Book contents
- Frontmatter
- Contents
- Preface
- 1 The equations of motion
- 2 Dimensionless parameters and stability
- 3 Turbulence
- 4 Degrees of freedom, dynamical systems, and attractors
- 5 On the existence, uniqueness, and regularity of solutions
- 6 Ladder results for the Navier-Stokes equations
- 7 Regularity and length scales for the 2d and 3d Navier-Stokes equations
- 8 Exponential decay of the Fourier power spectrum
- 9 The attractor dimension for the Navier-Stokes equations
- 10 Energy dissipation rate estimates for boundary-driven flows
- Appendix A Inequalities
- References
- Index
1 - The equations of motion
Published online by Cambridge University Press: 02 February 2010
- Frontmatter
- Contents
- Preface
- 1 The equations of motion
- 2 Dimensionless parameters and stability
- 3 Turbulence
- 4 Degrees of freedom, dynamical systems, and attractors
- 5 On the existence, uniqueness, and regularity of solutions
- 6 Ladder results for the Navier-Stokes equations
- 7 Regularity and length scales for the 2d and 3d Navier-Stokes equations
- 8 Exponential decay of the Fourier power spectrum
- 9 The attractor dimension for the Navier-Stokes equations
- 10 Energy dissipation rate estimates for boundary-driven flows
- Appendix A Inequalities
- References
- Index
Summary
Introduction
The Navier-Stokes equations of fluid dynamics are a formulation of Newton's laws of motion for a continuous distribution of matter in the fluid state, characterized by an inability to support shear stresses. We will restrict our attention to the incompressible Navier-Stokes equations for a single component Newtonian fluid. Although they may be derived systematically from the microscopic description in terms of a Boltzmann equation, albeit with some additional fundamental assumptions, in this chapter we present a heuristic derivation designed to illustrate the elements of the physics contained in the equations.
Euler's equations for an incompressible fluid
First we consider an ideal inviscid fluid. The dependent variables in the so-called Eulerian description of fluid mechanics are the fluid density ρ(x, t), the velocity vector field u(x, t), and the pressure field ρ(x, t). Here x ∈ Rd is the spatial coordinate in a d-dimensional region of space (d typically takes values 2 or 3, with a default value of 3 in this chapter). An infinitesimal element of the fluid of volume δ V located at position x at time t has mass δm = ρ(x,t)δV and is moving with velocity u(x,t) and momentum δmu(x,t). The normal force directed into the infinitesimal volume across a face of area nda centered at x, where n is the outward directed unit vector normal to the face, is —np(x, t)δa. The pressure is the magnitude of the force per unit area, or normal stress, imposed on elements of the fluid from neighboring elements. These definitions are illustrated in Figure 1.1.
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- Applied Analysis of the Navier-Stokes Equations , pp. 1 - 22Publisher: Cambridge University PressPrint publication year: 1995