1
00:00:16,500 --> 00:00:21,340
Welcome to this lecture number 9 on this N
P T E L course in fluid mechanics for chemical
2
00:00:21,340 --> 00:00:26,630
engineering undergraduate students. In the
last lecture we started discussing a new topic
3
00:00:26,630 --> 00:00:31,130
and the topic relates to description of motion
in fluids.
4
00:00:31,130 --> 00:00:38,130
So, this topic is called fluid kinematics
or flow kinematics, as I told you in the last
5
00:00:42,390 --> 00:00:49,390
lecture in any branch mechanics there are
two aspects to it, one is dynamics and the
6
00:00:53,880 --> 00:01:00,880
other is kinematics. Kinematics refers to
description of motion without reference to
7
00:01:07,720 --> 00:01:14,720
forces that cause the motion, these two forces
and dynamics is the next branch which will
8
00:01:18,090 --> 00:01:25,090
come to little later. Where we worry about
the forces and the motion, that are caused
9
00:01:25,340 --> 00:01:32,340
by applied forces that are caused by forces.
So, the first job in any mechanical subject
10
00:01:33,840 --> 00:01:40,840
is to understand how to describe flow. So,
in this topic we are going to discuss kinematics
11
00:01:52,140 --> 00:01:57,700
and we are going to just describe flows how
to describe motion in fluids.
12
00:01:57,700 --> 00:02:04,530
So, in this context we introduced the notion
of what is called a fluid particle? It is
13
00:02:04,530 --> 00:02:11,530
use full in the continuum hypothesis to identify
what are called fluid particles, these are
14
00:02:12,180 --> 00:02:18,940
not real particles as in an as in case of
molecule of a fluid, but they are high idealized
15
00:02:18,940 --> 00:02:24,040
hypothetical objects which are useful in describing
flows. What are these fluid particles? Well
16
00:02:24,040 --> 00:02:31,040
imagine you have fluid the container of a
fluid and a time t equal to zero there is
17
00:02:31,830 --> 00:02:37,489
no motion in this fluid the fluid is static.
And let us say you can mark various points
18
00:02:37,489 --> 00:02:44,430
in the fluid using various color dye. So,
I am just let me draw it slightly bigger so,
19
00:02:44,430 --> 00:02:49,420
that it is clearer.
So, you can mark various points in the fluid
20
00:02:49,420 --> 00:02:56,420
using various colored dye and this is a time
t equal to 0 so, in principle you can do this
21
00:02:58,319 --> 00:03:03,099
for a continuously for all points in a fluid,
because of fluid is a continuous medium within
22
00:03:03,099 --> 00:03:09,660
the continuum hypothesis. But for the sake
of illustration, I am showing few points.
23
00:03:09,660 --> 00:03:16,660
So, what this points will do upon this is
a time t equal to 0, I am using closed circles
24
00:03:21,120 --> 00:03:27,680
and at the later time upon application of
force this some kind of force we need not
25
00:03:27,680 --> 00:03:32,760
worry at this point what causes the motion
the some kind of force it causes a motion.
26
00:03:32,760 --> 00:03:39,760
At a later time t, all these points would
move to some other positions in general. So,
27
00:03:41,560 --> 00:03:48,560
I am using open circles to denote these locations
of these points at a later time.
28
00:03:50,080 --> 00:03:55,799
So, this point for example, can move here
this point could have moved here, this point
29
00:03:55,799 --> 00:04:00,700
could have moved here, this point could have
moved here, and this point could have moved
30
00:04:00,700 --> 00:04:07,700
here. So, that we are assuming that the dye
molecules are not diffusing or the diffusivity
31
00:04:08,040 --> 00:04:13,459
of the dye molecules are so small, that for
our time scales of inter estimate material
32
00:04:13,459 --> 00:04:19,840
that the dye is diffusing. So, the dye faithfully
represents a point in a fluid to the extent
33
00:04:19,840 --> 00:04:24,700
that you can resolve a point by the help of
a dye molecule with the help of the dye molecule
34
00:04:24,700 --> 00:04:31,400
dye drop. And, this dye drop which you are
using to identify a point will in general
35
00:04:31,400 --> 00:04:35,380
evolve in time up on application of forces,
because of the fluid is moving.
36
00:04:35,380 --> 00:04:42,380
So, this is roughly realization of this mathematical
or abstract idea of fluid particle so, in
37
00:04:44,440 --> 00:04:51,440
principle you can identify. At time t equal
to 0, the position of various particles so,
38
00:04:56,810 --> 00:05:00,570
with respect to our co ordinate system, as
usual; whenever you analyze any problem in
39
00:05:00,570 --> 00:05:06,180
flow fluids, you have to put a coordinate
system with respect to co ordinate system.
40
00:05:06,180 --> 00:05:12,160
You can label various points at time t equal
to 0. How do you label the points instead
41
00:05:12,160 --> 00:05:19,160
of using a dye molecule as I argued in this
thought experiment, we can label each point
42
00:05:20,400 --> 00:05:27,400
by their initial locations, the idea is you
are identifying each particle fluid particle
43
00:05:31,330 --> 00:05:35,560
on the basis of the initial locations with
the respect to your co ordinate system. And,
44
00:05:35,560 --> 00:05:42,560
at a later time, what will happen is that,
these initial locations the points that are
45
00:05:45,759 --> 00:05:51,840
identified by the initial locations will in
general evolve to a current location. So,
46
00:05:51,840 --> 00:05:58,840
schematically, I will take one point and then
at later time this will be like this. So,
47
00:06:00,000 --> 00:06:06,380
the trajectory of this particle at time t
equal to 0 is here and then a time t is here.
48
00:06:06,380 --> 00:06:12,550
So, this particle may move like this and at
a later time all this so, this is the position
49
00:06:12,550 --> 00:06:18,740
at time t equal to 0 is denoted as x p 0 and
this is the position of a fluid particle at
50
00:06:18,740 --> 00:06:25,740
time t. So, the position at time t will of
course, of a particle will be a function of
51
00:06:27,180 --> 00:06:34,180
where it was time t equal to 0. This is called
and of course, times because this is you have
52
00:06:36,750 --> 00:06:43,199
to follow this particle and at any time the
location that this fluid particle occupies
53
00:06:43,199 --> 00:06:47,360
at time t will be a function of time itself
as well as, where it was at time t equal to
54
00:06:47,360 --> 00:06:54,360
0. This is called a particle path or a particle
trajectory.
55
00:07:00,720 --> 00:07:07,720
Now, what is the use of having such information?
