WEBVTT
00:25.689 --> 00:32.360
today i will talk about the fluid mechanics
in fluid mechanics let me start what does
00:32.360 --> 00:37.430
it mean my fluids and then i will go on explaining
each and every each and everything in the
00:37.430 --> 00:51.790
flow in the subject fluid mechanics is basically
it is just it is the study of fluids and they
00:51.790 --> 00:55.940
are and the forces on them
00:55.940 --> 01:05.040
it includes liquids gases plasmas fluid mechanics
can be divided into fluid kinematics which
01:05.040 --> 01:11.210
is the subject of fluid motion and the fluid
dynamics which is the study of the effect
01:11.210 --> 01:20.110
of forces on fluid motions it can further
be divided into fluid statics means this is
01:20.110 --> 01:28.330
the study of fluids at rest and fluid kinetics
which is the study of fluids in motion so
01:28.330 --> 01:36.860
in fact in fact in terms of mechanics problem
it is in fact it is the branch of continuum
01:36.860 --> 01:39.440
mechanics which is a sub sect
01:39.440 --> 01:46.990
it models matter without using the information
that it is made up of atoms rather it models
01:46.990 --> 01:54.479
macroscopic point of view rather than from
the microscopic point of view fluid mechanics
01:54.479 --> 01:59.700
is an active field of research with many salt
or unserved problems
01:59.700 --> 02:06.979
sometimes it becomes mathematically complex
so in those circumstances it can best be handled
02:06.979 --> 02:15.690
using some numerical methods typically using
computers nowadays a modern discipline in
02:15.690 --> 02:21.739
computer simulation studies become popular
which is devoted to solve the fluid mechanics
02:21.739 --> 02:29.459
problem which is known as computational fluid
dynamics cfd there are some experimental methods
02:29.459 --> 02:38.590
also for visualize and analyzing the fluid
flow which is known as particle image velocimetry
02:38.590 --> 02:42.980
by taking advantage of highly visual nature
of fluid flow
02:42.980 --> 02:51.640
historically fluid mechanics has a long story
study of fluid mechanics goes back to the
02:51.640 --> 03:00.510
days of ancient greece who in archimedes studies
fluid statics and buoyancy and formulated
03:00.510 --> 03:09.790
a theory which is known as archimedes principle
medieval persien and arab natural philosophers
03:09.790 --> 03:18.540
including abu rayhan al biruni and al khazini
combined that earlier work with dynamics to
03:18.540 --> 03:24.570
foretell precedes the later development of
fluid dynamics rapid development in fluid
03:24.570 --> 03:31.630
mechanics began with leonardo da vinci with
the observation as well as he has performed
03:31.630 --> 03:36.860
many simple experiment to demonstrate different
aspects of fluid mechanics
03:36.860 --> 03:44.740
torricelli who is who invents the barometer
isaac newton which invented the viscosity
03:44.740 --> 03:54.020
concept and pascal who deals the hydrostatic
spots and was fire finally continued to by
03:54.020 --> 04:02.121
bernoulli with the introduction of mathematical
fluid dynamics in hydrodynamica in 1738 there
04:02.121 --> 04:10.209
are basically two kinds of fluid flow we can
call one is viscous flow and another is non
04:10.209 --> 04:16.729
viscous flow non viscous flow is known as
the ideal flow and viscous flow is known as
04:16.729 --> 04:17.849
non-ideal flow
04:17.849 --> 04:25.979
so non viscous fluid flow was further analyzed
by various mathematicians just leonardo euler
04:25.979 --> 04:33.780
d’ alembert lagranges laplace poisons etcetera
and viscous flow was explored by multitude
04:33.780 --> 04:41.800
of engineers including portcullis and ludwig
hagen further mathematical justification was
04:41.800 --> 04:50.050
provided by navier and stroke which is known
as famous navier-stokes equations and we will
04:50.050 --> 04:51.750
talk about on latter
04:51.750 --> 05:01.410
and boundary layers who had investigated ludwig
prantle while various scientists renauld kolmogarov
05:01.410 --> 05:09.100
taylor they advanced the understanding of
fluid viscosity and turbulences
05:09.100 --> 05:14.580
fluid mechanics basically it is a sub discipline
of continuum mechanics as illustrated in the
05:14.580 --> 05:20.500
following table basically it is the continuum
mechanics which is the study of the physics
05:20.500 --> 05:26.729
of continuous materials it can further divide
it into two things one is known as mechanics
05:26.729 --> 05:33.870
of rigid body which tells you the study of
physics of continuous materials with a defined
05:33.870 --> 05:41.190
rest shape second one is the fluid mechanics
it is the study of physics of continuous materials
05:41.190 --> 05:45.490
which take the shape of their container
05:45.490 --> 05:52.870
so again rigid body dynamics can be divided
can be thought of in two ways one is elasticity
05:52.870 --> 06:00.650
which describes the material that return to
their rest shape after an applied stress second
06:00.650 --> 06:09.740
one is plasticity which describe materials
and that permanently deform after a sufficient
06:09.740 --> 06:18.350
applied stress fluid dynamics again can be
divided into two fluids one is known as non-newtonian
06:18.350 --> 06:21.190
