# Coursebooks

## Similarity and Transport Phenomena in Fluid

Ancey Christophe

English

Every 2 years

#### Remarque

Next Time Fall 2020

#### Summary

The course is an introduction to symmetry analysis in fluid mechanics. The student will learn how to find similarity and travelling-wave solutions to partial differential equations used in fluid and continuum mechanics. The course covers mathematical and physical aspects

#### Content

Chapter 1: The concept of similarity

'          Geometrical similarity

'          Invariance by affine transformation, rotation, translation

'          Fractal similarity

'          Scaling law

'          Physical similarity

'          Complete similarity: drag force

'          Incomplete similarity: flow resistance

Chapter 2:  Transport phenomena in fluid dynamics

'          Transport phenomena

'          Diffusion Heat equation

'          Wave

'          Shocks and conservation equations

'          Boundary problems: fixed boundary, boundary layer, free boundary problem

'          Classification of partial differential equations

'          First-order equation, characteristic form

'          Second order equation, hyperbolic, elliptic, parabolic

Chapter 3: One-parameter groups, Lie groups

'          Groups of transformation

'          Group invariants

'          Invariant curves

'          Transformation of derivative

Chapter 4: First-order differential equations

'          Phase portrait

'          Singular point

'          Separatrix

'          Integrating factor

'          Invariant integral curves

'          Singular solution

'          Change of variables

Chapter 5: Second-order differential equations

'          Invariant differential equations

'          Lie's reduction theorem

'          Stretching group

'          Singularities

Chapter 6: Similarity solutions to partial differential equation

'          Similarity solutions

'          Associated stretching group

'          Asymptotic behavior

'          Determining equations

Chapter 5: Travelling wave solution

'          Translation groups

'          Example: diffusion with source

'          Propagation velocity

'          Approach to travelling waves

Chapter 8: Hyperbolic problems

Hyperbolic problems

'          One dimensional problems

'             Characteristic equations

'             Shock formation

'            The Riemann problem

Generalization to multidimensional problems

'             Linear systems

'             Nonlinear systems

'            The shallow-water equations

Chapter 9: Parabolic problems

'          Linear diffusion

'          Nonlinear diffusion

'          Stefan problem

'          Boundary layer problem

#### Keywords

partial differential equation, diffusion, advection, similarity solutions, travelling wave solution, hyperbolic problems

#### Resources

##### Bibliography

Bibliography is provided on the webpage

### In the programs

• Semester
• Exam form
Project report
• Credits
2
• Subject examined
Similarity and Transport Phenomena in Fluid
• Number of places
20
• Lecture
20 Hour(s)
• Exercises
8 Hour(s)

### Reference week

Lecture
Exercise, TP
Project, other

### legend

• Autumn semester
• Winter sessions
• Spring semester
• Summer sessions
• Lecture in French
• Lecture in English
• Lecture in German