Suppose, you have to experimentally measure
56
00:07:13,229 --> 00:07:19,479
such information and you can do it for not
just one point as I have shown here, but you
57
00:07:19,479 --> 00:07:24,330
can do it for many points. So, another point
will generally in general move like this so
58
00:07:24,330 --> 00:07:30,259
it can so this point is at a slightly different
initial position and the later time of course,
59
00:07:30,259 --> 00:07:34,919
it will occupy different position. But in
general the idea is you can label all points
60
00:07:34,919 --> 00:07:40,630
based on the initial positions. You can follow
the motion and represent it mathematically
61
00:07:40,630 --> 00:07:47,630
in this form. Usually, this function and form
is written as x p at time t, is a function
62
00:07:48,340 --> 00:07:55,340
of x p 0 and time so, instead of writing a
function like this you use the variable itself
63
00:07:58,669 --> 00:08:03,319
as a function so, this is a concise notation.
So, once I have this information and I can
64
00:08:03,319 --> 00:08:10,319
find what is the velocity of a particle, which
was at time t equal to 0 at x p 0 at a later
65
00:08:12,300 --> 00:08:19,300
time t this is the velocity of a fluid particle,
which was at time t equal to 0 at x p 0. Velocity
66
00:08:33,979 --> 00:08:39,940
from the fundamental definition in mechanics
is that rate of change of position. So, we
67
00:08:39,940 --> 00:08:46,940
will have to simply take the position that
the particle trajectory, which is given by
68
00:08:48,550 --> 00:08:55,550
this functional form, takes the particle trajectory
and differentiate to time. By keeping the
69
00:08:57,019 --> 00:09:02,330
initial the particle label concept what you
are keeping instant, that you are following
70
00:09:02,330 --> 00:09:09,330
the same particle and then you are measuring
its trajectory as well as you can find quantities
71
00:09:09,740 --> 00:09:13,250
like velocity.
So, what is being kept constant in this time
72
00:09:13,250 --> 00:09:19,740
differentiation is very important, what is
being kept constant is that the initial position
73
00:09:19,740 --> 00:09:25,050
of this particle or the particle label is
kept constant in other words.
74
00:09:25,050 --> 00:09:32,050
We are following the same fluid particles,
that is the meaning of keeping x p naught
75
00:09:41,100 --> 00:09:48,100
constant. This description is called fluid
motion is called the Lagrangian or some time
76
00:09:51,000 --> 00:09:58,000
it is called as material description. So,
what this description means is that you are
77
00:10:04,720 --> 00:10:10,950
following properties of a fluid by following
the position of each and every particle as
78
00:10:10,950 --> 00:10:17,950
fluid is moving. So, the independent variables
in the Lagrangian description are the initial
79
00:10:25,220 --> 00:10:30,899
position of the particles and time t. So,
initial positions these are the particle labeled
80
00:10:30,899 --> 00:10:36,450
remember that the particle labeled mathematically,
but with the help of the initial position
81
00:10:36,450 --> 00:10:40,010
and this is time.
So, for example, I need not just worry about
82
00:10:40,010 --> 00:10:46,269
purely kinematic quantities like velocity
or position, but I can also say things like
83
00:10:46,269 --> 00:10:53,269
temperature as the function in the Lagrangian
description. The temperature of in a fluid
84
00:10:55,230 --> 00:11:00,800
flow will be depicted as the function of position
in the initial position of fluid particle
85
00:11:00,800 --> 00:11:05,860
and time t so, what it means, physically is
that suppose the particle is here at t equal
86
00:11:05,860 --> 00:11:12,860
to 0. And it is moving later on to time t,
to some other location what this Lagrangian
87
00:11:15,209 --> 00:11:21,190
description of temperature means is that you
are attaching a thermometer to the particle.
88
00:11:21,190 --> 00:11:26,370
And you are moving along with the particle
and you are recording the temperature in the
89
00:11:26,370 --> 00:11:31,130
thermometer as you follow the particle so,
that is the meaning of keeping x p naught
90
00:11:31,130 --> 00:11:36,450
constant, you are attaching yourself, tagging
yourself along with the fluid particle for
91
00:11:36,450 --> 00:11:42,779
example, we are interested in measuring temperature.
So, we put a thermometer in our minds along
92
00:11:42,779 --> 00:11:48,320
with the fluid particle and measure the temperature
of the fluid particle as you go along with
93
00:11:48,320 --> 00:11:55,320
the fluid particle. So, this is the Lagrangian
description or Lagrangian temperature field.
94
00:11:59,269 --> 00:12:06,269
So, what is the advantage of Lagrangian description?
We have the motion, the current position of
95
00:12:08,010 --> 00:12:14,959
particle as a function of it is initial position
and time once, I have this is called the particle
96
00:12:14,959 --> 00:12:21,639
path or particle trajectory or simply the
motion of the particle. Now, I can calculate
97
00:12:21,639 --> 00:12:28,149
the velocity of the particle at the later
time by taking the partial derivative of the
98
00:12:28,149 --> 00:12:33,209
particle trajectory by keeping x p naught
constant. Because x p is the function of x
99
00:12:33,209 --> 00:12:39,370
p naught and time so, various points in the
fluid will move differently to various other
100
00:12:39,370 --> 00:12:44,820
locations. So, if I want the velocity of this
particle I have to simply follow this particle
101
00:12:44,820 --> 00:12:50,800
at time t and then take its time derivative
at time t. So, t is this is the rate of change
102
00:12:50,800 --> 00:12:56,220
of particle is kept constant the rate of change
of position with respect to time at a time
103
00:12:56,220 --> 00:13:01,110
t. Now, this is the velocity of a fluid particle.
104
00:13:01,110 --> 00:13:07,839
Now, acceleration in mechanics or kinematics
is simply the rate of change of velocity that
105
00:13:07,839 --> 00:13:12,220
of a particle so, that is also very simple.
Simply take the rate of change of velocity
106
00:13:12,220 --> 00:13:16,829
of particle, by keeping the particle identity
to be the same, that is you are following
107
00:13:16,829 --> 00:13:23,060
the same particle and you are finding the
rate of change of its velocity that will be
108
00:13:23,060 --> 00:13:29,110
the acceleration. Now, while this is a very
similar to what is being what is normally
109
00:13:29,110 --> 00:13:35,880
done in Newtonian particle mechanics of point
particles and objects, this is not really
110
00:13:35,880 --> 00:13:41,269
suited ideally for fluid flow problems, because
in a fluid for example, if you are interested
111
00:13:41,269 --> 00:13:46,019
in flow in a pipe.
