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Chapter 6
My Hero: Sir G. I. Taylor
Sir Geoffrey Ingram Taylor (1886-1975) was a classical physicist at heart,
though he worked in modern times. He worked on many diverse areas in physics from
hydrodynamic instability and turbulence to Electro-hydrodynamics, to the locomotion
of small organisms. That's why MIT has rightly designed a course on "Classical
Physics through the work of G. I. Taylor"[1].
He has done a vast amount of work in Fluid Dynamics. His name accompanies
3 major instabilities which are exhaustively studied even today: Taylor-Couette
instability (in rotating coaxial cylinders), Saffman- Taylor instability (at the interface of
two liquids when one displaces the other) and Rayleigh-Taylor instability (which
occurs, whenever a heavier fluid is placed on top of a lighter fluid in a constant
gravitational field).
He was possibly the only 20th century physicist who could strike such a fantastic
balance between the theory and experiments. Look at his work on swimming of long
and narrow tailed aquatic animals in 1952. He carried the fascination about that subject
from his friends and followed it right till he found the exact mechanism behind
propulsion of such animals. And in his very unique style he goes on to build a working
model of such species and obtains the experimental data. As usual, it agrees with his
theoretical calculations! He also verifies the calculations with actual photographs of the
aquatic animals. So much of the work but he manages to do it with just 3 references:
his earlier paper, Lamb's Hydrodynamics, G. N. Watson's Bessel functions. This is the
mark of the down to earth approach of this genius [2].
I get amazed looking at his list of publications. He has produced classic,
pioneering papers in a few months time, while solving lots of smaller problems.
He had a natural insight into the physics of the situation in hand. He could
easily single out the parameters of highest significance. He had the magical power of
converting the physical situation in hands into mathematical equations, with appropriate
approximations and assumptions.
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He was the first physicist to apply the theory of linear stability theory to
hydrodynamic instabilities. He used it with success in case of Taylor vortices and
experimentally verified the stability analysis curve. That was the starting point for
literally hundreds of experimental and theoretical studies in this field.
His work with Prof. Proudman marks the beginning of the study of flows in
atmosphere. These geostrophic flows as they are called, occur due to the rotating
reference frame of earth. These flows are essentially dominated by the Coriolis force.
The Taylor- Proudman theorem says that the flow in a rotating system for steady slow
motion of obstacles is two-dimensional. This gave rise to the famous Taylor columns:
Photo 6.1: 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. [1]
We can easily perceive the delight and surprise Taylor had when he saw these
columns, in his original paper.
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.
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He doesn't seem to be tired a little bit even at a later age in life:
Photo 6.2: Sir Geoffrey Ingram
Taylor (right) at age 69, in his
laboratory with his assistant
Walter Thompson. (AIP Emilio
Segrè Visual archives.)
His work will always be a great source of inspiration for me.
References:
1. Physics Today Article on “Classical Physics Through Work of G. I. Taylor”.
2. “Life at low Reynolds number”- E. M. Purcell Am. J. Phy. Vol. 45, 3-11,1997.
3. “The action of waving cylindrical tails in propelling microscopic organisms”- Sir G.
I. Taylor, Proc. Roc. Soc. (London) A 214, 158 (1952);
“Analysis of the swimming of microscopic organisms” Sir G. I. Taylor, Proc. Roy.
Soc. (London) A 209, 447 (1951).
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Conclusions
The Taylor vortices in coaxial rotating cylinders have been studied
experimentally for water and paraffin oil, as well as theoretically. All the images were
recorded using a CCD camera connected to VCR and analyzed using a standard image
processing software on computer. It was verified that the width of the vortices is equal
to the gap between the two cylinders. The analogy with a spring twisted in circular
shape is given for the trajectory of particles in these vortices. A C program was written
to visualize these trajectories.
Further studies of instabilities in the coaxial rotating cylinders, at higher rotation
rates is planned, including the rotations of outer cylinder. A brand new apparatus has
been recently built for the same.
A broad review of some of the related topics in fluid dynamics was taken during
the project work. Studies of motion of various living organisms, flows in atmosphere,
accretion processes on massive stars, dynamical interactions of a deformable body with
the surrounding fluid are also planned in near future.
“The behavior of fluids is in many ways very
unexpected and interesting. The efforts of a child
to dam a small stream of water on the street and
