentals of the Gasdynamic Design Technology.- 6.5.1 The Basic Algorithm.- 6.5.2 The Superposition Procedure.- 6.5.3 The Complement Procedure.- References.- 7 Multiphase Media.- 7.1 Mathematical Models of Multiphase Media.- 7.1.1 General Equations of the Mechanics of Multiphase Media.- 7.1.2 Equations of a Two-Phase Medium of the Type of Gas-Solid Particles.- 7.1.3 Equations of a Bicomponent Medium of Gas Mixture Type.- 7.2 Correctness of the Cauchy Problem: Relations at Discontinuities in Multiphase Media.- 7.2.1 The Characteristics of a System of Equations for Gas-Particle Mixtures and Correctness of the Cauchy Problem.- 7.2.2 Jump Relations.- 7.3 Quasi-One-Dimensional Flows of a Gas-Particle Mixture in Laval Nozzles.- 7.3.1 The Equations of the Quasi-One-Dimensional Flow of a Gas-Particle Mixture.- 7.3.2 The Flow of a Gas-Particle Mixture in the Laval Nozzle with Small Velocity and Temperature Lags of Particles.- 7.4 The Continual-Discrete Model and Caustics in the Pseudogas of Particles.- 7.4.1 The Equations of the Continual-Discrete Model of a Gas-Particle Mixture at a Small Volume Concentration of Particles.- 7.4.2 Investigation of Caustics in the Pseudogas of Particles.- 7.5 Nonstationary Processes in Gas-Particle Mixtures.- 7.5.1 Interaction of a Shock Wave with a Cloud of Particles.- 7.5.2 Acoustic Approximation in the Problem of Shock Wave Interaction with a Particle's Cloud at a Small.- Volume Concentration.- 7.6 The Flows of Heterogeneous Media without Regard for Inertial Effects.- 7.6.1 The Brownian Motion of Particles in a Fluid..- 7.6.2 Fluid Fi…