So, this pipe will be in general connected
112
00:13:46,019 --> 00:13:53,019
to some reservoir, it is figuratively shown
like this so, if you are interested in pressure
113
00:13:53,930 --> 00:14:00,170
drop across the pipe that is required to pump
a specific flow rate. Then you are not truly
114
00:14:00,170 --> 00:14:04,579
varied about various fluid particles, the
identity of fluid particles that are entering
115
00:14:04,579 --> 00:14:09,410
and leaving. Because all you are interested
in is what is the force that is being experienced
116
00:14:09,410 --> 00:14:14,980
by this pipe what is the drag force and consequently
what is the pressure drop? So, the Lagrangian
117
00:14:14,980 --> 00:14:21,810
description where in we follow the same particle
as a function of time is really not very useful
118
00:14:21,810 --> 00:14:27,260
especially, when you are considering fluid
flow problems.
119
00:14:27,260 --> 00:14:32,329
So, instead of Lagrangian description what
is normally followed? So, the Lagrangian description
120
00:14:32,329 --> 00:14:39,329
so, let me just write this not suited for
fluid flow problems. So, what is normally
121
00:14:54,980 --> 00:15:01,980
done in fluid mechanics is what is called
the EULERIAN description or SPATIAL
description of motion so, what this we mean
122
00:15:18,070 --> 00:15:19,810
is the following.
123
00:15:19,810 --> 00:15:26,810
In the Eulerian description we again put the
coordinate system x, y, z with respect to
124
00:15:29,410 --> 00:15:32,940
which we measure coordinate laboratory coordinates
system let us say the coordinate system is
125
00:15:32,940 --> 00:15:39,940
fixed in the lab. Now, here quantities such
as temperature are measure as a function of
126
00:15:40,190 --> 00:15:47,190
the three fixed coordinate x y z and time
so, what we do here is that you take a thermometer
127
00:15:48,389 --> 00:15:54,339
and then keep the thermometer at like given
point, then move the thermometer to various
128
00:15:54,339 --> 00:16:01,339
points and then keep measuring the temperature.
So, if you want the temperature at a given
129
00:16:01,610 --> 00:16:07,320
time so, you have to if the temperature is
changing as the function of both x y z as
130
00:16:07,320 --> 00:16:13,060
well as time, then what you have to do is
you have to put many thermometers at various
131
00:16:13,060 --> 00:16:16,730
locations.
And each thermometer it wills each thermometer
132
00:16:16,730 --> 00:16:23,430
will locater will read will indicate the temperature
at that location as the function of time.
133
00:16:23,430 --> 00:16:30,430
So, the in the Eulerian description the independent
variables are the spatial position of a point
134
00:16:36,240 --> 00:16:42,829
x y z and time. So, what is the fundamental
difference between Lagrangian and Eulerian
135
00:16:42,829 --> 00:16:49,480
descriptions in a fluid flow problem, in a
fluid flow context? Suppose, fluid is flowing
136
00:16:49,480 --> 00:16:56,410
and in the Lagrangian description you will
follow this material point or fluid particle
137
00:16:56,410 --> 00:17:00,180
and then you will measure its temperature
as a function of time.
138
00:17:00,180 --> 00:17:07,180
Suppose, you are just worrying about a small
time interval t and little later to t plus
139
00:17:09,270 --> 00:17:16,270
delta t. In the Lagrangian description if
you put a thermometer, this thermometer will
140
00:17:16,800 --> 00:17:23,610
measure the temperature of a fluid particle
which was here at t time t and which was here
141
00:17:23,610 --> 00:17:29,590
at a time t plus delta t. So, you are keeping
x p naught constant and you are measuring
142
00:17:29,590 --> 00:17:36,590
temperature as a function of time so, this
is the Lagrangian. In the Eulerian description
143
00:17:39,350 --> 00:17:46,350
in contrast suppose, you have well let us
fix the location suppose you fix the location
144
00:17:47,390 --> 00:17:54,390
so, this is the fixed location in space
at time t. Let us say this location is occupied
by a particle colored with blue that is this
145
00:18:02,400 --> 00:18:08,090
particle was residing at this point at time
t, but at a later time what is going to happen?
146
00:18:08,090 --> 00:18:14,799
So, this let me just write that this is Eulerian.
At a later time what is going to happen is
147
00:18:14,799 --> 00:18:21,419
that a same location which is denoted by a
star here; this point will not will locate
148
00:18:21,419 --> 00:18:24,710
the blue point will not be locating will not
be residing at the same location. Because
149
00:18:24,710 --> 00:18:31,710
at a later time this blue point in general
moves to, this is a time t at time t plus
150
00:18:33,070 --> 00:18:40,070
delta t, but this location will be occupied
by some other point at time t so, the thermometer
151
00:18:41,870 --> 00:18:48,870
at a fixed location in space records. It does
not record the history of a same fluid particle
152
00:18:49,770 --> 00:18:54,480
rather it records the temperature is same
the temperatures of various fluid particles
153
00:18:54,480 --> 00:18:58,549
that happen to be at a given location at various
times.
154
00:18:58,549 --> 00:19:05,549
So, the sense of history of temperature history
or velocity history or that this historical
155
00:19:06,630 --> 00:19:13,630
information of a given fluid particle is lost.
Historical information is lost, because we
156
00:19:17,539 --> 00:19:23,720
are not following or tagging along with the
same fluid particle instead, we are sitting
157
00:19:23,720 --> 00:19:29,440
at a same point and we are merely measuring
the temperatures that various fluid particles
158
00:19:29,440 --> 00:19:33,590
are going to occupy, that point at various
times.
159
00:19:33,590 --> 00:19:40,549
But, the Eulearian description even though
it has lost by its construction it has lost
160
00:19:40,549 --> 00:19:47,549
the sense of history of information of a given
fluid particle. But it is still very useful,
161
00:19:48,360 --> 00:19:55,360
because in laboratory it is easier to measure
temperature, easier to measure properties
162
00:20:05,890 --> 00:20:12,890
at fixed locations, rather than follow fluid
particles or material particles. So, in fluid
163
00:20:24,530 --> 00:20:31,530
flow problem there are two reasons why Eulerian
description is preferred over Lagrangian description.
164
00:20:31,530 --> 00:20:38,530
Firstly even if it was feasible having the
information about what are the various fluid
165
00:20:39,220 --> 00:20:44,100
particles that are occupying let us say a
given section of pipe is not relevant to many
166
00:20:44,100 --> 00:20:46,880
practical questions, such as what is the pressure
drop.
