Download study of fluid dynamics with emphasis on couette flow

STUDY OF FLUID DYNAMICS WITH EMPHASIS ON
COUETTE FLOW AND TAYLOR VORTICES
Project Report submitted to the
FERGUSSON COLLEGE
In Partial Fulfillment for the Degree of
BACHELOR OF SCIENCE
IN
PHYSICS
BY
ABHAY MADHUSUDAN KARNATAKI
(Exam No. 01038)
T. Y. B. Sc.
DEPARTMENT OF PHYSICS
FERGUSSON COLLEGE,
PUNE.
FERGUSSON COLLEGE, PUNE – 411004.
This is to certify that
Mr. Abhay Madhusudan Karnataki
(Exam No. 01038) has satisfactorily completed the project entitled
“STUDY OF FLUID DYNAMICS WITH EMPHASIS ON
COUETTE FLOW AND TAYLOR VORTICES”
under our guidance in partial fulfillment of Degree of Bachelor of Science
(B. Sc.) in Physics as per described by University of Pune during the
academic year 2000-2001.
A. G. Banpurkar
Dr. Mrs. R. Joshi
External Guide
Internal Guide
Prof. D. M. Kulkarni
Prof. P. T. Purandare
Project Incharge
Head, Department of Physics,
Fergusson College, Pune.
To the people who
shaped my life,
my Father, Mother and
Brother and
my beloved teacher
Prof. M. Prakash…
Contents
Preface
Ii
Acknowledgements
iv
1. What is fluid dynamics?
1
2. Toolkit of fluid dynamics
3
3. Fluid dynamics in daily life
13
4. Canned rolls
21
4.1 Introduction
22
4.2 While I was canning the rolls
22
4.3 Observations
26
4.4 Theoretical explanations
27
4.4.1 Argument for inviscid fluids
27
4.4.2 Plane and circular couette flow
29
4.4.3 Computer generated trajectory
34
4.4.4 Linear stability theory
37
4.5 Other interesting things observed during the project work
38
4.6 Recent work and further prospects
42
4.7 Applications
43
5. Playing with soap films
45
6. My hero - Sir G. I. Taylor
58
Conclusions
61
Appendices:
A. ‘C’ program for the particle trajectories
62
B. Accreting on stars
64
C. Fascinating BZ reaction
68
D. More about Navier Stokes equations
73
E. Some useful properties of common fluids
75
Suggestions for further reading
76
Acknowledgements
I express my sincere thanks to my external guide Shri. Arun Banpurkar for taking
out time from his busy schedule for Ph.D. thesis and helping me in all possible ways.
Many of the ideas in this project report originated and took a better shape in number of
exciting discussions with him. Arun sir essentially taught me how to set up an experiment
right from the scratch and how to optimize it for the best results with patience and efforts.
I thank my internal guide Dr. Mrs. R. S. Joshi for always showing me the right
direction to proceed and for many important suggestions.
It gives me a great pleasure to thank Dr. A. V. Limaye for helpful discussions at
several stages, which enabled me to analyze various situations in minute details.
I take this opportunity to express my thanks to Prof. S. B. Ogale for providing me
a strong impetus for hard work and future studies. I thank Dr. K. P. Adhi for raising
many important questions and for providing many suggestions. I thank all the members
of CLAMP; Dr. S. I. Patil, Dr. Bathe, Narhe, Khandkar, Mandar and Sadakale for making
the working environment so wonderful and for their help while performing the
experiments.
I thank Dr. Tapas K. Das from IUCAA for explaining me in detail the analysis of
the process of accretion of matter on stars and related things.
I thank the Heads of Departments of physics in Pune University and Fergusson
College for providing me the required infrastructure facilities. I thank the Center for
Non-Linear Dynamics, Texas, for sending me the reprints of some research papers on
Taylor vortices. I thank our elderly lab assistant Haribhau for his numerous suggestions
while making the apparatus.
I thank my friend Sandeep for many exciting discussions on mathematics and
especially for his book “What is mathematics?”, which provided me lot of things on soap
films. I thank all my friends from chemistry discipline, Ashutosh, Devayani, Sarita and
Mrunalini and teachers from department of Chemistry, Pune University, Dr. Avinash
Kumbhar and Dr. P. K. Choudhary for their patient help in flow visualization methods
and the fascinating BZ reaction. Special thanks to Ashutosh for always sharing the
excitement in science.
Because this project work marks the end of my undergraduate studies and as this
work was motivated from my previous studies in Physics, I would like to express my
thanks to all those who helped me in the last three years. First of all I would like to thank
Desai sir from Exploratory for giving me a free access to the lab facilities there, in these
years. I thank all the staff members of Physics department in Fergusson College for
making the studies of physics so joyful and exciting. Special thanks to Ogale madam,
Alawani madam and Dabhade madam for always encouraging me for extra curricular
activities. I also thank my Mathematics teachers, especially Acharya sir, Kulkarni sir for
