# STUDY OF FLUID DYNAMICS WITH EMPHASIS ON COUETTE FLOW AND TAYLOR .. - DOC by malj

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```									                                   Chapter 4

Canned Rolls

In this chapter the Taylor vortices in rotating coaxial cylinders are
described which I call as “canned rolls” [1]. In initial sections the apparatus and
experimental results are described. Then I give the argument for instability in case of
inviscid liquids, which is Rayleigh’s criterion. We then consider the plane and
circular Couette flows for viscous fluids. The trajectories of particles in Taylor
vortices are modeled by a spring twisted in the form of a circle. Such trajectories are
generated using computer. Then we consider the reasons for instability in case of
viscous fluids. Some other interesting things observed during the project work are
then described. I conclude this chapter with recent work on Taylor vortices and future
prospects.
4.1 Introduction

The hydrodynamic instabilities have puzzled the physicists for centuries. Simplest
example is the transition from one flow pattern to another in case of tap water, as the flow
rate is increased. Similarly the flow past a circular cylinder has been studied in detail by
many researchers. The flow past cylinder goes through many transitions as the flow rate
is increased. Initially the flow is streamlined along the sides of cylinder. Then the vortices
develop at the back of the cylinder and they are periodically shed out, to form what is
called as Karman vortex street. At even higher flow rates, the turbulence sets in. This
cascade of transitions in flow patterns is universal that is, it is independent of the system
under consideration. There may be differences in actual flow patterns, but the overall
transition is from laminar flow to onset of turbulence. These transitions are difficult to
describe mathematically, and turbulence remains an unsolved problem till today.
The first attempt to account for these transitions was done by Sir G. I. Taylor,
who used the linear stability theory to explain the structure of Taylor vortices and the
transition from circular Couette flow to Taylor vortices in the fluid between the rotating
coaxial cylinders. His approach as well as the physical reasons behind the instability will
be explained in the following sections. But the instability was first predicted by Rayleigh
for inviscid fluids. Mallock, who used the apparatus to measure viscosity, first
experimentally inferred it [2].

4.2 While I was Canning the Rolls
The journey towards obtaining the Taylor vortices wasn't a smooth and simple
one. It was not at all clear to start with, how I would manage to construct the apparatus
with all its requirements like the outer cylinder should be transparent, the cylinders
should rotate independent of each other and so on. So to start with we decided to modify
the available viscometer apparatus in our college. The fact that a drinking water glass
suited as an outer cylinder was a happy coincidence. To keep the inner cylinder in it and
allow it to rotate freely, I put a rubber sheet at the base of it and made a small hole in it,
so that the cone of the cylinder easily fits in. Then it was time to look for the ways to
make the inner cylinder rotate. After lot of search I got a dc motor of 12 volts for 55 Rs,
from Pioneer Electronics, which had sufficient power to rotate the inner cylinder. But
now the problem was that of a suitable pulley. Unfortunately the motor dealers don't
provide a pulley. After some more searches I found a suitable pulley, but its bore size
was less. So I had to get it drilled from Haribhau, our elderly lab assistant. After the
pulley the natural question was that of the belt. But we decided to work with all available
rubber bands and they worked pretty well. Later I replaced them with tape recorder belts.
Finally my initial apparatus took the form as shown in Fig. 4.1.

Figure 4.1: My earlier apparatus.

Having constructed the apparatus was not all. I was not sure which fluid to take
as working fluid, how to visualize the streamlines and see patterns. First I tried dropping
ink with a syringe, but it tends to get diffused. Then I tried several combinations of
working fluids and suspensions, which are described in Table 4.1.
Table 4.1: Description of flow visualization methods.

