Graduate Fluid Mechanics


Lecture notes:

In order to orient you about the scope and level of this class, these are left online from the past.

The order may change as we go on, but it is unlikely that we will have time to add more topics,

despite the desire to do so.

Introductory set of notes (in pdf format; to make things a bit more organized, I’ll be always giving a

“table of content”, including relevant corrections and comments).

  1. p.1-5: fluid mechanics as field theory, physical examples of fluids, validity of the continuum approximation and

“Knudsen number”

  1. p.6-10: velocity field and streamlines (note that immersed died particles will follow streamlines only if flow is

stationary, otherwise they follow “streaklines”), Eulerian description and material derivative

  1. p.11-15: forces on a fluid element, “volume” and “surface” forces, equation of motion of fluid element

Conservation laws and the general structure of the fluid equations of motion:

p. 16-20.1: mass conservation and continuity equation 

  1. p.21-27: momentum conservation and the stress tensor 

  2. p.28-29.2: interlude on tensors, more on tensors

  3. p.30-37:  diagonalization of the stress tensor, trace and traceless parts (30-32); pressure and the equations of

hydrostatic equilibrium (33-35); the Euler equatuions of an ideal fluid (36-37)

  1. p.38-49: kinetic energy balance of an ideal compressible fluid (38-40); thermodynamic relations for a fixed-mass

element (41-43); rate of change of internal energy of fluid and total energy balance (44-47); constancy of entropy per

unit mass (48); equations for a stationary flow of  an ideal fluid (49) 

  1. p.50-53:  viscosity and kinematic viscosity (50-52); an interlude on simple kinetic theory (52.1-52.4);

thermal conductivity (52.4-52.6); viscosity (52.6-52.8); Newtonian and non-Newtonian fluids (53)

A piece of extra reading---a recent Physics Today article on shear-thickening in colloidal suspensions; an

``experiment” was done also in class (for the recipe, google “oobleck”).

  1. p.54-67: strain tensor (54-55); symmetric strain tensor (55-58); antisymmetric strain tensor (58-61); decomposing a

general fluid motion (62-63); finally, the Navier-Stokes equation, shear and bulk viscosities (64-67)

For those of you who are curious, see the “Search for the perfect fluid” article. These are the fluids with the smallest

viscosity to entropy density ratio (according to a conjectured quantum lower bound). The fluids in question are either

the quark-gluon plasma produced at RHIC or quantum atomic gases “at unitarity”, i.e. strongly interacting (and very

quantum) ones.

  1. p.73-87 (missing page numbers are due to rearrangement of notes): energy balance of a fluid with dissipation (73-76);

the complete set of equations of fluid mechanics (77); second viscosity, pressure, and compressibility (78-81);

Bernoulli’s law (82-87)

  1. p.88-95: Toricelli’s law and Pitot tube (88.1-88.2); speed of sound and Mach number (88-92)

[note that discussion in notes including a uniform g-field is not precise, in class we will skip g as this complication is not necessary];

conditions allowing neglect of changes of density due to dissipation (93-95)

prompted by in-class discussion: pp. 95.1-95.5: correcting the estimate of dissipation’s contribution to changes

of density (95.1-.2); scale of time variation of flow and changes of density (95.2-95.3); examples of nonstationary

flows where time variation scale is not equal to L/U (95.5-.5)

Finally, for this portion of the class, see the Clay Institute “Millennium Problems” entry regarding the proof

of existence of solutions of the Navier-Stokes equation.

In class, there were also questions about fluid dynamics in relativistic case and the definition of the entropy and other

thermodynamic quantities. See Landau and Lifshitz as well as these recent lectures for a more modern

pedagogical discussion.

Various viscous flows, Reynolds number, Stokes flow:

The “efluids” gallery has a lot of pictures, videos, and simulations of fluid flows. Check it out!

pp. 96-103: Incompressible viscous flows: parallel plates (96-98); forces balance (99-99.1);  the ``forgotten” energy

equation (99.2,99.3), cylindrical pipe (Poiseulle) (100-103) [note that last eqn. on 99.2 and first eqn. on 99.3 are missing R^2 in

denominator; estimate does not change much as I took R^2~R~1 in units s.t. 1cm =1 - will be, hopefully, done correctly in class]

  1. pp.104-111: Reynolds number and similarity; relevance of small- and large-R flows (104-111)

See the closely related recent article on dynamical similarity in Physics Today, which also discusses some

more numbers and concrete applications than in class.

Read the classic article ``Life at small Reynolds number” by E. M. Purcell (an atomic physicist, as it comes) and

appreciate how the absence of inertia at small R requires quite a bit of inventiveness on the part of small organisms

to propel themselves.

Also see this video (there are three parts) of a rubber ribbon driven mechanical fish trying

to swim at small R (in corn syrup).

