Coursebooks

Similarity and Transport Phenomena in Fluid

ME-716

Lecturer(s) :

Ancey Christophe

Language:

English

Frequency

Every 2 years

Remarque

Next time: Fall 2018

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

Websites

In the programs

  • Mechanics (edoc), 2018-2019
    • 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

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