his surprise at the strange way the water works
its way out has its analog in our attempts to
understand the flow of fluids. We have tried to
dam the water up - in our understanding - by
getting the laws and equations that describe the
flow…[but] water has broken through the dam and
escaped our attempts to understand it.”
R. P. Feynman
In his Lectures on Physics.
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Appendix A
The C program for visualization of Taylor vortices
Following is the C program I wrote forvisualizing the trajectories of particles in Taylor
Vortices:
#include<conio.h>
#include<stdio.h>
#include<math.h>
#include<graphics.h>
#include <dos.h>
#define a 35
#define b 1
#define w 30
#define theta 3
#define width 10
#define angle 360
void main()
{
int i,j,gd=DETECT ,gm=DETECT;
float x=0,y=0,z=0,x1=0,y1=0,ang=0,xact=0,yact=0;
initgraph(&gd,&gm,"C:\\bcp\\bgi");
setcolor (WHITE);
line(300,225,300,0);
line(300,225,300-300*cos(theta*3.14/180),225+300*sin(theta*3.14/180));
line(300,225,300+300*cos(theta*3.14/180),225+300*sin(theta*3.14/180));
for(j=0;j<6;j++)
{
ang=-40*3.14/180;
for(i=0;i<angle;i++)
{
setcolor(YELLOW);
ang=ang+2*3.14/720;
z=-155+j*60+width*cos(w*ang);
xact=a*sin(ang)*(5+b*sin(w*ang));
yact=a*cos(ang)*(5+b*sin(w*ang));
x=(-yact+xact)*cos(theta*3.14/180);
y=(xact+yact)*sin(theta*3.14/180)-z;
if(i!=0)
lineto(x+300,y+225);
moveto(x1+300,y1+225);
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setcolor(WHITE);
z=-130+j*60+width*sin(w*ang);
xact=a*sin(ang)*(5+b*cos(w*ang));
yact=a*cos(ang)*(5+b*cos(w*ang));
x1=(-yact+xact)*cos(theta*3.14/180);
y1=(xact+yact)*sin(theta*3.14/180)-z;
if(i!=0)
lineto(x1+300,y1+225);
if(i%75==0)
getch();
moveto(x+300,y+225);
}
}
getch();
getch();
getch();
closegraph();
}
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Appendix B
Accreting on stars
The study of accretion of matter like stellar dust on a star began with the
discussions of Hoyle and Lyttleton [1]. The problem was that of the occurrence of ice
age and such changes in earth’s climate. The geological and terrestrial studies could not
account for such observations. Hoyle and Lyttleton concluded that the reasons must lie
in the extra terrestrial environment. At the same time it was known from astronomical
observations that there are large dust clouds in the extra-stellar regions. These clouds
were found to have sizes comparable to the galaxies themselves. Also the distribution of
such clouds was very irregular and they had varied shapes from strips to circles. The
time period required for passage of a star through one such cloud was comparable to the
various changes observed in climate of earth. Hoyle suggested that the sun must have
gone through a dust cloud in the ages when the earth was in the warm period. He put
forth the hypothesis that the accretion of dust on the sun must have increased the
radiation from the sun, due to conversion of kinetic energy to heat, which in turn heated
the earth.
To support this, they modeled the dust cloud as a fluid through which the sun
moves and then analyzed the situation for the falling matter. On the first thoughts, we
may feel that the area of cross section for which the matter accretes on a star is the area
“faced” by the infalling dust i.e. just   r 2 , where r is the radius of the star. But the dust
is accreted not only due to motion of the star through the cloud, but also due to
gravitation of the star. Let’s see how. To simplify the matters, let’s go to the frame of
reference of the moving star. Then the fluid is moving across the stationary star, with
some constant velocity at a large distant, say along negative x-axis. The trajectories of
the fluid elements are changed due to gravitational field of the star. As in the scattering
of particles in the central force field, these fluid particles will follow a trajectory like
parabola or hyperbola or ellipse, depending on their impact parameter. The impact
parameter is the closest distance the fluid element would have come, if the star was not
present. The two fluid elements incident symmetrically with respect to the motion of the
star will collide with each other on the axis of symmetry:
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Figure B.1: collisions in dust
After collision, the angular momentum of individual elements cancels out. There
is only the radial component of velocities that remains. Now, if this radial velocity is
insufficient for the particles to escape from the gravitational field of the star, then these
particles have to fall straight onto the star. So there is accretion of matter not only in the
direction in which the star moves, but at the back of the star as well!
The highest impact parameter for which the matter can be captured by the stars
is calculated form the known parameters like mass and velocity of the star. This
corresponds to the parabolic trajectory of the individual fluid elements. This can be
expected from the fact that the hyperbolic trajectories are followed by those particles,
which have velocities greater than the escape velocity. The effective radius of the star
galloping all this matter then becomes,