fluids another is known as newtonian fluids
06:21.190 --> 06:27.650
however we will discuss these parts non-newtonian
fluids and newtonian fluids in the latter
06:27.650 --> 06:30.520
part of my talks
06:30.520 --> 06:36.190
as you know like any mathematical model of
the real world you need some assumptions on
06:36.190 --> 06:44.880
which your model relies on so same way fluid
mechanics make some basic assumptions about
06:44.880 --> 06:52.280
the materials being studied these assumptions
are turn to into equations that must be satisfied
06:52.280 --> 06:58.910
if the assumptions are to be held true for
example consider an incompressible fluid in
06:58.910 --> 07:00.820
three dimensions ok
07:00.820 --> 07:07.400
incompressible means where the density does
not change this is known as incompressible
07:07.400 --> 07:14.050
fluid though the assumptions that mass is
conserved means that for any fixed flow surface
07:14.050 --> 07:21.180
such as a sphere the rate of mass passing
from outside to inside the surface must be
07:21.180 --> 07:29.020
the same as rate of mass passing the other
way that means mass inside remains constant
07:29.020 --> 07:37.250
as does the mass outside which can be turned
into an integral equation over the surface
07:37.250 --> 07:43.240
in fact if we will summarize the assumptions
fluid mechanics assumes that every fluid obeys
07:43.240 --> 07:50.930
the following rules conservation of mass conservation
of energy conservation of momentum and the
07:50.930 --> 07:52.360
continuum hypothesis
07:52.360 --> 08:00.630
further it is often useful at subsonic conditions
to assume a fluid is incompressible that means
08:00.630 --> 08:06.949
the density of the fluid does not change liquid
can often be modeled as incompressible fluid
08:06.949 --> 08:14.460
whereas gases cannot similarly it can sometimes
we assumed that the viscosity of the fluid
08:14.460 --> 08:21.930
is 0 that means flow is is known to be non
viscous some gases can often be assumed to
08:21.930 --> 08:28.910
be non viscous if the fluid is viscous and
its flow content in some way example in a
08:28.910 --> 08:32.760
pipe then the flow at the boundary must have
zero velocity
08:32.760 --> 08:40.079
for a viscous fluid if the boundary is not
porous the shear forces between the fluid
08:40.079 --> 08:47.740
and the boundary results also in zero velocity
for the fluid at the boundary this is called
08:47.740 --> 08:54.980
the no slip condition for a porous media otherwise
in the frontier of the containing vessel the
08:54.980 --> 09:02.130
slip condition is not zero velocity and the
fluid has a discontinuous velocity field between
09:02.130 --> 09:06.750
the free fluid and the fluid in the porous
media
09:06.750 --> 09:11.910
now what does it mean by continuum hypothesis
we need to understand very well depending
09:11.910 --> 09:17.510
on that whether the fluid can be dealt in
the conventional fluid mechanics or using
09:17.510 --> 09:22.870
this statistical mechanics so let me try to
understand what does it mean by continuum
09:22.870 --> 09:25.690
hypothesis in the fluid mechanics
09:25.690 --> 09:31.950
fluids are composed of molecules that collide
with one another and solid objects that means
09:31.950 --> 09:38.940
container of the fluid the continuum assumptions
however considers fluid to be continuous that
09:38.940 --> 09:47.330
is properties such as density pressure temperature
and velocity are taken to be well defined
09:47.330 --> 09:54.870
at infinitely small point known as the fluid
element at the geometric order of the distance
09:54.870 --> 09:57.500
between two adjacent molecules of the fluid
09:57.500 --> 10:06.060
properties are assumed to vary continuously
from one point to another and are averaged
10:06.060 --> 10:13.190
over in a fluid element that means averaged
over in delta v the fact that that fluid is
10:13.190 --> 10:21.510
made up of discrete molecule is ignored the
continuum hypothesis it is basically an approximation
10:21.510 --> 10:28.470
in the analogy that the planets are approximated
by point masses point center of a point particle
10:28.470 --> 10:36.029
where when dealing with celestial mechanics
and therefore results in an approximate solutions
10:36.029 --> 10:42.330
consequently some source of the continuum
hypothesis can lead to results which are of
10:42.330 --> 10:49.490
not desired accuracy however under these under
the right circumstances the continuum hypothesis
10:49.490 --> 10:56.350
produces extremely accurate tracer now in
a couple of minutes we will try to understand
10:56.350 --> 11:03.310
at what circumstances continuum hypothesis
hold good at what circumstances it does not
11:03.310 --> 11:05.890
hold good the continuum hypothesis
11:05.890 --> 11:11.790
in that case we need to resort to the more
fundamental theory which is known as statistical
11:11.790 --> 11:14.130
mechanics
11:14.130 --> 11:20.899
these problems for which the continuum hypothesis
does not allow solution of desired accuracy
11:20.899 --> 11:28.230
are solved using statistical mechanics to
determine whether or not to use conventional
11:28.230 --> 11:34.709
fluid dynamics or statistical mechanics we
need to evaluate some number for the given
