Modern Classical
Physics Through the Work of G. I. Taylor
One
scientist’s work provides material for an
entire course, covering topics ranging from
hydrodynamic stability and turbulence to
electrohydrodynamics and the locomotion of small
organisms.  Michael P. Brenner and Howard A.
Stone 

A
water bell
forms when a water jet hits the top of a
closed cylinder. The impact of the jet
creates a thin fluid sheet, which then
wraps around the cylinder to form a
beautiful “bell.” This flow
configuration was first analyzed by Felix
Savart in 1833. Taylor provided a
theoretical description of the shape of
the bell. This photograph was taken by
Robert Buckingham in the fluid dynamics
laboratory at MIT’s mathematics
department, under the supervision of John
Bush. 

During the
spring of 1998 we cotaught a graduate course on
modern classical physics that aimed to cover the
fundamentals while also conveying the directions
and sense of current research. As we talked about
the subject, we realized that many of the
important discoveries underlying a wide range of
topics of current interest in physics and
engineering were made by a single individual, the
British scientist Geoffrey Ingram (G. I.) Taylor
(1886–1975). Although many researchers are
familiar with one or another of Taylor’s
contributions, few seem to be aware of the
incredible breadth of his scientific publications
and their relevance to important research
questions today. The same person who is commonly
remembered as the namesake for several basic
fluid flow instabilities (Taylor–Couette,
Rayleigh–Taylor, and Saffman–Taylor)
also was the first to show experimentally that a
diffraction pattern produced by shining light on
a needle does not change when the intensity of
light is decreased. And these topics are only the
beginning. Taylor made fundamental contributions
to turbulence, championing the need for
developing a statistical theory, and performing
the first measurements of the effective
diffusivity and viscosity of the atmosphere.
He wrote one of the first scientific papers using
random walks; gave the first consistent theory of
the structure of shocks in gases; and explained
the importance of dislocations for determining
the strength of solids. He also described the
counterintuitive physics of fluid motion in a
rotating environment, providing the basic
principles for important aspects of atmospheric
and oceanic dynamics. 
Taylor studied all of these topics during the first 30
years of his career, between his 20th and 50th years.
During the next 30 years, among other achievements, he
quantitatively described dispersion of solute in fluid
flow; elaborated the basic principles for how
microorganisms can swim; and predicted, by dimensional
analysis, the energy of the atomic bomb explosion from a
series of US government publicity photographs. He also
recognized that accelerating an interface between two
fluids can lead to instability, and did seminal work on
the interaction between fluids and electric fields,
providing the foundation for electrohydrodynamics and the
basic principles for a slew of presentday industrial
processes and devices. Taylor did much of this research
involving electric fields between his 70th and 80th
years.
The
remarkable depth and breadth of Taylor’s research
impacts in one way or another much of modern research in
classical physics. Therefore, we decided that our ends
would be well served by structuring the course
exclusively around Taylor’s scientific papers. In
this article we summarize the structure and content of
our course, and in the process describe a few of
Taylor’s discoveries that are perhaps not widely
known outside of the disciplines that they impact most
substantially.
Course structure
Throughout the semester, it became increasingly clear
that there were many advantages to structuring a course
around Taylor’s published papers.^{1} First
of all, Taylor’s research interests provide an
excuse to cover a much wider range of topics than is
normally justifiable in a single course. Second, a
careful study of his papers inevitably draws attention to
his style, which is to compare theoretical arguments and
scaling analyses directly and quantitatively with
experimental results. The value of investigating science
and engineering questions in this way, while on the one
hand rather obvious, is on the other hand extremely
difficult both to teach and to learn, especially when
considering complicated nonequilibrium problems as Taylor
routinely did.
As anyone who has
tried to make a prediction about such a system knows too
well, the greatest difficulty is posing questions that at
the same time have simple quantitative answers and prove
insightful. Taylor’s great talent was to repeatedly
find ways of extracting a simple feature from a
complicated process or experiment. Not only did this lead
to direct, quantitatively testable predictions, but later
researchers tended to identify Taylor’s extractions
as the most important quantitative aspects for
understanding the system. In “teaching Taylor,”
there are endless opportunities to draw attention to the
value of this approach to scientific and engineering
questions and to compare and contrast it with more
modern, bruteforce approaches such as direct computation
of every aspect of a system. Although there is clearly
much to be said for both approaches, it is vastly easier
to teach the latter, as the examples of the former are
few and far between.
Course Outline 
Introductory
remarks 
Overview of G.I. Taylor's
research
State of fluid mechanics in 1900 
Taylor's
first two papers 
Diffraction at low light
levels
Regularization of shocks 
Instabilities 
Taylor–Couette flow
Saffman–Taylor problem
Rayleigh–Taylor instability 
Turbulence 
Eddy diffusivity in the
atmosphere
Diffusion by continuous movements
Statistical theory of turbulence
Vortex breakdown 
Rotating
flows 
Taylor–Proudman
theorem
Particle motion and Taylor columns 
Dispersion
in laminar flows 
Taylor–Aris dispersion
Measurement of molecular diffusivities 
Solid
mechanics 
Dislocations and the
strength of solids 
Swimming at
low Reynolds numbers 

