Similarity and Transport Phenomena in Fluid Dynamics
Course in English (28 hr, 2 credits): The course is an introduction to Lie groups and symmetry analysis, with applications to conservation equations in fluid mechanics. The course aims to provide the student with mathematical and physical tools to deal with transport equations in continuum mechanics. Emphasis is given to similarity and travelling-wave solutions.
Room: CM 1 113. Time: 12h15-14h every Friday as of 23 September 2022.
It is highly recommended that students bring a laptop (with Mathematica or Matlab installed) for the exercises.
Course material
- Course notebook
- Schedule: tentative schedule
- Contents:
- Chapter 1: The concept of similarity
- Chapter 2: Transport phenomena in fluid dynamics
- Chapter 3: One-parameter groups, Lie groups
- Chapter 4: First-order differential equations
- Chapter 5: Second-order differential equations
- Chapter 6: Similarity solutions to partial differential equation
- Chapter 7: Travelling wave solution
- Chapter 8: Hyperbolic problems
- Chapter 9: Parabolic problems
- Lecture notes: the course support is provided by the slides above. There are no lecture notes. A former version (more oriented toward the mathematical aspects) is still available (in French): Analyse différentielle
- Numerical solutions to hyperbolic problems: introduction to ClawPack (work in progress)
Examination
Bibliography
- Symmetry, similarity solutions
- Barenblatt, G.I., Scaling, Self-Similarity, and Intermediate Asymptotics, 386 pp., Cambridge University Press, Cambridge, 1996
- Barenblatt, G.I., Scaling, Cambridge University Press, Cambridge, 2003.
- Bluman, G.W., and S.C. Anco, Symmetry and Integration Methods for Differential Equations, Springer, New York, 2002.
- Cantwell, B.J., Introduction to Symmetry Analysis, Cambridge University Press, Cambridge, 2002.
- Dresner, L., Similarity Solutions of Nonlinear Partial Differential Equations, 123 pp., Pitman, Boston, 1983.
- Dresner, L., Applications of Lie’s Theory of Ordinary and Partial Differential Equations, Institute of Physics Publishing, Bristol, 1999.
- Fazio, R., A similarity approach to the numerical solution of free boundary problems, SIAM Review, 40, 616-635, 1998.
- Germain, P., Méthodes asymptotiques en mécanique, in Dynamique des Fluides, edited by R. Balian, and J.-L. Peube, pp. 1-148, Gordon and Breach Science Publishers, London, 1973.
- Gratton, J., Similarity and self similarity in fluid dynamics, Fundamentals of Cosmic Physics, 15, 1-106, 1991.
- Gratton, J., and C. Vigo, Self-similar gravity currents with variable inflow revisited: plane currents, Journal of Fluid Mechanics, 258, 77-104, 1994.
- Hydon, P., Symmetry Methods for Differential Equations — A Beginner’s Guide, Cambridge University Press, Cambridge, 2000.
- Holmes, P., J.L. Lumley, and G. Berkooz, Turbulence, coherent structures, dynamical systems and symmetry, Cambridge university press, 1998.
- Ibragimov, N.H., CRC Handbook of Lie Group Analysis of Differential Equations, CRC-Press, Bocan Rota, 1995.
- King, A.C., J. Billingham, and S.R. Otto, Differential Equations: Linear, Nonlinear, Ordinary, Partial, 541 pp., Cambridge University Press, Cambridge, 2003.
- Olver, P.J., Application of Lie Groups to Differential Equations, Springer, New York, 1993.
- Sachdev, P.L., Self-Similarity and Beyond, Chapman & Hall, Boca Raton, 2000.
- Sedov, L., Similarity and Dimensional Methods in Mechanics, CRC Press, Boca Raton, 1993.
- Zohuri, B., Dimensional Analysis and Self-Similarity Methods for Engineers and Scientists, Springer, Basel, 2015.
- Parabolic and hyperbolic differential equations
- Courant, R., and K.O. Friedrich, Supersonic Flow and Shock Waves, 455 pp., Intersciences Publishers, New York, 1948.
- Smoller, J., Shock waves and reaction-diffusion equations, 581 pp., Springer, New York, 1982.
- LeVeque, R.J., Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002.
- LeVeque, R.J., Numerical Methods for Conservation Laws, Birkhäuser, Basel, 1992.
- Ketcheson, D.I., R.J. LeVeque, and M. del Razo, Riemann Problems and Jupyter Solutions, Society for Industrial and Applied Mathematics, Philadelphia, 2020. The contents of this book are available online.
Additional material
- Additional Mathematica scripts
- Chapter 1: solving the Stokes’ first problem using DSolve.
- Chapter 4: plotting a phase portrait using StreamPlot.
- Chapter 4: plotting a phase portrait using StreamPlot and NDSolve.
- Chapter 4: solving the population dynamics equation using NDSolve.
- Chapter 5: exercise 5.1 using NDSolve.
- Chapter 9: solving the Blasius equation using NDSolve and exact shooting.
- Chapter 9: solving the Stefan problem using the method of lines.
- Additional Matlab scripts
- Chapter 1: solving the Stokes’ first problem using dsolve.
- Chapter 4: plotting a phase portrait using quiver.
- Chapter 4: solving the population dynamics equation using ode45.
- Chapter 6: solving the Hppert equation using pdepe or an implicit scheme (Adams Moulton).
Exercises
Solutions to exercises :