APPH E4200 Site Information

Physics of Fluids

Prof. Michael Mauel

General Textbook Grading Syllubus Homework TA Links


Welcome to the APPH E4200 Physics of Fluids class information site. Classroom: MW 8:40 -9:55 Room 524 S. W. Mudd Building

APPH E4200x Physics of Fluids 3 pts. Prerequisite: APMA E3102 Partial Differential Equations or equivalent; PHYS 1401 Introduction to Mechanics and Thermodynamics or 1601 Mechanics/Relativity or equivalent. An introduction to the physical behavior of fluids for science and engineering students. Derivation of basic equations of fluid dynamics: conservation of mass, momentum, and energy. Dimensional analysis. Vorticity. Laminar boundary layers. Potential flow. Effects of compressibility, stratification, and rotation. Waves on a free surface; shallow water equations. Turbulence.

The physics of fluids is a fascinating subject. Everyone has experienced the wonder of fluid phenomena, like ripples in a lake, violence and variability of weather, flight of airplanes, and mixing of cream in coffee. Fluid behavior is both complex and familiar. Because of this, the physics of fluids is one of the best subjects to apply our understanding of mathematics and our basic laws of physics to describe non-trivial phenomena and important applications.

APPH E4200 is an introductory course in fluids.  We approach the subject as physicists, as opposed to engineers or mathematicians (though elements of mathematics and engineering enter naturally).  Our goal is for students to attain a solid grasp of the fundamentals of the subject. 

In the first part of the course we will understand the basic equations and introduce principles that are essential to all aspects of fluid mechanics.  Before long we will move on to study specific types of fluid flow. 

The minimum prerequisites for this course are a year of college level physics (including mechanics and basic thermodynamics), and mathematics through multivariable calculus.  However, the class will be considerably easier if you have had more physics and mathematics than that. Prior experience with partial differential equations is particularly useful.  I will also assume at least a minimal knowledge of complex variables. If you are at all concerned about your level of preparation, come see me.


Kundu and Cohen, Fluid Dynamics, 4rd edition.  (3rd and 5th edition are fine also.) In the opinion of Prof. Adam Sobel, this is one best textbooks available on the subject!

An electronic version of this textboo is available to Columbia University students. Please follow this link, and also this link to the university ebook.

Fluid mechanics, the study of how fluids behave and interact under various forces and in various applied situationswhether in the liquid or gaseous state or bothis introduced and comprehensively covered in this widely adopted text. Fully revised and updated with the addition of a new chapter on biofluid mechanics, Fluid Mechanics, Fourth Edition is suitable for both a first or second course in fluid mechanics at the graduate or advanced undergraduate level. The leading advanced general text on fluid mechanics, Fluid Mechanics, 4e guides students from the fundamentals to the analysis and application of fluid mechanics, including compressible flow and such diverse applications as hydraulics and aerodynamics.

Hardcover: 840 pages
Publisher: Academic Press; 4 edition (November 27, 2007)
ISBN-10: 0123737354
ISBN-13: 978-0123737359 Customer Review: 4.5 stars; 16 reviews.


Problem sets will be assigned throughout the term, approximately once per week and a half (i.e. every 3rd lecture).  You are welcome to work on them in small groups, as long as you write up your answers yourself and make sure that you understand what you are writing.  If you try, you may be able to find old copies of the solutions from previous years.  You are not allowed to use these. 

There will be a midterm and a final exam

The grading for the course will be, approximately: either 25% problem sets, 50% final, 25% midterm, or 66% final, 33% midterm, whichever is greater.  In other words, doing the problem sets cannot hurt your grade, and is, strictly speaking, optional, though I highly recommend it. 


This Web Site is a basic resource for APPH 4200. Copies of my lecture notes will be available for download in Adobe PDF formats.

A preliminary lecture plan is llisted below. I anticipate changes as we move along. Some topics may require more lecture time, and some will require less. Depending on your interests and comments, we may change some of the topics in the last third of the course.

Lecture Dates Topics
Sept 5, 10,12

Introduction to Fluids.
Mathematical Preliminaries:  Cartesian tensors.
Chapters 1 and 2

Lecture 1 Notes
See discussion about lift generated by fluid flow around a rotating cylinder at NASA's Glenn Research Center.