Drops and
bubbles 
Drop deformation and
breakup
Viscosity of mixtures; emulsions 
Electrohydrodynamics 
Leaky dielectric model 
Surface
tension 
Thin films, peeling, water
bells 
Shocks 

Explosions 

The outline for our course is shown in the box above.
Typically there were two 90minute lectures per week, in
which we critically discussed a single paper, or
sometimes a group of two or three papers. The papers were
distributed in advance and students were expected to have
read them. In several instances, we distributed recent
review articles or closely related research papers as
well. We also organized a number of special seminars
given by local faculty and visitors; we asked these
lecturers to frame their remarks as: “subject X
since Taylor.”
Introductory ideas
To set the stage for Taylor’s research, we used the
first lecture to summarize the state of fluid mechanics
in the early years of the 20th century, before Taylor
became involved. We based this presentation on the
excellent review by Sydney Goldstein, published as the
first article in the first issue of Annual Reviews of
Fluid Mechanics.^{2} Although much was known
about fluid motion in the early years of the 20th
century, much discord and debate existed over the
relation of the theories to experiments. In 1916, Lord
Rayleigh wrote a review for Nature of the fourth
edition of Horace Lamb’s Hydrodynamics, in
which he said “Perhaps the time for [comparing
theoretical hydrodynamics with experiments] has not yet
come . . . . We may hope that before long [experiments
may be] brought into closer relation with theoretical
hydrodynamics.”
A
major problem at the time was that there was still
uncertainty about the correct boundary conditions on the
fluid velocity at solid surfaces, and whether these
boundary conditions could be independent of the state of
motion of the fluid. Although Ludwig Prandtl’s 1904
work introducing viscous boundary layers pointed toward
the resolution, his ideas were only gradually being
disseminated and understood. Goldstein writes that by the
mid20th century these problems were largely resolved.
“Several factors . . . contributed to this, but the
greatest influence has been the example of G. I.
Taylor.”
We
then turned to a discussion of Taylor’s papers. Our
choice of ordering, summarized in the box, was an attempt
to be pedagogical. We started with Taylor’s first
two scientific papers, written when he was less than 25
years old, and proceeded to read his work on
instabilities, turbulence, rotating flows, and so on.
The
rest of this article gives brief summaries of some of the
topics. Taylor contributed so much to fluid and solid
mechanics that it is both impossible and beyond our
competence to do justice even to his qualitative ideas in
a single course, much less in a single article, and so in
both cases there are egregious omissions. Our choice of
topics for this article was motivated by our desire to
show the breadth and continued relevance of Taylor’s
research, as well as to highlight those topics that we
found to be the most useful pedagogically. For more
detailed information about Taylor’s work and life,
we recommend George Batchelor’s recent biography of
Taylor,^{3} and recent review articles.^{4,5}
Interference
fringes by feeble light
We began our tour of Taylor’s research by discussing
his first scientific paper, which was published in 1909.
This was his only paper that was not classical physics,
but it nonetheless bore the experimental characteristics
that were to appear throughout his later work. At the
request of J. J. Thomson, Taylor performed an experiment
(in the children’s room of his parent’s house!)
to determine whether there was a qualitative change in a
diffraction pattern when the intensity of the light is
reduced greatly.^{3} Taylor indicates that
Thomson believed that there would be a change in the
pattern. Taylor took photographs of the shadow of a
needle, varying the intensity of light by shielding the
light source with smoked glass screens. When decreasing
the intensity he increased the exposure time to keep the
total amount of light on the photograph constant. The
longest experiment took three months, corresponding to
the intensity of a candle more than a mile away; some of
the experiments even took place while Taylor was on a
yachting trip. Taylor observed no change in the
diffraction pattern, wrote a twopage paper describing
this result, and then dropped this line of research.

