﻿ Laboratoire d'Hydraulique Environnementale - LHE

# Environmental Hydraulics Laboratory LHE

## Similarity and Transport Phenomena in Fluid Dynamics

Course in English (28h, 2 credits): 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.
• Form: in-person lectures. See Moodle. (course ME 716).
• Course notebook
• 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 using ClawPack: Introduction to ClawPack(work in progress).
• Examination : homework (20 Jan. 2023, 6 pm). Here is the Mathematica notebook for plotting phase portrait
• Schedule: tentative schedule
• Literature
• 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.E., 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.