Flow, Deformation and Fracture Book contents
- Frontmatter
- Dedication
- Contents
- Foreword
- Preface
- Introduction
- 1 Idealized continuous media: the basic concepts
- 2 Dimensional analysis and physical similitude
- 3 The ideal incompressible fluid approximation: general concepts and relations
- 4 The ideal incompressible fluid approximation: analysis and applications
- 5 The linear elastic solid approximation. Basic equations and boundary value problems in the linear theory of elasticity
- 6 The linear elastic solid approximation. Applications: brittle and quasi-brittle fracture; strength of structures
- 7 The Newtonian viscous fluid approximation. General comments and basic relations
- 8 The Newtonian viscous fluid approximation. Applications: the boundary layer
- 9 Advanced similarity methods: complete and incomplete similarity
- 10 The ideal gas approximation. Sound waves; shock waves
- 11 Turbulence: generalities; scaling laws for shear flows
- 12 Turbulence: mathematical models of turbulent shear flows and of the local structure of turbulent flows at very large Reynolds numbers
- References
- Index
4 - The ideal incompressible fluid approximation: analysis and applications
Published online by Cambridge University Press: 05 June 2014
- Frontmatter
- Dedication
- Contents
- Foreword
- Preface
- Introduction
- 1 Idealized continuous media: the basic concepts
- 2 Dimensional analysis and physical similitude
- 3 The ideal incompressible fluid approximation: general concepts and relations
- 4 The ideal incompressible fluid approximation: analysis and applications
- 5 The linear elastic solid approximation. Basic equations and boundary value problems in the linear theory of elasticity
- 6 The linear elastic solid approximation. Applications: brittle and quasi-brittle fracture; strength of structures
- 7 The Newtonian viscous fluid approximation. General comments and basic relations
- 8 The Newtonian viscous fluid approximation. Applications: the boundary layer
- 9 Advanced similarity methods: complete and incomplete similarity
- 10 The ideal gas approximation. Sound waves; shock waves
- 11 Turbulence: generalities; scaling laws for shear flows
- 12 Turbulence: mathematical models of turbulent shear flows and of the local structure of turbulent flows at very large Reynolds numbers
- References
- Index
Summary
Physical meaning of the velocity potential. The Lavrentiev problem of a directed explosion
We now must clarify the direct physical meaning of the velocity potential: without understanding this it is impossible to formulate the Dirichlet boundary value problem: we have to prescribe the velocity potential at the boundary, but we do not know yet what the potential is.
Consider a body in a continuous medium which at t = t0 is at rest. Assume that at t = t0 each particle experiences a pressure pulse such that the pressure varies according to the law
Here θ(x) is a function of the position of the particle, and δ(z) is the generalized Dirac function. According to the simplest definition of this function, which is all we need for now,
for arbitrarily small positive ∊.
The motion begins from a state of rest before the pressure pulse starts. Therefore uf tge system of mass forces acting on the medium is a potential one, the Lagrange–Cauchy integral holds in the ideal incompressible fluid approximation:
We put (4.1) into (4.3) and integrate from t = t0 − ε to t = t0 + ε.
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- Flow, Deformation and FractureLectures on Fluid Mechanics and the Mechanics of Deformable Solids for Mathematicians and Physicists, pp. 63 - 78Publisher: Cambridge University PressPrint publication year: 2014