Geoffrey
Ingram Taylor
(right) at age 69, in his laboratory with
his assistant Walter Thompson. (AIP
Emilio Segrč Visual Archives.) 

Motion of
discontinuities in gases
Taylor’s second scientific paper, published
in 1910 when he was 25 years old, was awarded the
Smith Prize for senior mathematics students at
Cambridge University. This paper solved a
longstanding, fundamental problem in fluid
mechanics. George Gabriel Stokes had noticed that
there was the real possibility that the velocity
in a gas could form discontinuities in a finite
time, if a slower region of gas were moving ahead
of a faster region. Such discontinuities, now
called “shocks,” are easily predicted
from the equations of ideal (inviscid) fluid
dynamics. They represent singularities, in that
velocity gradients diverge at the discontinuity.
At the time, it was not known what happened after
such shocks formed. Taylor demonstrated that in a
real gas the discontinuity would be eliminated by
dissipative effects (both viscosity and thermal
heating). This solution (realized qualitatively
in 1908 by Rayleigh, then 66 years old) is one of
the most basic features in gas dynamics. 
The
Taylor–Couette paper
The first topic we treated in detail was Taylor’s
1923 paper on instabilities of Couette flow—the flow
between concentric rotating cylinders. An interesting
feature is the paper’s motivation. Taylor begins by
observing that “A great many attempts have been made
to discover some mathematical representation of fluid
instability, but so far they have been unsuccessful in
every case.”^{6} The concept of stability
had been well formulated by this time, and many authors
(among them Lord Kelvin, Rayleigh, Heinz Hopf, and Arnold
Sommerfeld) had attempted to predict the instability of a
solution to the equations of fluid dynamics.
Unfortunately, however, no calculation agreed with
experiments. The failure to predict instabilities led to
great consternation and confusion. For example, Hopf
suggested that perhaps it was necessary to take account
of the rigidity of the boundaries to explain the
instability of shear flows in channels. (Taylor
commented: “There seems little to recommend this
theory as an explanation of the observed turbulent motion
of fluids.”6)
Taylor’s
paper is a major intellectual accomplishment,
representing the first example where a stability
calculation quantitatively matches an experiment. The
fact that the comparison worked is due in large part to
Taylor’s insight that among the different possible
experiments, the rotating cylinder apparatus is best
suited for quantitative comparison between theory and
experiment. The work demonstrated unambiguously that both
the approach used in the stability calculation, and its
underlying assumptions (the boundary conditions), were
correct. As Goldstein states in his review article,
“Simplifications of the mathematics . . . were to
follow, but there could be no [more] controversy.”^{2}
Taylor’s
paper was equally remarkable for its technical detail,
both theoretical and experimental. The calculations
leading to an instability threshold for inner and outer
cylinders of arbitrary radii are tedious, producing
formulas that are each about a page long, involving
determinants of Bessel functions. (In lecture, we avoided
the algebra by using the thingap limit, first introduced
by Harold Jeffreys in 1928, and expanded on at length by
Subrahmanyan Chandrasekhar.^{7}) At the time,
determining the numerical values of the formulas was
itself a significant challenge. Designing an experiment
consistent with the assumptions of the calculation was
equally delicate—in particular, end effects of the
cylinder could not influence the onset of the
instability. The results for the instability boundary as
a function of the rotation rates of the two cylinders
were in beautiful agreement with the theory, as the figure on page 35
shows, and several of Taylor’s photographs of the
flow are still reproduced. Rather amusingly, Taylor
actually measured more points on the stability boundary
experimentally than he calculated theoretically,
presumably due to the tediousness in evaluating the
Bessel function determinants! At the end of the paper,
Taylor described his observations of the panoply of
nonlinear states that exists in the rotating cylinder
apparatus above the instability threshold. As the
relative speed of the cylinders is increased, the flow
goes from steady, to a time varying
“barberpole” pattern of vortices, to a
turbulent irregular flow. As summarized by Richard
Feynman in his lectures:
The
main lesson to be learned from [Taylor’s work]
is that a tremendous variety of behavior is hidden in
the [Navier–Stokes equations]. All the solutions
are for the same equations, only with different
values of the [rotation speed]. We have no reason to
think that there are any terms missing from these
equations. The only difficulty is that we do not have
the mathematical power today to analyze them . . . .
That we have written an equation does not remove from
the flow of fluids its charm or mystery or its
surprise.^{8}