If you are theoretically inclined, see the article of Al Shapere and Frank Wilczek (the latter won the 2004

Nobel Price for the discovery of asymptotic freedom in Quantum Chromodynamics, the theory of the strong nuclear

force) on the mathematical description of swimming at low Reynolds number using gauge potentials and fibre bundles.

p. 112-117.3: qualitative behaviour of the drag at small and large R (117.1-117.3); flow through a pipe with a slowly

varying cross section (111-114); flow through a moving channel with varying cross section

and ``lubrication” (115-117) [will go back to omitted 117.1-117.3]

On fluid flow against pressure gradient, prompted by a question in class 

  1. p.118-134: “Stokes’ 1st problem” - drag on a sphere at small R: consequences of linearity of small-R equations

(118-119); necessary interlude on ``2d” incompressible flows (120-122); setting up the Stokes problem (123-124);

continuity equation and the stream function (125); vorticity equation in terms of the stream function and boundary

conditions (126-127); solution for the stream function and the velocity field (128-132);

the stress tensor and ``Stokes’ law” for the drag force (133-134)

Notes on uniqueness of solutions of Stokes’ equations (small-R NS, that is), by popular demand; this is really an old

(mid 1800’s)  argument due to Helmholtz.

  1. p.135-143.1: using Stokes’ law to measure viscosity (135); streamlines of the Stokes flow in

instantaneous rest frame (136-141); streamlines in frame where body is at rest and fluid moves (141-143.1)

Many aspects of the theory of the Stokes flow were only understood recently. See this article discussing the use

of the renormalization group in this context.

Here’s the video of another fun demonstration of creeping flow, mentioned in class and on p. 119 of notes.

On an unrelated topic, here’s a presentation on the fluid mechanics of beer foam (it involves discussion of topics

that we have not touched upon such as surface tension, solubility, etc., but also things we’ve been through).

One of you pointed out to me this recent PRL article on spontaneous symmetry breaking and the generation of

a lift force (and torque, without increasing drag) due to a free flexible filament attached to the downstream side of

a cylinder immersed in a steady incompressible 2d flow. Check it out, the physics is fun! (Notice that this concerns

lift at reasonably large R, our next subject.)

Large-R flows, properties of inviscid flows, “lift theorem”, and boundary layers:

Notes on surface tension and boundary condition at interfaces: molecular origin of surface tension, estimate of

surface tension of water (1-3); surface tension as a ``line” force (3-4); Laplace’s formula from line force and

energy balance; estimate of water droplet capillary radius (5-6); momentum conservation and the surface

force (7); math interlude (8-9); momentum conservation across interface and general boundary conditions (9-10);

another derivation of Laplace’s formula (10); boundary condition neglecting surface tension (10-11).

[Later, if we have time, we may talk about capillary waves.]

  1. p.144-157: qualitative discussion of large-R behaviour of Stokes flow (144-145); importance of vorticity and the

NS equation for vorticity (146-149); circulation and vorticity (150-152); implications of circulation

conservation in ideal flows (152-157)

For those of you who are curious, here is some extra reading about the theory of ideal fluids: an article on

the Hamiltonian description of ideal fluids from a fluid dynamicist, another one from field-theorists, and finally,

a recent paper on the difficulties in quantization of ideal fluids.

Speaking of the relevance of vortices, see this 2006 article, by Dan Green and Bill Unruh, about their role in the

destruction the Tacoma bridge in WA state, in the 1940s. There are a few simulations as well, click here and here;

courtesy of Dan Green (undergraduate at the time).

  1. p.158-173: relevance of incompressible, inviscid, irrotational flows (158-159); 2-dimensional incompressible inviscid

flows and the complex potential and complex velocity (160-163); boundary conditions

and the construction of complex velocity fields for some simple flows (164-167); Blasius’ theorem  (168-170)

and Zhukovski’s “lift theorem” (170-172); the Riemann mapping theorem (173) 

Notes on D’Alembert’s paradox (theorem) on the absence drag on a body in an ideal steady fluid flow

(with all coeffts and signs correct; to be discussed briefly on Friday).

New notes on the symmetrical airfoil, the Zhukovsky/Kutta/Chaplygin condition, and the lift force.

Watch this simulation of the formation of a ``trailing vortex” near the sharp edge of a wing. The simulation starts

with a configuration where the stagnation point is on the upper portion of the wing, far from the rear tip. The

velocity of an ideal flow near the tip in such a case (where circulation is not that required by the

Zhukovksy/Kutta/Chaplygin condition) would be infinite; in real viscous fluids, it is simply large, which causes

effects of friction to be most pronounced near the tip, resulting in the creation of vorticity. Observe the

tendency of establishing a flow like the one predicted by ideal fluid (the simulation ends before the vortex is shed

and an ideal-like flow with the right circulation is established---but this is seen in experiments). A very short 3d

simulation can be seen here. Another one which shows different shapes and angles of attack (usually large, so aft

flow is not laminar) and the von Karman vortex sheet in the wake is here. If you can muster the time (25 mins),

this US War Department 1941 video is fun. Check out the tools they use to analyze their data-have you even

seen these?

