This photo sequence shows the "gobbling droplets" phenomenon. A jet of liquid is unstable because of surface tension and usually breaks into small droplets. The addition of minute quantities of polymeric molecules provides an additive elastic stress which stabilizes the liquid column. In this situation the terminal droplet has the time to gobble many of its incoming neighbors before its detachment. (Photo by Jose Bico and Christian Clasen, used courtesy of Prof. Gareth McKinley.)
Course Highlights
This course features a unit of interactive problems in the assignments section, and extensive study materials and related resources.
Course Description
This course surveys the principal concepts and methods of fluid dynamics. Topics include mass conservation, momentum, and energy equations for continua, the NavierStokes equation for viscous flows, similarity and dimensional analysis, lubrication theory, boundary layers and separation, circulation and vorticity theorems, potential flow, an introduction to turbulence, lift and drag, surface tension and surface tension driven flows. The class assumes students have had one prior undergraduate class in the area of fluid mechanics. Emphasis is placed on being able to formulate and solve typical problems of engineering importance.
Special Features
 Video demonstration
Prerequisites
2.006; 18.075 or 18.085
Topics Covered

Continuum Viewpoint and the Equation of Motion

Static Fluids

Mass Conservation

Inviscid Flow (Differential Approach): Euler's Equation, Bernoulli's Integral, and the Effects of Streamline Curvature

Control Volume Theorems (Integral Approach): Linear Momentum Theorem, Angular Momentum Theorem, First and Second Laws of Thermodynamics

NavierStokes Equation and Viscous Flow

Similarity and Dimensional Analysis

Boundary Layers, Separation and the Effect on Drag and Lift

Vorticity and Circulation

Potential Flow, Lift, Drag, and Thrust

Surface Tension and its Effect on Flows

Introduction to Turbulence (if time)
Textbooks
The suggested text is: Kundu, Pijush K., and Ira M. Cohen. Fluid Mechanics. 3rd ed. Burlington, MA: Elsevier, 2004. ISBN: 9780121782535.
This book is strongly recommended and readings from it will be assigned.
Another excellent book is: Fay, James A. Introduction to Fluid Mechanics. Cambridge, MA: MIT Press, 1994. ISBN: 9780262061650.
Fay's book is at the advanced undergraduate level, but covers most of the topics dealt with in the lectures. The lectures will cover some material with greater rigor or different emphasis; special notes are provided for selected topics, as indicated in the outlines. Students are responsible for material covered in class or indicated in the course outlines.
The following book is required for all students, as the source of most assigned homework problems: Shapiro, Ascher H., and Ain A. Sonin. Advanced Fluid Mechanics Problems. (Selfpublished manuscript.)
For Section 5 (Control Volume Theorems), a revised set of problems is posted in assignments, with hints and answers as well as some full solutions and examples. This is part of an ongoing updating of the problems and the way they are presented.
Knowledgeable students will be able to read equally profitably from alternative texts and readings, provided they are in the habit of reading broadly and searching out references that satisfy them on the fundamentals. For instance:
Tritton, D. Physical Fluid Dynamics. 2nd ed. New York, NY: Oxford University Press, 1988. ISBN: 9780198544937.
Schlichting, H. Boundary Layer Theory. 7th ed. New York, NY: McGraw Hill, 1979. ISBN: 9780070553347.
For excellent reviews of the state of the art in all areas of fluid mechanics, see Annual Review of Fluid Mechanics, Vol. 1 (1969) to Vol. 37 (2005).
Finally, there is an excellent CDROM available with English, French and German language content: Homsey, G. M., et al. Multimedia Fluid Mechanics. New York, NY: Cambridge University Press, 2004. ISBN: 9780521604765.
Homework Problems and Tutorial Sessions
Homework problems from Shapiro and Sonin's Advanced Fluid Mechanics Problems are indicated in the course outline for each topic. The homework problems are not to be turned in. Instead, they will be discussed in the tutorial sessions. Three tutorials are scheduled per week, but the idea is that each student should come to one session each week, usually the same one. Three sessions are scheduled, partly to accommodate a variety of student schedules and partly to reduce the class size and allow for a more informal atmosphere for discussion.
New for 2005; detailed solutions for some of the problems will be prepared by the Teaching Assistant and will be posted online weekly.
The main purpose of this course is not so much to feed the students with "advanced" material (the topics covered do not in fact appear terribly advanced) as to help students develop a mastery of the underlying principles and the ability to solve, quickly and efficiently, a variety of real fluid mechanics problems from first principles. The lectures present and illustrate the fundamental laws and the methods and modeling approximations that form the basis of fluid mechanics. The problems and tutorials help the students gain a mastery of the material and to develop, by practice and trial and error, the mindset of an effective problem solver in fluid mechanics.
Both the assigned problems and the tutorials are entirely voluntary. No problem sets are collected, nor is roll call taken, except perhaps to help the instructors remember the students' names. However, based on repeated experience over many years, you may take our word that your chances of doing well in this course are minimal if you do not independently do at least the assigned problems before the tutorials, and use the tutorials to repair weaknesses and develop new insights. We are ready to help you in every way to master the course material. There is, however, a profound difference between being taught and learning.
Examinations
There will be two onehour quizzes during the term, announced well in advance. In order to minimize time pressures, we prefer to give the (nominally) onehour quizzes in the evening starting at 7 pm, and give students until 9 pm to complete the problems.
There will be a threehour final exam.
Quizzes and the exam will permit a limited number of pages of open notes (and a calculator and a book of mathematical formulas and tables). No other books will be allowed. The quizzes and the exam will not present you with routine problems, but will probe for mastery of the underlying material and for skill in modeling problems in the simplest possible realistic terms.
Grading
ACTIVITIES 
PERCENTAGES 
Quiz 1 
25% 
Quiz 2 
25% 
Final Exam 
50% 
You can gain extra credit by turning in, at the conclusion of the final exam, a notebook in which you have reworked and amplified your lecture notes in cohesive, clearly reasoned form. This is not obligatory, and that your grade will not suffer if you do not do it: grades will be assigned before the notebooks are examined, and only upward adjustments will be made thereafter. However, thinking through and rewriting the lecture notes, preferably on the same day as the lectures and in consultation with a text, is one of the most effective forms of study, and well worth the effort. Please do not bother to turn in a pretty version of what is on the blackboard: extra credit will be given only when it is apparent that thought has gone into the rewriting.
Calendar
The calendar below provides information on the course's lecture (L) and tutorial (T) sessions.
Course schedule.

