Caution, these contents corresponds to the coursebooks of last year

Similarity and Transport Phenomena in Fluid


Lecturer(s) :

Ancey Christophe




Every 2 years


Next time: Fall 2018


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


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


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



Bibliography is provided on the webpage


In the programs

  • Mechanics (edoc), 2019-2020
    • Semester
    • Exam form
      Project report
    • Credits
    • Subject examined
      Similarity and Transport Phenomena in Fluid
    • Number of places
    • Lecture
      20 Hour(s)
    • Exercises
      8 Hour(s)

Reference week

Exercise, TP
Project, other


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