 # MA 5019: Mathematical Continuum Mechanics

## News

• 2. lecture on Friday, 10:15-11:45 clock, MI 00.09.022

## Content

This course contains the essential conservation laws of continuum mechanics, i.e. mass, momentum and energy conservation. These laws are also satisfied for the classical Newtonian mechanics, therefore the knowledge of distributions is important. Also elastic materials are introduced in physical coordinates. The entropy principle is defined in a differential form and the free energy inequality is considered as a special case.

After this general part, we will focus on specific models, for example: Fluid Flows, Chemical and biological Reactions, Self-Gravitation, Liquid Crystals. At the request of the audience, other models will be treated.

• Conservation Laws
• Distributions
• Mass and Momentum Equations
• Coordinate Transformation
• Elasticity
• Observer Transformation
• Objectivity of Differential Equations
• Frame Indifference
• Entropy Principle
• Free Energy Inequality
• Navier-Stokes Equations
• Chemical and biological Reactions
• Self-Gravitation
• Liquid Crystals

## Lecture Schedule

Lecture   Date      Topic       Script
1. 9th Oct
Overview / Introduction
Mass and Momentum
(Definitions, Figure 1)
Conservation laws

Cont
Cont

Notations of Derivatives

Conservation law (I.1.1)
I.1.1 Representation of the divergence operator
2. 10th Oct
Conservation laws

Cont
Cont
Cont
I.1.3 Plane polar coordinates, I.1.4 Plain divergence
I.1.6 Relativity of velocity
I.1.7 Example (Mass conservation)
3. 16th Oct
Distributions

Distr
Distr
Distr
2.1, 2.2 Definition, 2.3 Property
2.4 Derivative and multiplication, 2.5 Measure and functions
2.7 Fundamental solution of the Laplace operator
4. 16th Oct
Gravitation

Cont
Cont
Cont
I.2.5 Moving mass point, I.2.6 Lemma
Newton's gravitation (I2.9), General gravitational law (I2.10)
I.2.10 Garvitational field of a point-shaped star, I.2.11 Uniqueness
5. 23th Oct
Gravitation  Cont
Cont
Cont
I.2.12 Theorem (Jump behaviour on a surface)
I.2.13 Gravitational fi eld of a globe
I.2.14 Convergence to a mass point
6. 23th Oct
Conservation of momentum
Cont
Cont
Cont
General mass-momentum equation (I3.1) and (I3.3)
I.3.1 Mass point, I.3.2 Collision of mass points
I.3.3 Multiple mass points
7. 30th Oct
Flow problems
Cont
Cont
Cont
Cont
(Compressible) Navier-Stokes equations (I3.19)
I.3.5 Centrifuge, I.3.6 Different Materials (2),(3)
Incompressible Navier-Stokes equation (I3.24)
I.3.8 Gravity field of an incompressible planet
8. 30th Oct
Flow problems
Interfaces
Cont
Cont
Cont
Cont
I.3.7 Poiseuille flow in a pipe
General equations (I4.1)
I.4.3 Stationary liquid with a surface
I.4.4 The parabolic shape of the surface
9. 6th Nov
Interfaces  Cont      Example: Moving Planet
10. 6th Nov
Change of coordinates  Cont
Cont
General transformation rule (I.5.1 Theorem, I.5.2 Property)
Invariance of the divergence system with respect to Z (I5.11) (or (I5.8))
11. 13th Nov

Reference coordinates
Cont
Cont
Example: I.5.5 Air flow on the earth
Reference coordinates (I6.2)
12. 13th Nov
Reference coordinates

Cont
Cont
Cont
I.6.1 Theorem, Mass and momentum (I6.4)
(I6.7) Nonlinear elasticity
I.6.2 - I.6.4 Rigid bodies
13. 20th Nov
Observers transformations

Objectivity
Cont
Cont
Cont
II.1.1 Galilei transformation, II.1.2 Group property
Newton's transformation (II1.3), II.1.4 Linear approximation
II.3.2 Scalar equation, II.3.3 Objective tensors, II.3.4 Velocity (Definition)
14. 20th Nov
Constitutive relations
Cont
Cont
II.4.1 Definition (Objective constitutive function), II.4.2 Example
II.4.4 Lemma (Objectivity of \hat{J}), II.4.7 Lemma (Objective representation of \Pi)
15. 27th Nov
Objectivity of balance laws

Entropy
Cont
Cont
Cont
II.3.7 Mass-momentum equation, II.3.10 Classical force (Definition)
II.3.12 Mass-momentum-energy equation (Definition)
III.1.1 Entropy principle, III.1.2 Property
16. 27th Nov
Entropy
Cont
Cont
III.1.3 Example from gas theory, III.1.4 Gibbs relation
III.1.5 Energy and temperature, III.1.6 Absolute temperature
17. 11th Dec
Entropy

Cont
Cont
III.2.1 Energy identity, Mass-momentum-energy system (III2.5)
Lemmata III.2.2 and III.2.3, III.2.6 Thermometer
18. 11th Dec
Applications  Cont      IV.2 Fluids and gases
19. 18th Dec
Applications  Cont      IV.5 Nonlinear elasticity
20. 18th Dec
Applications  Cont      IV.5 Nonlinear elasticity
21. 15th Jan
Applications  Cont      IV.6 Tissue growth
22. 15th Jan
Applications  Cont      IV.7 Navier-Stokes equation
23. 22th Jan
Applications  Cont      IV.3 Euler's equation
24. 22th Jan
Applications  Cont      IV.8 Reaction-diffusion systems (isothermal case)
25. 29th Jan
Applications  Cont      IV.8 Reaction-diffusion systems (isothermal case)
26. 29th Jan
Applications  Cont      IV.8 Reaction-diffusion systems (general case)

Exercise Schedule

Exercise   Date      Content
1. 15th Oct
Cylinder Coordinates Explanation distributions (I.1.9)
2. 22th Oct
Explanation distributions
Exercise I.7.11 (Script: Cont)
3. 29th Oct
Model of a rocket
Kepler's laws of planetary motion (I.3.4, Classical mechanics)
4. 5th Nov
Archimedes' Principle
5. 12th Nov
Gravitational field of a moving shell (I.2.15)
General transformation rule for cylindrical coordinates
6. 19th Nov
Rigid bodies (some properties)
7. 26th Nov
Description of inhomogeneous materials (I.4.8,I.4.9)
Elastic body and objectivity
6. 10th Dec
Legendre-Fenchel transform
Absolute temperature (see III.1.6)

## Material

-- GabrieleWitterstein - 09 Oct 2014