always promoting my interest in Mathematics. I also thank my teachers from Electronics
department, especially Bhide sir and Khedkar sir for introducing me to the excitement in
the world of Electronics. I would like to thank the Professors in Pune University Physics
Department Prof.. P. V. Panat, Prof. A.W. Joshi, Dr. C. V. Dharmadhikari, Dr. Mrs. A.
Kshirsagar, Dr. R. K. Pathak for their ever-welcoming nature towards my Physics
queries.
My special thanks to Dr. A. D. Gangal from Pune University Physics Department
for his guidance for reading extra curricular books and for resolving many mysteries of
fluid dynamics as well as other topics in “Feynman Lectures on Physics” in a delightful
manner. He introduced me the joy of theoretical physics in the simplest ways.
My special thanks also to Dr. S. V. Dhurandhar for teaching me the concepts of
special and general theory of relativity in an elegant manner; which has given me a deep
satisfaction of studying physics.
I thank all my seniors who have become my close friends, Deepti, Harshad,
Prasad, Anuradha, Aditi, Subhangi, Anamica, Aparajita, Deepanjan and Devraj. They
have always shown me the opportunities lying ahead and have provided the glimpses of
advanced physics. I thank all my friends in colleges for sharing the joy of doing physics.
Special thanks to Priya, Vidyut, Sonal, Shashank, Sourabh, Abhijit, Maitreyi, Sheetal,
Sulbha and Rahul for many exciting discussions. I thank my chess friends Nivedita,
Ashwini and Udayan among others and my chess teacher Mr. Joseph D’ Souza for
keeping the chess player in me alive, which was essential for my good studies after
leaving chess as a profession. My special thanks to my friend Sneha for her constant
support and encouragement, and to Abhijit and Virendra for always being with me in my
adventures.
Last but not the least; I thank my friend Vaibhav for helping me in typing this
project report and also for being a “test student”! Some names might have got omitted in
this acknowledgement by mistake, and the people I have mentioned here have helped me
in ways more than I can describe in words. I thank them all with all my sincerity.
Abhay Karnataki.
Chapter 1
What Is Fluid Dynamics?
We think of fluids as liquids or gases only. But the subject of fluid dynamics
deals with a large class of systems, than just liquids and gases e.g. Certain types of
glasses can be thought of as a fluid. Why? Because they ‘yield’ under the force of
gravity. Glasses of windows in old houses are found to be thicker at bottom than at the
top. Because glass can flow! So fluids are those substances which cannot stand the
shearing forces.
If you push the upper end of a thick book resting on a table, then its pages tilt – its
pages slide upon each other. This is the action of shearing stress. Similarly, layers of
fluid slide over each other. Now, sliding involves friction, and this sideways force
between two layers is called the Viscous Force. The measure of amount of viscous force
is the property of fluids called Viscosity. More the viscosity of the fluid, more is the
viscous force. And this viscous force stops the fluid layers from moving indefinitely on
each other. But, if the external force is applied for sufficiently long time, the fluid layers
have to move.
Thus, we may define fluid as, “A Fluid is matter in a readily distortable form,
so that the smallest external unbalanced force on it causes an infinite change of
shape, if applied for a time long enough” [1].
Fluid dynamics can be used to model a variety of physical situations. E.g. it can
be used to study the rate of accretion of stellar dust on a star moving through dust clouds
in a galaxy. Now, such dust particles may be a few kilometers apart, but on astronomical
scales, we can still treat the dust cloud as a ‘Continuous Fluid’. Fluid dynamics may also
be used to model traffic on Mumbai Pune Highway. Here the particles would be of
different types, corresponding to various vehicles. But as a whole, they can be treated as
a fluid.
But for all purposes, we may study fluid dynamics with the working fluid as
water, to start with. Water can be used as a standard example of fluid while studying
fluid dynamics.
Having understood what a fluid is, we should try to see what are the various
physical parameters associated with the flows of the fluid. It is the wide diversity of
these parameters that makes the subject of fluid dynamics so challenging and difficult to
understand.
Simplest parameters we can think of are, density of fluid and velocity of fluid at
each point inside the fluid. Also, from Bernoulli’s Equation and Hydrostatistics we are
familiar with pressure at a point in fluid, which is another important parameter. In
addition there can be external forces imposed by the experimenter using pumps, or the