11:34.709 --> 11:43.680
problem that number is known as knudsen number
the knudsen number which is denoted as kn
11:43.680 --> 11:49.510
is a dimensionless number defined as the ratio
of the molecular mean free path
11:49.510 --> 11:55.920
to a representative physical length scale
for an example this length scale could be
11:55.920 --> 12:03.470
the radius of a body in a fluid this number
is named after danish physicist martin knudsen
12:03.470 --> 12:10.180
now let us try to understand how to calculate
the knudsen number and from the knudsen number
12:10.180 --> 12:17.470
how to understand that when i allow when i
should apply the conventional fluid mechanics
12:17.470 --> 12:24.350
when i should apply the statistical mechanics
to understand the fluid dynamics
12:24.350 --> 12:32.310
so the knudsen number is a dimensional number
which is defined as kn =to lambda upon l where
12:32.310 --> 12:39.040
lambda is the molecular mean free path and
l is the representative physical length scale
12:39.040 --> 12:45.090
so that justifies that it is a dimensionless
number
12:45.090 --> 12:53.959
for an ideal gas the mean free path as you
know can be calculated easily where kn = kbt
12:53.959 --> 13:04.410
by square root pi sigma square pl where kb
is the boltzmann constant t is the temperature
13:04.410 --> 13:11.760
of the system sigma is the particle hertz
cell diameter and p is the total pressure
13:11.760 --> 13:17.980
for for an example for particle dynamics in
the atmosphere and assuming standard temperature
13:17.980 --> 13:23.019
and pressure that is 25 degree centigrade
and one atmosphere pressure
13:23.019 --> 13:30.200
we can calculate the value of the mean free
path which is of which comes about 8 into
13:30.200 --> 13:37.589
10 to the power - 8 meter or approximately
26 into 10 to the -9 feet mean free path that
13:37.589 --> 13:45.570
means this is the distance during which particle
you will not collide to each other okay now
13:45.570 --> 13:52.410
let us calculate the let us try to get the
relationship between mac and reynold number
13:52.410 --> 13:58.130
in the gaseous in the fluids itself because
it is a very interesting relation who is tells
13:58.130 --> 14:03.920
you under which circumstances you have to
deal the fluid dynamics originally conventional
14:03.920 --> 14:05.360
fluid mechanics
14:05.360 --> 14:10.450
and under what circumstances you have to deal
with the statistical mechanics let me try
14:10.450 --> 14:14.370
to understand what is the relations between
them
14:14.370 --> 14:20.240
the knudsen number can be related to the mach
number and the reynolds number which you are
14:20.240 --> 14:29.740
going to calculate now dynamic viscosity is
defined as mu is half rho c bar lambda where
14:29.740 --> 14:35.110
c bar is nothing but the average molecular
speed which can be obtained from the maxwell
14:35.110 --> 14:40.620
boltzmann distribution which i have already
talked in my earlier talks where c bar is
14:40.620 --> 14:44.380
defined as root over 8 kbt by pi m
14:44.380 --> 14:54.930
thus the mean free path lambda is mu by rho
mu upon rho root over pi m by 2 kbt so that
14:54.930 --> 15:00.300
means mean free path can be defined in terms
of the dynamic viscosity which is different
15:00.300 --> 15:06.550
by the relations that mu upon rho root over
pi m by 2 kbt
15:06.550 --> 15:14.579
dividing through l the knudsen number is defined
as lambda upon l which is nothing but mu upon
15:14.579 --> 15:21.930
rho root over pi m by 2 kbt where c bar is
the average molecular speed which i have already
15:21.930 --> 15:28.019
told it can be obtained from the maxwell boltzmann
distribution t is the thermodynamic temperature
15:28.019 --> 15:35.140
if the system is maintained at thermal equilibrium
then in that case t is known as the temperature
15:35.140 --> 15:42.370
mu is the dynamic viscosity m is the molecular
mass kb is the boltzmann constant and rho
15:42.370 --> 15:52.610
is the density of the fluid before going to
the relations between the knudsen number and
15:52.610 --> 15:57.209
mach number etcetera let us try to understand
what does it mean by mach number
15:57.209 --> 16:04.389
mach number is the speed of an object moving
through air or any other fluid substance divided
16:04.389 --> 16:11.620
by the speed of sound as it is as it is in
that substance for its particular physical
16:11.620 --> 16:17.389
conditions including those of temperature
and pressure it is commonly used to represent
16:17.389 --> 16:24.890
the speed of an object when it is traveling
close to or above the speed of sound so mach
16:24.890 --> 16:29.520
number is defined as the ratio u infinity
upon cs
16:29.520 --> 16:36.510
where cs is the speed of sound which is defined
as root over gamma rt upon m or root over
16:36.510 --> 16:42.540
in terms of boltzmanns constant root over
gamma kv upon t where gamma is the ratio of
16:42.540 --> 16:48.740
specific heat cp upon cv this relation we
have already derived in my earlier lectures
16:48.740 --> 16:55.940
in the kinetic theory of gases and u infinity
is the free stream speed and r is the universal
16:55.940 --> 17:03.390
gas constant m is the molar mass and as usually
gamma is the sphere ii ratio of specific heats