Lecture 2a Notes
Lecture 2b Notes
Download useful vector identities (and plasma and hydro info)
Naval Research Lab's Plasma Formulary.

In class problem (Sept 10): Fluid Circulation around a Cylinder

In class problem (Sept 12): Vector Calculus Identities

Sept 17
(No Class Wed 9/19)

Mathematical Preliminaries:  Cartesian tensors.
Chapter 2
Lecture 3 Notes

Fluids in the news (9-2-2014): Soliton Turbulence seen for the first time on North Carolina’s Coast. See "Soliton Turbulence in Shallow Water Ocean Surface Waves," PRL, 113, 108501. By Andre Costa, a PhD student at Aix-Marseille Université, France.

Fluids in the news (9-10-2018): "Leidenfrost wheels," Nature Physics (2018) Beautiful movies illustrating "particle image velocometry (PIV); see supplemental materials.

Basic kinematics:  Lagrangian vs. Eulerian,
Streamlines & trajectories, total derivative,
Mass continuity.
Chapter 3
Lecture 4 Notes

In class problem (Sept 17): Axisymmetric steady compressible flow

Sept 24, 26

Derivation of Navier-Stokes equation.  
Balance of forces.  Bernoulli's Equation.
Stress and strain tensors, definition of a Newtonian fluid. 
Effects of a rotating reference frame.
Chapter 4

Lecture 5 Notes
Lecture 6 Notes

Excellent video about Eulerian and Lagrangian Descriptions of Fluids (by Prof. John Lumley, Penn State, 1962). Part 2 is located here. The mathematics is very well explained in Part 2. Finally, Part 3 illustrates a more general vector field and the acceleration of a material element of a fluid.

Read about the Clay Mathematics Institute's Navier-Stokes $1,000,000 challenge at this link.

Oct 1, 3

Fluids in the news:

Using the conservation equations…
Chapter 4
Lecture 7 Notes

Oct 8, 10

Chapter 5
Lecture 8 Notes

Fluid Video Gallery from Melbourne, AU, with colliding vortex rings.

Line Vortex

See wonderful work by UCSD Pure Electron Plasma Group, including "Vortex crystals from 2D Euler flow: Experiment and simulation", Phys. Fluids, 11(4) 905 (April 1999) and an early news article in Science highlighting both Columbia and the UCSD plasma research.

"Anatomy of a Bathtub Vortex," by A. Andersen and co-authors, Physical Review Letters, Vol. 91, p. 104502 (2003).

More Chapter 5
Lecture 9 Notes

Excellent Vorticity videos from the 1960's Fluid Mechanics Films series. See Vorticity Part 1 and Vorticity Part 2.

Oct 15 Cancelled


Oct 17

Simple calculations of 2D motion of line vortices. Mathematica notebook is here Line Vorticies.nb. A video is here ring_vortex.mp4

Potential flow theory. 
Irrotational flow. 
Potential flow around obstacles:  lift and drag.
Chapter 6
Lecture 10 Notes

Link to "complex analysis" at Wolfram's MathWorld.
Mathworld's page on Cauchy-Riemann Equations.

Nice biography of Paul Richard Heinrich Blasius (1883-1970).

Lecture 11 Notes (Ch. 6)

Numerical example prepared using Mathematica. Mathematica is available in all Columbia University computer labs, and a student license is available.

Mathematica notebook and print-out in pdf.

Oct 22

Dimensional analysis, similarity. Good overview: "Dynamic similarity, the dimensionless science," by Bolster, Hershberger, and Donnelly, Physics Today, Sept, 2011.
Reynolds number. 
Laminar boundary-layer theory, drag on thin flat plate.
Chapter 9

Lecture 12 Notes (Ch. 8 & 9)
Lecture 13 Notes
(Ch. 9)

Lecture 15 Notes (Midterm Review)

Oct 24

(Appendecies and Equations attached to exam: here.)
Results are summarized here.