forces like gravity, which are omnipresent. On a free surface of a fluid, e.g. on the
surface of flow in river, surface tension will be an important factor. To complicate the
matter, there may be charges present in the fluid e.g. ions of some chemicals. Moving
charges constitute current. So if external electric and magnetic fields are applied, the
charges will interact with them. Also, internally the charges will interact with each other.
This will modify the flow of the fluid. These branches of study are called
‘Magnetohydrodynamics’, and ‘Electrohydrodynamics’. To further complicate the
things, there may be temperature gradient inside the fluid, which will in turn induce
density variations in the fluid. There may be density variations produced due to
impinging sound waves [2].
Fluid Dynamics is the study of fluid under these various conditions put separately
or together. It also deals with interaction of fluid with solid boundaries near it, e.g. body
of a fish swimming in a pool of water.
This may give an impression that Fluid Dynamics is a hopelessly complicated
subject. But in many real life situations only few parameters are of prime importance.
And over the centuries, physicists, with their unique insight into natural phenomena, have
discovered many rules that regulate the fluids.
References:
1. A textbook of fluid dynamics- Fransis.
2. The Feyman Lectures on Physics -Vol. II
Chapter 2
Toolkit of Fluid Dynamics
So, how do we formulate a theory of such a nasty fluid, flowing with its own will
- so to say, interacting with so many physical parameters?
The central and starting assumption of fluid dynamics is that we can describe the
fluid under the consideration as a continuous medium. This is the so-called “Continuum
Hypothesis ”.
But we know that there is even in case of water, a lot of void space between two
molecules of water. So how can we possibly treat water or any other practical fluid as
continuous?
The point is that we will be observing phenomena, which are sufficiently
macroscopic in nature. And ‘sufficiently macroscopic’ means that the characteristic
length scales over which the flow is examined, are larger than the mean free path of the
molecular collisions. E.g. we might be interested in knowing how the presence of an
obstacle like an airplane causes change in the flow velocity of the air near it. Now, these
changes occur with typical length scales of a few meters, and mean free path of air
molecules may be of a few millimeters. Or we may put some obstacle like cylinder in
flowing water. Here flow patterns change over a few millimeters, but the mean free path
of water molecules is a few micrometers. So, the fluid in consideration can be treated as
continuous. Similarly, the dust particles accreting on stars, which are a few kilometers
apart, as in case of upper atmosphere of earth, can be treated as continuous medium over
the length scales of tens of lakhs of kilometers.
So, when I say that "consider a fluid element of volume dV", I mean a cube of
length greater than the mean free path of fluid under consideration, although
mathematically we treat dV as an infinitesimal element.
When I say that "a fluid particle" has moved from A to B, I mean the ensemble of
particles in a fluid element of volume dV situated at A has moved, on an average, to B.
Or the " fluid particle" has a velocity V at (x, y, z), I mean the ensemble of particles in a
fluid element of volume dV at (x, y, z) has, on an average, the velocityV.
The particles in that fluid element may have various different velocities and some
of the molecules may not go from A to B or some of the molecules may not have
velocityV, but the averaged out quantities are important. Same reasoning applies when
we say that a layer of fluid moves over another layer.
The continuum hypothesis and the results derived based on it have been tested in
the laboratories in various experiments. And they have been proved to be successful
since centuries. But we should be aware that it has certain limits imposed by length scales
under consideration. In the days of nanotechnology, fluid flowing through nanotubes
may not be treated as continuous, and molecular size effects will have to be taken under
consideration.
Having made the Continuum Hypothesis, we now describe the equation, which
expresses conservation of mass. The mass is conserved means that the change in mass
contained in a closed surface per unit time is equal to the total mass that goes in or comes
out of the same surface. The component of velocity perpendicular to the surface
determines the flux of mass coming in or going out of a surface. The component parallel
to the surface just causes the mass to move on the surface. So we find the mass flux
through a closed surface ‘S’ as:

 u .ds ,
S
Where I have written -the density of the fluid, instead of mass ‘m’, because we will be
considering physical quantities per unit volume. This is because the “fluid ‘particle”
itself is of a finite volume and not a point particle, as explained previously.
The rate of change of mass enclosed by the same surface is
volume enclosed by the surface.

 where V is the
t V
Conservation of mass implies that

 u .ds   t  
S
…..
(1)
V
where negative sign is introduced to account for the fact that outward flux is considered
positive. So, when  decreases with time, i.e. fluid goes out of the surface ‘S’, the flux
remains positive.
Left-hand side of (1) can be written as:
 .u dV   u .ds ,
V
S
by Gauss’s divergence theorem. And so, as the volume ‘V’ enclosed by the surface ‘S’
becomes very small, we can write equation 1 as in differential form as:
.u   

t
..… (2)
Equation (2) is called as the ‘Continuity Equation’. This equation is completely
analogous to the continuity equation in electrodynamics. There the roll of mass is played
by charges and the velocity charges gives rise to currents. But, with essentially the same
reasoning, there it expresses the conservation of charges. The continuity equation in this
most general form is required for the treatment of the gas coming out of a small nozzle,
under high pressure.
In that case the gas expands, so density variations become
important. But in most of the cases of liquids, we will consider the density of the fluid to
be constant. E.g. density of water can be taken to be constant unless, sound waves are
passed through it. So, liquids like water, glycerin, paraffin oil etc. will be treated as
incompressible. Then the equation of continuity takes a simple form
.u  0
…..(3)
Equation (3) says that the amount of fluid entering a closed surface is same as that
leaving the surface. Or the flux of the fluid i.e. surface integral of u.ds is zero. So
essentially there are no sinks in the region of concern.
So the continuity equation puts some restrictions on the velocity distribution in
the fluid. But, what governs how the fluid moves? The fluid moves under the action of
forces like pressure, gravity etc. Thus, 'Newton's second law' decides how the fluid
moves.
Let me first show you how the Newton’s second law is written in case of fluids,
and then explain the significance of each term.
u
 u .  u     P   2 u  Fext
t
These vectorial equations are called as “Navier Stokes equations”.
.….(4)
The sum of all forces acting on a fluid particle is written as the right hand side,
and the left-hand side of the equation gives acceleration of the particle. Let’s see why.
The first term on the right hand side of equation 4 is the force due to pressure
changes, (either caused by the motion of the fluid itself or applied externally) in the fluid.
Let us consider component in the X-direction of  P . It is 
P
. Consider a layer
x
of fluid perpendicular to x axis as shown below:
z
y
x
F=P. Ax
F=P. Ax
Ax
Figure 2.1: Pressure forces on a thin fluid layer perpendicular to x-axis. P is the
new pressure at x +x.
The
force
in
X-direction
on
a
fluid
layer
of
thickness
x
is

 P  
PAX   P    x  AX , where AX = Area of cross section of the layer perpendicular
 x  

to X – axis. So, the net force per unit volume acting along the X – axis is: 
Hence, the force per unit volume along X –axis is 
P
. AX .X .
x
P
P 
i is
. In vectorial notation, 
x
x
the force per unit volume in X – direction. Similarly, 
P 
P
j and 
k are the forces
x
x
per unit volume in Y & Z – direction.
The second term on RHS is the force due to viscosity of the liquid.  is the
coefficient of viscosity. The term  2 u is derived from a detailed analysis of stress
tensor, and I will not discuss it here. But we can see how this expression reduces to the
familiar expression of Newton’s law for viscous force:
F  A
du
dr

Consider a fluid moving in X – direction only. Then u  u X i .
For example,
consider fluid moving in an open channel parallel to X – axis.
z
y
ux
x
ux=z2
Figure 2.2: Open, Wide, Shallow channel.
u X is same for all x, because of the continuity equation, since the same amount
of fluid passes through any 2 cross sections perpendicular to X – axis. Taking Y-axis
along width and Z – axis along the height of the channel, if channel is wide enough
compared to its depth, change in velocity with respect to Y can be neglected. So
 2 u  
 2u
 2u
 2u