17:03.390 --> 17:06.169
cp upon cv
17:06.169 --> 17:13.659
second one which is more interesting in the
fluid mechanics many people have already heard
17:13.659 --> 17:17.669
this name which is known as the reynold number
17:17.669 --> 17:23.980
the dimensionless reynold number is a dimensionless
number that gives a measure of the ratio of
17:23.980 --> 17:30.509
the internal forces to the viscous forces
and consequently quantifies the relative importance
17:30.509 --> 17:37.710
of the two type of forces for given flow condition
the concept was first introduced by gabrielle
17:37.710 --> 17:48.590
stokes in 1851 but the reynold number is named
after reynolds who popularized and it is used
17:48.590 --> 17:56.309
in 1883 it is defined as re = rho into u infinity
l upon mu
17:56.309 --> 18:04.239
dividing the mach number by the reynolds number
you can get the ratio mu upon rho l root over
18:04.239 --> 18:14.779
m upon gamma kbt into gamma pi upon 2 which
is nothing but mu by rho l root over pi m
18:14.779 --> 18:21.640
by 2 kbt and if you will multiply this equation
by root over gamma pi by 2 you will get ma
18:21.640 --> 18:31.299
by re = u infinity by cs and rho u infinity
l upon mu which is nothing but mu upon rho
18:31.299 --> 18:38.769
l cs and if you will substitute the values
of cs you will get mu upon rho l into gamma
18:38.769 --> 18:48.669
kbt by m which turns out to be mu by rho l
into root over m gamma kbtand the knudsen
18:48.669 --> 18:50.529
number is obtained
18:50.529 --> 18:56.690
so finally you can get the relation of mach
raynald and knudsen number are related as
18:56.690 --> 19:07.570
kn = ma by re into gamma pi by 2 the knudsen
number is useful for determining whether statistical
19:07.570 --> 19:13.539
mechanics or continuum mechanics formulation
of fluid dynamics should be used for an example
19:13.539 --> 19:21.010
if the knudsen number is near or greater than
1 the mean free path of a molecule is comparable
19:21.010 --> 19:23.979
to the length scale of the problem
19:23.979 --> 19:30.360
and the continuum assumption of fluid mechanics
is no longer a good approximation in that
19:30.360 --> 19:38.549
case statistical mechanics must be used similarly
other way around if it is less than 1 then
19:38.549 --> 19:46.799
continuum assumptions of fluid mechanics will
be valid approximation in that case the usual
19:46.799 --> 19:54.950
fluid mechanics formulation is good enough
to calculate any quantities in a fluid dynamics
19:54.950 --> 20:00.129
problem for an example problems with high
knudsen number include the calculation of
20:00.129 --> 20:03.970
motions of a dust particle through the lower
atmosphere
20:03.970 --> 20:10.860
or the motion of a satellite through the exosphere
one of the most widely used application of
20:10.860 --> 20:18.019
the knudsen real number is in micro fluids
and mems device design the solution of the
20:18.019 --> 20:25.019
fluid around an aircraft has a low knudsen
number making it firmly in the realm of continuum
20:25.019 --> 20:33.139
mechanics however in that case in a aircraft
it goes from low knudsen number to high knudsen
20:33.139 --> 20:34.139
number
20:34.139 --> 20:42.590
so that makes the complicated nature of the
fluid flow across an aircraft so now let me
20:42.590 --> 20:50.769
talk in a small summary what does it mean
by navier stokes theorem which is one of the
20:50.769 --> 20:54.129
most beautiful equation in fluid dynamics
20:54.129 --> 21:01.009
the navier-stokes equation named after claudia
lewis navier and george gabriel stokes are
21:01.009 --> 21:09.409
the set of equations that describe the motion
of fluid substances such as liquids and gases
21:09.409 --> 21:17.309
how are this equation is in the non relativistic
fluid equation these equations state that
21:17.309 --> 21:24.499
change in momentum of fluid particles depend
only on the external pressure and internal
21:24.499 --> 21:28.450
viscous forces similar to friction acting
on the fluid
21:28.450 --> 21:36.429
thus the navier stokes equation described
the balance of forces acting any given region
21:36.429 --> 21:42.710
of the fluid the navier stokes theorem are
differential equation who is describe the
21:42.710 --> 21:49.539
motion of a fluid such equation established
relation among the rates of change of the
21:49.539 --> 21:52.340
variable of interest
21:52.340 --> 21:58.599
for example the navier stokes equation for
an ideal fluid with 0 viscosity states that
21:58.599 --> 22:04.729
that acceleration is proportional to the derivative
of the internal pressure okay in case of ideal
22:04.729 --> 22:11.090
fluid that means there is no viscosity okay
this means that the solution of navier-stokes
22:11.090 --> 22:17.239
equation for a given physical problem must
be sort with the help of calculus in practical
22:17.239 --> 22:22.070
terms only the simplest cases can be solved
exactly in this way
22:22.070 --> 22:28.659
these cases generally involved non turbulent
steady flow flow does not change with time
22:28.659 --> 22:36.600
in which the number is small as we know that
if the reynolds number is very small then
22:36.600 --> 22:44.000
the flow is the laminar flow non turbulent