167
00:20:46,880 --> 00:20:50,250
Because here we are not really worried about
which fluid particle is coming and exerting
168
00:20:50,250 --> 00:20:55,559
a drag force. We are merely interested in
the force that is experienced by involves
169
00:20:55,559 --> 00:21:02,559
of the pipe or shear that is moving or on
so. So, the historical information is not
170
00:21:04,720 --> 00:21:11,720
practical importance in general, in fluid
practical applications. And also even if you
171
00:21:13,230 --> 00:21:19,960
want to measure such historical information
in the Lagrangian sense it is not easy to
172
00:21:19,960 --> 00:21:24,350
measure in lab, because you have to really
follow the same fluid particle and it is not
173
00:21:24,350 --> 00:21:29,840
easy. Rather, it is easier to fix probes such
as velocity probes or pressure probes, temperature
174
00:21:29,840 --> 00:21:35,840
probes at a given point, in special location
or at various fix points in a spatial locations
175
00:21:35,840 --> 00:21:41,840
rather than moving along with a particle.
So, in the Eulerian description the independent
176
00:21:41,840 --> 00:21:48,840
variables are the fixed locations of various
points in space and time so, the Eulerian
177
00:21:51,039 --> 00:21:55,860
description is very easy to measure in laboratory.
178
00:21:55,860 --> 00:22:02,860
So, if you are interested in kinematic quantity
such as velocity of a fluid flow is described
179
00:22:04,240 --> 00:22:09,510
in the Eulerian description as a function
of various points in space and time suppose,
180
00:22:09,510 --> 00:22:16,510
you have a flow in a pipe and you put a coordinate
system x y z. So, you can sit at this point
181
00:22:17,940 --> 00:22:22,820
and measure its velocity as a function of
time, then change the location of observation
182
00:22:22,820 --> 00:22:27,460
and then or you can put multiple probes for
velocity and then measure the velocity at
183
00:22:27,460 --> 00:22:33,090
various locations, fixed locations in space
as well as time. That is the main crocks of
184
00:22:33,090 --> 00:22:39,030
Eulerian description. But the there is the
problem with Eulerian description in the sense,
185
00:22:39,030 --> 00:22:46,030
that suppose; I have this information temperature
as function spatial location and time. And
186
00:22:46,330 --> 00:22:53,330
suppose, I take this partial derivative temperature
is a function of x vector is a combination
187
00:22:53,770 --> 00:23:00,400
of 3 variables so, x y z and time.
So, when I take when I say partial temperature
188
00:23:00,400 --> 00:23:07,400
by partial time, I am keeping the location
x y z constant, I will figuratively denote
189
00:23:07,620 --> 00:23:09,520
this as vector x.
190
00:23:09,520 --> 00:23:16,520
So, suppose, I have given this given temperature
field let us say from an experiment so, the
191
00:23:18,059 --> 00:23:24,640
temperature field is given as T x y z time
or in short form I will simply write this
192
00:23:24,640 --> 00:23:31,640
as T x t given this information. Suppose,
if we calculate if I calculate at a spatial
193
00:23:37,980 --> 00:23:44,980
location, this is not telling me how the temperature
is changing so, this is the partial derivative
194
00:23:46,490 --> 00:23:52,409
so, if this fluid particle is moving from
time t to t plus delta t, its temperature
195
00:23:52,409 --> 00:23:57,940
can in general change as you follow the particle.
But this is not giving that information what
196
00:23:57,940 --> 00:24:03,130
this derivative giving you is that if you
sit at a point what is, that rate of change
197
00:24:03,130 --> 00:24:06,919
of temperature that you will feel at that
point locally.
198
00:24:06,919 --> 00:24:13,850
So, the sense of history is lost in the Euleruian
description, while it is not critical in probably
199
00:24:13,850 --> 00:24:17,659
applications like temperature suppose, you
are interested in kinematic quantities like
200
00:24:17,659 --> 00:24:24,659
acceleration. What is acceleration? In the
Lagrangian description, if you recall, this
201
00:24:30,919 --> 00:24:37,919
is basically the rate of change of its position
by keeping the label of the particle constant.
202
00:24:39,250 --> 00:24:46,250
But in the Eulerian description v is written
now not as a function of the initial positions,
203
00:24:46,669 --> 00:24:53,669
but rather the spatial locations in a coordinate
system. So, first of all I do not have this
204
00:24:55,360 --> 00:25:00,250
particle path information. Even, if I have
the velocity information in the Eulerian in
205
00:25:00,250 --> 00:25:06,350
the sense, I cannot calculate acceleration
as partial v partial t, because now I am not
206
00:25:06,350 --> 00:25:10,700
keeping I am not following the same particle,
I am merely sitting at a same particle point
207
00:25:10,700 --> 00:25:16,340
in space. Where as in the Lagrangian description,
acceleration of the particle is the rate of
208
00:25:16,340 --> 00:25:23,340
change of it is velocity, because you are
following the same particle. So, what is being
209
00:25:27,140 --> 00:25:34,140
kept constant is this so, in the Lagrangian
description it is very clear; what is acceleration,
210
00:25:37,460 --> 00:25:42,130
because you are merely keeping the initial
location of the particle constant you are
211
00:25:42,130 --> 00:25:46,700
following the same particles.
But in the Eulerian description, if you have
212
00:25:46,700 --> 00:25:50,840
the velocity field like this you cannot compute
acceleration by simply taking the partial
213
00:25:50,840 --> 00:25:57,840
derivative. Because this does not have information
as to how a given particle is moving, given
214
00:25:58,270 --> 00:26:02,909
particle is moving as function of time this
information is not there in the Eulerian description.
215
00:26:02,909 --> 00:26:08,500
So, we cannot compute accelerations from the
Eulerian velocity fields. So, you may ask
216
00:26:08,500 --> 00:26:12,890
why is this is an issue the reason why this
is an issue is that when you want to eventually
217
00:26:12,890 --> 00:26:18,080
go to dynamics you are going to apply the
Newton’s second law of motion to continues
218
00:26:18,080 --> 00:26:23,890
fluid. The Newton’s second law of motion
says that the force on a particle an identifiable
219
00:26:23,890 --> 00:26:29,820
piece of matter this mass times acceleration.
So, we need the acceleration when you want
220
00:26:29,820 --> 00:26:35,539
to eventually write down equations of motion
for the flow. But if you also want to work
221
00:26:35,539 --> 00:26:42,539
simply with the Eulerian frame work, because
it is much simpler it is more useful. But
222
00:26:43,820 --> 00:26:48,220
fundamentally we need accelerations so, we
need acceleration, but acceleration cannot
223
00:26:48,220 --> 00:26:55,220
be obtained from velocities, like this. So,
how do I connect the acceleration to the velocity
224
00:26:56,470 --> 00:27:03,470
field Eulerien velocity field? That is the
question is given the Eulerian velocity field
225
00:27:03,539 --> 00:27:10,539
v as a function of x t how do I compute accelerations?
So, there is a very nice frame work for doing
226
00:27:13,630 --> 00:27:19,200
this and this is the reason why we introduce
the Lagrangian description although we are
227
00:27:19,200 --> 00:27:24,809
not going to use the Lagrangian description.