x 2
y 2
z 2
becomes, as ux is a function of Z alone,

d 2 u X z i
 u  
dz 2
2
As in the case of pressure force,  2 u gives the force between two layers, per
unit volume. Consider a layer of area ‘A’ parallel to XY plane and of infinitesimal
thickness dz. Then the force between two such layers is:
 2 u . A.dz  

d 2u X

A

dz
i
dz 2
  2 u . A.dz  A
du X 
i.
dz
which is the same expression as given by Newton’s second law based on experimental
observations.
The third term on RHS is the sum of all other external forces such as that of
gravity, or the forces due to externally applied electric or magnetic field, if the fluid
contains charged particles.
So, RHS describes the sum of all forces acting on the fluid particle. Now, that is
equal to the acceleration into mass per unit volume of the fluid element at that point.
That is represented by LHS as follows:
Consider a fluid particle situated at (x, y, z) at time ‘t’. Let its velocity be
u t , x, y, z  . In time t the particle would move to a position (x+x, y+y, z+z), where
x  u x t ,
y  u y t ,
z  u z t .
The
velocity
of
fluid
particle
is
then u t  t , x  x, y  y, z  z  .
At ( t  t , x  x, y  y, z  z ) the momentum per unit volume will be
 (t  t , x  x, y  y, z  z )  u (t  t , x  x, y  y, z  z ) . Hence, the rate of
change of momentum per unit volume, with respect to time will be:
lim
t 0
   u t  t , x  x, y  y, z  z      u t , x, y, z 
 lim
t
   u t  t , x  u x t , y  u y t , z  u z t      u t , x, y, z 
t 0

t
u
 u 
 u 
 u 
which is the LHS.
 uX
 uy
 uz
t
x
y
z
Note that we have considered the rate of change of “momentum per unit volume”,
and not “rate of change of momentum” per unit volume. That is we have included ‘’
before taking the derivatives. The reason lies in the fact that the fluid elements are finite
in size and not infinitesimal. And also, due to the fact that the same fluid element at (x, y,
z, t) may occupy a different volume (due to expansion or contraction) at a later instant
t  t and at a new position ( x  x, y  y, z  z ). So we have to consider change in
volume before taking derivative.
We have now written down the two laws that govern all of fluid dynamics:
i.
The “Equation of motion” or the “ Navier Stokes Equation ”.
u
 u . u     P   2 u  Fext
t
And
ii.
The “ Conservation Of Mass ” or the “ Continuity Equation ”
. v   

t
Within these two equations is contained all the swirling, moving, rotating or
turbulent fluid motion. But, these are extremely difficult to solve analytically except in a
few situations. So fluid dynamics is studied hand in hand with the experiments and
theory as well as numerical solutions using computers.
The Navier Stokes equations and the continuity equation are second order,
nonlinear, partial differential equations in four variables, x, y, z, t and four unknowns
u x , u y , u z and . The non-linearity implies that you can't superpose two solutions. That
is, the solution of one situation, with certain boundary conditions can't be added to
another solution with boundary condition, to get the solution of a problem with combined
boundary conditions. For example, the flow patterns for a cylindrical obstacle can be
easily found out, but from that we can't find the flow across to such obstacles, just by
adding the solutions separately. The problem of two cylinders is altogether different.
This is because the one cylinder interacts locally with the flow disturbances produced by
other cylinder and vice versa. This is in great contrast with the electrodynamics situation,
where Maxwell’s equations are linear and the fields can be superposed on each other. The
existence of fields due to one charge is independent of the presence of other charges.
Here the solutions have to be obtained in a self-assistant manner. That is why fluid
dynamics is much harder to understand than electrodynamics.
Since the Navier Stokes equations are vectorial in nature, they have three
component equations and hence we have a consistent system of variables and equations
describing the fluids. As these are differential equations, their solutions will obey certain
boundary value conditions. Let us see what these boundary value conditions are.
At any solid boundary, the fluid cannot enter inside that surface. Neither can it
come out of the solid. So the component of the velocity perpendicular to the solid must
be zero. Also, it is experimentally tested that the fluid layers near the boundary of a solid
are carried by the solid with it, so that the velocity of fluid at these boundaries is also
equal to the velocity at that boundary itself. This is called as the “no slip” condition.
As an illustration, let’s find out the velocity profile in the shallow, wide, open
channel, we considered for deriving the Newton’s formula for viscous force from the
viscous term in the Navier Stokes equations. The various components of Navier Stokes
equations are given in appendix D. 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. The flow along the x-axis is due to some external
pressure gradient along x-axis. We may take this gradient to be k. Then the Navior
Stokes equations just reduce to
p d 2 u

