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
Dans les plans d'études
- Nombre de places: 20
- Matière examinée: Similarity and Transport Phenomena in Fluid
- Cours: 20 Heure(s)
- Exercices: 8 Heure(s)
- Type: optionnel