We do need the motion of the Lagrangian description
228
00:27:24,809 --> 00:27:31,809
in order to compute acceleration of fluid
particles. So, instead of doing this for acceleration
229
00:27:33,029 --> 00:27:37,429
I am going to illustrate this for a derivative
like temperature.
230
00:27:37,429 --> 00:27:44,429
That is the question is given the spatial
description or Eulerian description of temperature
231
00:27:45,620 --> 00:27:52,620
how do I compute partial T as I follow a particle,
this is the Lagrangian time derivative. See
232
00:27:55,570 --> 00:28:00,070
notice, that the Lagrangian description and
Eulerian description in differ merely by what
233
00:28:00,070 --> 00:28:07,070
are independent variables. This is the Lagrangian
description, this is the Eulerian description
234
00:28:08,429 --> 00:28:15,429
so, they change not only by the independent
variables that we chose to describe the problem
235
00:28:16,909 --> 00:28:23,909
with. So, I am going now, do this thing called
Substantial or material derivative.
236
00:28:33,090 --> 00:28:40,090
This will help us
to calculate the time derivative as we follow
particle from and Eulerian description. So,
237
00:28:49,580 --> 00:28:56,100
to motivate this it will take a very simple
context so; imagine you have a channel in
238
00:28:56,100 --> 00:29:02,260
which fluid is flowing with constant uniform
velocity. The velocity is constant in the
239
00:29:02,260 --> 00:29:09,260
sense that suppose, you call this x and y
the velocity is constant in the y direction
240
00:29:09,720 --> 00:29:15,539
it is not realistic, but this is just for
the sake of our illustration. So, you should
241
00:29:15,539 --> 00:29:19,419
do not vary about this part that y the velocity
is uniform let us assume that the velocity
242
00:29:19,419 --> 00:29:26,419
is largely uniform. Now, let us imagine that
there is a location so, let me introduce a
243
00:29:32,029 --> 00:29:39,029
location this is x equal to 0 and here a fluid
particle, let us focus on fluid particles
244
00:29:43,919 --> 00:29:50,919
blue and red.
So, here there is a red fluid particle at
245
00:29:51,240 --> 00:29:58,240
time t equal to t 0 let us say we had a time
t equal to t 0. And then we have another fluid
246
00:30:00,120 --> 00:30:07,120
particle which is blue in color at the location
x is minus v naught delta t since; the fluid
247
00:30:09,429 --> 00:30:16,429
is flowing at a constant velocity. And so,
we are looking at two fluid particles, which
248
00:30:19,399 --> 00:30:26,399
are separated by distance delta x and that
is delta x is v naught delta t, where delta
249
00:30:28,520 --> 00:30:33,679
t is a time interval, that we are going to
introduce at just shortly. So, imagine two
250
00:30:33,679 --> 00:30:40,549
fluid particles identified by their colors
blue and red and we are following the motion
251
00:30:40,549 --> 00:30:43,390
of these two fluid particles as a function
of time.
252
00:30:43,390 --> 00:30:50,390
So, this is the time t equal to t 0 this is
the situation the red particle is situated
253
00:30:51,870 --> 00:30:58,870
at x equal to 0 and the blue particle is situated
slightly behind and since it is x is positive
254
00:31:00,039 --> 00:31:05,890
in this direction. So, this is at a distance
minus delta x and since delta x is v 0 delta
255
00:31:05,890 --> 00:31:12,890
t x is 0 minus v 0 delta t so, this is minus
v 0 delta t this is slightly behind the red
256
00:31:14,730 --> 00:31:21,730
particle. Now, at a later time, imagine after
time t naught plus delta t what would happen?
257
00:31:27,350 --> 00:31:34,350
Let us try to draw this is x equal to 0 this
is x equal to v 0 delta t. Since, the fluid
258
00:31:38,490 --> 00:31:45,490
is moving this particle will move eventually
so, the red particle move from 0 to v 0 delta
259
00:31:47,110 --> 00:31:54,000
t and the blue particle will move from minus
v 0 delta t to x equal to 0, this is a time
260
00:31:54,000 --> 00:32:01,000
t equal to t 0 plus delta t.
So, let us mark also at this point the Lagrangian
261
00:32:02,830 --> 00:32:09,830
labels of this particles is that initial position
at time t equal to 0 so, the red particle
262
00:32:16,409 --> 00:32:23,130
is denoted by x equal to 0. That is the position
of the particle at time t equal to t naught
263
00:32:23,130 --> 00:32:30,130
and the blue particle is denoted by x equal
to minus v 0 delta t. So, the Lagrangian so,
264
00:32:31,500 --> 00:32:37,730
x naught I mean sense Lagrangian variables
are denoted with this is the position at time
265
00:32:37,730 --> 00:32:44,260
t equal to t 0.
So, this particle while it is present it is
266
00:32:44,260 --> 00:32:51,260
also x naught is minus x naught is minus v
0 and x here the red particle x naught is
267
00:33:00,130 --> 00:33:07,130
0. Now, even at a later time t 0 plus delta
t this blue particle is still denoted by the
268
00:33:09,039 --> 00:33:16,039
same Lagrangian label, because the Lagrangian
description uses the position of the particles
269
00:33:16,159 --> 00:33:21,080
at an initial time, let us say t naught, to
label them. So, x naught is still minus v
270
00:33:21,080 --> 00:33:28,080
0 delta t and x naught for the red particle
is still 0. So, this is the initial position
271
00:33:32,070 --> 00:33:36,980
of the particle which is currently at x equal
to 0 this is the initial motion of this red
272
00:33:36,980 --> 00:33:42,480
particle, which is currently at x equal to
v 0 delta t. So, these are the Lagrangian
273
00:33:42,480 --> 00:33:49,480
coordinates these are the Lagrangian labels
or coordinates.
274
00:33:53,809 --> 00:34:00,809
Now, let us say we are having, if we are measuring
temperature at this point we are measuring
275
00:34:07,570 --> 00:34:14,159
temperature at this point x, x equal to 0.
So, we are measuring temperature by putting
276
00:34:14,159 --> 00:34:21,159
a thermometer at x equal to 0, and so here
as well so, we are placing the thermometer
277
00:34:24,370 --> 00:34:31,370
at x equal to 0 this is the thermometer. What
this thermometer is measuring as a function
278
00:34:31,609 --> 00:34:36,960
of time? At time t equal to t 0 it will measure
the temperature of the red particle, while
279
00:34:36,960 --> 00:34:41,740
at time t equal to t 0 plus delta t it will
measure the temperature of the blue particle
280
00:34:41,740 --> 00:34:46,399
this is the key to our derivation.