flow if the reynold number become large if
22:44.000 --> 22:49.479
the velocity will become too large in that
case reynold numbers will be very large in
22:49.479 --> 22:55.480
that case the flow will no more be the laminar
flow it will be the turbulent
22:55.480 --> 23:03.099
for more complex situation such as global
weather systems like el nino problem or lift
23:03.099 --> 23:09.070
in a wing solution of the navier stokes equation
can currently only be found with the help
23:09.070 --> 23:18.220
of computers this is the field of sciences
by its own called computational fluid dynamics
23:18.220 --> 23:25.139
general let us try to we are not going to
derive the navier stokes equation let us try
23:25.139 --> 23:30.690
to understand the different terms in navier-stokes
equation what does it tell
23:30.690 --> 23:38.590
easily the general form of navier stokes equation
for the conservation of momentum is rho dv
23:38.590 --> 23:47.820
by dt = divergence of p + rho time save where
rho is the fluid density and d upon d by dt
23:47.820 --> 23:54.509
is the substance of substantive derivative
also called the material derivative that means
23:54.509 --> 24:00.700
this is the rate of change of time in the
fluid frame itself and v is the velocity vector
24:00.700 --> 24:08.320
f is the body force vector and p is the tensor
that represents the surface forces applied
24:08.320 --> 24:09.869
on fluid particle
24:09.869 --> 24:16.969
which is the which is the stress tensor we
are going to give its form now unless the
24:16.969 --> 24:23.389
fluid is made up of spining degrees of spinning
degrees of freedom like vertices p is a symmetric
24:23.389 --> 24:24.389
tensor
24:24.389 --> 24:34.360
in that case p has the form of 3 cross 3 matrices
where each element sigma xx tau xy tau xz
24:34.360 --> 24:42.159
similarly as you can see its form where sigma
are the normal stresses and tau are the tangential
24:42.159 --> 24:50.389
stresses or shear stresses the above equation
is usually a set of three equations one per
24:50.389 --> 24:55.750
dimension if you will open up it for three
dimension so it gives the three differential
24:55.750 --> 24:58.450
equation one part each dimension
24:58.450 --> 25:04.460
by themselves they are not sufficient to produce
a solution however adding conservation of
25:04.460 --> 25:10.769
mass and appropriate boundary conditions to
the system of equation produces a solvable
25:10.769 --> 25:17.669
sets of equation so naevia stokes theorem
itself is not a solvable with the help of
25:17.669 --> 25:23.169
other conservation equation conservation of
mass and appropriate boundary condition then
25:23.169 --> 25:30.100
in that case it leads to a solvable sets of
equations however we are not going to discuss
25:30.100 --> 25:32.249
it now
25:32.249 --> 25:39.179
now let us take as we told you that fluid
mechanics fluids can be divided in terms of
25:39.179 --> 25:44.719
the newtonian fluid and non newtonian fluid
let us try to understand what is newtonian
25:44.719 --> 25:47.289
fluid and what is non-newtonian fluids
25:47.289 --> 25:54.269
in newtonian fluid named after isaac newton
is defined to the fluid whose shear stress
25:54.269 --> 26:00.159
is linearly proportional to the velocity gradient
in the direction perpendicular to the plane
26:00.159 --> 26:07.629
of shear so this definition means regardless
of the forces acting on a fluid it continues
26:07.629 --> 26:14.989
to flow for example water is a newtonian fluid
because it continues to display fluid properties
26:14.989 --> 26:18.299
no matter how much it is stirred or mixed
26:18.299 --> 26:25.629
a slightly less rigorous definition is that
the drag of a small object being moved slowly
26:25.629 --> 26:32.009
through the fluid is proportional to the force
applied to the object important fluids like
26:32.009 --> 26:38.730
water as well as most gases behave up to a
good approximation as in newtonian fluid under
26:38.730 --> 26:41.600
normal conditions on the earth
26:41.600 --> 26:49.840
by contrast starring a non-newtonian fluid
can leave a hole behind this will gradually
26:49.840 --> 26:58.599
fill up over time these behavior is seen in
materials such as bodying only wake or sand
26:58.599 --> 27:06.090
all those sand is not a is not strictly a
fluid alternatively starring a non-newtonian
27:06.090 --> 27:11.830
fluid can cause the viscosity to decrease
so the fluid appears thinner this is seen
27:11.830 --> 27:14.739
in non drip paints
27:14.739 --> 27:21.200
there are many types of non-newtonian fluids
as they are defined to be something that fails
27:21.200 --> 27:28.210
to obey a particular propertyfor example most
fluid with long molecular chains can react
27:28.210 --> 27:31.879
in a non-newtonian manner
27:31.879 --> 27:39.340
so now let us define what is newtonian fluid
in a mathematical form the constant of proportionality
27:39.340 --> 27:44.990
between the shear stress and the velocity
gradient as we told that if the relation is
27:44.990 --> 27:52.739
linear then it is known as the newtonian fluid
so if the proportionality constant is viscosity
27:52.739 --> 28:02.229
in that case a simple relation to describe
to define the newtonian fluid is tau = - mu
28:02.229 --> 28:09.219