Instability
during the peeling of adhesive tape. G. I. Taylor
studied this problem in 1964 (at the age of 78),
and demonstrated that viscous stresses in the
adhesive fluid contribute significantly to its
“stickiness.” When the adhesive is
peeled from a solid surface (the blue region),
competition between applied pressure and surface
tension leads to an instability with a
welldefined wavelength (squiggles). Interest in
the relevance of fluid mechanical instabilities
to adhesion continues to this day. (For a review,
see the article by Cyprien Gay and Ludwik
Leibler, Physics Today, November 1999,
page 48.) (Image © Felice Frankel, Massachusetts
Institute of Technology; from F. Frankel, G. M.
Whitesides, On the Surface of Things,
Chronicle Books, San Francisco, 1997.) 
Diffusion by
continuous movement
Taylor’s
work on turbulence centered on relentless attempts to
describe turbulence by formulating mathematical theories
that could be directly and quantitatively compared with
experimental data. During the semester, we discussed five
of Taylor’s papers on turbulence, starting with his
monumental (and largely unreadable) 1915 paper,
“Eddy motion in the atmosphere,” and ending
with his 1939 paper introducing what is now known as the
Taylor–Greene vortex. In the latter paper, Taylor
constructs a solution to the Navier–Stokes equations
that demonstrates the turbulent energy cascade.
In
general terms, Taylor’s contribution to our
understanding of turbulence was his observation that
“by analogy to the kinetic theory of gases” one
should find a statistical description. He therefore aimed
to find ways of predicting statistical properties of the
flow. His most penetrating contribution was probably the
formula (given in a 1923 paper):
where
denotes a time average, x denotes
position, and C(t  x) = <v(t)v(t
 x)>/<v(t)^{2}>
is the velocity correlation function.
At
one level this formula is a trivial mathematical identity
and is independent of the details of how an actual fluid
moves. However, the formula represents two different
types of experimental measurements: The lefthand side
gives the dispersion of tracers in the flow and can be
measured by observing the diffusivity of dye in a
turbulent flow; the righthand side can be measured by
sampling the velocity field at different times, and
measuring the correlations. Taylor demonstrated that the
correlation function is sufficient to specify the
statistical properties of a stationary random function,
an idea that has had great influence beyond the realm of
fluid mechanics. For example, Norbert Wiener writes,
describing his beginning research on random functions:
I
was an avid reader of the journals, and in particular
of the Proceedings of the London Mathematical
Society. There I saw a paper by G. I. Taylor, later
to become Sir Geoffrey Taylor, concerning the theory
of turbulence . . . . The paper was allied in my own
interests, in as much as the paths of air particles
in turbulence are curves and the physical results of
Taylor’s papers involve averaging or integration
over families of curves.^{9}
Wiener
goes on to say that Taylor “represents a peculiarly
English type in science: the amateur with a professional
competence.” The above formula has had tremendous
impact on developing the theory of turbulence: To this
day, it is believed that the fundamental quantities to be
predicted from the governing equations are correlation
functions.
Taylor dispersion
One of Taylor’s most useful results concerns the
dispersion of a solute in a flowing fluid stream. The
motivation for this project was to understand the manner
in which drugs are dispersed in blood flow; other
applications abound. The idea is to consider the steady
laminar flow in a straight circular pipe of radius a, and
understand how an initially localized solute disperses
with time.
If
there were no molecular diffusion, the solute would be
spread out considerably by the flow, because of the large
velocity gradient across the pipe. Taylor recognized that
molecular diffusion actually impedes this
dispersion: Molecular diffusion forces the solute in the
center of the pipe to diffuse near the walls, where it
moves more slowly. Taylor demonstrated that if the
concentration is denoted c(r,z,t), where z
lies along the pipe axis, and the areaaveraged
crosssectional concentration is <c>(z,t),
then the average concentration evolves according to the
convectivediffusion equation
and
D is the molecular diffusion constant.^{10}
The solute center of mass moves with the mean velocity
<u> and has a Gaussian spread about the mean
that increases in proportion to . The largest
contribution to the dispersion typically comes from the 1/48(<u>^{2}a^{2}/D)
term, which is inversely proportional to the diffusion
coefficient! Taylor even used this idea to measure the
molecular diffusion constant, an approach that is used to
this day.^{11}

