 # MA5019: Mathematical Continuum Mechanics

## News

• The exercises begin in the second week on October 24, 2019.

## 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 the 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. 17th Oct
Overview / Introduction
Mass and momentum

Cont
Cont
Cont
Main principles (Conservations Law, Objectivity, Entropy Principle)
Conservation law (I1.1), I.1.2 Notations of derivatives (Definitions, Figure 1)
I.1.3 Representation of the divergence operator (without proof),
General mass conservation (I1.7), I.1.7 Example: Mixture - Air
2. 18th Oct
Conservation laws
Distributions
Cont
Cont
Cont
Cont
I.1.9 Example: A particle in a fluid
I.2.1 Definition, I.2.2 Derivatives, Multiplications
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
3. 24th Oct
Gravitation

Cont
Cont
Cont
Newton's gravitation (I2.11), I.2.12 Fundamental solution for the 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
4. 25th Oct
Conservation of momentum

Cont
Cont
General mass-momentum equation (I3.1) and (I3.3)
I.3.1 Mass point, I.3.2 Collision of mass points
5. 7th Nov
Cont      Subsection: Gravity applied to space objects
6. 8th Nov

Cont
Cont
Subsection: Momentum of a single planet, I.3.3 Keplers laws of planetary motion
I.3.4 Multiple mass points (without proof)
7. 14th Nov
Flow problems

Cont
Cont
Cont
(Compressible) Navier-Stokes equations (I3.32)
Incompressible Navier-Stokes equation (I3.37), I.3.5 Centrifuge
I.3.6 Different Materials (1),(2), I.3.7 Poiseuille flow in a pipe
8. 15th Nov
Interfaces

Cont
Cont
Cont
I.4.1 Normal velocity (without proof), General equations (I4.3)
I.4.3 Stationary liquid with a surface, I.4.4
I.4.5 Gravity field of an incompressible planet
9. 21st 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. 22nd Nov

Observers transformations
Cont
Cont
Example: I.5.5 Air flow on the Earth
II.1.1 Galilei transformation, II.1.3 Newtonian transformation
11. 28th Nov

Cont
Cont
II.1.4, II.1.5 Doppler effect, II.1.7 Theorem
II.2 Lorentz transformation (not in exam)
12. 29th Nov
Objectivity
Objectivity of balance laws
Cont
Cont
II.3.1 Scalar equation, II.3.2 Objective tensors, II.3.3 Velocity
II.3.4 Mass equation, II.3.5 Gravitation law
13. 12th Dec

Cont
Cont
II.3.6 Mass-momentum equation (Definition), II.3.7 Theorem
II.3.8, II.3.9 Classical Force
14. 13th Dec

Constitutive relations
Cont
Cont
Cont
II.3.10 Inertial frame, II.3.11 Example
II.3.12 Mass-momentum-energy equation (Definition)
II.4.1 Definition, II.4.2 Objective scalars, II.4.3., II.4.4, II.4.6 Diffusion (Example)
15. 19th Dec

Entropy
Cont
Cont
Cont
II.4.7 Orthonormal system, II.4.12 Lemma (Objective tensor)
II.4.13 Pressure tensor for liquids, II.4.14 (without proof)
III.1.1 Entropy principle, III.1.2 Property
16. 20th Dec

Cont
Cont
Cont
III.1.3 Example from gas theory, Subsection: Consequences of the entropy principle
III.1.4 Gibbs relation (proof (1),(2),(3) only)
III.1.5 Energy and inverse absolute temperature
17. 9th Jan
Energy

Cont
Cont
Cont
III.2.1 Energy system (Definition), Mass-momentum-energy system (III2.5)
III.2.2 Lemma (kinetic term) with proof, 2.3 Lemma with proof
III.2.4 Theorem: Residual inequality for Mass-momemtum-energy system
18. 10th Jan
Mixtures

Cont
Cont
III.3 Mixtures, III.3.1 Corresponding reference velocities
Mixture Case I, Case II, Case III
19. 16th Jan
Applications  Cont      IV.2 Fluids and gases
20. 17th Jan
Applications  Cont      IV.4 Eulers equation - from (IV4.1) to IV.4.4
21. 23rd Jan
Applications  Cont      IV.4 Eulers equation - IV.4.5, IV.4.6, and Compressible case
22. 24th Jan
Applications  Cont      IV.1 Tidal period - the whole theme
23. 30th Jan
Applications  Cont      IV.16 Self-gravitation: Star as a point, IV.16.1 - IV.16.2
Stationary case, IV.16.4, IV.16.5 (without proof);
Chandrasekhars compressible stars, IV.16.7 - IV.16.9
24. 31st Jan
Applications  Cont      Chandrasekhars compressible stars: IV.16.10 - IV.16.12
IV.8 vr-Vortices: IV.8.1 Couette flow
25. 6th Feb
Applications  Cont      IV.8 vr-Vortices - IV.8.2 Theorem
26. 7th Feb
Applications  Cont      IV.12 Chemical reactions - IV.12.1 - IV.12.9

Exercise Schedule

Exercise   Date      Content 1. 24th Oct
2. 7th Nov
3. 14th Nov
4. 21st Nov
5. 28th Nov
6. 12th Dec
7. 19th Dec
8. 9th Jan
9. 16th Jan
10. 23rd Jan
3.       I.1.10 Cylinder coordinates, I.2.17 (Distributional) Convergence to a mass point (proof partly)
2.       I.1.4 Examples, I.1.8 Relativity of velocity
1.       I.1.5 Plane polar coordinates, I.1.6 Plain divergence
4.       I.2.18 Hollow sphere (with proof)
5.       I.4.2 Archimedes principle (1. part)
6.       I.4.2 Archimedes principle (2. part), About: Constitutive relations - Examples up to now
7.       II.4.8 Lemma - Objective representation of J, II.4.11 Lemma - Objective representation of \Pi,
Example: Layered material, II.4.9, (II.4.10, without proof)
10.       Parts of IV.4.7 Rankine vortex (Example)
8.       Parts of Lemma II.4.12, Part of the proof of II.4.13, Frobenius inner product with properties
9.       Proof of IV.2.5 Lemma (2),(3),(4), IV.2.6 Theorem