MAT 221A-2: MATHEMATICAL FLUID DYNAMICS - FALL 2003

     TR 3:10-4:40, Wellman 103

HOMEWORK PROBLEMS AND SOLUTIONS:

COURSE OUTLINE: This course will focus on the incompressible and compressible Navier-Stokes and Euler equations. In particular, we shall merge mathematical analysis, numerical analysis, and scientific computation to understand the motion of fluids in physical, biological, and engineering applications. Topics that will be covered in the first quarter include:

  • Weak and classical solutions for the Navier-Stokes and Euler equations
  • Basic energy laws and circulation theorems
  • Galerkin methods, approximation schemes, a priori estimates
  • Spectral schemes and finite element methods
  • Short-time existence and uniqueness for the Navier-Stokes and Euler equations
  • Bounded domains and exterior problems
  • PDE methods for convergence of Galerkin schemes
  • Numerical methods: accuracy, stability, and consistency of numerical schemes
  • Well-resolved and convergent schemes
  • Global weak (Leray) solutions for the 3D Navier-Stokes equations
  • Regularity and uniqueness in 2D
  • Runga-Kutta and time-stepping schemes
  • Projection methods
  • Lagrangian vs. Eulerian representations
  • Vorticity formulation and vortex methods
  • Flow over aerofoils

    The second quarter course 221B will deal with the important area of compressible flows, moving interfaces, free boundaries, and shock waves.

    INSTRUCTORS:

    TEXTS (Optional):

  • Deville, M. O.; Fischer, P. F.; Mund, E. H. High-order methods for incompressible fluid flow. Cambridge Monographs on Applied and Computational Mathematics, 9. Cambridge University Press, Cambridge, 2002.
  • Durran, Dale R. Numerical methods for wave equations in geophysical fluid dynamics. Texts in Applied Mathematics, 32. Springer-Verlag, New York, 1999.
  • Acheson, D. J. Elementary fluid dynamics. Oxford Applied Mathematics and Computing Science Series. The Clarendon Press, Oxford University Press, New York, 1990.
  • Batchelor, G. K. An introduction to fluid dynamics. Second paperback edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1999.
  • Meyer, Richard E. Introduction to mathematical fluid dynamics. Corrected reprint of the 1971 original. Dover Publications, Inc., New York, 1982.
  • Lions, Pierre-Louis Mathematical topics in fluid mechanics. Vol. 1. Incompressible models. Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996.
  • Van Dyke, M. Album of Fluid Motion. Parabolic Press, 1982.

    PREREQUISITES: Undergraduate analysis; experience with MATLAB will be helpful, but not mandatory.

    ATTENDANCE: The first day of lecture is Thursday, September 25 and the last day of lecture is Thursday, December 4. Regular attendance to the lectures is strongly advised. You will be responsible for all the material presented in class, regardless of whether or not you were present.

    GRADING SCALE: Grades will be assigned on the basis of your performance on homework and two in-class exams as follows:

    HOMEWORK AND EXAMS: Homework assignments will be a marriage of analysis and MATLAB code development. Homework will be assigned every Tuesday and collected at the start of class the following Tuesday. If you cannot attend class, then slip the homework under the door of the instructor at least 20 minutes before class time (on the day it is due). No late homework will be accepted, and there will be no makeup exams. In the case that an exam is missed due to a medical emergency, it will not count toward the final grade.

    You may discuss the homework assignments with classmates, but you are each expected to do your own assignments which include the MATLAB programs that you will be asked to design. Please write legibly. We strongly recommend that you do your homework every day, and not wait until the night before the assignment is due.