Figure 4.7: Unstable velocity profile in circular Couette flow.
Because 2  0 , we can find the outer radius a2 from the graph as shown. For
convinience, I have also plotted the derivative of (ur) on the same graph. The fact that it
is –ve for all values of a1<a2 shows , by Rayleigh’s criterion, that the flow is unstable. So
if the outer cylinder is not stable, then the flow is unstable. When both the cylinders are
rotating such that A and B both are positive, the velocity distribution is as shown below:
10
u = r + 1/r
derivative(ur) = 2r
Tangential velocity u
8
6
4
2
0
0
2
4
6
Distance from the axis r
Figure 4.8: Stable velocity profile in circular Couette flow.
Here we see that the derivative of (ur) is positive for any values of a1 and a2. To make
both A and B positive, we require 1   2 and  2 a22 .  1a12 [3].
The major use of this apparatus has been, traditionally, to measure the viscosity. The
torque transferred to the outer cylinder can be measured by mechanical methods and the
angular velocity of the inner cylinder can be measured by various finding the revolutions
per minute or second. From this we can find the viscosity as explained in the following
calculations:
The torque 1 acting on the inner cylinder (per unit length in the z direction) is given by
 u
   r
the viscous stress  r 

r



 

multiplied by the circumference 2a1 and by the


 r a
1
radius a1 i.e.
4(1   2 )a12 a22
.
1 
(a22  a12 )
Similarly the torque on the outer cylinder is
4(1   2 )a12 a22
.
2  
(a22  a12 )
We notice that 1 and 2 are equal and opposite, as they must be since the total
angular momentum of the fluid is not changing.
From the measured value of torque, rotation rate of inner cylinder and other
geometrical factors, viscosity of liquid can be determined. It must be mentioned here that
this calculation is valid only when rotation rates are sufficiently low so that the circular
Couette flow is maintained. At higher rotation rates, the flow breaks into vortices and the
torque experienced by the walls will not be the same as above. So, while using this
apparatus we must take the precaution to keep the rotation rates low enough so that the
flow is stable. This is especially important if we try to find out the viscosity of low
viscosity fluids like water, using this apparatus. For highly viscous fluids like castor oil,
the instability occurs at very high rotation rates, so we don’t need to worry about the
rotation rates.
4.4.3 Computer generated trajectory
I wrote a program in C to visualize the trajectories of the particles in the Taylor
vortices. The trajectory of such a particle is like a spring twisted in the form of a circle.
To plot this trajectory we must find the parametric equation of the spring twisted into
circle. To do this, again consider the cylindrical coordinate system. The trajectory is like
a circle rotating about an axis with the angular velocity say  , and a point moving on the
circumference of this revolving circle with angular velocity say  . Now imagine a
spring folded and fitted between the two cylinders.
Z
Y
R
X
a
br
Figure 4.9: Vector diagram for the particle trajectory.

Set up a vector pointing to the center of the circle say a , where a is the average
of two radii a1 and a2. Then set up another vector pointing to the point on the
circumference, from the center of this revolving circle. Note that the revolving circle
changes its orientation while rotating, so that it will always be in the plane containing the

z-axis and the radial vector  . So the vector towards the point on the circumference will



be bcost z  sin t   br where b is the gap between the two cylinders. The




resultant position vector is Rt   a  bcost z  sin t   . Now, as the circle revolves,

