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.
    • 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).