1. kinematics of fluid flow,
   
2. conservation theorems,
   
3. scaling and simplification procedures, and,
   
4. problem definition, solution methods and interpretation for simple flows.

Solution methods will include elementary numerical methods for ODEs and PDEs. Example problems will be taken from engineering as well as geophysical fluid dynamics.

Prerequisites
It is presumed that students will have had a course in classical mechanics and partial differential equations, but no previous fluid dynamics. Students who suspect that they may be significantly under or over prepared should contact the instructor.

Schedule
The class will meet on Monday, Wednesday and Friday, 8:30-9:45, Rm 24-411 at MIT (subject to change once the inevitable conflicts with other classes become evident). Students who reside in Woods Hole should contact the instructor as soon as possible regarding video tape and P-tel options.

Resources
The course has three main components: 1) class meetings three times per week, mostly lectures, but including some films. 2) reading assignments that will repeat some lecture material, though with a different format or emphasis and 3) homework problem sets.

Lectures
The lectures are intended to introduce key concepts and to set them into the context of geophysical problems. Formal lectures on fluid mechanics have the potential to be stupefying. The instructor will do his best to make the lectures interesting; the student's part will be to ask questions and offer comments whenever a clarification would be helpful. The lectures by themselves would not constitute an adequate introduction to fluid mechanics and must be supplemented by reading and by the laboratory session.

Reading
We will make use of four textbooks. The primary source will be Fluid Mechanics by P. Kundu and I. Cohen (2001) (hereafter, KC01) which is recommended for purchase. Three other useful texts are Atmosphere-Ocean Dynamics by A. E. Gill (1982) (G82), Physical Fluid Dynamics by D. J. Tritton (T88) and Mathematics Applied to Deterministic Problems in the Natural Sciences by C. C. Lin and L. A. Segel (1974) (LS74). For a few topics there will be an additional, short reading assignment that can be recovered from the web page (the first one is ELreps.pdf). The reading assignments listed here are the minimum needed for a given section or topic. We do not follow the order of KC01 and the result is a rather chopped-up tour through the text. You will find it more efficient to read larger pieces of the text than are indicated here.

Homework
Homework will be assigned most weeks and will be a significant factor in the grade. Homework assignments will be made during the lecture and will be listed on a web page, HWlog, that will be updated about one day after each assignment. All aspects of an assignment may be discussed with classmates; indeed, collaboration is encouraged, but the final homework paper must be completed by each class member individually. Five days will usually be allowed to complete an assignment. If for any reason a homework assignment can not be turned in on time, then the student will have to make arrangement with the instructor. Some assignments will require the use of a computer to calculate and plot solutions. Students can use any programming language, however Matlab is is recommended for those who do not already have some facility with a programming language. Matlab is widely used at MIT and WHOI and is a powerful, flexible and very highly developed for scientific use. MATLAB can also be very frustrating on first encounter and novices are urged to spend a few hours with a Matlab tutorial. Further help will be provided as needed to overcome programming problems.

Exams and Grading
There may be a mid-term examination, and there will be a final examination to be scheduled during the final exam period. Grading will be apportioned roughly 40% exam(s) and 60% homework.

Syllabus

0. First Class Meeting. (Wednesday, 5 Sep at 8:30, room 24-411 at MIT.)

1. Goals of this class.
   
2. Class administration; homework, exams, grades, etc.
   
3. Scheduling options.
   
4. Math review assignment.

Reading: K90, 2.1-2.3, 2.7-2.10, 2.13-2.16 Appendix B2. For much more detail on these topics, see Boas, 1983.

1. A First Look at Fluid Mechanics. (approx. one meeting.)

1. Definition of a fluid and the continuum hypothesis.
   
2. The laws of classical physics applied to fluid mechanics.
   
3. Some classical and geophysical fluid problems; what makes fluid dynamics challenging?

Reading: K90, 1.1-1.7. Highly recommended, T88 1, 5.1, 5.2, 26.

2. Kinematics of Fluid Flow. (approx. six meetings.)

1. Eulerian and Lagrangian representations of fluid flow.
   
2. Trajectories, streaklines and streamlines.
   
3. The material derivative.
   
4. Reynolds transport theorem.
   
5. Conservation of mass; divergence.
   
6. Velocity gradient tensor and the Cauchy-Stokes theorem.

Reading: ELreps.pdf; KC01, 3.1-3.5, 4.1-4.3, 3.12-3.14, 2.4-2.5, 2.11-2.12, 3.6-3.14
Films: Flow Visualization

3. The Euler Fluid. (approx. six meetings.)

1. Forces in fluids, pressure and buoyancy.
   
2. Newton's laws applied to a perfect (Euler) fluid.
   
3. Bernoulli functions.
   
4. The circulation theorem and vorticity balances; Helmholtz vortex theorems.
   
5. Potential flows; Interacting potential vortices.
   
6. Potential flow around an obstacle. §

Reading: KC01 4.7, 4.8, 4.16 - 4.18, 5.1, 5.2, 5.4, 5.5, 5.7.
Films: Pressure Fields and Flow Acceleration, Vorticity.

