Chapter 6
My Hero: Sir G. I. Taylor
Sir Geoffrey Ingram Taylor (18861975) 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 Electrohydrodynamics, 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: TaylorCouette instability (in rotating coaxial cylinders), Saffman Taylor instability (at the interface of two liquids when one displaces the other) and RayleighTaylor 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 20^{th} 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. 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 twodimensional. This gave rise to the famous Taylor columns:
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. He doesn't seem to be tired a little bit even at a later age in life:
His work will always be a great source of inspiration for me.
References: Physics Today Article on “Classical Physics Through Work of G. I. Taylor”. “Life at low Reynolds number” E. M. Purcell Am. J. Phy. Vol. 45, 311,1997. “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).
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.
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 #include #include #include #include #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(300,225,300300*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 { 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);
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(); }
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 extrastellar 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 , 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 xaxis. 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: 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, ....(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: .....(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:
....(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: ….(B.4) We also have to take into consideration the formula for the speed of sound, ‘a’: ….(B.5) Using the above equations, the rate of accretion was obtained by Bondi as …..(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: .….(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 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 , 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 , 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: “The effect of interstellar dust on climatic variation” Hoyle and Littleton, Proc. Camb. Phil. Soc. 35, 405,1939. “On spherically symmetric accretion” H.Bondi, 1952. Available on NASA Astrophysics Data System. 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 BelusovZabotinsky 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: Malonic acid (3.575 gm) Potassium Bromate (KbrO_{3}) (1.305 gm) Ammonium Ceric Sulphate (ACS) (0.137 gm) 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: Add above first 3 reagents in 1010ml 1 M H_{2}SO_{4} separately. Dissolve them completely. Mix them in the order. Do not change the sequence of addition. Then add 812 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...! Figure C.1: the color changes in BZ reaction. The first 3 photos are taken after 2 seconds each, the 4^{th} 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 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 preypredator 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 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 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. Appendix D
More about Navier Stokes equations The vectorial forms of the continuity equation and NS equations are ..…(D.1) ..…(D.2) These have to be reduced in the various components as follows (assuming to be a constant)[1]: In Cartesian coordinates: …..(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 …..(D.5)
The r component is …..(D.6)
The component is …..(D.7)
The z component is …..(D.8) Observe the nonlinearity 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
Appendix E
Some useful properties of common fluids
Fluids  Viscosity (cP)  Density (gm/cc)  Air (20 ^{o}C)  0.018  1.2 x 10^{3}  Water  15.6 ^{o}C  1.13  1.0  54.4 ^{o}C  0.55  1.0 
 Paraffin oil (20 ^{o}C)  170  0.8  Castor oil (37.8 ^{o}C)  248.64 – 312  0.96  Glycerine (20.3 ^{o}C)  816.4  1.26  Mercury (20 ^{o}C)  1.56  13.6 
Liquids  Surface tensions in (mN/m) at 20 ^{O}C 
Water  20 ^{o}C  72.75  25 ^{o}C  72  100 ^{o}C  58  Hexane  18.4  Methanol  22.8  Carbon tetrachloride  27.0  Benzene  28.88  Paraffin oil  35  Mercury  472 
Suggestions for further reading
“Life at low Reynolds number” E. M. Purcell Am. J. Phy. Vol. 45, 311,1997. A very interesting lecture by a Nobel laureate. G. I. Taylor, Proc. Roy. Soc., A 104, 213, (1923). Fascinating paper describing the discovery of Taylor columns. 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. “Physical Fluid Dynamics”  D. J. Tritton A fantastic account of fluid dynamics, with lot of insight and experimental details. “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. “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, selfaggregation and so on. Strongly recommended for any student of science.