the vector  itself changes its direction with angular velocity  . Hence we



have   cost i  sin t  j . Using this you can find the position vector as a function of
time in Cartesian coordinates.
To visualize the three dimensional motion on two dimensions, I have projected
the three dimensional trajectory on a plane.
To do this, we need to calculate the
projection of actual x, y, z coordinates to a new set of x and y coordinates, which can be
done using some simple trigonometry.
After managing to draw one vortex, I went further to draw the adjacent ones. Since
adjacent vortices counter rotate, I had to introduce a phase difference of 180 degrees in
the original vortex. Similarly I managed to draw the remaining vortices. The result is the
following:
Figure 4.10: The computer generated trajectory of particles in Taylor vortices.
In the second apparatus, the particle density was found to be alternately maximum
and minimum at the boundaries of the vortices. This can be explained as follows: the
particles from alternate vortices are rotating in opposite sense. As discussed in the chapter
two, the fluid elements constitute an ensemble of particles in the fluid. These particles
collide on the boundary and their vertical velocity components get cancelled. Since only
the horizontal velocity components remain, particles tend to settle in the circular region. I
have tried to account for this extra particle density by adjusting the spacing between the
bands.
The crux of the matter lies in the simple model of the spring, which we can use to
describe the motion of particles in Taylor vortices. It implies that the width of the bands
is the same as the gap between the cylinders, no matter which working fluid is used.
In fact the significance of these trajectories goes much beyond the Taylor
vortices. The electrons in a Tocamac, a toroidal particle accelerator, follow the same
trajectories. There are two types of magnetic fields in a Tocamac, an axial magnetic field,
which runs along the axis of Tocamac and the toroidal magnetic field, which goes round
the torus. These magnetic fields are also used to confine hot plasma, in which the ions
will follow the trajectory shown above.
4.4.4
Linear stability theory
The exact details of the onset of instability can’t be obtained by simple arguments
as in case of inviscid fluid. They have to be worked out from a detailed analysis of
circular Couette flow using a method called as linear stability analysis. In linear stability
analysis, we consider first the possible stable flow and then arbitrarily add a small
perturbation to it. The goal is to see which of these perturbations persist and grow, and
which die out. The reason for introducing these perturbations in theory is that, such
disturbances are produced in the experiments, due to some local irregularities and any
other arbitrary disturbances. Note that we have to keep the perturbations on the
boundaries to be zero. This is because the velocities on the boundary can’t be perturbed
by any physical means. We then substitute the perturbed velocity in the Navier Stokes
equations. Then we keep the terms in first order of perturbations, that is, we linearise the
Navier Stokes equations. The perturbations introduced contain a multiplying factor of
e st . The perturbations with negative s die out as time proceeds. The perturbations with
positive s increase with time. The case s = 0 gives a perturbation which can persist. These
perturbations can then be superposed on the basic flow. The result of such analysis in
case of the Couette flow is a circular trajectory on which sinusoidal variations are
superposed in radial and vertical directions. This is the trajectory essentially described by
the spring twisted in a circle. Such a computer-generated trajectory was shown in last
section. The basic flow profile in circular Couette flow was obtained in the previous
sections. Sir G. I. Taylor performed the linear stability analysis of this velocity profile
and obtained the following stability curve:
Figure 4.11: Taylor’s stability curve based on linear stability analysis. The
experimental results he got nicely fitted into it.
The
Rin 
Reynold
 in

