Fiches de cours

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

    • Semester
    • Exam form
       Project report
    • Credits
    • Subject examined
      Similarity and Transport Phenomena in Fluid
    • 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