So, it is very simple, because the thermometer
281
00:34:46,399 --> 00:34:52,950
is fixed at the same spatial location x equal
to 0, which I am denoting by this green color.
282
00:34:52,950 --> 00:34:57,720
But the points that are occupying the same
spatial location at different the material
283
00:34:57,720 --> 00:35:01,109
particle that are occupying the fluid particles
are different, because the fluid is continuously
284
00:35:01,109 --> 00:35:02,730
flowing.
285
00:35:02,730 --> 00:35:09,730
So, let us understand what is what we will
do by calculating at the partial t of the
286
00:35:12,930 --> 00:35:19,220
temperature with respect to time at x equal
to 0 at the same location. What is this is
287
00:35:19,220 --> 00:35:26,220
from fundamental definition of calculus this
is t 0 plus delta t minus x equal to 0, t
288
00:35:29,750 --> 00:35:36,160
equal to t 0 divided by delta t in the limit
as delta t goes to 0 this is the fundamental
289
00:35:36,160 --> 00:35:41,520
definition of partial derivative. Since, it
is a constant we are keeping x is constant
290
00:35:41,520 --> 00:35:48,520
at 0 so, this label is the function of two
variables t is a function of x and t is Eulerian
291
00:35:48,550 --> 00:35:53,339
description since, x is kept constant at 0,
we see simply have to take the derivative
292
00:35:53,339 --> 00:36:00,339
with respect to time. Now, let us try to understand
what this means, at x equal to 0 and t equal
293
00:36:05,010 --> 00:36:12,010
to t 0 plus delta t let us look picture, this
is time t 0 plus delta t at x equal to 0 the
294
00:36:13,579 --> 00:36:17,130
particle that occupied is the blue particles.
So, the temperature that the thermometer will
295
00:36:17,130 --> 00:36:24,130
measure is nothing but limit delta t tends
to 0 temperature of the blue particle which
296
00:36:25,380 --> 00:36:31,160
is identified by its Lagrangian variable.
So, which is x naught is minus v naught delta
297
00:36:31,160 --> 00:36:38,160
t and the time is t 0 plus delta t minus here,
the temperature at x equal to 0 time t equal
298
00:36:39,880 --> 00:36:46,880
to 0 is namely that of the red particle. Because
at x equal to 0 at time t equal to 0 notice
299
00:36:48,369 --> 00:36:54,020
that it is the red particle that is occupying
and the thermometer will measure at time t
300
00:36:54,020 --> 00:37:00,280
equal to delta t, t zero the temperature of
the red particle. And the red particle is
301
00:37:00,280 --> 00:37:07,280
identified by its Lagrangian variable, which
is nothing but, t x naught is 0 at time t
302
00:37:07,310 --> 00:37:14,310
0 divided by delta t. So, essentially we are
trying to measure the temperature at the given
303
00:37:20,240 --> 00:37:23,070
spatial location in this example, in this
illustration.
304
00:37:23,070 --> 00:37:30,070
So, we are putting thermometer at the position
x equal to 0 the same positions same spatial
305
00:37:30,270 --> 00:37:36,599
location with respect to this coordinate system.
And but at x equal to 0 at time t is equal
306
00:37:36,599 --> 00:37:43,599
to t 0 the red particle is occupying the location
spatial location x equal to 0. At a later
307
00:37:44,650 --> 00:37:51,650
time, the same spatial location x equal to
0 is occupied by the blue particle. So, when
308
00:37:52,329 --> 00:37:57,380
you take this measurement and when you take
the partial derivative of the temperature
309
00:37:57,380 --> 00:38:02,250
using these measurements. Partial derivative
of temperature with respect to time is nothing
310
00:38:02,250 --> 00:38:09,250
but the partial derivative of temperature
is limit time at that spatial location x equal
311
00:38:09,650 --> 00:38:15,560
to 0 is temperature at later time minus temperature
is t 0 divided by delta t as delta t goes
312
00:38:15,560 --> 00:38:19,210
to 0.
But the key realization that we must have
313
00:38:19,210 --> 00:38:26,210
is that the temperature at x equal to 0 at
later time corresponds to that particle which
314
00:38:26,770 --> 00:38:31,390
was there at a later time which is near to
the blue particle. Now, the blue particle
315
00:38:31,390 --> 00:38:38,390
is denoted by it is Lagrangian labels x naught
is minus v 0 delta t. Now, the temperature
316
00:38:40,099 --> 00:38:47,099
at x equal to 0 at time t 0 is due that of
the red particles so, we can change from x
317
00:38:47,829 --> 00:38:53,050
equal to 0 x naught equal to 0, because the
red particle is identified by it is Lagrangian
318
00:38:53,050 --> 00:38:57,950
variable which is nearly x naught equal to
0. So, this is the key realization when going
319
00:38:57,950 --> 00:39:04,950
from Eulerian to Lagrangian so, that we can
change the labels from Eulerian to Lagrangian
320
00:39:06,960 --> 00:39:11,680
by knowing which particle was occupying the
current position and the previous position
321
00:39:11,680 --> 00:39:18,440
and so on.
So, having done this, we will just do simple
322
00:39:18,440 --> 00:39:25,440
we are still having on the left side the spatial
derivative the Eulearian time derivative of
323
00:39:25,579 --> 00:39:32,579
the temperature field. So, I am going to do
a small mathematical simplification by adding
324
00:39:33,190 --> 00:39:39,560
and subtracting. So, let me write first two
terms x 0 minus v 0 delta t time is t 0 plus
325
00:39:39,560 --> 00:39:46,560
delta t let me subtract time temperature at
x o is minus v 0 delta t and t is t 0, and
326
00:39:50,230 --> 00:39:57,230
add the same thing again x 0 is minus v 0
delta t, t is t 0 minus T x 0 is 0 and then
327
00:40:03,520 --> 00:40:10,520
t 0 and then divided by delta t so, let we
put 1 over delta t here that is of course
328
00:40:14,070 --> 00:40:21,070
stays. Now, here we are keeping x 0 the same.
So, what is this term? So, let us let me mark
329
00:40:24,810 --> 00:40:31,810
this with red so, that you can see what is
this term? This term is nothing but the red
330
00:40:35,680 --> 00:40:42,680
term, that here is nothing but so let me write
this from separately the red term is nothing
331
00:40:43,780 --> 00:40:47,849
but this write in red color so that it is
clear.