No. Fluids Suspension / Method             Observation                    Comments
1   Water   Aluminum Powder                Sticks    to      walls   of Though         these   combinations
cylinders.       Blurred out form a good suspension, they
white solutions.               don't     provide     any       good
2   Water   Ferrous Oxide        Powder    Sticks    to      walls   of visualization.
cylinders.       Blurred out
reddish solutions
3   Water   Local     pH    change    dye The solution turns yellow Suitable for a closed system.
method:        add     Bromo near       one       of     the We didn't use it much because
Thymol Blue indicator and electrodes.           This yellow the change wasn't prominent
place the electrodes in is carried away by the and                      large       concentrated
solution. Apply 3-4 volts. flow,            producing     a quantities would be required.
Bubbles should be avoided streakline.                         Advantage of this method is
[3].                                                          that the dye doesn't spoil the
whole solution, it changes back
to its original color after some
time.
4   Water   Fluorescent dye method a In UV light the dye turns The glow is uniform, so it was
due called Ruthenium Tris orange from yellow.                 not clear how it could help in
Bipyridine.                                                   visualizing the vortices.
5   Water   Aluminum Oxide Powder The powder aggregates, This                     method       was     initially
6   Water   Aluminum Oxide Powder Good suspension, low Best combination. The addition
+ little soap solution         settling rate.                 of      soap      solution      stops
aggregation, an ingenious idea
indeed. There is a drastic
amount of soap solution.
7 Paraffin Aluminum Oxide Powder Good                        suspension, Good combination, this gave
oil                                     appreciable settling rate.   us the clearest results in the
beginning
8 Paraffin Aluminum Oxide Powder                                         This       gave     even       better
oil      colored with blue ink and Good suspension, low visualization                  due     to   color
dried under lamp.              settling rate.               contrast     with       the    inner
cylinder.
9    Castor Aluminum Oxide Powder           Good suspension, very For all available rotation rates,
oil                                    low settling rate.           only circular flow was seen.
This combination will require
very high rotation rates.

We got the best results with the combinations 6, 7, 8 in the above list. Now the
reader would have realized the amount of efforts involved in constructing a new
apparatus and setting up an experiment. But this work had rewarding outcomes for me.
Firstly it gave me confidence to build an apparatus. Secondly I came to know the various
resources involved. Thirdly, I experienced the joy of a successful experiment.
Having secured some good results, we confidently proceeded to make a
professional apparatus, for more rigorous study in the near future.            With available
infrastructure and some more investment, we could prepare a well-finished apparatus,
with innovative ideas and efforts on part of the workshop persons.
The newer experimental setup for studying the Taylor vortices is given in Fig. 4.2.
The two cylinders are placed coaxially one inside the other. The inner cylinder rests on
the outer one with a cup and cone arrangement. The outer cylinder is a glass pipe, which
is fixed in a groove at the base by Araldite. The inner cylinder is held at the upper side
using a ball bearing arrangement. Both the cylinders are provided with tape recorder
belts and can be rotated independent of each other using separate motors.
Figure 4.2: Taylor vortices apparatus.

4.3 Observations
When the inner cylinder is rotated slowly, the fluid moves in circular lines. But as
we go on increasing the rotation rate, something interesting happens: some horizontal
lines are seen in the flow and when the inner cylinder is rotated above a certain critical
rate, the flow breaks up into horizontal vortices. The particles in the vortex help in
identifying the sense of rotation of the vortex. It can be easily seen that the adjacent
vortices are counter rotating. Apart from 2-3 vortices near top and near bottom, which are
distorted due to end effects, all the other vortices have same width.

4.3.1 Taylor Vortices Using Old Apparatus

This apparatus is shown in Fig. 4.1. With the working fluid as paraffin oil and
suspension powder as blue colored Al2O3 Powder, we saw nicely separated bands (see
Fig. 4.3). The width of bands was observed to be equal to the gap between the cylinders.
Figure 4.3: Taylor Vortices in paraffin oil.

4.3.2 Taylor Vortices Using New Apparatus

Figure 4.3: Taylor vortices in water.

Since for this apparatus the gap was smaller, the vortices seen in this figure are
smaller.