x dz 2
From this we find,
u x ( z )  Az 2  B
i.e. the velocity changes quadratically with the height:
10
x = z2
Height z
8
6
4
2
0
0
20
40
60
80
100
Velocity in x direction
Figure 2.3: Velocity profile of flow in shallow, wide, open channel in x direction as a
function of z.
This velocity profile was indicated in figure 2.2 .The velocity at the upper surface is
decided by the pressure gradient in x direction.
To gain some confidence in using the Navier Stokes equations, let’s see how they
reduce to more familiar Bernoulli’s equation. The Bernoulli’s equation is a statement of
conservation of energy, so it can be applicable only to the flows which are not
dissipative, that is the viscosity is very low. So the viscous term in Navier Stokes
Equation drops to zero. Also we take only those external forces into consideration which
can be written as the gradient of some potential say  . Then the in Navier Stokes
Equations become
u
P
 u . u   
 
t

We can rearrange this equation using the identity u   u    u   u 
…..(5)
1
  u  u  .
2
If we now define a new vector field     u , this vector identity can be written
as u   u    u 
1
 u 2 . Using these the equation 5 becomes
2
u
1
P
   u  u 2  
  .
t
2

…..(6)
Now we limit ourselves to the flows, which are steady, that is the flows which do not
change with time. These flows may be very complicated and changing in space, but the
steady flow means that the velocities at a point do not change with time. A fluid at any
point is replaced by a new fluid moving in exactly the same way. The velocity field is
then static, i.e. it is the same for all times. So
u
 0 . Then if we take the dot product of u
t
with equation of motion (6), the u  (  u ) term drops out, and we get
p
1 
u      u 2   0
2 

…..(7)
The quantity in the bracket is a scalar function of position. The equation (7) says that
gradient of this scalar function is perpendicular to the velocity, at all points. That means,
the equipotentials of the bracketed quantity are along the lines of velocity. Since the lines
of velocity are nothing but streamlines, we have
p
1
   u 2  constant (for that streamline)

