Mathematical Fluid Dynamics

AMCS 334. Mathematical Fluid Dynamics, Fall 2012, KAUST
Instructor: Aslan Kasimov
Lectures:  9:00-10:30 every Monday and Wednesday in Room 9-xxxx

Prerequisites: AMCS 231 or AMCS 201. Recommended: AMCS 331. Equations of fluid dynamics; inviscid flow and Euler equations; vorticity dynamics; viscous incompressible flow and Navier-Stokes equations; existence, uniqueness, and regularity of solutions of Navier-Stokes equations; Stokes flow; free-surface flows; linear and nonlinear instability and transition to turbulence; rotating flows; compressible flow and shock dynamics; detonation waves.

Note. I follow my own notes (which are handed out to students), but I frequently consult the following texts. In style, the lectures are close to the presentations of Serrin and Aris, and also of Drazin, when discussing stability. The books by Acheson and Chorin & Marsden can be useful as elementary introductions. Lai, Rubin, and Krempl is used mostly for its introduction to tensors. Various relevant papers from Annual Reviews of Fluid Mechanics, Journal of Fluid Mechanics, and other sources are also used in presenting the subject.

Texts:

  1. J. Serrin, Mathematical Principles of Classical Fluid Mechanics.
  2. W. Lai, D. Rubin, E. Krempl, Introduction to Continuum Mechanics.
  3. P. Drazin, Introduction to Hydrodynamic Stability.
  4. R. Aris, Vectors, Tensors and the Basic Equations of Fluid Mechanics.
  5. S. Childress, An Introduction to Theoretical Fluid Mechanics.
  6. D. Acheson, Elementary Fluid Dynamics.
  7. A. Chorin and J. Marsden, A Mathematical Introduction to Fluid Mechanics.

The final grade will be based on homework (~40%), one midterm exam (~30%), and a project paper (~30%). There will be no final exam. The project paper should be a 10-15 page analysis of a problem chosen by a student with the help or consent of the instructor. An end-term presentation on the project and its defense is also required.

Sample topics: Riemann problem for real materials; shock-wave stability; fluid flow in flexible tubes; MHD shock waves; radiative shock waves and their instability; biological propulsion; vortex patterns and their stability; shock implosion; blow-up in incompressible Euler equations; square-wave detonations; the flapping-flag problem; Faraday instability; quantum hydrodynamics; ...

The following software can be very helpful and is recommended:

  1. Matlab/Mathematica/Maple for symbolic/numerical computations and visualization
  2. LyX/LaTex/TeXShop for writing papers
  3. BidDesk for organizing references
  4. Keynote/Powerpoint for presentations

Week

Topics

Notes

1: Sep 3, 5

Lecture 1. Introduction. Course overview. Review of tensor algebra and calculus.

Lecture 2. Fluid as a continuum. Eulerian and Lagrangian descriptions. Reynolds transport theorem. Equations of motion. Law 1: Conservation of mass in Eulerian and Lagrangian forms. Incompressibility. The momentum equation. Cauchy's stress principle. Cauchy's theorem. Law 2: Cauchy's equation of motion.

 

2: Sep 10, 12

Lecture 3. Equations of motion. Law 2': conservation of angular momentum. Symmetry of the stress tensor. Kinetic energy equation. The deformation tensor and the local velocity field. Translation, deformation, and rotation of a fluid element.

Lecture 4. Equations of motion. Law 3: the energy equation. The first law of thermodynamics. Law 4: the second law of thermodynamics, the Clasius-Duhem inequality. The dissipation function. Definition of an ideal fluid. The meaning of pressure in compressible and incompressible fluids.

Sep 7, last day to add

3: Sep 17, 19

Lecture 5. The constitutive equations of viscous fluids. Stokesian fluid. Isotropy and material homogeneity. Rivlin-Ericksen theorem. The Newtonian fluid. Stokes assumption. The Navier-Stokes equations.

Lecture 6. Summary of the main equations and special cases: ideal, barotropic, irrotational, creeping, high-speed compressible, free-surface, boundary-layer, rotating, reacting, etc. Bernoulli's theorem. Vorticity, vortex lines and tubes. The vorticity equation. Vortex stretching.

Sep 12, last day to drop w/o W

4: Sep 24, 26

Lecture 7. Material vector fields. Kelvin's circulation theorem. Helmholtz' theorems. Velocity from vorticity, Bio-Savart law.

Lecture 8. The problem of blow-up in incompressible Euler equations. Beale-Kato-Majda theorem. Potential flow in 2D. Use of complex variables. Basic potential flows. Theorems of Blasius, Kutta-Joukowski, and D'Alembert.

 

5: Oct 1, 3

Lecture 9. Interaction of point vortices. Von Karman vortex street. Stability of vortex patterns. Vortex methods for numerics. The Hamiltonian structure of point-vortex dynamics.

Lecture 10. Boundary conditions. Free-surface problems. Surface tension.

 

6: Oct 8, 10

Lecture 11. Static problems with surface tension: drops, bubbles, menisci, jets and sheets. Effects of surfactants. Marangoni flows.

Lecture 12. Flow of thin films. Lubrication theory.

 

7: Oct 15, 17

Lecture 13. Thin-film flow. Reynolds lubrication equation.

Lecture 14. Basic viscous flow problems: Couette flow, Poiseuille flow, flow down an incline, flow around a sphere, stagnation-point flow, flow past a flat plate.

Oct 20. Mid-term exam

8: Oct 22, 24

Lecture 15. Stokes flow. Fundamental solution of the Stokes equation. Stokeslets. Reversibility of Stokes flow. Stokes flow around a sphere and a cylinder.

Lecture 16. Boundary-layer theory. Prandtl's equations.

Oct 24, last day to drop w/ W

Oct 25, Nov 2

Fall Break

 

9: Nov 5, 7

Lecture 17. Blasius solution for a flow over a flat plate. Thickness of a boundary layer. Eckman layer. Taylor-Proudman theorem.

Lecture 18. Introduction to flow instability. Linear and nonlinear stability. Asymptotic stability. Lyapounov stability. Normal mode analysis. Serrin's theorem. Thomas-Hopf theorem.

 

10: Nov 12, 14

Lecture 19. Kelvin-Helmholtz and Rayleigh-Taylor instability.

Lecture 20. : Rayleigh-Plateau instability.

 

11: Nov 19, 21

Lecture 21. Stability of parallel flows. Equations of Orr-Sommerfeld and Squire. Square's theorem. Inviscid instability. Rayleigh's and Fjortoft's theorems.

Lecture 22. Introduction to compressible flow. Compressible Navier-Stokes and Euler equations. Elements of thermodynamics. Ideal gas.

 

12: Nov 26, 28

Lecture 23. Smooth compressible flow. Crocco's equation. Flow in stream tubes and nozzles. Shocks and Rankine-Hugoniot conditions. Contact discontinuities.

Lecture 24. Normal shocks. Detonations and deflagrations. The theory of Zel'dovich, von Neumann, and Doering.

 

13: Dec 3, 5

Lecture 25. The theory of Zel'dovich, von Neumann, and Doering.

 

Dec 10:

Project Presentations