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.
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:
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:
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).
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<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);
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 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 , 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:
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.F 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.
2. “On spherically symmetric accretion”- H.Bondi, 1952. Available on NASA Astrophysics Data System.