dv by dy where tau is the shear stress exerted
by the fluid or drag force
28:09.219 --> 28:16.269
and mu is the fluid viscosity which is a constant
of proportionality and dv by dy is the velocity
28:16.269 --> 28:22.879
gradient perpendicular to the direction of
this shear for a newtonian fluid the viscosity
28:22.879 --> 28:28.809
by definition depends only on temperature
and pressure not on the forces acting upon
28:28.809 --> 28:29.940
it
28:29.940 --> 28:37.649
if the fluid is incompressible and viscosity
is constant across the fluid the equation
28:37.649 --> 28:46.629
governing the shear stress in cartesian coordinates
is tau ij = mu since it is a constant del
28:46.629 --> 28:55.200
vi by del xi + del vj by del xi where tau
ij is the shear stress on the ij ith phase
28:55.200 --> 29:01.769
of a fluid element in the jth direction i
stands for phase of the fluid j stands for
29:01.769 --> 29:09.190
the direction of the fluid and vi is the velocity
of the ith direction xj is the jth direction
29:09.190 --> 29:10.190
coordinate
29:10.190 --> 29:17.340
if a fluid does not obey this relation it
is termed as non-newtonian fluid of which
29:17.340 --> 29:25.479
there are several types among fluids to rough
broad divisions can be made ideal non-ideal
29:25.479 --> 29:32.830
fluids and ideal fluid really does not exist
but in some calculations the assumptions is
29:32.830 --> 29:40.899
made such a way so that it will behave like
an ideal fluid and ideal that means an ideal
29:40.899 --> 29:47.649
fluid is a non viscous which offers no resistance
whatsoever to a shearing force
29:47.649 --> 29:54.259
one can group real fluid into newtonian and
non-newtonian newtonian fluids agree with
29:54.259 --> 30:01.630
newtons law of viscosity non-newtonian fluids
can be either plastic dilettante etcetera
30:01.630 --> 30:11.719
now let us try to understand from the what
does it mean by fluids liquid gases from a
30:11.719 --> 30:16.779
very grassroot level now let us try to understand
30:16.779 --> 30:24.649
so hydrostatics deals with the mechanics of
fluids in equilibrium and our first step therefore
30:24.649 --> 30:32.889
is to understand clearly as to what exactly
do we mean by a fluid unlike a solid in which
30:32.889 --> 30:38.830
the strain set up under a shearing stress
lasts throughout the period of application
30:38.830 --> 30:46.960
of this stress whereas if fluid may be defined
as that state of matter which cannot indefinitely
30:46.960 --> 30:51.969
or permanently oppose or register shearing
stress
30:51.969 --> 31:00.629
that is the basic difference between solid
and in fact it constantly and continuously
31:00.629 --> 31:07.489
yields to it though the yield may be rapid
in some cases and slow in others
31:07.489 --> 31:14.470
in the former case the liquid is said to be
mobile like water alcohol etcetera in the
31:14.470 --> 31:21.809
latter viscous like honey trickle etcetera
in either case however a fluid has no definite
31:21.809 --> 31:29.499
shape of its own and assumes ultimately the
shape of the containing vessel and yet with
31:29.499 --> 31:35.739
all this seemingly clear cut distinctions
between a solid and a fluid it is not quite
31:35.739 --> 31:42.139
so easy to distinguish between the two in
many a borderline cases
31:42.139 --> 31:47.899
there are many borderline cases where you
cannot exactly distinguish which one is solid
31:47.899 --> 31:48.950
and which one is fluid
31:48.950 --> 31:59.639
the fundamental distinctions between two nevertheless
remains and we declare a substance to be a
31:59.639 --> 32:06.499
fluid or a solid according as the does or
does not yield to a shearing stress applied
32:06.499 --> 32:16.369
to it over a long enough period fluids two
are further divided into two classes liquids
32:16.369 --> 32:24.119
and gases a liquid is a fluid which although
it has no safe of its own occupies a definite
32:24.119 --> 32:30.209
volume which cannot be altered however great
the force applied to it
32:30.209 --> 32:38.269
a liquid is a fluid which is quite incompressible
and has a free surface of its own as for example
32:38.269 --> 32:47.789
water alcohol ether honey trickling etcetera
a gas on the other hand is a fluid which cannot
32:47.789 --> 32:54.330
only be easily compressed when subject to
pressure but with a progressive reduction
32:54.330 --> 33:01.389
of the pressure on it it can also be made
to expand indefinitely occupying all the space
33:01.389 --> 33:06.989
made available to it thus the whole of the
gas will escape out from a vessel if they
33:06.989 --> 33:11.739
are the tiniest efforts are in it somewhere
33:11.739 --> 33:19.899
if you will summarize these distinctions between
fluid solid and within a fluid liquids and
33:19.899 --> 33:28.169
gases a gas if you will summarize all these
concept together then a gas is a fluid which
33:28.169 --> 33:36.479
has neither shape nor a free surface of its
own example oxygen hydrogen carbon dioxide
33:36.479 --> 33:45.649
air which is a mixture of gases etcetera but
liquid is a fluid which is quite incompressible
33:45.649 --> 33:51.119
and has a free surface of its own unlike the
gases
33:51.119 --> 33:58.940
now let us try to understand how a rate of