and Rout 
numbers
 out

for
inner
and
outer
cylinder
are
defined
as
respectively, where  is the kinematic viscosity, which is the
ratio of viscosity to density. For lower reynold numbers, the flow is laminar, i.e. the
Couette flow. So below the stability curve, the flow is stable. The inviscid flow is stable
only in the shaded region below the line of stability predicted by Rayleigh’s criterion. We
can see the extra stability provided by the viscosity.
4.5
Other interesting things observed during the project work
We also tried rotating the outer cylinder. For that we used the available circular
base. We bought an outer transparent cylinder, we got the base grooved from the
university workshop, fitted the base with Araldite and with all the excitement we had, as
soon as the Araldite dried, we added the oil, kept a glass tube as inner cylinder and
started rotating the outer cylinder. And whoosh…! The Araldite hadn't done its job of
sealing the glass at the bottom completely! All oil spilled over the apparatus. Next time I
had to clean it for about half an hour. Never mind, even in that small show of rotating
outer cylinder, we saw the spiraling waves we were looking for. The rotation of outer
cylinder is planned with the new apparatus, but is yet to be carried out.
But thereafter the base assembly of outer cylinder alone was available, after better
sealing. We rotated water in it at higher velocities and obtained the paraboloid of
revolution predicted by the equations:
Figure 4.12: Paraboloid of water surface in a rotating cylinder. The cylinder is
progressively slowed down from a maximum rate.
An interesting thing we observed was that the particle impurities in such rotating fluid
tend to form a nice heap at the center when the rotation is stopped. Why? Initially, when
outer cylinder is rotating, the outer layers are moving faster than the layers at the center.
So we would expect the impurity particles to go away at the boundary, as is observed. In
fact the whole fluid is thrown away, giving the shape of a paraboloid. But when the
cylinder stops rotating, the inner layers are moving faster than the outward layers. By no
slip condition, the outer layers are brought to rest immediately after the outer cylinder has
stooped. By Bernoulli’s Principle, the high velocity region has lower pressure, so the
impurity particles move towards the center and form a heap as in case of sand piles.
Another interesting thing we observed was the following:
When we added the Al2O3 powder in the apparatus while the inner cylinder was
rotating, it didn’t mix immediately with the liquid. It formed a thick uniform layer on the
surface.
The distribution was very uniform because the powder fell on all the
circumference of the surface, while the liquid rotated. Then, to mix the powder, we
stopped the cylinder. And we observed a very beautiful growth pattern as the layer of
powder descended. It was in the form of a Mushroom.
Figure 4.13: Mushroom growth patterns in our apparatus.
This Mushroom growth is an example of instability at the interface of two
liquids when one medium displaces the other medium.
penetration of liquid into the layer of powder.
The amazing thing is the
A striking similarity of patterns is
observed in case of a plate heated at the bottom:
Figure 4.14: The Mushroom kind of patterns in a plate heated at the bottom [3].
In both these cases the morphology of the patterns is the same, though the
physical reasons behind the instability are different. In case of hot plate, the temperature
difference gives rise to density variations, whereas in our case the density variations are
put up externally. Because the pattern formation is due to the growth of disturbances at
the interfacial boundaries, the patterns found are the same.
The following interesting question was raised during the experimentation:
Why do the suspension particles remain suspended in a circular flow when the fluid
rotates? Whereas they settle when the rotation stops?
The reason behind this is related to the rotors in amusement parks.
Figure 4.15: Rotor in an amusement park [5].
When the rotor rotates very fast, the person standing near the wall remains
transfixed to the walls even after the base is removed. The reason for this is the
following: the wall provides the necessary centripetal force for the circular motion of the
person with the rotor in the form of normal reaction force. The frictional force between
the body of the person and the wall is this normal force times the coefficient of friction.
As the rotation speed increases, the required centripetal force and hence the normal force
both increase. Hence the frictional force increases, and after a certain rotation rate, this
force due to friction can balance the weight of the person and he doesn’t fall even if the
base is removed.
Coming back to the question of suspension of particles, now consider the various
coaxial cylindrical layers in the fluid. The outer layers will act like the walls in the rotor.
The inner layers and the particles contained in it will behave like the standing person in
that rotor. So the inner layers and the particles therein are held from falling down by a
vertical frictional force. This frictional force is in addition to the tangential viscous force
considered between the two cylindrical layers, in the Navier-Stokes equations.
4.6 Recent work and further prospects
Ever since Sir G. I. Taylor published his work in 1923 [6], there has been a
tremendous study in this field by various researchers. Being the simplest geometry, the
fluid dynamical transitions have been easier to analyze. At higher and higher rotation
rates, many interesting flow patterns are seen. The Taylor vortices become wavy, the
frequency of these waves goes on increasing. At even higher rotation rates, some more
frequencies are superimposed on the vortices. This cascade of patterns finally leads to
turbulence. Researchers have also worked using various aspect ratios  
where l is the length of the cylinders and various radius ratios  
a2  a1  ,
l
a2
. The researchers
a1
Andrek et al. [7] have done experiments by varying the angular velocity and plotted the
following phase portrait of patterns in rotating coaxial cylinders:
Figure 4.16: Patterns in rotating coaxial cylinders as function of rotation rates [7].
They have found around 20 distinct flow patterns. This shows the diversity of
phenomena contained in Navier Stokes equations, even in a geometrically simple system.
Earlier researchers reported another astonishing fact that the patterns in rotating coaxial
cylinders are not just the functions of rotation rates, but they are path dependent on
rotation rates. That is, even if the cylinders are given the same final velocities, the
patterns found in them need not be the same. In fact they are dependent on the history of
the fluid in recent past! It was shown that around 15-20 different flow patterns are
possible for the same rotation rates. So while plotting the diagram of flow patterns shown
above, the researchers have taken care to follow the same path from resting cylinders. So,
that is the state of affairs in patterns in rotating coaxial cylinders. A large number of
phenomena remain unexplained [8,9].
4.7 Applications
Recently the Taylor vortices have been put to a practical use for direct industrial
application by the scientist Gretchen Baeir he has studied the vortices that are formed in a
system of two fluids mixed into each other [2]. These emulsified fluids are difficult to
separate by other mechanical means and the previous methods were not as effective as
this newly developed method:
Figure 4.17: Schematic of a two fluid Taylor Couette extractor [2].
The idea in this apparatus is the following: the two fluids are separated into two
layers in the gap between the cylinders when the inner cylinder is rotated at high speeds.
Each of these phases shows the Taylor vortices. Due to the sensitive interface formed
between the two phases, the mass transfer is very active. The two fluids are then
extracted using the two outlets, and a fresh mixture is introduced from another inlet.
These kind of fluid separators are useful in case of bio-separators.
References:
1. “Self Made Tapestry: Pattern Formation in Nature”, by - Philip Ball.
2. “Liquid-Liquid extraction Based on a New Flow Pattern: Two fluid Taylor-Couette
Flow”, Ph. D. thesis - Gretchen Beir.
3. “Physical fluid dynamics”, by - D. J. Tritton.
4. MIT course available on net (Plane Couette flow).
5. “Fundamentals of Physics”, by - Resnik and Halliday.
6. G. I. Taylor, Phil. Tran. Roy. Soc. (Lodon) A 223 (1923);
Feynman Lectures on Physics- Vol. II.
7. C.D. Andreck, S. S. Liu, H. L. Swinney, J. Fluid Mech. 164, 155 (1986).
The
8. “Hydrodynamic Instability and Transition to Turbulence” – edited by H. L. Swinney
and Gollub.
9. “Instabilities and Chaos in Rotating Fluids” H. L. Swinney – from “Non-linear
Evolution and Chaotic Phenomena”- edited by G. Gallavatti and P. F. Zweifel.