As for real airplanes, see the ``lift formula” used in this NASA educational website. It is very similar to the one

we derived for a 2d airfoil except that all the dependence on shape/angle of attack is in the “CL” coefficient of

lift, determined by the “velocity relationship curve graph” (!) given.

Aerodynamic studies of the A380 using what they call a “high-lift wing design”  are the topic of this presentation.

Ignore the few slides in German (this appears to be a job talk but has more info than I could find on the Airbus’

website). Notice the devices, p.9-12, on the front and rear part of the wings used to control its geometry and

reduce/enhance lift; the graph of pressure distribution on p.13 as well as the various simulations of exact or

approximate equations used. The wind tunnels used and their scales are shown on p. 14, the Reynolds number

varies between 1.5 - 12 million (notice also the provocative comments about computer simulations vs experiment

on p.15). The rather complex wing profiles are shown on p.16, along with various pressure distributions. The CL

vs angle graph is on p.22, notice the linear relationship obtained for small angles; the numerical value for CL is,

unfortunately, not given---but it should be O(1) (only a theoretical physicist would be happy with this estimate,


  1. p.174-189: the lift on a wing at small angle of attack; comparison with experiment and importance

of vortex shedding (174-177); the boundary layer equations (178-182); examples of boundary layers (183-184.1);

boundary layer separation (185-189).

Notes on surface waves: assumptions and kinematic condition on surface (1-2); pressure condition without

surface tension (2-3); ``small” amplitude expansion and boundary conditions on surface and bottom (3-4);

solution for wave and dispersion relation (4-6); meaning of ``small” (6-7); including surface tension (8);

capillary radius and competition between gravity and surface tension (9). Here is the explanation of

the formula for delta-p on p.8, small slope approximation. Also, prompted by a question, explaining the

relation of the kinematic boundary condition to mass conservation.

Now, I hope you realize that, as Feynman put it: ``water waves that are easily seen by everyone and

which are usually used as an example of waves in elementary courses are the worst possible example.

They have all the complications that waves can have”. There is a multitude of phenomena that can be

explained using the dispersion relation: the trace after a boat,  waves in a pond, tsunami, etc. The trajectories

of physical fluid elements can also be studied using our formulae  for the velocity field.

While many interesting issues (instabilities, turbulence...) are left for future (self-) study,

I hope you got some appreciation of the beauty and complexity of fluid mechanics!

Guest lectures:

Nov. 22, 2012, Thursday, 9:10-10:00: Prof. Sabine Stanley, Dept. of Physics,

Fluid dynamics in planetary cores

Nov. 30, 2012, Friday, 9:30-10:30: Prof. Chris Matzner, Dept. of Astronomy and Astrophysics,

“Shock waves in astrophysics”

Prof. Erich Poppitz

office: 1113A, 416-946-7546

email: poppitz physics utoronto ca

Syllabus and literature

This is an introduction to fluid mechanics aimed at graduate students in physics. No previous

familiarity with fluid mechanics is assumed.

Topics include: Eulerian description of fluids, body and surface forces, energy and momentum

conservation, Newtonian and non-Newtonian fluids, the complete set of equations of fluid

mechanics, sound waves and conditions for incompressibility, simple incompressible viscous

flows, dynamical similarity, flows at small and large Reynolds numbers, Stokes formula,

Kelvin/Helmholtz theorems, 2 dimensional flows and Zhukovski's "lift theorem", boundary

layers and their separation.

Time permitting: waves, instabilities, and turbulence may be also mentioned.

Lecture notes will regularly be posted on this website. I will not follow any particular book and will

attempt to balance formalism with discussion of fluid mechanics phenomena. There are many

good books on the subject that you can study from. Note that none of these are required for the

class and are given here for reference only:

G.K. Batchelor, ``An Introduction to Fluid Dynamics"

A classic of the subject: very detailed and pedagogical introduction to the subject. Has much

more detail that we could possibly go into.

D.J. Acheson, ``Elementary Fluid Dynamics"

This is a relatively new book, more accessible than the previous one, with pictures and

discussions of phenomena.

L.D. Landau and E.M. Lifshitz, ``Fluid Mechanics" (2nd ed)

This is a rather theoretical book and is not easy for beginners. However, if you are serious

about the subject (and are theoretically inclined), in the future you will be grateful for having

been introduced to it.

NOTE: The notes below are left here for reference. They contain unfixed errors, incl. undiscovered ones.