flow of a liquid can be estimated fluids as
33:58.940 --> 34:05.240
you know include both liquids and gases they
are main characteristics being that they cannot
34:05.240 --> 34:12.820
permanently withstand any searing stresses
however it is small we shall concern ourselves
34:12.820 --> 34:20.810
here only with ideal liquids or gases that
is with liquids which are perfectly mobile
34:20.810 --> 34:28.859
that means there is no viscosity that means
it is ideal and incompressible means 0 compressibility
34:28.859 --> 34:32.480
or infinite bulk modulus
34:32.480 --> 34:41.010
and with gases which perfectly obey the boyle’s
or charles law so now let us calculate the
34:41.010 --> 34:47.960
rate of flow so the rate of flow of a fluid
is defined as the volume of it that flows
34:47.960 --> 34:56.919
across any section of a pipe in unit time
that means volume of it that flows across
34:56.919 --> 35:04.589
any section of the power per unit it is really
the volume rate of flow of the fluid or its
35:04.589 --> 35:09.520
discharge usually represented by the symbol
q or v
35:09.520 --> 35:17.269
now let us see this picture of a pipe considering
the fluid to be incompressible if it is velocity
35:17.269 --> 35:25.730
of flow v be in a direction perpendicular
the two sections a and b as shown in the figure
35:25.730 --> 35:34.920
1 of area a and distance l apart and if t
is the time taken by the liquid to flow from
35:34.920 --> 35:41.029
a to b we have vt should be the length of
this tube which is l
35:41.029 --> 35:47.599
so obviously the volume of the liquid flowing
through the section ab in this time is equal
35:47.599 --> 35:55.470
to the cylindrical column ab which is nothing
but the length time area l times a l is nothing
35:55.470 --> 36:04.200
but the vt so vt times a this is therefore
the volume of the liquid flowing across the
36:04.200 --> 36:11.390
section in time t so which is nothing but
the vt times a so the volume rate of flow
36:11.390 --> 36:21.430
of liquid into discharge q or v which is nothing
but the vt times a by t tt will cancel which
36:21.430 --> 36:27.960
is nothing but the a times v which is nothing
but the velocity of the liquid v and a is
36:27.960 --> 36:34.420
the area of the cross section of the tube
if you will measure in a pear system this
36:34.420 --> 36:36.710
is known as cubic feet per second
36:36.710 --> 36:45.109
this unit is cusec which is known as cusec
which is nothing but the cubic feet per second
36:45.109 --> 36:50.730
sometimes the rate of flow of a liquid is
also expressed in terms of the mass of the
36:50.730 --> 36:57.690
liquid flowing across any section in unit
time and is referred to as mass rate of flow
36:57.690 --> 37:03.380
when you dis described in terms of mass so
it is known as mass rate of flow
37:03.380 --> 37:09.690
thus the mass rate of flow of liquid equal
to mass of liquid flowing across any section
37:09.690 --> 37:17.799
per unit time which is nothing but the velocity
of the liquid into area of cross section into
37:17.799 --> 37:24.529
density of the liquid to get in terms of mass
which is nothing but the v times a times rho
37:24.529 --> 37:31.010
v is the velocity of the liquid a is the area
of cross section and rho is the density of
37:31.010 --> 37:34.200
liquid
37:34.200 --> 37:41.829
now let us try to understand the lines and
tubes of flow which are very similar to the
37:41.829 --> 37:49.720
lines of force in electrostatic and magnetic
fields so it is very easy to understand the
37:49.720 --> 37:54.190
lines of lines and tubes of flow in fluid
mechanics
37:54.190 --> 38:01.210
in a simple flow of liquid that is when it
is slow and steady the velocity at every point
38:01.210 --> 38:08.099
in the fluid remains constant in magnitude
as well as in detail this is the most important
38:08.099 --> 38:16.369
concept in the fluid mechanics so let me repeat
it again when the velocity of the liquid is
38:16.369 --> 38:24.240
very slow and steady in that case the velocity
at every point throughout the fluid remains
38:24.240 --> 38:28.640
constant in magnitude as well as indeed x
38:28.640 --> 38:36.931
the energy needed to drive it used up in overcoming
the viscous drag between its layers in other
38:36.931 --> 38:46.410
words each particle in the fluid follows exactly
the same path and has the same velocity as
38:46.410 --> 38:54.420
its predecessor and the fluid is said to have
an orderly or streamline ok this is the meaning
38:54.420 --> 39:00.980
of streamline there are two kinds of flow
one is streamline flow another is turbulent
39:00.980 --> 39:09.950
in streamline flow if every particle in the
fluid egg follows exactly the same path and
39:09.950 --> 39:16.789
has the same velocity as its predecessor in
that case that flow is known as streamline
39:16.789 --> 39:17.789
flow
39:17.789 --> 39:23.849
deviation of this streamline flow is known
as turbulent flow which is always very tough
39:23.849 --> 39:31.539
to handle in such a case if we consider a
line in the streamline case if you will consider
39:31.539 --> 39:39.250
a line along which the particles of the liquid
move the direction of the line at any point
39:39.250 --> 39:47.559
is the direction of the velocity of the fluid