332
00:40:47,849 --> 00:40:54,849
T at x naught is minus v 0 delta t at time
t 0 plus delta t minus T at x naught is minus
333
00:41:00,200 --> 00:41:07,200
v 0 delta t at t 0 divided by delta t, that
is of course there here common and limit delta
334
00:41:09,510 --> 00:41:16,510
t going to 0 is nothing but see here we are
keeping x naught constant so, this is nothing
335
00:41:20,310 --> 00:41:27,310
but the time derivative as you keep x naught
constant. So, this is the time derivative
336
00:41:29,780 --> 00:41:36,780
as we keep x naught constant, this is the
Lagrangian time derivative as you follow the
337
00:41:37,670 --> 00:41:44,670
same particle. Here originally, we are measuring
the time temperature at the same location,
338
00:41:46,560 --> 00:41:53,560
but here this part of this expression corresponds
to the rate of change of temperature with
339
00:41:53,680 --> 00:41:57,369
time as you follow the same particle, because
the particle is being fixed here.
340
00:41:57,369 --> 00:42:03,730
So, this is what we are often we want to calculate
the rate of change of temperature with time,
341
00:42:03,730 --> 00:42:08,530
as you follow the same particle. But, there
is one more piece here which we will have
342
00:42:08,530 --> 00:42:14,220
to tackle so, what is that piece let me write
it in blue color. So, we still have this additional
343
00:42:14,220 --> 00:42:21,220
piece limit delta t tending to 0 one over
delta t T x naught is minus v naught delta
344
00:42:21,690 --> 00:42:28,690
t, t equal to t 0 minus T at x naught is 0
at the same time divided by delta t. Now suppose,
345
00:42:35,450 --> 00:42:42,450
you consider the change in position delta
x, delta x is nothing but v 0 delta t recall
346
00:42:43,980 --> 00:42:50,980
that this point, the 2 points x the 2 points
is separated by at distance delta x, x equal
347
00:42:54,190 --> 00:43:01,150
to 0 and this earlier distance which is nothing
but minus v naught delta t, that so, delta
348
00:43:01,150 --> 00:43:05,240
x is separating distance is v 0 delta t that
is so, delta x is separating distance is v
349
00:43:05,240 --> 00:43:12,240
0 delta t. So, this is nothing but we can
write this as so, one over delta t.
350
00:43:19,910 --> 00:43:26,910
T x naught is minus delta x t 0 minus T x
naught 0. Now, if you look at what is partial
351
00:43:38,420 --> 00:43:45,420
derivative of partial t partial x at x equal
to 0 you can write this as T at x is 0, but
352
00:43:47,200 --> 00:43:54,200
we can also before I do that we can also change
the labels now. When x naught is 0 x is also
353
00:43:58,490 --> 00:44:05,490
0 this is nothing but so, let us go back to
the figure in the previous slide. When x naught
354
00:44:09,869 --> 00:44:16,869
is 0 x is 0, when x naught is minus v 0 delta
t x is also minus v 0 delta t. So, we can
355
00:44:18,720 --> 00:44:25,720
change this to the Eulerian description as
limit delta t going to 0 1 over delta t. When
356
00:44:28,460 --> 00:44:35,460
x naught is minus v 0 delta T x is also minus
v 0 delta t, t 0 minus T when x naught is
357
00:44:37,450 --> 00:44:44,450
also 0, x is also 0 at t 0 this is what we
have.
358
00:44:44,859 --> 00:44:50,040
But if you recall what is the fundamental
definition of the rate of change of temperature
359
00:44:50,040 --> 00:44:57,040
with respect to position at a given time?
Let us say this is T at x equal to 0 minus
360
00:44:57,270 --> 00:45:04,270
T at x equal to 0 minus delta x divided by
delta x, but in our case delta x is nothing
361
00:45:06,930 --> 00:45:13,930
but v 0 delta t. So, I can write this as so,
instead of delta x and write v 0 delta T here
362
00:45:15,010 --> 00:45:22,010
so, T at x equal to 0 minus T at minus delta
x divided by v 0 delta t this is partial t,
363
00:45:28,020 --> 00:45:31,540
partial x at constant time.
364
00:45:31,540 --> 00:45:38,540
So, I can pull this v 0 up here and realize
that what I have here in this expression is
365
00:45:42,410 --> 00:45:49,410
nothing but I have here T at x equal to minus
delta x t 0 minus t at 0 this is nothing but
366
00:45:52,069 --> 00:45:59,069
minus partial T by partial x times v 0 delta
t. So, if I go back to my original expression
367
00:46:02,030 --> 00:46:09,030
where I have two terms, if you remember let
us go back to this expression I have two terms.
368
00:46:10,560 --> 00:46:15,319
So, let me just simplify here itself partial
T, partial of temperature with respect to
369
00:46:15,319 --> 00:46:22,319
time at x equal to 0 so this term is simplified
as so, we have still one over delta t, limit
370
00:46:22,339 --> 00:46:29,190
delta t going to 0 well we are taken the limiting
process. So, let us remove the limits now
371
00:46:29,190 --> 00:46:36,190
this is nothing but you have the first term
we already simplified it in to partial T,
372
00:46:36,690 --> 00:46:41,220
partial t at x equal to 0.
The second term is now simplifying to minus
373
00:46:41,220 --> 00:46:48,220
v 0 delta t partial T partial x. Now, what
we are after is naught so, let me just write
374
00:46:53,079 --> 00:47:00,079
on this result once again partial T partial
t at constant x is nothing but partial T partial
375
00:47:05,599 --> 00:47:12,599
t at constant x 0 minus v 0 partial T partial
x at constant time. So, what we are after
376
00:47:17,359 --> 00:47:22,920
is this term, because this is something that
is easier to measure experimentally where
377
00:47:22,920 --> 00:47:26,800
this is something is difficult, because this
is the rate of change of temperature as you
378
00:47:26,800 --> 00:47:31,400
follow a point, where as this is a rate of
change of temperature at a fixed point.
379
00:47:31,400 --> 00:47:38,400
So, finally we have rate of change of temperature
when you follow a particular fluid particle
380
00:47:39,470 --> 00:47:44,750
is equal to rate of change of temperature
at a fixed point this term will go to the
381
00:47:44,750 --> 00:47:51,750
other side, if you take the negative sign
to the other side, becomes positive at constant
382
00:47:52,180 --> 00:47:57,829
time. I am sorry this is x now, because this
is in the other side this is x.
383
00:47:57,829 --> 00:48:04,829
So, this is called the Lagrangian or substantial
time derivative or in fact sometimes it is
384
00:48:11,650 --> 00:48:18,650
called the material derivative this is the
usual partial derivative of time with respect
385
00:48:21,520 --> 00:48:28,520
to time. When I take partial derivative of
temperature with respect to time, I have to
386
00:48:30,369 --> 00:48:35,650
keep the spatial location constant so, this
is the spatial location at a given point space
387
00:48:35,650 --> 00:48:42,650
.We can call it x equal to 0 in this example
but in general it can be x now, this one is
388
00:48:45,180 --> 00:48:51,000
what is called the convicted rate of change
it is called the convicted rate of change
389
00:48:51,000 --> 00:48:58,000
of temperature. So, here what this is telling
is that particle that was here will move from
390
00:48:59,740 --> 00:49:03,700
one location to another by virtue of flow
that is where this velocity is coming in.
