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MA5019: Mathematical Continuum Mechanics



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 Sorted ascending
1. 17th Oct      
2. 18th Oct      
3. 24th Oct      
4. 25th Oct      
5. 7th Nov      
6. 8th Nov      
7. 14th Nov      
8. 15th Nov      
9. 21st Nov      
10. 22nd Nov      
11. 28th Nov      
12. 29th Nov      
13. 12th Dec      
14. 13th Dec      
15. 19th Dec      
16. 20th Dec      
17. 9th Jan      
18. 10th Jan      
19. 16th Jan      
20. 17th Jan      
21. 23rd Jan      
22. 24th Jan      
23. 30th Jan      
24. 31st Jan      
25. 6th Feb      
26. 7th Feb      
7.       Flow problems
    (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
24.       Applications  Cont      Chandrasekhar’s compressible stars: IV.16.10 - IV.16.12
    IV.8 vr-Vortices: IV.8.1 Couette flow
    Observers transformations
    Example: I.5.5 Air flow on the Earth
    II.1.1 Galilei transformation, II.1.3 Newtonian transformation
4.       Conservation of momentum
    General mass-momentum equation (I3.1) and (I3.3)
    I.3.1 Mass point, I.3.2 Collision of mass points
9.       Change of coordinates
    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
2.       Conservation laws
    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
8.       Interfaces
    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
    II.1.4, II.1.5 Doppler effect, II.1.7 Theorem
    II.2 Lorentz transformation (not in exam)
12.       Objectivity
    Objectivity of balance laws
    II.3.1 Scalar equation, II.3.2 Objective tensors, II.3.3 Velocity
    II.3.4 Mass equation, II.3.5 Gravitation law
    Constitutive relations
    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)
    II.3.6 Mass-momentum equation (Definition), II.3.7 Theorem
    II.3.8, II.3.9 Classical Force
    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
    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.       Energy
    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.      Mixtures
    III.3 Mixtures, III.3.1 Corresponding reference velocities
    Mixture Case I, Case II, Case III
22.       Applications  Cont      IV.1 Tidal period - the whole theme
26.       Applications  Cont      IV.12 Chemical reactions - IV.12.1 - IV.12.9
23.       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
19.       Applications  Cont      IV.2 Fluids and gases
20.       Applications  Cont      IV.4 Euler’s equation - from (IV4.1) to IV.4.4
21.       Applications  Cont      IV.4 Euler’s equation - IV.4.5, IV.4.6, and Compressible case
25.       Applications  Cont      IV.8 vr-Vortices - IV.8.2 Theorem
1.       Overview / Introduction
    Mass and momentum

    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
3.       Gravitation

    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
5.        Cont      Subsection: Gravity applied to space objects
    Subsection: Momentum of a single planet, I.3.3 Kepler’s laws of planetary motion
    I.3.4 Multiple mass points (without proof)

Exercise Schedule

Exercise   Date      Content      
1. 24th Oct  
      I.1.5 Plane polar coordinates, I.1.6 Plain divergence
2. 7th Nov  
      I.1.4 Examples, I.1.8 Relativity of velocity
3. 14th Nov  
      I.1.10 Cylinder coordinates, I.2.17 (Distributional) Convergence to a mass point (proof partly)
4. 21st Nov  
      I.2.18 Hollow sphere (with proof)
5. 28th Nov  
      I.4.2 Archimedes’ principle (1. part)
6. 12th Dec  
      I.4.2 Archimedes’ principle (2. part), About: Constitutive relations - Examples up to now
7. 19th Dec  
      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. 9th Jan  
      Parts of Lemma II.4.12, Part of the proof of II.4.13, Frobenius inner product with properties
9. 16th Jan  
      Proof of IV.2.5 Lemma (2),(3),(4), IV.2.6 Theorem
10. 23rd Jan  
      Parts of IV.4.7 Rankine vortex (Example)

Time and Location