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MA5019: Mathematical Continuum Mechanics

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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.

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):

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

Time and Location

Material