4. Earth's rotation. (approx. six meetings.)

1. Earth's rotation and the Coriolis acceleration.
   
2. Inertial motion, geostrophic motion, gradient flow and Rossby number.
   
3. Potential vorticity for a shallow fluid layer.
   
4. Taylor-Proudman theorem.

Reading; KC01, 4.12, 5.6, 14.1 - 14.5, 14.8, 14.13. Highly recommended, T88 16.1-16.5.
Films: Rotating Flows, Secondary Flows

5. Waves. (approx. five meetings.)

1. Dispersion relation for inertia-gravity waves.
   
2. An initial value problem.
   
3. Energy propagation and group speed.
   
4. Geostrophic adjustment.

Reading: KC01, 7.1-7.3; 7.9-7.10; 7.18-7.20; 14.8-14.15. Highly recommended for c) and g) G88 5, 7.
Films: Waves in Fluids

6. Viscous Effects and Boundary Layers. (approx. five meetings.)

1. Tangential stress, the stress tensor.
   
2. Constitutive relations, Newtonian fluids and the Navier-Stokes eqns.
   
3. Stokes first and second problems and Couette flow.
   
4. Energy and vorticity balances.
   
5. Scale analysis and dimensional analysis.
   
6. Boundary layers; Blasius flow.
   
7. Ekman layers and Ekman pumping.
   
8. A first look at turbulence and Reynolds averaging.

Reading: KC01 4.5, 4.6, 4.10, 4.11, 8, 9.1-9.11, 9.15, 10.1-10.5. 14.6, 14.7 Highly recommended, T88 2, 5.6, 5.7, 8, 9 and for part e), LS74 6 and 7.
Films: Fundamentals of Boundary Layers, Flow Instabilities, Turbulence

7. Final Examination, XXX.

References

Primary References: Reading assignments will be made from these four texts. It is recommended that you have a personal copy of at least K90.

KC01; Kundu, P. K. and I. M. Cohen, 2001. Fluid Mechanics, Academic Press. (A complete and well balanced introduction to fluid mechanics. Homework problems will be assigned from this text.)

T88; Tritton, D. J., 1988. Physical Fluid Dynamics. Oxford Univ. Press. (A physically motivated discussion of fluid phenomenon that emphasizes the role of experimentation and observation. Introduces turbulence from the outset. )

G82; Gill, A. E., 1982. Atmosphere-Ocean Dynamics. Academic Press. (A scholarly and extensive introduction to ocean and atmosphere dynamics. Example problems are motivated by observations, and carried through to interesting results. Many tables of useful data.)

LS74; Lin, C. C. and L. A. Segel, 1974. Mathematics Applied to Deterministic Problems in the Natural Sciences. MacMillan Pub. (Part C is devoted to the theory of continuous media, and is a precise, mathematical treatment of continuum mechanics. A bit more formal than we want here, but very valuable as a second, advanced source. Chapters 6 and 7 on scale analysis and simplification procedures are superb.)

Fluid Dynamics/Applied Mathematics: There are many useful textbooks and monographs that are relevent to fluid dynamics. A few of those that are most likely to be of use for this class and that are available from the instructor or the Lindgren Library are listed below.

Acheson, D. J., 1990. Elementary Fluid Dynamics. Oxford Univ. Press. (Concise and elegant treatment of mainly engineering fluid dynamics.)

Aris, R., 1962. Vectors, Tensors and the Basic Equations of Fluid Mechanics, Dover Pub. (Some of the opening chapters are very clear and precise treatments of vector calculus, etc. A bit advanced for a first course.)

Boas, M. L., 1983. Mathematical Methods in the Physical Sciences. 2nd Ed., John Wiley. (This is perhaps the best applied math reference at the level of senior undergraduate. Excellent chapters on vector and tensor analysis and coordinate transformations.)

Cushman-Roisin, B., 1994. Introduction to Geophysical Fluid Dynamics. Prentice Hall. (Very clear introduction to GFD.)

Farlow, S. J., 1993. Partial Differential Equations for Scientists and Engineers. Dover. (This is a valuable resource for those needing to refresh or extend their working knowledge of PDEs.)

Holton, J. R., 1992: An Introduction to Dynamic Meteorology, 3rd Ed., Academic Press. (An excellent text covering many aspects of atmospheric science. The early chapters are a clear and concise introduction to fluid dynamics at just about the level one can expect to remember. A good book for ocean and atmospheric science students to own.)

Pedlosky, J., 1987. Geophysical Fluid Dynamics, 2nd Ed., Springer-Verlag. (This is the standard for GFD, though a little advanced for a first course in fluid mechanics.)

Wilcox, D. C., 2000. Basic Fluid Mechanic, 2nd Ed. DCW Industries. (This is an excellent introduction to fluid mechanics. Where it overlaps K90 and KC01 it is often superior in clarity and depth. The emphasis is on engineering (very little on waves and geophysical flows) or this would be our primary reference.)