1
00:00:16,619 --> 00:00:21,039
Welcome to this lecture number 10, on the
NPTEL course on fluid mechanics for under
2
00:00:21,039 --> 00:00:25,730
graduate students in chemical engineering.
In the last lecture, we discussed in detail
3
00:00:25,730 --> 00:00:32,730
the notions of Eulerian versus Lagrangian
description of motion. So, just to recall
4
00:00:39,530 --> 00:00:44,519
very briefly in the Lagrangian description,
we imagine that a fluid is compressed. Suppose,
5
00:00:44,519 --> 00:00:51,519
you have a box of fluid, a fluid is compressed
of various points, which are called material
6
00:00:53,960 --> 00:00:58,839
points or fluid particles. And these points
can be followed, as a function of time. The
7
00:00:58,839 --> 00:01:04,909
position or location of these various points
can be followed as a function of time. Each
8
00:01:04,909 --> 00:01:11,909
point in the Lagrangian approach is denoted,
is labeled by fluid particles or material
9
00:01:15,039 --> 00:01:22,039
points are labeled by their positions at time
t is equal to 0, which we called x 0 T, where
10
00:01:30,910 --> 00:01:33,580
p stands for particle.
11
00:01:33,580 --> 00:01:40,580
So, the motion of the fluid is described by,
how the position of various particles, fluid
12
00:01:42,700 --> 00:01:47,390
particles changes function of time. And they
are given as the function of that initial
13
00:01:47,390 --> 00:01:54,390
positions x 0 p. So, this is there is no time
here, where x 0 p is definition, x p let us
14
00:01:58,619 --> 00:02:05,619
call this x 0, in the super script x p at
time t equal to 0. So, again we will have
15
00:02:08,410 --> 00:02:15,410
superscript here 0 and time. So, each point
here moves in a specific way and at after
16
00:02:20,069 --> 00:02:25,310
this is a time t is equal to 0 after a time
T, the trajectory of all these points in principle
17
00:02:25,310 --> 00:02:29,159
and infinite set of points are known. And
once they are known, you can compute quantity
18
00:02:29,159 --> 00:02:35,129
such as velocity, velocity of a fluid particle.
Instead of writing v p, we will write velocity
19
00:02:35,129 --> 00:02:42,129
of a fluid particle x p of t at a time t or
other. Velocity of a fluid particle, which
20
00:02:43,650 --> 00:02:50,650
is dented by x p 0, that is the particle which
was there at x p 0 and time t equal to 0 that
21
00:02:52,040 --> 00:02:55,670
is the label of the particle.
And after time t this particle would have
22
00:02:55,670 --> 00:03:02,590
moved to some point here, from here. So, this
is the velocity of a fluid particle, which
23
00:03:02,590 --> 00:03:07,370
was at x p 0 at time t is equal to 0. The
velocity of such particle at time t is given
24
00:03:07,370 --> 00:03:14,370
by the rate of change of it is position by
keeping, the identity constant the label constant.
25
00:03:16,389 --> 00:03:23,389
So, in the Lagrangian a description of motion,
the independent variables the Lagrangian description,
26
00:03:29,680 --> 00:03:36,680
independent variables are the initial positions
and time. This is these are the initial positions
27
00:03:40,379 --> 00:03:47,379
of various fluid particles that serve as their
identity, they serve as the label and time.
28
00:03:48,969 --> 00:03:54,779
So, not just these are not just restricted
to velocity, you can calculate acceleration
29
00:03:54,779 --> 00:04:01,779
of a fluid particle at time T, which is nothing
but the rate of change of it is velocity as
30
00:04:03,389 --> 00:04:09,549
you keep, the identity of the particle the
same and so on. So, once you have the trajectory
31
00:04:09,549 --> 00:04:16,549
of the motion, as described by this equation.
This is the motion, the trajectory of the
32
00:04:18,780 --> 00:04:25,280
particles one can compute, kinematic quantities
like velocity and accelerations. One can also,
33
00:04:25,280 --> 00:04:32,060
extend this to other field, such as temperature
field, temperature of a fluid particle at
34
00:04:32,060 --> 00:04:39,060
a later time. This is imagine, this is a Lagrangian
description, imagine you follow a motion of
35
00:04:39,740 --> 00:04:44,930
a particle, the particle was here at time
t equal to 0 and it is moving at a later time
36
00:04:44,930 --> 00:04:50,030
here.
So, imagine putting a thermometer, attaching
37
00:04:50,030 --> 00:04:55,120
a thermometer to a particle and then you follow
the same particle and then you measure the
38
00:04:55,120 --> 00:05:01,490
temperature history of a given fluid particle.
