# Similarity and Transport Phenomena in Fluid

## Frequency

Every 2 years

## 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

• Advection

• 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

## Ressources en bibliothèque

- Introduction to Symmetry Analysis / Cantwell
- Symmetry and Integration Methods for Differential Equations / Bluman
- Applications of Lie's theory of ordinary and partial differential equations / Dresner
- Turbulence, coherent structures, dynamical systems and symmetry / Holmes
- Similarity Solutions of Nonlinear Partial Differential Equations / Dresner
- Symmetry Methods for Differential Equations -- A Beginner's Guide / Hydon
- Scaling / Barenblatt
- Differential Equations: Linear, Nonlinear, Ordinary, Partial / King
- Application of Lie Groups to Differential Equations / Olver
- Self-Similarity and Beyond / Sachdev
- Scaling, Self-Similarity, and Intermediate Asymptotics / Barenblatt

## Websites

## Moodle Link

## In the programs

**Number of places:**20**Subject examined:**Similarity and Transport Phenomena in Fluid**Lecture:**20 Hour(s)**Exercises:**8 Hour(s)**Type:**optional