 # MA5019: Mathematical Continuum Mechanics

## News

• The next session of exercise group 2 will take place on 22.11.2018, 10:00 - 12:00 -- presumably exceptionally in room MI 01.13.007.
• As announced, the lecture on Friday the 2.11.2018 is canceled.

## 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. 18th Oct
Overview / Introduction
Mass and momentum

Cont
Cont
Main principles (Conservations Law, Objectivity, Entropy Principle)
Conservation law (I1.1), I.1.2 Notations of derivatives (Definitions, Figure 1)
General mass conservation (I1.7)
2. 19th Oct
Conservation laws

Distributions
Cont
Cont
Cont
I.1.7 Example: Mixture - Air, I.1.8 Relativity of velocity,
I.1.9 Example: A particle in a fluid, I.1.10 Cylindrical coordinates
Introduction to distributions (not relevant in exam)
3. 25th Oct

Moving mass point
Gravitation
Cont
Cont
Cont
Cont
I.2.1 Definition, I.2.3 Derivatives, Multiplication), I.2.3 Functions as distribution,
Conservation law in the distributional sense (I2.4)
I.2.7 & I.2.8 Mass conservation of moving mass point
Newton's gravitation (I2.11)
4. 26th Oct

Cont
Cont
Cont
I.2.12 Fundamentallösung für den Laplace Operator
I.2.13 Gravitational potential of a point-shaped star, I.2.14 Uniqueness
I.2.15 Theorem (Jump condition), I.2.16 Gravitational potential of a globe
5. 8th Nov
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
(I3.11) Newton's force density
6. 9th Nov

Cont
Cont
Momentum for planets (I3.14), (I3.16), (I3.17)
I.3.3 Keplers laws of planetary motion (proof not in exam)
7. 15th Nov

Flow problems
Cont
Cont
Cont
I.3.4 Collection of mass points
(Compressible) Navier-Stokes equations (I3.32), (I3.33)
I.3.5 Centrifuge
8. 16th Nov

Cont
Cont
Cont
I.3.6 Different Materials
Incompressible Navier-Stokes equation (I3.37), (I3.38)
I.3.7 Poiseuille flow in a pipe
9. 22nd Nov
Change of coordinates

Observers transformations
Cont
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.5.5. Air flow on the earth
II.1.1 Galilei transformation, II.1.3 Newtonian transformation
10. 23rd Nov
Objectivity

Objectivity of balance laws
Cont
Cont
Cont
Cont
Transformation rule: (II3.3) and (II3.4),
Different possibilities of choosing matrix Z.
II.3.1 Scalar equation, II.3.2 Objective tensors, II.3.3 Velocity (Definition)
II.3.4 Mass equation, II.3.5 Gravitation law
11. 29th Nov

Cont
Cont
II.3.6 Mass-momentum equation (Definition), II.3.7 Theorem
II.3.8 Classical Force, II.3.9 Inertial systems, II.3.10 Example
12. 30th Nov

Constitutive relations
Cont
Cont
II.3.12 Mass-momentum-energy equation (Definition), II.3.13 Theorem (without proof)
II.4.1 Definition, II.4.2 Objective scalars, II.4.3, II.4.5 Objective vectors
13. 6th Dec

Cont
Cont
II.4.6 Diffusion, II.4.7 Lemma (Objective representation of \Pi)
II.4.10 without proof, II.4.11 Lemma (Objective tensor)
14. 7th Dec

Entropy
Cont
Cont
II.4.12 Constitutive function for liquids.
III.1.1 Entropy principle.
15. 14th Dec

Cont
Cont
III.1.2 Property, III.1.3 Example from gas theory
III.1.4 Gibbs relation (without proof), III.1.7. Example
16. 20th Dec
Energy

Cont
Cont
III.2.1 Energy system (Definition), Mass-momentum-energy system (III2.5)
III.2.4 Theorem: Residual inequality for Mass-momemtum-energy system
17. 21st Dec

Cont
Cont
III.5.1 Dissipation inequality, III.5.2 Free energy inequality (θ variable)
III.5.4 Free energy inequality (θ = const), III.5.5 Conclusion
18. 10th Jan
Applications  Cont      III.6 Distributional entropy
19. 11th Jan
Applications  Cont      IV.2 Fluids and gases
20. 17th Jan
Applications  Cont      IV.2 Fluids and gases
21. 18th Jan
Applications  Cont      IV.1 Tides
22. 24th Jan
Applications  Cont      IV.1 Tides, IV.3 Navier-Stokes equation
23. 25th Jan
Applications  Cont      IV.3 Navier-Stokes equation
24. 31st Jan
Applications  Cont      IV.3 Navier-Stokes equation
25. 1st Feb
Applications  Cont      IV.8 vr-Wirbel
26. 7st Feb
Applications  Cont      IV.16 Self-gravitation
27. 8st Feb
Applications  Cont      IV.16 Self-gravitation

Exercise Schedule

Exercise   Date      Content
1. 25th Oct, 8th Nov
I.1.3 Representation of the divergence operator - proof, I.1.5 Plane polar coordinates
2. 8th Nov,     --
I.1.5 Plane polar coordinates, Construction of the gravitational field in the outer and inner space
3. 16th Nov, 10th Jan
Regular <-> singular distribution, I.2.18 Gravitational potential of the hollow sphere
4. 22nd Nov, 22nd Nov
I.5.4 Example: Cylindrical coordinates
5. 29th Nov, 29nd Nov
I.5.4 Example: Cylindrical coordinates, II.7.5 Example: Objective vector
6. 6th Dec, 20th Dec
II.7.5 Example: Objective vector, II.4.9 Lemma (Objectivity of J) (Script: Cont), Layered material
7. 20th Dec, 20th Dec
III.7.2 Ideal gas, III.7.3 Lemma
8. 10th Jan, 17th Jan
Cyclone, siehe Kapitel 8 aus Exercises
9. 17th Jan, 31th Jan
Proof of Theorem IV.1.3
10. 24th Jan,     --
Isothermal limit
11.-12. 7th Feb, 7th Feb
Summary and repetition