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MA 5019: 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 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.

Lecture Schedule

Lecture   Date      Topic       Script
1. 9th Oct      
      Overview / Introduction
    Mass and Momentum
    (Definitions, Figure 1)
    Conservation laws



 Cont 
 Cont 

    Notations of Derivatives

    Conservation law (I.1.1)
    I.1.1 Representation of the divergence operator
2. 10th Oct      
      Conservation laws

   
 Cont 
 Cont 
 Cont 
    I.1.3 Plane polar coordinates, I.1.4 Plain divergence
    I.1.6 Relativity of velocity
    I.1.7 Example (Mass conservation)
3. 16th Oct      
      Distributions

   
 Distr 
 Distr 
 Distr 
    2.1, 2.2 Definition, 2.3 Property
    2.4 Derivative and multiplication, 2.5 Measure and functions
    2.7 Fundamental solution of the Laplace operator
4. 16th Oct      
      Gravitation
   
 Cont 
 Cont 
 Cont 
    I.2.5 Moving mass point, I.2.6 Lemma
    Newton's gravitation (I2.9), General gravitational law (I2.10)
    I.2.10 Garvitational field of a point-shaped star, I.2.11 Uniqueness
5. 23th Oct      
      Gravitation  Cont 
 Cont 
 Cont 
    I.2.12 Theorem (Jump behaviour on a surface)
    I.2.13 Gravitational fi eld of a globe
    I.2.14 Convergence to a mass point
6. 23th Oct      
      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. 30th Oct      
      Flow problems
 Cont 
 Cont 
 Cont 
 Cont 
    (Compressible) Navier-Stokes equations (I3.19)
    I.3.5 Centrifuge, I.3.6 Different Materials (2),(3)
    Incompressible Navier-Stokes equation (I3.24)
    I.3.8 Gravity field of an incompressible planet
8. 30th Oct      
      Flow problems
    Interfaces
 Cont 
 Cont 
 Cont 
 Cont 
    I.3.7 Poiseuille flow in a pipe
    General equations (I4.1)
    I.4.3 Stationary liquid with a surface
    I.4.4 The parabolic shape of the surface
9. 6th Nov      
      Interfaces  Cont      Example: Moving Planet
10. 6th Nov      
      Change of coordinates  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))
11. 13th Nov      
     
    Reference coordinates
 Cont 
 Cont 
    Example: I.5.5 Air flow on the earth
    Reference coordinates (I6.2)
12. 13th Nov      
      Reference coordinates
   
   
 Cont 
 Cont 
 Cont 
    I.6.1 Theorem, Mass and momentum (I6.4)
    (I6.7) Nonlinear elasticity
    I.6.2 - I.6.4 Rigid bodies
13. 20th Nov      
      Observers transformations
   
    Objectivity
 Cont 
 Cont 
 Cont 
    II.1.1 Galilei transformation, II.1.2 Group property
    Newton's transformation (II1.3), II.1.4 Linear approximation
    II.3.2 Scalar equation, II.3.3 Objective tensors, II.3.4 Velocity (Definition)
14. 20th Nov      
      Constitutive relations
 Cont 
 Cont 
    II.4.1 Definition (Objective constitutive function), II.4.2 Example
    II.4.4 Lemma (Objectivity of \hat{J}), II.4.7 Lemma (Objective representation of \Pi)
15. 27th Nov      
      Objectivity of balance laws

    Entropy
 Cont 
 Cont 
 Cont 
    II.3.7 Mass-momentum equation, II.3.10 Classical force (Definition)
    II.3.12 Mass-momentum-energy equation (Definition)
    III.1.1 Entropy principle, III.1.2 Property
16. 27th Nov      
      Entropy
 Cont 
 Cont 
    III.1.3 Example from gas theory, III.1.4 Gibbs relation
    III.1.5 Energy and temperature, III.1.6 Absolute temperature
17. 11th Dec      
      Entropy
   
 Cont 
 Cont 
    III.2.1 Energy identity, Mass-momentum-energy system (III2.5)
    Lemmata III.2.2 and III.2.3, III.2.6 Thermometer
18. 11th Dec      
      Applications  Cont      IV.2 Fluids and gases
19. 18th Dec      
      Applications  Cont      IV.5 Nonlinear elasticity
20. 18th Dec      
      Applications  Cont      IV.5 Nonlinear elasticity
21. 15th Jan      
      Applications  Cont      IV.6 Tissue growth
22. 15th Jan      
      Applications  Cont      IV.7 Navier-Stokes equation
23. 22th Jan      
      Applications  Cont      IV.3 Euler's equation
24. 22th Jan      
      Applications  Cont      IV.8 Reaction-diffusion systems (isothermal case)
25. 29th Jan      
      Applications  Cont      IV.8 Reaction-diffusion systems (isothermal case)
26. 29th Jan      
      Applications  Cont      IV.8 Reaction-diffusion systems (general case)

Exercise Schedule

Exercise   Date      Content      
1. 15th Oct  
      Cylinder Coordinates Explanation distributions (I.1.9)
2. 22th Oct  
      Explanation distributions
    Exercise I.7.11 (Script: Cont)
3. 29th Oct  
      Model of a rocket
    Kepler's laws of planetary motion (I.3.4, Classical mechanics)
4. 5th Nov  
      Archimedes' Principle
5. 12th Nov  
      Gravitational field of a moving shell (I.2.15)
    General transformation rule for cylindrical coordinates
6. 19th Nov  
      Rigid bodies (some properties)
7. 26th Nov  
      Description of inhomogeneous materials (I.4.8,I.4.9)
    Elastic body and objectivity
6. 10th Dec  
      Legendre-Fenchel transform
    Absolute temperature (see III.1.6)

Time and Location

Material

-- GabrieleWitterstein - 09 Oct 2014