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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 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):

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  Sorted ascending    Content      
1.       I.1.5 Plane polar coordinates, I.1.6 Plain divergence
2.       I.1.4 Examples, I.1.8 Relativity of velocity
3.       I.1.10 Cylinder coordinates, I.2.17 (Distributional) Convergence to a mass point (proof partly)
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)
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
10.       Parts of IV.4.7 Rankine vortex (Example)
2. 7th Nov  
8. 9th Jan  
6. 12th Dec  
3. 14th Nov  
9. 16th Jan  
7. 19th Dec  
4. 21st Nov  
10. 23rd Jan  
1. 24th Oct  
5. 28th Nov  

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