So, that is t as a function of time. So, there
39
00:05:01,490 --> 00:05:05,960
are Lagrangian function description tells
you the historical information, such as temperature
40
00:05:05,960 --> 00:05:10,710
of a fluid particle as you follow it or velocity
of a fluid particle as you follow it and so
41
00:05:10,710 --> 00:05:11,180
on.
42
00:05:11,180 --> 00:05:18,180
So, this gives you the temperature of history,
of a tagged fluid particle. In contrast, we
43
00:05:22,710 --> 00:05:28,000
also mentioned, this Lagrangian description
is not very practical, because measurements
44
00:05:28,000 --> 00:05:34,919
in labs are often, done based on keeping probes
at fixed location space. For example, we will
45
00:05:34,919 --> 00:05:41,080
keep thermometers or various temperature sensing
devices at various points in space or rather
46
00:05:41,080 --> 00:05:44,180
then follow it along to the fluid particle
normally.
47
00:05:44,180 --> 00:05:51,180
So, in the other description of a used in
the fluid mechanics or the description that
48
00:05:51,879 --> 00:05:58,680
is often used in the fluid mechanics is the
Eulerian description or the spatial description.
49
00:05:58,680 --> 00:06:01,990
The Lagrangian description is also, called
as the material description as I mentioned
50
00:06:01,990 --> 00:06:08,139
in the last lectures, here you imagine you
having a fixed reference frame in your lab.
51
00:06:08,139 --> 00:06:15,139
And suppose have a pipe in which fluid I following,
you can measure, you keep probes with in the
52
00:06:15,740 --> 00:06:22,360
pipe, you can say keep thermometers with in
the pipe. And you can measure for example,
53
00:06:22,360 --> 00:06:29,360
the temperature as a function of x y z coordinates
and time, at fixed at various fixed locations
54
00:06:33,580 --> 00:06:38,050
in space.
So, you can measure, so this is often condensed
55
00:06:38,050 --> 00:06:43,960
in short hand and t as function of x vector
t. So, the independent variables in the Eulerian
56
00:06:43,960 --> 00:06:50,960
vector description of the spatial positions
and time. So, keeping the differentiates,
57
00:06:54,330 --> 00:06:58,770
Eulerian and Lagrangian description of motion
is that the independent variables, that are
58
00:06:58,770 --> 00:07:04,699
used to characterize the motion. And 1, in
1 case you are following the particles in
59
00:07:04,699 --> 00:07:08,710
rather case, in the Lagrangian case you are
following in fluid particles, where as in
60
00:07:08,710 --> 00:07:12,580
the Eulerian, you are sitting at the various
points in space and measuring quantities and
61
00:07:12,580 --> 00:07:13,400
function of time.
62
00:07:13,400 --> 00:07:20,400
So, in the Eulerian description, this is the
Eulerian description by nature of it is of
63
00:07:21,860 --> 00:07:28,860
the description, we will use notion of historical
information. That is, if you point the temperature
64
00:07:29,689 --> 00:07:36,689
at a given point in space at time t. And,
if you find at a later time, t plus delta
65
00:07:39,680 --> 00:07:45,150
t. Suppose, you have flow and you measure,
suppose you have flow you measure temperature
66
00:07:45,150 --> 00:07:51,310
at a point this is the fixed point, this is
the point x. You measure temperature using
67
00:07:51,310 --> 00:07:58,310
a thermometer and at a time x, this point
will be occupied by let us say red particle
68
00:08:00,729 --> 00:08:07,729
at time t at a later time. The same point
x, special location x will be occupied this
69
00:08:10,530 --> 00:08:15,270
red particle, will move at a t time t plus
delta t this red particle move elsewhere and
70
00:08:15,270 --> 00:08:22,069
may be some other particle will come and occupy.
And t plus delta t, this red particle would
71
00:08:22,069 --> 00:08:28,599
have moved from the location x to some other
point. And the location x itself, it will
72
00:08:28,599 --> 00:08:35,140
be occupied by some other fluid particle.
So, what we are merely measuring in the Lagrangian,
73
00:08:35,140 --> 00:08:41,289
in the Eulerian description. I am sorry, that
at a given point is what are the properties
74
00:08:41,289 --> 00:08:46,670
of such as temperature velocity or acceleration
or pressure as a function of time without
75
00:08:46,670 --> 00:08:53,490
worrying about, which particles, belong I
mean without worrying about what is the material
76
00:08:53,490 --> 00:08:57,360
particle, a fluid particle that is occupying
that location.
77
00:08:57,360 --> 00:09:02,700
This is usually ok, except in the case of
acceleration. Suppose, the Eulerian velocity
78
00:09:02,700 --> 00:09:09,700
field
the Eulerian velocity field
79
00:09:21,410 --> 00:09:28,410
is given by v as a function of x t. Once you
have this information, we cannot compute the
80
00:09:30,700 --> 00:09:37,550
partial derivative of v as a function of x
t with respect due to t and call it acceleration.