2
which is the Bernoulli’s theorem.
......(8)
With this, we have the toolkit of fundamentals in fluid dynamics in our hands and
we are ready to analyze various situations in fluid dynamics. But before we do that, let
us see the variety of instances in daily life where fluid dynamics can be used.
…That we have written an equation [of
motion] does not remove from the flow
of fluids its charm or its mystery or its
surprise.
R. P. Feynman
In his Lectures on Physics.
References:
1.
The Feynman Lectures on Physics- Vol. II.
2.
The physical fluid dynamics – D. J. Tritton.
Chapter 3
Fluid Dynamics in Daily Life
Anywhere the water or any liquid is, fluid dynamics is there. In fact one of the
most beautiful phenomenon I have ever perceived is that of a liquid drop falling in a
reservoir of that liquid! The way it splits up into small droplets forming a crown like
shape is simply wonderful:
Photo 3.1: A falling drop of water [1].
Let's start with a cup of tea. You can see some particles floating on the surface of
your tea. Look at the surface in the reflected light of a tube. When you blow air on it to
cool the tea, you can see surface convection currents being set up. There are these
circulating loops on the surface.
The process of making tea also shows interesting fluid dynamics. Basically there
is convection in vertical loops. Hot water at the bottom is lighter and hence rises above
and it has to be replaced by the cool water above. From the position where the bubbles
form, you can specify the distribution of flames below it. From size of the bubbles and
the time required to leave the bottom, you can guess the temperature gradient on the
bottom. As the temperature goes high, things become very bubbly and you can't make
out what is happening. But now you turn of the gas and are very much eager to drink the
tea, but wait! If you start pouring the tea immediately, you will require a filter. If you
wait for some time, the tea powder will settle down completely and you can just tilt the
utensil to pour it without tea powder entering in your cup. The reason tea powder takes
some time to settle is that the bottom is still hotter than the surface and convection hasn't
stopped. When a roughly uniform temperature is attained throughout the container, no
convection occurs and the powder easily settles down.
Convection of similar kind occurs in atmosphere when clouds are formed. But it
is not necessary that whenever you heat the liquid from the bottom, a convective flow
would set up. It seems that it doesn't, when you heat, e.g. tomato soup. That's why it
starts making noise and would soon burn at the bottom if you don't stir it. For onset of
convection, the viscous forces are to be overcome by the rising fluid.
All of us have seen the various transitions the flow of tap water goes through, as
you increase the flow rate.
Initially, at very low speeds, the flow is laminar and
streamlined. But have you seen that it narrows downward? Why does it become narrow?
Microscopically, there are internal cohesive forces which keep the molecules bound
together. But as the water falls, it acquires extra velocity. So more liquids would flow
through a horizontal cross section. So, by continuity equation, the area of cross section
must decrease, as the flux of mass crossing each horizontal cross section must remain the
same. At higher flow rates, the flow changes to a very irregular, turbulent one. This
laminar to turbulent transition takes place in almost all fluid dynamical systems as some
parameter is varied.
Sometimes you also see, at low flow rates, that the flow breaks up into small
spherical droplets. The reason for this is the following: there may be some irregularities
at the end of the tap or anywhere in the flow, which introduce the sinusoidal disturbance
on the stable flow. Then the surface tension forces come into picture. They make the
convex surface more convex and the fluid nearby acquires a spherical shape [2]. Thus we
see an example where a small initial disturbance is enhanced to give rise to an instability
in the flow. It is also interesting to see how surface tension causes folds on a flow of milk
as it is poured from a tilted vessel.
As we have seen, gases are also fluids. Smoke rising from a cigarette is an
example of transition from laminar flow to turbulence. The hurricanes and typhoons and
tornadoes are examples of vortices. A vortex can be seen by making a hole at the bottom
of the tank or as water gushes out through basin. Here we see the significance of
Reynolds number: for the same Reynolds number, we get the vortices in air and in water.
But because the density and viscosity of air are so less than that of the water, the
velocities in case of air is very high. Also the distance scale is very large in case of air.
But for same Reynolds number, the patterns in flow are same.
You might have tried throwing some drops of water on a very hot frying pan.
They move all over the pan and evaporate in a dramatic fashion. Viewed on a minute
scale this show will be comparable to large vortices in air driving the dust all around or
large roaring waves in the middle of the sea. The evaporating drops form a cushion of air
at their base on which they float and then due to curvature of the pan they keep moving
throughout the pan.
Fluid dynamics is most closely related to life of aquatic animals [2,3]. How do the
fish swim? How do the microorganisms swim? These questions form an interesting
multidisciplinary study. Understanding of swimming of humans is useful in the world of
sports of swimming.
Just leave a paper up in the air and see how it falls. This age-old problem of a
deformable body moving through fluid is yet unsolved completely. It involves dynamic
interactions between fluid and the falling mass. Researchers have studied this
phenomenon by restricting the motion of falling body to two dimensions [4]. They have
photographed how a metallic rectangular disc falls between two parallel plates containing
a fluid. They have also visualized the vortices that are created while the disk falls. But
the equations are still too complicated to be solved completely. A similar mystery of
flapping of flags remains unsolved. I will describe recent experiments in this field in
chapter 5.
Try out this experiment sometime: Keep an egg floating in a glass of water [5].
You can do this either by taking the rotten egg or adding some salt to the water. Then
keep it under the tap water. (Or if you have used the salt water in the container, put it
