## The course

This page describes my notes from the winter 2012 session of the University of Toronto Continuum Mechanics course (PHY454H1S), taught by Prof. Kausik S. Das, which I took as a non-degree student.

The official course description at the time was:

The theory of continuous matter, including solid and fluid mechanics. Topics include the continuum approximation, dimensional analysis, stress, strain, the Euler and Navier-Stokes equations, vorticity, waves, instabilities, convection and turbulence.

That said, this was, at least at the time, a course that was really only about *fluid dynamics*. Anything related to solids was really just to introduce the stress and strain tensors as lead up to expressing the Navier-Stokes equation. There was nothing in this course about beam deformation, Euler stability, or similar topics that were touched on back when I took CIV102. Given the title and description, I’d expected some of those topics to be covered in more depth in this course.

My notes for the class are available

- in black and white, for $10 USD,
- or as a PDF, for $0,
- or as latex (see below.)

The paper version is softcover, published using amazon’s kindle direct publishing (aka createspace) and has a 6×9″ format (358 pages.)

Mathematica notebooks associated with the class are also available.

If you download the free PDF, compile the latex version, or buy a copy and feel undercharged, feel free to send some bitcoin my way.

## Contributing.

Should you wish to actively contribute typo fixes (or additions, editing, …) to these notes, you can do so by contacting me, or by forking your own copy of the associated git repositories and building the book pdf from source, and submitting a subsequent merge request.

git clone git@github.com:peeterjoot/latex-notes-compilations.git peeterjoot cd peeterjoot submods="figures/phy454-continuumechanics phy454-continuumechanics mathematica latex" for i in $submods ; do git submodule update --init $i (cd $i && git checkout master) done export PATH=`pwd`/latex/bin:$PATH cd phy454-continuumechanics make

I reserve the right to impose dictatorial control over any editing and content decisions, and may not accept merge requests as-is, or at all. That said, I’ll probably not refuse reasonable suggestions or merge requests.