2G  M
v2
....(B.1)
Which, for the star like our Sun and for velocity v = 20 km per sec, is as large as
1000 solar radii!
Thus the work of Hoyle and Lyttleton showed the importance of studies of
accretion. The credit of Hoyle and Lyttleton lies in the insightful manner they have
connected a terrestrial problem to an observation in astronomy, and analyzed the
situation using fluid dynamics and classical mechanics.
Based on their calculations, Hoyle and Lyttleton obtained the following formula
for accretion of matter on a star moving through the dust cloud with velocity V:
dM
2
 2      GM  V 3  
dt
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.....(B.2)
where,   is the density of matter at very large distances, M is the mass of the
star, and  is a numerical constant between 1 and 2.
Hoyle and Lyttleton had considered the dynamic effects of motion of the star
through the dust clouds. If a star is at rest in a dust cloud then the effects of pressure and
gravitational energies of infalling matter become more important. Bondi [2] studied this
problem for a spherical star at rest in a large dust cloud. He assumed that the velocity of
infalling matter is only radial. When the viscous forces are neglected and the flow is
assumed to be steady, the Navier Stokes equations take the form:
u. u   P  
....(B.3)
This equation is called as Euler’s equation. is the gravitational potential
energy. Due to spherical symmetry, we choose spherical polar coordinates to analyze
this problem. Also the quantities like pressure, velocity are only a function or radius. To
connect these fluid dynamical variables with the thermodynamical properties we have to
introduce an equation of state. We adopt the adiabatic equation of state:
P  K 
….(B.4)
We also have to take into consideration the formula for the speed of sound, ‘a’:
a2 
P

