 # MA 5019: Mathematical Continuum Mechanics

## 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 appreciated, but is not required. Also elastic materials are introduced in physical coordinates. The entropy principle is defined as a differential inequality and the free energy inequality is considered as a special case.

• Conservation Laws
• Mass and Momentum Equations
• Coordinate Transformation
• Observer Transformation
• Objectivity of Differential Equations
• Frame Indifference
• Entropy Principle
• Free Energy Inequality

As stated above, the knowledge of distributions is not necessary for successfully passing of the lecture, but distributions lead to a better understanding.

After this general part, we will focus on specific models, for example the following applications (only a limited number of applications has to be understood):

• Fluid Flows
• Elasticity
• Navier-Stokes Equations
• Chemical Reactions
• Biological Reactions
• Self-Gravitation
• Liquid Crystals

## Lecture Schedule

Lecture   Date      Topic       Script
1. 22nd Oct
Overview / Introduction
Mass and momentum
(Definitions, Figure 1)
Conservation laws

Cont
Cont
Main principles
Notations of derivatives

Conservation law (I.1.1), General mass conservation (I.1.6)
I.1.1 Representation of the divergence operator
I.1.6 Exapmle: Mixture - Air
2. 23rd Oct
Conservation laws
Distributions

Cont
Cont
Cont
I.1.7 Relativity of velocity, I.1.8 Example (Mass conservation)
I.2.1 Definition (Derivatives, Multiplication), I.2.2 Functions as distribution
Conservation law in the distributional sense (I.2.2)
3. 29th Oct
Gravitation

Cont
Cont
Cont
Newton's gravitation (I2.10), I.2.13 Gravitational field of a globe
I.1.12 Theorem (Jump condition)
I.2.14 Convergence to a mass point
4. 30th Oct
Conservation of momentum

Cont
Cont
Cont
General mass-momentum equation (I3.1) and (I3.3)
I.2.5 Moving mass point, I.2.6 Lemma (Mass equation)
I.3.1 Mass point, I.3.2 Collision of mass points
5. 5th Nov
Conservation of momentum

Cont
Cont
Cont
Momentum for planets
I.3.3 Keplers laws of planetary motion
Collection of mass points
6. 6th Nov
Conservation of momentum
Flow problems
Cont
Cont
Cont
I.3.4 Multiple mass points
(Compressible) Navier-Stokes equations (I3.19)
Incompressible Navier-Stokes equation (I3.24)
7. 12th Nov
Flow problems

Cont
Cont
I.3.5 Centrifuge, I.3.6 Different Materials (2),(3)
I.3.7 Poiseuille flow in a pipe
8. 13th Nov
Interfaces

Cont
Cont
Cont
I.4.4 Gravity field of an incompressible planet
General equations (I4.1)
I.4.1 Principle of Archimendes
9. 19th Nov
Change of coordinates

Cont
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)
Example: I.5.4 Cylindrical coordinates
10. 20th Nov

Reference coordinates
Cont
Cont
Example: I.5.5. Air flow on the earth
Reference coordinates (I6.2)
11. 26th Nov
Reference coordinates

Observers transformations
Cont
Cont
Cont
Cont
I.6.1 Lemma, I.6.2 Theorem: Mass and momentum equation
Nonlinear elasticity (I6.9),(I6.10), Rigid bodies: Lemma I.6.4
II.1.1 Galilei transformation, Newtonian transformation (II1.3)
II.1.4 Linear approximation
12. 27th Nov
Observers transformations
Objectivity

Cont
Cont
Cont
Characterization of Newtonian transformation: II.1.6 Theorem
II.3.2 Scalar equation, II.3.3 Objective tensors, II.3.4 Velocity (Definition)
II.3.5 Mass equation, II.3.6 Gravitation law, II.3.7 Mass-momentum equation
13. 10th Dec
Objectivity of balance laws

Constitutive relations
Cont
Cont
Cont
II.3.7 Mass-momentum equation, II.3.9 Classical force (Definition)
II.3.12 Mass-momentum-energy equation (Definition)
II.4.1 Definition (Objective constitutive function), II.4.2 Example
14. 11th Dec
Constitutive relations

Cont
Cont
Cont
Cont
II.4.3 Example, II.4.4 Inequality, II.4.5 Lemma (Objectivity of \hat{J})
II.4.8 Lemma (Objective representation of \Pi)
II.4.9 Lemma (Layered material), II.4.14. Lemma (Objective tensor)
II.4.15. Constitutive function for liquids (without proof)
15. 17th Dec
Entropy

Cont
Cont
III.1.1 Entropy principle, III.1.2 Property
III.1.3 Example from gas theory
16. 18th Dec
Entropy

Cont
Cont
Cont
III.1.4 Gibbs relation, 1.7 Example
III.2.1 Energy identity, Mass-momentum-energy system (III2.5)
III.2.2 Lemma
17. 7th Jan
Entropy

Cont
Cont
III.2.3 Lemma
III.2.4 Theorem: Residual inequality for Mass-momemtum-energy system
18. 8th Jan
Entropy
Applications
Cont
Cont
III.2.5 Theorem: Gases and liquids
IV.1 Tidal period
19. 14th Jan
Applications  Cont      IV.1 Tidal period
20. 15th Jan
Applications  Cont      IV.1 Tidal period
21. 21st Jan
Applications  Cont      IV.2 Fluids and gases
22. 22nd Jan
Applications  Cont      IV.8 Navier-Stokes equation
23. 28th Jan
Applications  Cont      IV.9 Vorticity
24. 29th Jan
Applications  Cont      IV.9 Vorticity
25. 4th Feb
Applications  Cont      IV.15 Self-gravitation
26. 5th Feb
Applications  Cont      IV.10 Reaction-diffusion systems

Exercise Schedule

Exercise   Date      Content
1. 29th Oct / 30th Oct
I.1.3 Plane polar coordinates, I.1.4 Plain divergence
2. 5th Nov / 6th Nov
Cylinder Coordinates Explanation distributions, I.1.10
3. 12th Nov / 13th Nov
Gravitational field of a hollow sphere and convergence result
Script Exercises Mathematical Continuum Mechanics, see Section 6
4. 19th Nov / 20th Nov
Gravitational field of a sphere shell
Exercise I.7.16: Free energy
Exercise I.7.17: Earth's atmosphere
5. 26th Nov / 27th Nov
Low-pressure area
Script Exercises Mathematical Continuum Mechanics, see Section 3: Cyclone
6. 10th Dec / 11th Dec
Exercise II.5.3
Exercise II.5.6: Material derivative
7. 17th Dec / 18th Dec
II.4.11 Elastic body and objectivity
II.4.12 Constitutive function for elastic bodies
8. 7th Jan / 8th Jan
Thermometer, III.2.6