Taylor
columns. When
an object moves in a rotating flow, it
drags along with it a column of fluid
parallel to the rotation axis. This
photograph shows the flow when a dyed
drop of silicone fluid (radius 2 cm)
rises through a large tank of water
rotating at 56 rpm (From ref. 17.)


Viscous
hydrodynamics
The subject of viscous hydrodynamics was
popularized in the physics community by Edward
Purcell’s article, “Life at low
Reynolds numbers,” in which he describes his
work with Howard Berg on understanding bacterial
propulsion.^{12} What is perhaps not so
well known is that the first widely recognized
work on this topic was Taylor’s.^{13}
Purcell wrote
But at
that time G. I. Taylor’s paper in the Proceedings
of the Royal Society could conclude with
just three references: H. Lamb, Hydrodynamics;
G. I. Taylor (his previous paper); G. N.
Watson, Bessel Functions. That is
called getting in on the ground floor.
Taylor’s
interest in this subject was apparently
stimulated by his interaction with the zoologist
James Gray of Cambridge University. The basic
difficulty of lowReynoldsnumber propulsion is
that motion is reversible: By reversing
kinematical motions one always ends up at the
same starting place. Purcell popularized this
idea through his “scallop theorem,”
which states that a scallop (an object with only
one joint) in a viscous fluid cannot swim.
Taylor
investigated simple swimming situations where
reversibility is broken, to demonstrate how
motion is possible. For example, through explicit
calculation he demonstrated that transverse waves
propagating along a sheet submerged in a fluid
cause the sheet to translate with uniform
velocity. These ideas have found many recent
applications, from the design of micromechanical
machines to hypotheses about propulsion
mechanisms in unusual organisms. Also, Taylor
developed the stillavailable educational film
“Low Reynolds Number Flows,” which is
familiar to many and recommended to all as a
wonderful example of Taylor’s creativity and
clarity.