….(B.5)
Using the above equations, the rate of accretion was obtained by Bondi as
dM
2
 2      GM  a 3  
dt
…..(B.6)
There is a striking similarity between the two accretion rates given by B. 3 and
B.6. These two cases considered so far may be called as velocity limited and
temperature limited. The intermediate range of cases is difficult to analyze. Bondi
suggested the following formula for the general case:


3 / 2
dM
2
 2      GM  V 2  a 2

dt
.….(B.7)
From this formula we see that when V exceeds ‘a’, the dynamical effects are
important, and when ‘a’ is much larger than V, then the thermodynamic effects are more
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important. Thus we expect this formula to give the correct order of magnitude of the
accretion rate.
Another interesting feature obtained from the analysis of the flow velocities is
the following: the gas is assumed to be at rest at large distances. Its velocity goes on
increasing as it approaches the star. At a certain critical radius rc , it crosses the speed of
sound and falls on the star at a supersonic speed. So around a star you have this sphere
of radius rc , at which the matter is falling onto the star at speed of sound. This sphere is
called as “Sound Horizon”. It is analogous to the event horizon in case of black holes,
the light cannot escape from the part inside the event horizon, similarly the sound or any
thermodynamic disturbances can’t come out of sound horizon. So you can’t hear outside
the sound horizon a person shouting on a star under accretion!
If the shape of the cloud is not spherically symmetric but is planar, then the
matter falls along a disc. Such accretion discs are more common and are widely studied.
The infalling matter need not have only a radial velocity, a small angular momentum
will make the matter spiral onto the star. Also, in dense clouds the viscosity comes into
picture. The infalling matter may also emit the heat in the form of radiation.
Works of Hoyle, Lyttleton and Bondi have initiated great interest in the study of
accretion of interstellar matter on a star. Now this process is known to be a major source
of energy to the massive stars. Accretion on black holes and other compact objects like
neutron stars is also being studied widely.
The most powerful single intellectual device
known in physics is the transformation of
the reference frame from which we observe
a process.
C. Kittel, W.D. Knight, M.A. Ruderman.
Mechanics, Berkeley Physics Course, Vol.1
References:
1. “The effect of interstellar dust on climatic variation”- Hoyle and Littleton, Proc.
Camb. Phil. Soc. 35, 405,1939.
2. “On spherically symmetric accretion”- H.Bondi, 1952. Available on NASA
Astrophysics Data System.
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Appendix C
Interesting BZ reaction and Cellular Automata
As an interesting activity, I performed with the help of my chemistry friends, a
chemical reaction known as Belusov-Zabotinsky reaction. So I wish to share the joy of
this reaction and convey various interesting things related to it in this appendix.
Following is one of the chemical recipes we performed. The chemicals required are:
1. Malonic acid (3.575 gm)
2. Potassium Bromate (KbrO3) (1.305 gm)
3. Ammonium Ceric Sulphate (ACS) (0.137 gm)
4. The indicator to be used is Ferroin. To prepare Ferroin, add Phenanthronin
(0.371gm) and hydrated ferrous Sulphate (0.17 gm) in 100 ml of water.
The procedure is as follows:
1. Add above first 3 reagents in 10-10ml 1 M H2SO4 separately. Dissolve them
completely.
2. Mix them in the order. Do not change the sequence of addition. Then add 8-12 drops
of Ferroin indicator.
And then, right before your eyes, you will see something surprising: the color of the
solution in the beaker changes from blue to red and back to blue to red and so on...!
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Figure C.1: the color changes in BZ reaction. The first 3 photos are taken after
2 seconds each, the 4th photo was taken 18 seconds after the third and the last 3
photos were taken at the interval of 2 seconds after the fourth.
This is an oscillating chemical reaction. When it was first discovered
accidentally by Belusov, no one believed him. The chemists were surprised to see a
reaction, which did not proceed to equilibrium. Is it violating the second law of
thermodynamics? The answer in no, it isn’t because the reaction does indeed go to an
equilibrium after half an hour, but before that it keeps on showing two products. The
reason for these oscillations is the competition between two opposite reactions. Without
going into details of the chemicals involved, we may describe the process with a
different example: consider a field containing ample grass. Suppose there are some
rabbits and lions in that field. The lions live by eating the rabbits and the rabbits live by
eating the grass. If the population of lions is large to begin with, then population of
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rabbits will have to decrease with time. But with the rapid decrease in population of
rabbits, the lions will start dying out of starvation. So the population of lions will
decrease. That will help in allowing the rapid growth of rabbits. That in turn will
increase the population of lions. Thus the cycle will keep going on and on. This kind of
model is known as the prey-predator cycle.
The role of rabbits and that of lions is played by two types of chemical species in
BZ reaction. The two colors are due to two different oxidation states of an ion. Though
the actual details are much more complicated than that, the essential characteristics are
well explained by analogy with prey predator model.
But the BZ reaction has many more things to offer than the changing colors. There
are many interesting spatial patterns formed in the BZ reaction. These patterns are
difficult to analyze in 3 dimensions, so we performed the reaction in a petridish instead
of a beaker:
Figure C.2: The spatial patterns in BZ reaction.
The spatial patterns arise due to the distribution of various chemicals in the
petridish. The direction in which the reaction proceeds is dependent on the local
concentration of the reactants in that place. This may vary from place to place
throughout the petridish. Then the reaction mixture shows different colors in the
petridish. But the growth of these patterns is systematic. When the reaction proceeds in
a certain direction at some point, it proceeds in the same direction in its neighborhood.
So a circular spot of that color grows. But after certain interval of time, the environment
at the center is suitable for the reaction to proceed in opposite direction. Then the color
of the solution flips. As time proceeds, this color develops a circular spot at the center.
So it’s like a circular wave front proceeding in a direction. Now if an obstacle comes in
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the path of this wave front it breaks in two parts. Each part grows in its own way and the
following spiral waves are seen:
Figure C.3: The spiral waves in BZ reaction.
The researchers in this area are studying these pattern formations. They have
also studied the changes in concentrations of various substances in the reaction
mixtures. These variations are found to have nonlinear relations with the experimental
parameters. They have studied the variations in period of oscillations. The equations
governing the reaction dynamics are basically some differential equations. The patterns
seen in concentration changes in the beaker reflect the nature of the solutions of these
differential equations.
The point of introducing this field at this place is that the patterns seen in nature
are independent of the system we are studying. The same growth mechanisms give rise
to same pattern formations. Just like the mushroom patterns we considered in chapter
four, the patterns in BZ reaction are not unique to the system, but are also found in
growth of bacterial colonies. So the mechanisms governing these growths are universal.
A model called Cellular Automata is introduced to account for the observed patterns in
BZ reaction. In this model, we divide the plane area of the petridish into number of tiny
regions called cells. With each cell we associate a color representing the state of
compounds in that cell. The color of one cell is affected in next instant by the color of
surrounding cells. The relations governing these are the rules for the game of cellular
automata. The evolution of the system can then be studied using these rules and setting
certain initial conditions. Indeed scientists have simulated the spiral waves in BZ
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reaction using such model. They have also used it for a variety of other situations and
found it to be useful.
Thus the patterns developed using the cellular automata are based on the
minimum spatial requirements to be satisfied by the system and are hence applicable to
a variety of phenomenona.
Reference:
“Self Made Tapestry: Pattern Formation in Nature.” – Philip Ball.
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Appendix D
More about Navier Stokes equations
The vectorial forms of the continuity equation and NS equations are
. v   

t
u
 u . u     P   2 u  Fext
t
..…(D.1)
..…(D.2)
These have to be reduced in the various components as follows (assuming to
be a constant)[1]:
In Cartesian coordinates:
u v w