4.4    Theoretical Explanations
4.4.1 Argument for instability in an inviscid liquid

Consider a fluid with zero viscosity. Practically no such fluid exists, except for
the Helium3 superfluid at very low temperatures. And it gives Helium3 some of its
astonishing properties. It has this property that it keeps rotating once it is set into motion!
It also has this amazing property to creep over the walls of the container, so just keeping
it in our apparatus won't do! We will require confining it into the concerned region. But
air can be taken as a practical substitute as a very low viscosity fluid. Now we can easily
imagine that if only the air is present in the gap between the cylinders, then even the
slightest rotation of inner cylinder would cause lot of disturbance in it. In fact the flow
will be a turbulent one. Actually even without rotation of the inner cylinder the flow in
air is not stable. I hope this gives the reader a sufficient judgement of the behavior of
fluids of low viscosity. But water has some significant viscosity. It will be unstable after
certain rotation rates, but it does show a stable circular flow for low rotation rates. The
reason for instability in an inviscid liquid is the following: the fluid near the inner
cylinder continues in its state of motion, by Newton's first law, as no force acts on it. But
the inner cylinder is rotating and the liquid has to stay in the confined region. So it hits
the outer wall and bounces in a different direction, since the walls are curved. The inner
cylinder's motion is required to throw the fluid near it towards the outer cylinder again
and again. Then, to replace this outgoing fluid, some fluid must come from beneath it.
The fluid coming towards the inner cylinder then occupies the place of this fluid. In this
manner the convection rolls are set up, and these canned rolls are nothing but the Taylor
vortices.
This can be quantitatively explained as follows: Suppose a fluid layer at distance
 , circulating with angular velocity  is displaced to a distance  , where  <  .

Because there is no external torque on this fluid particle, the angular momentum of this
fluid particle must remain the same. Hence its new angular velocity will be
2

                                                                                
. The centrifugal force of this fluid particle, with angular velocity  will
2


exceed the available centrifugal force at  , if     . Using the angular momentum

relation, we can write this condition as:
d
r 2  0 .                                          ..... (4.1)
dr
That is, the radial gradient of angular momentum r 2 is negative. This condition
is called as Rayleigh’s criterion for instability in inviscid Couette flow [3]. If the two
cylinders are rotating in same sense, then the flow is either every where stable or
everywhere unstable. Substituting the radii a1 and a2 in Rayleigh’s criterion, we get the
condition as
1a12   2 a2
2
.....(4.2)
When the two cylinders are counter rotating, the region near the inner cylinder is
unstable while the region near the outer cylinder is stable. So the inner cylinder rotations
have a destabilizing effect, while the outer cylinder’s rotations have a stabilizing effect.

4.4.2 Plane and circular Couette flow.

Our task is now to find the velocity profile in case of viscid fluid between the
rotating cylinders. Naturally, due to cylindrical symmetry, the best coordinate system to
choose is the cylindrical polar coordinate system. But the Navier Stokes equations in
cylindrical coordinate system are complicated compared to those in Cartesian coordinate
system. So, as any good Physicist would do, let’s first try to approximate the situation by
a simpler one and try to solve the Navier Stokes equations there. First let’s neglect the
end effects due to finite size and consider the infinitely long cylinders.           Then the
velocities will be independent of the positions on z-axis. Let’s restrict ourselves to low
rotation rates for the time being. Then there are no vortices and the fluid just moves in
circles as observed. So there is no dependence of velocity on the angle. At a given
radius, the velocity is same over the entire vertical cylinder at that radius.
If we consider a small vertical section of the cylinder and magnify the picture then
we will have the flow in 2 almost parallel plates, just like the earth looks flat in the
surroundings we see. This limiting case we are considering is equivalent to an apparatus
with large radii of cylinders and small gap between the cylinders. The outer plate is at
rest and inner moves at a constant velocity in one direction, let's say x-direction. Due to
no slip condition, the velocity of the fluid near the outer plate is 0 and at the inner plate it
will be equal to that of the inner plate. We can set up the coordinate system with x-y axes
parallel to the two plates and z-axis perpendicular to them. Then we can use Navier
Stokes equations to find the velocity distribution as a function of z. [4].
Z

u

a

X
Figure 4.4: Geometry of plane Couette flow.

