# EOSC 352 · Geophysical Continuum Dynamics

Introduction to tensor calculus and continuum mechanics. Stress, strain and strain-rate tensors. Mass, momentum and energy balance. Applications to problems of geophysical heat transport, elasticity and fluid dynamics illustrated using MATLAB. [3-0] Prerequisite: One of EOSC 250, MATH 317.

#### Course Availability & Schedule

**Instructor:** Christian Schoof

**TA:** Luke Brown

#### Textbook

Required: course lecture notes (see below)

Recommended:

- Continuum mechanics by G. E. Mase in the McGraw Hill Schaum Outline Series.
- Geodynamics by D. Turcotte and G. Schubert

I highly recommend the text by Mase as it gives a very succinct description of the basic concepts and the mathematics involved. The text by Turcotte and Schubert also gives an abbreviated introduction, and applies continuum mechanics to real Earth Sciences problems. Some of the material in Turcotte and Schubert goes beyond the present course.

#### Course Content

**Division of marks:**

Assignments - 40%

Mid-term 10 %

Quizzes/Class participation - 10%

Final - 30 %

*Mid-Term Examination - tba*

**Assignments:**

I expect to set 4-5 assignments in this course. They will be due at the end of class on the date indicated below (which I will generally try to make a Thursday). No marks for late assignments without a good reason

**Office hours**

Wednesday 2-3 pm, Canvas Collaboarte Ultra Course Room. Contact me if you can't make this but need to see me. I may make this a by-appointment office hour if there is little regular uptake.

#### Lecture Topics

- Review of continuum physics
- Mathematical tools: volume and surface integrals, gradients and divergences
- Transport of conserved quantities: fluxes
- Conservation laws and the divergence theorem
- Partial differential equations as continuum physics models
- Methods: steady states, similarity and wave solutions for the heat equation
- Length scales, non-dimensionalization and dimensionless parameters
- Continuum mechanics: extending scalar continuum physics to vectors
- Fluxes for vectors: tensors
- Tensor calculus and subscript notation
- The stress tensor and conservation of linear and angular momentum
- Viscous flow
- Time permitting, an introduction to convection
- Time permitting, principal strsses

**Background material:**

- course information sheet
- vector calculus practice
- vector calculus practice answers
- Differential equations lecture notes
- Density and volume integrals lecture notes
- Flux and surface integrals lecture notes
- Divergence theorem notes
- Conservation of enegry lecture notes
- Heat flux and gradient lecture notes
- Heat equation notes
- Line integrals notes

**Course materials for EOS 352:**

- EOS 352 Conservation laws notes
- EOS 352 Complex variables notes
- EOS 352 Temperature waves notes
- Temperature wave movie
- EOS 352 Scaling notes
- EOS 352 Similarity solution notes
- EOS 352 Subscript notation notes
- EOS 352 Momentum conservation notes
- EOS 352 Angular momentum conservation notes
- EOS 352 summary of conservation laws
- EOS 352 notes on scalars, vectors and tensors
- EOS 352 fluids notes
- EOS 352 porous media notes
- EOS 352 Fourier series notes
- EOS 352 notes on convection in porous media
- Convection movie
- EOS 352 expanded heat equation notes
- Galerkin solver for porous medium convection problem
- For supplementary reading on temperature waves, see Turcotte and Schubert, Geodynamics, chapter 4, especially section 4.14
- For supplementary reading on similarity solutions, see Turcotte and Schubert, Geodynamics, chapter 4, especially section 4.15
- For supplementary reading on nondimensionalization, see Middleton and Wilcox, Mechanics in the Earth and Environmental Sciences, chapter 3
- For supplementary reading on vector conservation laws / momentum conservation, see Turcotte and Schubert, Geodynamics, chapter 2
- HIGHLY RECOMMENDED: For supplementary reading on subscript notation and vector conservation laws / momentum conservation, see Mase, Continuum Mechanics, chapters on Mathematical Foundations, Analysis of Stress, Motion and Flow and Fundamental Laws of Continuum Mechanics (chapters 1, 2, 4 and 5)

**Sample exam materials:**

- Equation sheet
- Practice midterm
- Detailed midterm answers
- 2010 midterm answers
- Practice final exam
- Practice final exam answers
- 2010 final exam
- 2010 final exam answers
- 2011 final exam
- 2011 final exam answers
- 2012 final exam
- 2012 final exam answers

#### Homework

- Vector calculus and differential equations review homework, due January 20tth. 2 % of final grade
- Homework for Tuesday January 24th.
- Answers to homework for Tuesday January 24th.
- Homework for Tuesday February 14: read the notes on complex variables and do all exercises therein