 # MA 5019: Mathematical Continuum Mechanics

## News

• Lecture and Exercises: Scripts are updated.

## 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. 17th Oct
Overview
Mass and Momentum
(Definitions, Figure 1)
Conservation laws

Cont

Notations of Derivatives

Conservation of mass (I.1.7)
2. 18th Oct
Conservation laws

Distributions
Cont
Cont
Distr
I.1.6 Relativity of velocity, I.1.7 Example (Mass conservation),
I.1.9 Cylinder coordinates
2.1, 2.2 Definition, 2.3 Property
3. 24th Oct
Distributions

Distributional conservation law
Mass point
Distr
Distr
Cont
Cont
2.4 Derivatives, 2.5 Examples,
2.6 Dirac-Distribution
(*) Derivation
I.2.5 Moving mass point, I.2.6 Lemma
4. 25th Oct
Gravitation
Fundamental solution
Cont
Cont
Cont
Newton's gravitation (I2.8), General gravitational law (I2.9)
I.2.9 Fundamental solution of the Laplace-operator
I.2.11 Uniqueness, I.2.12 Theorem
5. 31th Oct
Gravitation  Cont
Cont
Cont
I.2.13 Gravitational fi eld of a globe
I.2.14 Convergence to a mass point
I.2.15 Example (Gravitational shell)
6. 7th 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
I.3.3 Multiple mass points
7. 8th Nov
Classical Mechanics
Flow Problems
Cont
Cont
Cont
Cont
I.3.4 Kepler's laws of planetary motion
(Compressible) Navier-Stokes equations (I3.19)
Incompressible Navier-Stokes equation (I3.24)
I.3.7 Poiseuille flow in a pipe
8. 14th Nov
Interfaces
Cont
Cont
I.4.2 Stationary liquid with a surface
I.4.3 The parabolic shape of the surface
9. 15th Nov
Change of coordinates

Reference Coordinates
Cont
Cont
Cont
Cont
Invariance of the divergence system with respect to Z (I5.11)
General transformation rule (I.5.1 Theorem, I.5.2 Property)
Reference coordinates (I6.2), I.6.1 Theorem
Mass and momentum in reference coordinates (I6.4)
10. 21th Nov
Elasticity  Cont
Cont
Cont
Cont
Nonlinear Elasticity (I6.7)
I.6.2 Lemma (Elementary property)
I.6.3 Transformation (resp. Reference coordinates)
I.6.4 Rigid bodies
11. 22th Nov

Objectivity
Observers transformations
Cont
Cont
Cont
I.5.5 Air flow on the earth
II.1.1 Galilei transformation, II.1.2 Group property
Newton's transformation (II1.3)
12. 28th Nov
Objectivity of balance laws
Cont
Cont
II.3.3 Objective tensors, II.3.4 Velocity (Defi nition)
II.3.7 Mass-momentum equation
13. 29th Nov
Objectivity of balance laws
Cont      II.3.10 Classical Force (De finition)
II.3.12 Mass-momentum-energy equation (Defi nition)
14. 5th Dec
Constitutive relations
Cont
Cont
Cont
II.4.1 De finition (Objective constitutive function)
II.4.2 Example, II.4.4 Lemma (Objectivity of \hat{J})
II.4.7 Lemma (Objective representation of \Pi)
15. 6th Dec

Fluid dynamics
Entropy
Cont
Cont
Cont
Example (II4.8 Lemma)
II.4.13 Lemma, II.4.14 Constitutive function for liquids
III.1.1 Entropy principle, III.1.2 Property
16. 13th Dec
Entropy

Cont
Cont
Cont
III.1.3 Example from gas theory, III.1.4 Gibbs relation
III.2.1 Energy identity, Mass-momentum-energy system (III2.5)
III.2.6 Thermometer
17. 9th Jan
Applications

Cont
Cont
IV.1 Tides
IV.2 Fluids and gases
18. 10th Jan
Applications

Cont
Cont
IV.2 Fluids and gases
IV.4 Nonlinear elasticity
19. 16th Jan
Applications  Cont      IV.5 Tissue growth
20. 17th Jan
Applications  Cont      IV.6 Navier-Stokes equation
21. 23th Jan
Applications  Cont      IV.7 Prandtl's boundary layer
22. 24th Jan
Applications  Cont      IV.8 Vorticity
23. 6th Feb
Applications  Cont      IV.9 Self-gravitation
24. 7th Feb
Concluding remark        Lorentz transfomation