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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. 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

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