SES #

TOPICS

KEY DATES

1. The Continuum Viewpoint and the Equation of Motion

L1

Introduction: Continuum Hypothesis

Homework 1 out


L2

The Material Derivative
Lagrangian and Eulerian Descriptions
Thermophysical Properties
Compressibility Effects in Gases



T1

Tutorial Session



L3

Forces Acting on a Continuum
The Inviscid Fluid



2. Static Fluids

L4

Static Fluids

Homework 2 out


T2

Tutorial Session



3. Mass Conservation in Flowing Media

L5

Mass Conservation in Flowing Media

Homework 3 out


4. Inviscid Flow

L6

Steady Bernoulli Equation

Homework 4 out


T3

Tutorial Session



L7

Unsteady/Generalized Forms of the Bernoulli Equation



5. Control Volume Theorems and Applications

L8

The Reynolds Transport Theorem

Homework 5 out


T4

Tutorial Session



L9

Conservation of Mass/Energy/Entropy



T5

Tutorial Session



L10

Conservation of Linear Momentum
Examples of Conservation of Linear Momentum



T6

Tutorial Session




Quiz 1



L11

Conservation of Angular Momentum



T7

Tutorial Session



6. NavierStokes Equation and Viscous Flow

L12

Kinematics of Deformation

Homework 6 out


L13

The NavierStokes Equation
Boundary Conditions for NavierStokes Equations



T8

Tutorial Session



L14

Fully Developed Flows, Stability of Viscous Flows



L15

Startup and Transient Flows Similarity Solution for a Flat Plate (The Rayleigh Problem)



T9

Tutorial Session



L16

QuasiFully Developed Flows: Lubrication Theory



T10

Tutorial Session



7. Dimensional Analysis

L17

The Buckingham Pi Theorem
Physical Significance of Dimensionless Variables

Homework 7 out


T11

Tutorial Session



L18

Asymptotic Limits of the Governing Equations and Scaling with Dimensionless Variables