under the flow of salt water.)
Figure 3.3: An egg in the flowing stream of water [5].
Then contrary to the expectation that the egg should be pressed downwards due to
the force of falling water, the egg rises above! The reason behind this strange behavior of
egg is the highly streamlined shape of egg. As a result, the force of the falling water on
the egg is far less than it would be, if it has, say, a flat shape. The water is simply
deflected aside by the egg, and its velocity is hardly reduced. However, when the falling
water reaches the water surface, its vertical velocity is suddenly reduced and it exerts a
large downward force that displaces the water underneath the egg and propels it upwards.
How does the soap solution help in removing the dirt in our clothes? One may think of
surface tension forces at first, but as it turns out, the surface tension actually decreases
with addition of soap. Actually, the soap solutions are not the solutions in true sense, the
particles of soap solution are not free to wonder around in the soap solution [6]. Instead,
soap is dispersed in the form of spherical clusters called "micelles", each of which may
contain hundreds of soap molecules.
Photo 3.2: “Micelle” of soap in water [6].
To see how this happens, we will have to look at a little bit to the chemistry of
soap molecules. The soap molecules are long chain organic compounds. One end of
these molecules is polar; it has e.g. a group like -COO-Na+, and the other one is a carbon
chain of 12 to 18 atoms. The polar end is water-soluble and is thus hydrophilic (meaning
water loving). The non-polar end is water insoluble and is hence hydrophobic (meaning
water hating). Molecules like this are called as amphipathic. They have both polar and
non-polar ends and are long enough to display both the properties. The dirt essentially
consists of fat and oils like grease. When we add oil to water, it tends to aggregate and
form bigger spheres. And these spheres are inert to water. So water can't drag the oil out.
When we add a soap solution, the non-polar part of it attaches itself to the oily part and
the polar molecule can stay freely in water.
So the soap molecules form a kind of layer on the oil drop and then the other end
of them provides a handle for the water molecules to carry away the oil spheres.
You must have seen heaps of grains at various places. A close observation shows
the angle at the vertex of these heaps or sand piles is always the same. There are grain
separators used in fields or industries. Since grains can flow, this topic comes under
study of fluid dynamics. But the behavior is drastically different, as you can see that the
continuum hypothesis doesn't hold. This granular flow is a hot topic of research today.
Why do we slip on a banana cover so easily? The banana cover has a thin coating
of a non-Newtonian fluid. These are the fluids, which show change in viscosity as the
velocity of neighboring layers increases.
As the velocity increases, the viscosity
decreases. The fluid becomes more and more slippery. So as the pressure of our feet
causes the layers of fluid move over each other, they move more and more easily. Mud is
another example of non-Newtonian fluid. The organic compounds with long molecules
show this kind of behavior. So study of such fluids is of importance to chemical
industries. Fortunately, all common liquids like water are Newtonian and hence we will
not consider the change of viscosity with velocity in this write-up.
When you through a stone on the surface of the lake water, for some angles it just
bounces up again and for other angles it just hits the surface and sinks [7]. What are those
special angles for these skipping stones? The ripples on the surface of the water are
another interesting things to observe. These surfaces are called as free surfaces, as there
is no solid boundary blocking the fluid there. The treatment of these free surfaces is
mathematically complicated, so we won't consider them in this project report.
Why do birds sometimes fly in a V shaped curve? This is because the first bird's
flapping creates a low-pressure zone behind it [8]. It is spread at an angle depending on
the speed of the birds. These bow waves are just like the waves created when a ship goes
at a speed greater than the speed of water waves in water or when a plane goes at a speed
greater than that of the sound in air. The low-pressure zone created behind the first bird
gives an easy flight to the rest of the birds. So they also fly in that very zone and in turn
make the flying easier for the rest of the birds. We many a times see a cloud of
mosquitoes swirling around in rapid motion below a street bulb, or sometimes on a
person's head. Why do they prefer such a cylindrically symmetric motion?
Even as you are reading this article, there is a lot of convection going on near
your face [2]. Because the ambient air is usually cooler than the human body, in absence
of other causes of appreciable motion, free convection boundary layers, similar to those
in a fluid heated at the bottom, are set up near our body.
Photo 3.1: Schlieren photograph of free convection from man's head [2].
In recent years it has been realized that this flow may be of considerable medical
significance. The concentration of airborne microorganisms is markedly higher in the
boundary layers than in the surrounding air. In particular, the body is constantly losing
skin scales (about 1010per day) through rubbing actions of limbs and clothes, and many of
these scales have microorganisms attached. It has been suggested that convection may
produce the observed connection between the skin diseases and the respiratory disease
(e.g. eczema and asthma), by transporting organisms from skin to nose. It may also
account for the increased respiratory infections after a fall in the temperature, which
would result in more vigorous convection.
References:
1. Photo taken from an Internet site for the book “A Drop of Water: A Book of Science
and Wonder” by Walter Wick.
2. “Physical Fluid Dynamics” – D. J. Tritton.
3. “Life at low Reynolds number”- E. M. Purcell Am. J. Phy. Vol.45, 3-11,1997.
4. “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.
5. “Why toast lands jelly-side down” – Robert Ehrlich, University Press, Physics.
6. “Organic chemistry” – Morrison and Boyd.
7. “Analyzing how the stone skipps” – C. L. Stong Sci. Am. August 1968, p112.
8. Prof. J.V. Narlikar’s explanation in “Surabhi” on DD1.