Swimming snakes
Gray also provoked Taylor’s interest in the swimming
of snakes. How do various types of deformations of the
snake produce forward thrust? At first sight, this
problem seems intractable, because the flow generated by
a snake is typically turbulent, and so theories do not
really exist. Taylor observed, however, that there is
much experimental data regarding the forces on cylinders
in a turbulent flow, and proceeded to use this data as
the basis for his theory. By modeling the snake as a sum
of cylinders, he computed the swimming velocity as a
function of the deformation. This allowed him to explain
quantitatively features of how snakes swim—for
example, the wave amplitude of the snake that makes it
move the fastest. Perhaps his most interesting discovery
is that a snake with a rough surface can swim forward by
sending waves in the forward direction. Taylor writes,
“On showing [the result] to Professor Gray, [he]
called my attention to a set of photographs he had taken
of a marine worm Nereis diversicolor which does in
fact swim in this way.”^{14} And, as
predicted, the worm has a rough surface.
Taylor columns
In a steady, rapidly rotating flow with angular velocity W, the dominant forces
are pressure gradients and Coriolis forces, and the
Navier–Stokes equations reduce to
where
r is the
fluid density and p is the pressure. Taking the
curl of this equation, it follows that the velocity is
independent of the coordinate along the rotation axis.
The flow is therefore effectively twodimensional. This
result, first demonstrated by Joseph Proudman in 1915, is
now called the Taylor–Proudman theorem.
Taylor’s name
got attached because he addressed the question of what
happens if one tries to disturb the twodimensionality of
the flow. In a paper published in 1923, he reported
placing a short cylinder in a rotating tank of fluid and
dragging the cylinder relative to the flow. Without
rotation, of course, the motion of the short cylinder
would disturb the flow in all directions. How can this be
reconciled with Proudman’s result? The experiments
demonstrated what Taylor called a “remarkable”
conclusion: The flow remains twodimensional! The solid
object (nearly) immobilizes an entire column of fluid
parallel to the rotation axis. Thus, in a rotating
environment, a slowly moving object behaves nearly as a
solid cylinder extended parallel to the rotation axis.
There are numerous applications of this idea to motions
in atmospheres and oceans, because surface topographic
features produce “columnar” disturbances that
interfere with, or block, the flow at substantial
elevations.

Taylor–Couette
stability diagram. This plot,
from Taylor’s 1923 paper on the instability
of flow between two coaxial rotating cylinders,
was the first example of a theoretical
calculation of a fluidflow instability that
quantitatively agreed with experiments. The
stability boundary as a function of the rotation
speed of the outer cylinder (ordinate) and inner
cylinder (abscissa) is shown. The dashed line W_{1}R^{2}_{1}
= W_{2}R^{2}_{2}
is a previous theory by Lord Rayleigh. The solid
points represent experimental measurements; the
open points, theoretical calculations of the
stability boundary. Due to the complexity of
evaluating numerically the formulas from the
theoretical calculations, there are more
experimental data points than theoretical points.