Refer to the appendix D for components of Navier Stokes equations in Cartesian
coordinates. For steady state, time derivative is zero. Also we may neglect any external
forces. Also u  u x ( z )i i.e., velocity is in x direction only and its magnitude changes
with z alone. Then the left-hand side of Navier Stokes equations is zero. On the right-
hand side we may take external forces to be zero, also there is no external pressure
applied, hence the pressure gradient is also zero. Then the Navier Stokes equations
reduce to just  2u  0. since only the x-component of velocity is nonzero, the solution
of this equation is , with the boundary conditions u=0 at z=0 and u= V at z=a,
V
ux ( z)      z.
a

10

Vz/a=u
height z from lower plate

8

6

4

2

0
0       20      40          60       80       100

velocity in x direction

Figure 4.5: Velocity profile in plane Couette flow.
Note that this flow profile is different from the flow considered in chapter 2, for an open,
wide, shallow channel. There, the velocity profile was quadratic and here it is linear.
There, the velocity was pressure driven. Here it is induced by the viscous action alone.
There, the velocity of upper layer was selected by the pressure gradient. Here it is
imposed as an external condition. So we see the difference between the flow profiles
even though the boundaries are same, due to different physical reasons for the flow.
Now we are ready to solve for the velocity profile in our cylindrical geometry.
We set our coordinates as follows: we choose the cylindrical polar coordinate system
such that the axis of coordinate system is that of our cylinders. As discussed before, the
flow is only tangential, that is the velocity is tangent to any circle drawn perpendicular to
the z-axis. So only u is nonzero. Also u is the function of r alone. We can expect the
solution to be of the form r + some function of r as the solution of plane Couette flow was
linear.

Figure 4. 6: The geometry of circular Couette flow.
Because u is the function of r alone, the continuity equation is automatically satisfied.
Referring to appendix D, we can easily see that the z component of Navier Stokes
equations also gets satisfied automatically. The  component Navier Stokes equations
reduces to
d 2u 1 du u
0  (              ).
dr 2 r dr r 2
The velocity at the inner cylinder is 1a1 and that at the outer cylinder is 2a2.This can be
u
easily solved using the variable      , with the above boundary conditions, to give [3],
r
B
u (r )  Ar      .
r
( 2 a 2  1 a12 )
2
(1   2 )a12 a 2
2
Where, A                         , and B 
a 2  a12
2
     
(a 2  a12 )
2
.

See how the 1/r term absorbs the curvature of the geometry compared to the plane
Couette flow. Also note that this velocity distribution is independent of ! That means,
for a given geometry of apparatus and for given rotation rates, the same flow exists for all
d (ur )
liquids! We can easily see from these expressions that                                              2 A . So according to
dr
rayleigh’s criterion, the flow will be unstable if A < 0 and flow will be stable if A > 0.
The r component reduces to
u2        dp
                .
r          dr
The important point to be noted here is that though there is no external pressure,
the pressure develops intrinsically, due to the velocity distributions. Here we are having
the pressure due to the centrifugal force. Putting the velocity as a function of r, we can
easily find out the pressure.
Now let's look at the plot of the velocity profile in this circular couette flow. In
our case,  2  0 . Taking the the constants to A and B to be 1 and –1 respectively, we get
the following velocity distribution.

10
u = -r +1/r
derivative(ur)= - 2r
tangential velocity u

5

0

-5

-10

0        2         4              6           8         10
a2
distance from axis r
Figure 4.7: Unstable velocity profile in circular Couette flow.

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