Cancelled Oct 29

Oct 31

More Chapter 9...
Lecture 14 Notes (More Ch. 9) Waves

Vortex Sheading Video (a great student project, entertaining to watch)

Nov 5, 7

APS Division of Plasma Physics Meeting (New Orleans). Classes cancelled.

Nov 6

Election Day Break

Nov 12, 14

More waves....

Lecture 16 Notes (Ch. 7, Surface Gravity Waves)

Lecture 17 Notes (Ch. 7, Internal Gravity Waves)

Outstanding Physical Oceanography Lab Demos from Univ Rhode Island

Linear surface gravity waves.  Continuously stratified flow.
Shallow and deep water equations.

Nov 19, 21

Fluids in the news (11-2014): Viscoelasticity Breaks the Symmetry of Impacting Jets

Lecture 18 Notes (Ch. 7, Capillary Waves & Sound)

Lecture 18 (more) Notes (Tides and Stellar Clouds)

Linear Instability; Benard Thermal Instability; Kelvin-Helmholtz problem: 
stability of a sheared flow with stable density stratification.
Chapter 12

Lecture 19 Notes
Mathematica notebook and print-out in pdf.

Nov 22-23

Thanksgiving Holiday

Nov 26, 28

Lecture 20 Notes Rotating and parallel shear flows (Chapter 12)

"Transport of solar wind into Earth's magnetosphere through rolled-up Kelvin-Helmholtz vortices", Hasegawa, et al., Nature, 430, 755 (2004).

Three relevant articles reviewing viscous dynamics:

Chapter 10: Ludwig Prandtl and viscous boundary layer

Lecture 22 Notes (Ch. 10)
(Mathematica Notebook and PDF file of numerical solution to boundary layer.)

Nice biographies of Ludwig Prandtl (1875-1953) and also of his graduate student, Paul Richard Heinrich Blasius (1883-1970).

Dec 3, 5

Chapter 13: Introduction to turbulence

Lecture 21 Notes
(Lorenz Model Mathematica Notebook and PDF file.)

Lorenz Model at Wikipedia Site.
Movie of Turbulent Fluid Mixing from Prof. Thomas Peacock at MIT.

Chapter 14 Geostrophic and quasi-geostrophic flow.

Lecture 23 Notes

UNISYS Weather site and atmospheric vorticity and flow maps.
Biography of Carl-Gustaf Rossby at American Meteorological Society.
Time magazine's cover story about Rossby and weather forecasting.
Biography of Jule Charney at AGU.
Good review of Rossby waves by Robert Dickinson (Annual Review of Fluid Mechanics, vol 10, 1978)
Rossby waves in oceans, Chelton and Schlax,Science, 1996.
Rossby waves on the sun, Kuhn, et al., Nature, 2000.
Biography of Jack Bjerknes.

Dec 10

Lecture 24 Notes Review for final exam.




Homework problem sets will be posted below. You will have one or two weeks to prepare your solutions. I will post homework solutions one the day after they are due; therefore, I will not grade or accept late homeworks.

(Some of these homework assignements are the same as last year's. The due dates are as listed below, submitted before class begins on Tuesdays.)

Homework Assigned Due Date
PS1 9-5-18 9-24-18 HW1-Solutions.pdf
PS2 9-17-18 10-1-18 HW2-Solutions.pdf
PS3 9-26-18 10-8-18 HW3-Solutions.pdf
PS4 10-31-18 11-12-18 HW4-Solutions.pdf
PS5 11-14-18 11-26-18 HW5-Solutions.pdf
PS6 11-21-18 12-2-18 HW6-Solutions.pdf

TA and Help

To be determined....

Fluid Dynamics Links

Wikipedia's Fluids sites:

eFluids a specialty web portal designed to serve as a one-stop web information resource for anyone working in the areas of flow engineering, fluid mechanics research, education and directly related topics. Check it out! Including Gallery of Flow Images, Experiments, and Problems.

Brown University's site on computational fluid mechanics and visulaizations.

NSF's site for Fluid Dynamics research funding.

CFD Online

MIT's Fluids Lab and Gallery

Professor Michael E. Mauel
Department of Applied Physics
Columbia University

Go to Prof. Mauel's HomePage