Electrohydrodynamics
Taylor spent much of his later life studying the
interaction of fluids with electric fields. His most
important contribution—made at age 80—is the
realization that the idealization of perfect conductors
or perfect dielectrics is misleading for electrically
dominated flows. There is always some residual free
charge present, typically residing on the interfaces
between different fluids. Thus, any electric field
tangential to the interface results in a tangential
stress, and this stress can be balanced only by a viscous
flow.
Taylor
discovered this basic notion when trying to explain an
experimental anomaly in the observed shapes of dielectric
drops in a uniform external electric field. Simple
energetics predicts that such a drop should elongate in
the direction of the field, whereas for some fluids the
drop actually shortened in the field direction. Because
the tangential electrical stresses described above
require a steady viscous flow for balance, the drop shape
cannot be obtained by energy minimization. The
characterization of liquids using both a conductivity and
a dielectric constant is referred to as the “leaky
dielectric model.”^{15}
Nuclear explosions
No article about Taylor would be complete without
including the often told story about his calculation of
the energy in a nuclear blast. Fables of this story
abound. As told by Taylor himself,^{16} during
the early years of World War II he was told by the
British government about the development of the atomic
bomb, and was asked to think about the mechanical effect
produced by such an explosion. He realized that the
energy released from the bomb would quickly lose memory
of its initial shape and distribution, and would produce
a strong shock in the air. The structure of the shock far
from the ground would be wellapproximated as spherical.
With
these simplifications Taylor recognized that the
parameters in the problem are the energy E, the
density r of air, the
pressure p in the air, the radius R(t) of
the blast wave, and the time t since the blast.
Because the blast is very strong, the air pressure will
not affect the wave very much, and so p is not a relevant
parameter. Taylor realized that this implies that there
is a single dimensionless number characterizing the
process; the reader can verify that Et^{2}/rR^{5}
is dimensionless.
Because
this quantity does not depend on any aspect of the
problem, it must be a constant. This implies that the
radius of the blast wave is given by
R(t)
= c(Et^{2}/r)^{1/5},
where
c is a constant. In fact, it turns out that for
air c~1.033 according to a calculation. Therefore, given
a picture that shows the radius of the blast, a reference
length scale, and the time since the blast, one can
deduce the energy.
Much
after the fact, Taylor analyzed photographs taken by J.
E. Mack of the first atomic explosion in New Mexico.
These pictures were taken at precise time intervals from
the instant of the explosion, and Taylor confirmed that
the scaling law agrees very nicely with the data. It is
interesting to note that, of the papers written in the
early 1950s reporting independent discoveries of the
blast scaling law (authors including John von Neumann and
Leonid Sedov), only Taylor’s paper took publicly
available data to show that the above equation agrees
with experiments.
There
are many topics that we have not been able to cover in
this article, or in our course—among them, the bulk
of Taylor’s contributions to solid mechanics.
Another effort to design a course around Taylor’s
papers would likely arrive at a completely different list
of topics. We encourage interested readers to browse
Taylor’s collected works and to design their own
course using his papers as a gateway to the modern
literature.
Many
people provided constructive criticism of an early draft
of this article. We thank Herbert Huppert for many
helpful suggestions that improved the final manuscript.
References
1. G.
K. Batchelor, ed., Scientific papers of G. I. Taylor,
Cambridge U. P., Cambridge, England (1971).
2. S. Goldstein, Ann. Rev. Fluid Mech. 1, 1
(1969).
3. G. K. Batchelor, The Life and Legacy of G. I.
Taylor, Cambridge U. P., Cambridge, England (1996);
reviewed in Physics Today, June 1997, p. 82.
4. J. K. Bell, Experimental Mechanics, 1
(1995).
5. J. S. Turner, Ann. Rev. Fluid Mech. 29, 1
(1997).
6. G. I. Taylor, Phil. Trans. Roy. Soc. Lond. A
223, 289 (1923).
7. H. Jeffreys, Proc. Roy. Soc. Lond. A 118,
195 (1928). S. Chandrasekhar, Hydrodynamics and
Hydromagnetic Stability, Oxford U.P., Oxford, England
(1961).
8. R. P. Feynman, R. B. Leighton, M. Sands, The
Feynman Lectures on Physics, AddisonWesley, Reading,
Mass. (1964), vol. 2, p. 41.11.
9. N. Wiener, I am a Mathematician, MIT
Press, Cambridge, Mass. (1956).
10. G. I. Taylor, Proc. Roy. Soc. Lond. A 219,
186 (1953).
11. M. S. Bello, R. Rezzonico, P. G. Righetti,
Science 266, 773 (1994).
12. E. M. Purcell, Am. J. Phys. 45, 3 (1977).
13. G. I. Taylor, Proc. Roy. Soc. Lond. A 209,
447 (1951).
14. G. I. Taylor, Proc. Roy. Soc. Lond. A 214,
158 (1952).
15. J. R. Melcher, G. I. Taylor, Ann Rev. Fluid
Mech. 1, 111 (1969). D. A. Saville, Ann. Rev.
Fluid Mech. 29, 27 (1995).
16. G. I. Taylor, Proc. Roy. Soc. 201, 11
(1949).
17. J. W. M. Bush, H. A. Stone, J. Bloxham, J. Fluid
Mech. 282, 247 (1995).
Michael
Brenner is an associate professor of applied
mathematics at the Massachusetts Institute of
Technology in Cambridge, Massachusetts. Howard
Stone is a professor of chemical engineering
and applied mechanics at Harvard University in
Cambridge, Massachusetts. 
© 2000 American
Institute of Physics
