Mathematical Models of Solids and Fluids

✍ Scribed by Pascal Grange

• Publisher: Liverpool University Press
• Year: 2021
• Tongue: English
• Leaves: 171
• Category: Library

✦ Table of Contents

Cover
Contents
Introduction
1 Description of solids and fluids
1.1 Tensors and continuum mechanics: motivations and history
1.1.1 Continuous media: solids and fluids
1.1.2 Systems of coordinates
1.2 Notations and conventions
1.2.1 Notation: the summation rule over repeated indices
1.2.2 Scalar product, Kronecker symbol
1.3 Changes of orthonormal basis
1.3.1 Transformation of the components of vectors
1.3.2 Transformation of the matrix of an endomorphism under a change of orthonormal basis
1.4 Generalisation: definition of tensors
1.5 Exercises
2 Kinematics of continuous media
2.1 Lagrange description of flows
2.2 Transformation of tangent vectors
2.3 Application to deformations
2.4 Exercises
3 Dynamics of continuous media
3.1 Body forces, surface forces, Cauchy postulate
3.2 Stokes' theorem
3.3 Surface forces in the normal vector
3.3.1 Balance equation of a thin cylinder
3.3.2 Balance equation of an infinitesimal tetrahedron
3.4 Balance equations for a continuous medium
3.4.1 Forces sum to zero at static equilibrium
3.4.2 Moments of forces sum to zero at static equilibrium
3.5 Exercises
4 Boundary conditions
4.1 The hydrostatic pressure
4.2 Boundary conditions
4.3 Statically admissible stress tensors
4.4 Exercises
5 Linear elasticity: material laws
5.1 Hooke's law (for a cylinder)
5.2 Small deformations
5.3 Strain as a function of stress
5.4 Stress as a function of strain
5.5 Exercises
6 Elasticity, elementary problems
6.1 The Navier equations of linear elasticity
6.2 Solution for a spherical shell
6.2.1 Explicit form of the Navier equations with spherical symmetry
6.2.2 Determination of the integration constants
6.3 Exercises
7 Viscous fluids
7.1 Description of continuous media
7.2 Description of fluids
7.2.1 The Euler description of fluids
7.2.2 Conservation of mass (the continuity equation)
7.2.3 Acceleration of a particle of fluid
7.3 Viscous fluids
7.3.1 Thought experiment on friction
7.3.2 Model: material law for Newtonian fluids
7.3.3 Boundary conditions on the velocity field for Newtonian fluids
7.4 Exercises
8 Viscosimetric flows
8.1 Navier–Stokes equations for Poiseuille flow
8.1.1 Functional form of the velocity field
8.1.2 Incompressibility
8.1.3 Explicit form of each term in the Navier–Stokes equations
8.1.4 Solution of the equations of motion
Separation of variables
Boundary conditions
8.2 The Poiseuille law
8.2.1 Derivation of the flow rate
8.2.2 The Poiseuille flow is viscosimetric
8.3 Exercises
9 Cylindrical Couette flow
9.1 Couette flow
9.2 Solution of the Navier–Stokes equations
9.2.1 Cylindrical coordinates
9.2.2 Navier–Stokes equations in cylindrical coordinates
9.2.3 Integration of the Navier–Stokes equations
9.3 Application: Couette flow as a viscosimeter
Exercises
10 Solutions to the exercises
Exercises in Chapter 1
Exercises in Chapter 2
Exercises in Chapter 3
Exercises in Chapter 4
Exercises in Chapter 5
Exercises in Chapter 6
Exercises in Chapter 7
Exercises in Chapter 8
Exercises in Chapter 9
Glossary of terms
Physical constants,orders of magnitude
Bibliography
Index