391
00:49:03,700 --> 00:49:08,460
And the temperature difference is going to
field is given by the Eulerian spatial derivative
392
00:49:08,460 --> 00:49:13,079
of temperature.
So, this is essentially that temperature difference
393
00:49:13,079 --> 00:49:18,240
between these two points as given by the Eulerian
description. And, if you multiply by the velocity
394
00:49:18,240 --> 00:49:23,410
of the particle by the velocity at which the
particle is moving, then that is going to
395
00:49:23,410 --> 00:49:28,859
give us the convicted rate of change. So,
this is sometimes called the local rate of
396
00:49:28,859 --> 00:49:35,859
change, this is called the convicted rate
of change. So, in general therefore, we can
397
00:49:48,440 --> 00:49:55,440
write so, this is the Eulerian derivative
this is the substantial derivative this is
398
00:49:56,000 --> 00:50:00,829
the normal partial derivative, sorry this
is the Lagrangian derivative, this is the
399
00:50:00,829 --> 00:50:04,130
normal partial derivative and this is the
connected rate of change.
400
00:50:04,130 --> 00:50:10,579
So, in general, the rate of change of any
property like temperature as a function of
401
00:50:10,579 --> 00:50:17,579
time for fixed Lagrangian particle is given
by the rate of change of temperature with
402
00:50:20,200 --> 00:50:25,770
respect to fixed point in space. Let us still
keep x as a single variable, I will generalize
403
00:50:25,770 --> 00:50:32,770
it to more three dimensions shortly plus the
velocity in the x direction the times partial
404
00:50:33,099 --> 00:50:40,099
t partial x at a given time. So, this right
side can be computed completely from Eulerian
405
00:50:41,530 --> 00:50:48,530
description whereas, this is inherently Lagrangian
quantity, because you are following the same
406
00:50:52,819 --> 00:50:59,819
particle. Now, Some times in text books you
will find that instead of having this same
407
00:51:02,559 --> 00:51:07,970
symbol with respect to different independent
variable being kept constant this is normally
408
00:51:07,970 --> 00:51:14,540
denoted by D T D t.
So, capital D is reserved for substantial
409
00:51:14,540 --> 00:51:19,980
derivative for a single, if the temperature
functions of only single coordinate x and
410
00:51:19,980 --> 00:51:26,980
time so, it is the velocity in the x direction
time’s partial T partial x constant time.
411
00:51:31,190 --> 00:51:38,190
Now, I can generalize this two more dimensions
instead of just having the one dimension.
412
00:51:41,559 --> 00:51:48,559
Suppose, the temperature is a function of
not just x, y, z and time, then what is the
413
00:51:48,920 --> 00:51:54,460
substantial derivative? Substantial derivative
is the time derivative of temperature as you
414
00:51:54,460 --> 00:52:01,460
follow a particle. So, what is it? So, we
can generalize very straight forward in a
415
00:52:03,550 --> 00:52:10,550
straight forward way is partial T partial
t at constant spatial location plus v x partial
416
00:52:13,200 --> 00:52:20,200
T partial x plus v y partial T partial y plus
v z partial T partial z. So, again so, here
417
00:52:25,740 --> 00:52:32,520
partial T partial x is calculated by keeping
y, z, t constant this is calculated by keeping
418
00:52:32,520 --> 00:52:39,520
x, z, t constant it is calculated by keeping
y, x and t constant.
419
00:52:40,300 --> 00:52:47,300
So, that is the basic idea of a substantial
derivative where in by purely having completely
420
00:52:48,440 --> 00:52:55,440
Eulerian information, we are able to calculate
the rate of change of a quantity like temperature,
421
00:52:55,990 --> 00:53:02,990
as you follow particle infinitely at a later
time delta t. Now, if you try to see whether
422
00:53:05,700 --> 00:53:11,710
we can write this in a slightly better form
now, this is let us look at this part of the
423
00:53:11,710 --> 00:53:18,710
equation
and see whether we can write this slightly
in a more compact form. So, this is like,
424
00:53:24,829 --> 00:53:31,829
if I have two vectors a and b and each vector
is given by a x in terms of its Cartesian
425
00:53:32,910 --> 00:53:39,910
coordinates b is similarly, given as b x i
plus b y j plus b z k, then a dot b is nothing
426
00:53:48,440 --> 00:53:55,440
but a x b x plus a y b y plus a z b z. Likewise,
if you look at this expression it look a dot
427
00:54:00,530 --> 00:54:07,410
product of two vectors.
One vector is the velocity; the other vector
428
00:54:07,410 --> 00:54:14,410
is the gradient of the temperature let me
just explain how this comes about.
429
00:54:14,460 --> 00:54:21,460
So, if you look at velocity is v x i plus
v y j plus v z k, look at gradient of temperature
430
00:54:26,589 --> 00:54:33,589
it is nothing but partial T partial x i plus
partial T partial y j plus partial T partial
431
00:54:34,410 --> 00:54:41,230
z k. If I take the two dot products, where
the two dot product of this two vectors v
432
00:54:41,230 --> 00:54:48,230
dot del t, then you get v x partial T partial
x plus v y partial T partial y plus v z partial
433
00:54:50,690 --> 00:54:56,410
T partial z. So, we can write the substantial
derivative of temperature in a more compact
434
00:54:56,410 --> 00:55:03,410
form is as the local time derivative of temperature
at a fixed spatial location plus v dot grad
435
00:55:04,069 --> 00:55:10,349
T, this is for three dimensions. This is the
local rate of change sometimes this is called
436
00:55:10,349 --> 00:55:17,349
the local rate of change and this is called
the convicted rate of change.
437
00:55:20,790 --> 00:55:27,790
So, this is the very important concept in
fluid mechanics, because this is a vehicle
438
00:55:30,770 --> 00:55:35,559
that allows us to calculate the substantial
derivative idea. It is a vehicle that allows
439
00:55:35,559 --> 00:55:42,559
to calculate the rate of change of many quantities,
as you follow a fluid particle from a given
440
00:55:43,290 --> 00:55:49,220
time to a later time purely based on Eulerian
description quantities based on Eularian description.
441
00:55:49,220 --> 00:55:54,869
So, we will stop here and we will see you
in the next lecture where we will continue
442
00:55:54,869 --> 00:56:01,869
future in fluid kinematics. Thank you.