## Contents:

- Copyright
- Document Version
- Dedication
- Preface
- Contents
- List of Figures
- 1 Introduction to continuum mechanics
- 1.1 Continuum Mechanics
- 1.2 Nomenclature and basic definitions
- 1.3 Texts
- 2 Strain Tensor
- 2.1 Deformations
- 2.2 Matrix representation, diagonalization, and deformed volume element
- 2.3 Strain in cylindrical coordinates
- 2.4 Compatibility condition compatibility condition for 2D strain
- 2.5 Compatibility condition for 3D strain
- 2.6 On the factor of two in the tensor definition
- 2.7 Summary
- 2.7.1 Strain Tensor
- 2.7.2 Diagonal strain representation
- 2.7.3 Strain in cylindrical coordinates
- 2.7.4 Compatibility condition
- 2.8 Problems
- 3 Stress tensor
- 3.1 Force per unit volume
- 3.2 Stress tensor in 2D
- 3.3 Stress tensor in 3D
- 3.4 Cauchy tetrahedron
- 3.5 Constitutive relation
- 3.6 Constitutive relation for Hydrostatic compression
- 3.7 Constitutive relation for uniaxial stress
- 3.8 Summary
- 3.8.1 Stress tensor
- 3.8.2 Constitutive relation
- 3.8.3 Uniform hydrostatic compression
- 3.8.4 Uniaxial stress. Young’s modulus. Poisson’s ratio
- 3.9 Problems
- 4 Elastodynamics
- 4.1 Elastic waves
- 4.2 P-waves
- 4.3 S-waves
- 4.4 Relative speeds of the p-waves and s-waves
- 4.5 Assuming a gradient plus curl representation
- 4.6 A couple summarizing statements
- 4.7 Phasor description of elastic waves
- 4.8 Some wave types described
- 4.9 Summary
- 4.9.1 Elastic displacement equation
- 4.9.2 Equilibrium
- 4.9.3 P-waves
- 4.9.4 S-waves
- 4.9.5 Scalar and vector potential representation
- 4.9.6 Phasor description
- 4.9.7 Some wave types
- 4.10 Problems
- 5 Navier-Stokes equation
- 5.1 Time dependent displacements
- 5.2 Comparing to elastostatics
- 5.3 Antisymmetric term, the vorticity
- 5.4 Symmetric term, the strain tensor
- 5.5 Newtonian Fluids
- 5.6 Dimensions of viscosity
- 5.7 Conservation of mass in fluid
- 5.8 Incompressible fluid
- 5.9 Conservation of momentum (Navier-Stokes equation)
- 5.10 Incompressible fluids
- 5.11 Boundary value conditions
- 5.12 Normals and tangents at fluid interfaces
- 5.13 Solutions by intuition
- 5.14 Summary
- 5.14.1 Vector displacements
- 5.14.2 Relative change in volume
- 5.14.3 Conservation of mass
- 5.14.4 Constitutive relation
- 5.14.5 Conservation of momentum (Navier-Stokes)
- 5.14.6 Observe the first order time derivative here
- 5.14.7 No slip condition
- 5.14.8 Traction vector matching at an interface
- 5.14.9 Flux
- 5.15 Problems
- 6 Hydrostatics
- 6.1 Steady state and static fluids
- 6.2 Height matching in odd geometries
- 6.3 Summary
- 6.3.1 Hydrostatics
- 6.3.2 Mass conservation through apertures
- 7 Bernoulli’s theorem
- 7.1 Derivation
- 7.2 Summary
- 7.2.1 Bernoulli equation
- 7.3 Problems
- 8 Surface tension
- 8.1 Traction vector at the interface
- 8.2 Surface tension gradients
- 8.3 Summary
- 8.3.1 Laplace pressure
- 8.3.2 Surface tension gradients
- 8.3.3 Surface tension for a spherical bubble
- 8.4 Problems
- 9 Nondimensionalisation
- 9.1 Scaling
- 9.2 Rescaling by characteristic length and velocity
- 9.3 Reynold’s number
- 9.4 Summary
- 9.4.1 Non-dimensionality and scaling
- 9.5 Problems
- 10 Boundary Layers
- 10.1 Time dependent flow
- 10.2 Unsteady rectilinear flow
- 10.3 Review. Impulsively started flow
- 10.4 Boundary layers
- 10.5 Universal behavior
- 10.6 Fluid flow over a solid body
- 10.6.1 Scaling arguments
- 10.7 Summary
- 10.7.1 Impulsive flow
- 10.7.2 Oscillatory flow
- 10.7.3 Blassius problem (boundary layer thickness in flow over plate)
- 10.8 Problems
- 11 Singular perturbation theory
- 11.1 Magnitude of the viscosity and inertial terms
- 11.2 Asymptotic solutions of ill conditioned equations
- 11.3 Summary
- 11.3.1 Singular perturbation theory
- 12 Thermal effects and stability
- 12.1 Stability
- 12.1.1 Stability. Some graphical illustrations
- 12.2 Characterizing stability
- 12.2.1 Case I. Oscillatory unstability
- 12.2.2 Case II. Marginal unstability
- 12.2.3 Case III. Neutral stability
- 12.3 A mathematical description
- 12.4 Thermal stability review. Rayleigh Benard Problem
- 12.5 Application of the perturbation to the energy equation
- 12.6 Non-dimensionalisation of the thermal velocity equation
- 12.7 Non-dimensionalization of the energy equation
- 12.8 Normal mode analysis
- 12.9 Back to our coupled equations
- 12.10 Multimedia presentations
- 12.11 Summary
- 12.11.1 Stability
- 12.11.2 Thermal stability: Rayleigh-Benard problem
- 12.12 Problems
- Appendixes
- A Strain Tensor in cylindrical and spherical coordinates
- A.1 Cylindrical coordinates
- A.2 For general coordinate representation
- A.3 Cartesian tensor
- A.4 Cylindrical tensor
- A.5 Spherical tensor
- A.6 Spherical tensor. Manual derivation
- B Non coordinate strain and traction vector representation
- B.1 Motivation
- B.2 Verifying the relationship
- B.3 Cylindrical strain tensor
- B.3.1 Outwards radial normal ncap equals rcap
- B.3.2 Azimuthal normal ncap equals phicap
- B.3.3 Longitudinal normal ncap equals zcap
- B.3.4 Summary
- B.4 Spherical strain tensor
- B.4.1 Outwards radial normal ncap equals rcap
- B.4.2 Polar normal ncap equals thetacap
- B.4.3 Azimuthal normal ncap equals phicap
- B.4.4 Summary
- C Poisson’s ratio and shear modulus relations
- D Surfaces
- D.1 Normals and tangents
- D.2 Review. Surfaces
- E Identities and proofs
- E.1 Error function properties
- E.2 A Fourier series refresher
- E.3 Vector identities
- F Attempt at general inclined flow problem
- F.1 Motivation
- F.2 Equations of motion
- F.3 Boundary value constraints
- F.4 Laplacian of Pressure and Vorticity
- F.4.1 Separation of variables?
- F.4.2 In terms of vorticity?
- F.4.3 Pressure and vorticity equations with the non-linear term retained
- F.4.4 Reworking slightly
- F.5 Now what?
- G Steady state velocity profile of stirred cup of non-bottomless coffee
- G.1 Motivation
- G.2 Navier-Stokes for the problem
- G.3 Working around the no-slip troubles at the base of the cup
- G.4 Spin down below the stir point
- H Mathematica notebooks
- Bibliography