0
x y z
  2u  2u  2u 
 u
u
u
u 
p
   u  u  u       2  2  2   Fx
x
y
z 
x
y
z 
 t
 x
…..(D.3)
…..(D.4)
together with similar equations for v and w.
In Cylindrical Polar coordinates(r= distance from axis,  = azimuthal angle
about the axis, z = distance along axis):
The continuity equation is
u r u r 1 u u z



0
r
r r 
z
…..(D.5)
The r component is
2
 u r
u r u u r
u r u 
p

 ur

 uz


r
r 
z
r 
r
 t
  2ur

2
 r

1 u r u r 1  2 u r  2 u r 2 u 



 2
  Fr
r r r 2 r 2  2
z 2
r  
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…..(D.6)
The  component is
u
u r u u u
u 
 u
p
 ur  

 uz    
r
r
r 
z 

 t

  2 u

2
 r

2
2
1 u u
1  u  u
2 u 
 2  2

 2 r   F
2
2
r r
r
r 
z
r  
…..(D.7)
The z component is
u u z
 u z
u
u 
p
 ur z  
 uz z   
r
r 
z 
z
 t

  2u z

2
 r

1 u z 1  2 u z  2 u z 


  Fz
r r r 2  2
z 2 
…..(D.8)
Observe the non-linearity of these equations due to the cross product terms of
velocities and velocities with derivatives.
The Navier Stokes Equations take a much more complicated form in spherical
polar coordinates, due complex forms of divergence and Laplacian operators. As we
haven’t used them in this report, I do not give those components here. They are useful
when the problem under consideration has a spherical symmetry, e.g. in case of flows
around the earth in atmosphere. The interested reader should see the reference.
Reference:
“Physical Fluid Dynamics” - D. J. Tritton
92
Appendix E
Some useful properties of common fluids
Fluids
Viscosity (cP)
Density (gm/cc)
Air (20 oC)
0.018
1.2 x 10-3
15.6 oC
1.13
1.0
54.4 oC
0.55
1.0
Paraffin oil (20 oC)
170
0.8
Water
Castor oil
(37.8 oC)
248.64 – 312
0.96
Glycerine
(20.3 oC)
816.4
1.26
Mercury
(20 oC)
1.56
13.6
Surface tensions in (mN/m) at 20 OC
Liquids
Water
20 oC
72.75
25 oC
72
100 oC
58
Hexane
18.4
Methanol
22.8
Carbon tetrachloride
27.0
Benzene
28.88
Paraffin oil
35
Mercury
472
93
Suggestions for further reading
1. “Life at low Reynolds number”- E. M. Purcell Am. J. Phy. Vol. 45, 3-11,1997.
A very interesting lecture by a Nobel laureate.
2. G. I. Taylor, Proc. Roy. Soc., A 104, 213, (1923).
Fascinating paper describing the discovery of Taylor columns.
3. Hoyle and Lyttleton:
“The evolution of the stars.” Proc. Camb. Phil. Soc. 35, 592,1939.
“On the accretion of interstellar matter by stars.” Proc. Camb. Phil. Soc. 36, 5,1940.
These are extremely lucid discussions by the authors. We see how a theory is
proposed and gradually accepted after lot of criticism.
4. “Physical Fluid Dynamics” - D. J. Tritton
A fantastic account of fluid dynamics, with lot of insight and experimental details.
5. “Flutter and Tumble in Fluids” – A. Belmonte and E. Moses,
“Chaotic dynamics of falling discs” – S. B. Field et al, Nature, Vol. 388,17 July
1997.
These are two nice papers on motion of a body through a fluid.
6. “Self Made Tapestry: Pattern Formation in Nature.” – Philip Ball.
An excellent description of patterns right from sandpiles, honeybee houses,
snowflakes, strips on zebra to flows of granular media, self-aggregation and so on.
Strongly recommended for any student of science.
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