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 Advanced Fluid Mechanics  posted by  member150_php   on 2/17/2009 Add To Favorites
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Abstract/Syllabus:

McKinley, Gareth, Ahmed F. Ghoniem, Ain Sonin, and Anette Hosoi, 2.25 Advanced Fluid Mechanics, Fall 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed 07 Jul, 2010). License: Creative Commons BY-NC-SA

## Advanced Fluid Mechanics

### Fall 2005

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 Navier-Stokes 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
• Navier-Stokes 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. (Self-published 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 on-going 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 CD-ROM 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 one-hour quizzes during the term, announced well in advance. In order to minimize time pressures, we prefer to give the (nominally) one-hour quizzes in the evening starting at 7 pm, and give students until 9 pm to complete the problems.

There will be a three-hour 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. Navier-Stokes Equation and Viscous Flow

L12

Kinematics of Deformation

Homework 6 out

L13

The Navier-Stokes Equation
Boundary Conditions for Navier-Stokes Equations

T8

Tutorial Session

L14

Fully Developed Flows, Stability of Viscous Flows

L15

Start-up and Transient Flows Similarity Solution for a Flat Plate (The Rayleigh Problem)

T9

Tutorial Session

L16

Quasi-Fully 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 Wall-Bounded 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. Navier-Stokes Equation and Viscous Flow L12 Kinematics of Deformation Homework 6 out L13 The Navier-Stokes Equation Boundary Conditions for Navier-Stokes Equations T8 Tutorial Session L14 Fully Developed Flows, Stability of Viscous Flows L15 Start-up and Transient Flows Similarity Solution for a Flat Plate (The Rayleigh Problem) T9 Tutorial Session L16 Quasi-Fully 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 Wall-Bounded Turbulence Kolmogorov Energy Cascade Homework 12 out L26 Turbulence (Conclusions) Course Review Final Exam

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