at that point such a line is called a streamline
39:47.559 --> 39:55.589
more correctly a streamline may be defined
as a curve the tangent to which at any point
39:55.589 --> 40:00.450
gives the direction of flow of the fluid at
that particular point
40:00.450 --> 40:09.640
it may be straight curved according as the
latter lateral pressure on it is the same
40:09.640 --> 40:16.599
throughout or different depending on the lateral
pressure whether it is same throughout the
40:16.599 --> 40:24.130
fluid or different the size shape of this
streamlined it could be straight or curved
40:24.130 --> 40:29.760
in the latter case when d the lateral pressure
is different the pressure being greater on
40:29.760 --> 40:39.140
the convex side than on the concave one no
two streamline can ever cross one another
40:39.140 --> 40:45.880
so if we consider let us try to understand
this thing in a given example in given figure
40:45.880 --> 40:53.059
figure 2 if you will consider two areas a
and b at right angles to the direction of
40:53.059 --> 41:00.650
flow of the fluid just you see this figure
2 and draw streamlines through their boundaries
41:00.650 --> 41:01.650
as soon
41:01.650 --> 41:09.349
we obtain a tubular space ab bounded by a
surface containing streamlines called a stream
41:09.349 --> 41:16.170
tube or a tube of flow so this is the definition
of tube of flow first we have understood what
41:16.170 --> 41:25.260
is streamline and using this streamline we
can construct a tube we can construct a stream
41:25.260 --> 41:33.269
tube or a tube of flow using two streamlines
the sides of the tube of flow being everywhere
41:33.269 --> 41:40.210
in the direction of the fluid flow no fluid
can cross the sides as though they are rigid
41:40.210 --> 41:47.930
and must enter or leave through the ends in
other words there is no intermingling of the
41:47.930 --> 41:54.000
fluid in the two adjacent tubes of flow across
the imaginary walls
41:54.000 --> 42:00.720
this holds good however only so long as the
velocity of the liquid does not exceed a particular
42:00.720 --> 42:08.809
limiting value called its critical velocity
beyond who is the flow of liquid loses all
42:08.809 --> 42:16.950
its steadiness and orderliness and becomes
zigzag or sinus acquiring what is called as
42:16.950 --> 42:23.890
turbulent motion so whatever descriptions
i have given those descriptions is only valid
42:23.890 --> 42:30.519
when the velocity of the liquid does not exceed
a particularly limiting value known as the
42:30.519 --> 42:31.700
critical
42:31.700 --> 42:37.810
this thing can also be understood in terms
of the reynold number if the critical velocity
42:37.810 --> 42:43.990
will be too large the reynold number will
be too greater than one in that cases the
42:43.990 --> 42:52.430
these concepts will no longer valid this may
be easily seen by introducing is a small jet
42:52.430 --> 42:59.180
of a coloring matter let us try to do this
experiment also in this figure like ordinary
42:59.180 --> 43:05.510
ink for example axially into a tube ab you
just see this figure ab
43:05.510 --> 43:12.819
through which water may be made to flow with
gradually increasing velocity by raising the
43:12.819 --> 43:18.869
water head responsible for the flow it will
be seen that as long as the velocity of water
43:18.869 --> 43:26.260
flow remains below its critical value there
is only a thin streak of the coloring matter
43:26.260 --> 43:35.619
along the axis of the tube as you can see
in figure 3 signifying a steady stream line
43:35.619 --> 43:36.619
motion
43:36.619 --> 43:45.210
but as soon as the velocity attains its critical
value the coloring matter takes a zigzag path
43:45.210 --> 43:53.339
which is shown in figure 3 indicating haphazard
change of velocity from point to point and
43:53.339 --> 44:00.900
consequent distortion of the tubes of flow
and later when this value is exceeded far
44:00.900 --> 44:07.660
apart the coloring matter spreads out in all
directions feeling the entire tube showing
44:07.660 --> 44:14.680
that the tubes of flow have now broken down
and the motion has become utterly disorderly
44:14.680 --> 44:16.800
or turbulent
44:16.800 --> 44:24.599
so this experiment even you can perform me
in your home also in your small lab also that
44:24.599 --> 44:32.400
that shows you the streamline flow and the
turbulent flow raynald all actually use this
44:32.400 --> 44:40.859
very method to determine the critical velocity
for water obviously the velocity of the water
44:40.859 --> 44:47.299
just before the thin streak of the coloring
matter becomes zigzak or a sinus directly
44:47.299 --> 44:53.690
gives the value of the critical velocity vcc
for it and is equal to the rate of flow of
44:53.690 --> 44:54.690
the liquid
44:54.690 --> 45:00.961
that is the volume of the liquid flowing out
of the tube per second divided by the area
45:00.961 --> 45:07.779
of the cross section of the tube so from there
renault calculated the critical velocity from
45:07.779 --> 45:14.860
there he calculated the reynolds number if
the velocity is less than it it will be the
45:14.860 --> 45:20.430
stream line flow if the velocity will be more
the critical velocity it will be the turbulent