8. Potential Flow Theory

L19

The Velocity Potential and Streamfunction
Complex Variable Formulation

Homework 8 out


T12

Tutorial Session



L20

Examples of Potential Flow Solutions




Quiz 2



9. Boundary Layers, Separation and Drag

L21

Boundary Layer on a Flat Plate
Effect of a Pressure Gradient
Separation

Homework 9 out


T13

Tutorial Session



10. Vorticity and Circulation

L22

Definition of Circulations
Kelvin's Circulation Theorems
Lift, Induced Drag

Homework 10 out


T14

Tutorial Session



11. Surface Tension and its Importance

L23

Free Surface Force Balance
Scaling and Dimensional Analysis

Homework 11 out


L24

Sample Flows



T15

Tutorial Session



12. Turbulence (v. Brief Introduction)

L25

Mean and Fluctuating Quantities
Reynolds Stresses, Eddy Viscosity, Taylor Microscale
Homogeneous and WallBounded Turbulence
Kolmogorov Energy Cascade

Homework 12 out


L26

Turbulence (Conclusions)
Course Review




Course schedule.

SES #

TOPICS

KEY DATES

1. The Continuum Viewpoint and the Equation of Motion

L1

Introduction: Continuum Hypothesis

Homework 1 out


L2

The Material Derivative
Lagrangian and Eulerian Descriptions
Thermophysical Properties
Compressibility Effects in Gases



T1

Tutorial Session



L3

Forces Acting on a Continuum
The Inviscid Fluid



2. Static Fluids

L4

Static Fluids

Homework 2 out


T2

Tutorial Session



3. Mass Conservation in Flowing Media

L5

Mass Conservation in Flowing Media

Homework 3 out


4. Inviscid Flow

L6

Steady Bernoulli Equation

Homework 4 out


T3

Tutorial Session



L7

Unsteady/Generalized Forms of the Bernoulli Equation



5. Control Volume Theorems and Applications

L8

The Reynolds Transport Theorem

Homework 5 out


T4

Tutorial Session



L9

Conservation of Mass/Energy/Entropy



T5

Tutorial Session



L10

Conservation of Linear Momentum
Examples of Conservation of Linear Momentum



T6

Tutorial Session




Quiz 1



L11

Conservation of Angular Momentum



T7

Tutorial Session



6. NavierStokes Equation and Viscous Flow

L12

Kinematics of Deformation

Homework 6 out


L13

The NavierStokes Equation
Boundary Conditions for NavierStokes Equations



T8

Tutorial Session



L14

Fully Developed Flows, Stability of Viscous Flows



L15

Startup and Transient Flows Similarity Solution for a Flat Plate (The Rayleigh Problem)



T9

Tutorial Session



L16

QuasiFully Developed Flows: Lubrication Theory



T10

Tutorial Session



7. Dimensional Analysis

L17

The Buckingham Pi Theorem
Physical Significance of Dimensionless Variables

Homework 7 out


T11

Tutorial Session



L18

Asymptotic Limits of the Governing Equations and Scaling with Dimensionless Variables



8. Potential Flow Theory

L19

The Velocity Potential and Streamfunction
Complex Variable Formulation

Homework 8 out


T12

Tutorial Session



L20

Examples of Potential Flow Solutions




Quiz 2



9. Boundary Layers, Separation and Drag

L21

Boundary Layer on a Flat Plate
Effect of a Pressure Gradient
Separation

Homework 9 out


T13

Tutorial Session



10. Vorticity and Circulation

L22

Definition of Circulations
Kelvin's Circulation Theorems
Lift, Induced Drag

Homework 10 out


T14

Tutorial Session



11. Surface Tension and its Importance

L23

Free Surface Force Balance
Scaling and Dimensional Analysis

Homework 11 out


L24

Sample Flows



T15

Tutorial Session



12. Turbulence (v. Brief Introduction)

L25

Mean and Fluctuating Quantities
Reynolds Stresses, Eddy Viscosity, Taylor Microscale
Homogeneous and WallBounded Turbulence
Kolmogorov Energy Cascade

Homework 12 out


L26

Turbulence (Conclusions)
Course Review




Final Exam



