Book contents
- Frontmatter
- Contents
- Preface
- 1 An Introduction to Vortex Dynamics for Incompressible Fluid Flows
- 2 The Vorticity-Stream Formulation of the Euler and the Navier-Stokes Equations
- 3 Energy Methods for the Euler and the Navier–Stokes Equations
- 4 The Particle-Trajectory Method for Existence and Uniqueness of Solutions to the Euler Equation
- 5 The Search for Singular Solutions to the 3D Euler Equations
- 6 Computational Vortex Methods
- 7 Simplified Asymptotic Equations for Slender Vortex Filaments
- 8 Weak Solutions to the 2D Euler Equations with Initial Vorticity in L∞
- 9 Introduction to Vortex Sheets, Weak Solutions, and Approximate-Solution Sequences for the Euler Equation
- 10 Weak Solutions and Solution Sequences in Two Dimensions
- 11 The 2D Euler Equation: Concentrations and Weak Solutions with Vortex-Sheet Initial Data
- 12 Reduced Hausdorff Dimension, Oscillations, and Measure-Valued Solutions of the Euler Equations in Two and Three Dimensions
- 13 The Vlasov–Poisson Equations as an Analogy to the Euler Equations for the Study of Weak Solutions
- Index
1 - An Introduction to Vortex Dynamics for Incompressible Fluid Flows
Published online by Cambridge University Press: 03 February 2010
- Frontmatter
- Contents
- Preface
- 1 An Introduction to Vortex Dynamics for Incompressible Fluid Flows
- 2 The Vorticity-Stream Formulation of the Euler and the Navier-Stokes Equations
- 3 Energy Methods for the Euler and the Navier–Stokes Equations
- 4 The Particle-Trajectory Method for Existence and Uniqueness of Solutions to the Euler Equation
- 5 The Search for Singular Solutions to the 3D Euler Equations
- 6 Computational Vortex Methods
- 7 Simplified Asymptotic Equations for Slender Vortex Filaments
- 8 Weak Solutions to the 2D Euler Equations with Initial Vorticity in L∞
- 9 Introduction to Vortex Sheets, Weak Solutions, and Approximate-Solution Sequences for the Euler Equation
- 10 Weak Solutions and Solution Sequences in Two Dimensions
- 11 The 2D Euler Equation: Concentrations and Weak Solutions with Vortex-Sheet Initial Data
- 12 Reduced Hausdorff Dimension, Oscillations, and Measure-Valued Solutions of the Euler Equations in Two and Three Dimensions
- 13 The Vlasov–Poisson Equations as an Analogy to the Euler Equations for the Study of Weak Solutions
- Index
Summary
In this book we study incompressible high Reynolds numbers and incompressible inviscid flows. An important aspect of such fluids is that of vortex dynamics, which in lay terms refers to the interaction of local swirls or eddies in the fluid. Mathematically we analyze this behavior by studying the rotation or curl of the velocity field, called the vorticity. In this chapter we introduce the Euler and the Navier–Stokes equations for incompressible fluids and present elementary properties of the equations. We also introduce some elementary examples that both illustrate the kind of phenomena observed in hydrodynamics and function as building blocks for more complicated solutions studied in later chapters of this book.
This chapter is organized as follows. In Section 1.1 we introduce the equations, relevant physical quantities, and notation. Section 1.2 presents basic symmetry groups of the Euler and the Navier–Stokes equations. In Section 1.3 we discuss the motion of a particle that is carried with the fluid. We show that the particle-trajectory map leads to a natural formulation of how quantities evolve with the fluid. Section 1.4 shows how locally an incompressible field can be approximately decomposed into translation, rotation, and deformation components. By means of exact solutions, we show how these simple motions interact in solutions to the Euler or the Navier–Stokes equations. Continuing in this fashion, Section 1.5 examines exact solutions with shear, vorticity, convection, and diffusion. We show that although deformation can increase vorticity, diffusion can balance this effect.
- Type
- Chapter
- Information
- Vorticity and Incompressible Flow , pp. 1 - 42Publisher: Cambridge University PressPrint publication year: 2001