81
00:09:37,550 --> 00:09:42,420
Because whenever, we take partial derivative,
we keep since the independent variables are
82
00:09:42,420 --> 00:09:46,160
x vector and t vector, when you partially
differentiate with respect to t, we are keeping
83
00:09:46,160 --> 00:09:51,839
the respecter same, that is we are merely
taking the at given point x. What is this
84
00:09:51,839 --> 00:09:56,560
spatial, what is the variation of velocity
at that point as a function of time? Whereas
85
00:09:56,560 --> 00:10:01,680
acceleration really speaking is what is the
velocity rate of change of velocity of a fluid
86
00:10:01,680 --> 00:10:07,730
particle? That is acceleration of a fluid
particle is inherently here like Lagrangian
87
00:10:07,730 --> 00:10:12,320
quantity.
So, in principle, what we must do to compute
88
00:10:12,320 --> 00:10:18,310
acceleration? Is that, suppose you are at
a point x, let us use blue color to denote
89
00:10:18,310 --> 00:10:25,310
the point, this is the physical location x
at time t. This red particle is here and at
90
00:10:26,399 --> 00:10:32,660
time t plus delta t this red particle, would
have in general more away from x. This is
91
00:10:32,660 --> 00:10:39,660
the point x, the red particle, which was originally
here would have moved more elsewhere. So,
92
00:10:41,490 --> 00:10:48,490
the acceleration in principle is defined as
limit delta t tending to 0, velocity of the
93
00:10:50,310 --> 00:10:56,560
particle. Which has which was that x at time
t, sorry velocity of the particle, which is
94
00:10:56,560 --> 00:11:03,560
at x plus delta x. At time t plus delta t
minus velocity of that particle, which was
95
00:11:06,940 --> 00:11:10,730
at x, because this particle was at x time
t divided by delta t.
96
00:11:10,730 --> 00:11:17,730
Whereas, normal partial derivative, such as
this would merely measure, v at x plus delta
97
00:11:18,820 --> 00:11:25,820
sorry, v at x t plus delta t minus v at x
t divided by delta t as limit delta t going
98
00:11:30,699 --> 00:11:36,839
to 0. So, this is clearly not acceleration,
because this is not acceleration because we
99
00:11:36,839 --> 00:11:41,420
are not following the same particle, whereas,
here we are recognizing explicitly the fact
100
00:11:41,420 --> 00:11:46,980
in this definition that, we are following
the same particle. And you have to realizing
101
00:11:46,980 --> 00:11:51,459
the fact that this particle, which was here
at x would have moved to some other location
102
00:11:51,459 --> 00:11:52,630
at a later time.
103
00:11:52,630 --> 00:11:59,630
Now, while this is all fine, normally in fluid
mechanics, we do not have the, we have only
104
00:12:02,570 --> 00:12:09,570
the Eulerian information at hand. That is
we have only v as a function of spatial coordinates
105
00:12:19,259 --> 00:12:26,259
and time. Now, the question is how to get
accelerations from this information, well
106
00:12:29,269 --> 00:12:36,269
we saw this notion of substantial derivative
that will help us to do this, we said this
107
00:12:40,769 --> 00:12:45,220
in the context of we explained in the context
of temperature. Because the temperature is
108
00:12:45,220 --> 00:12:47,300
the scalar and it is much simpler.
109
00:12:47,300 --> 00:12:54,300
So, imagine you have a channel and fluid is
flowing at constant velocity, uniform velocity,
110
00:12:54,410 --> 00:13:01,410
constant velocity and you are at a point x
equal to 0. And this is the point x equal
111
00:13:06,139 --> 00:13:13,139
to 0. And at this point, we had a particle
that was occupied, we had the blue particle
112
00:13:14,180 --> 00:13:21,180
and at an earlier location, x equal to minus
delta x, we had a red particle. So, x is minus
113
00:13:26,220 --> 00:13:33,220
delta x. Now, the distance is minus delta
x now since for simplicity, we will consider,
114
00:13:35,199 --> 00:13:40,300
only 1 dimension that is direction. So, we
will call this x, we will call it x equal
115
00:13:40,300 --> 00:13:47,300
to 0, here x equal to minus delta x. So, the
distance separates delta x this particles
116
00:13:49,420 --> 00:13:54,089
behind, because the co ordinate axis increases
in this direction. So, we have minus delta
117
00:13:54,089 --> 00:13:57,930
x here.
Now, imagine measuring temperature using a
118
00:13:57,930 --> 00:14:04,930
probe such as a thermometer at this point.
So, you are doing an Eulerian measurement,
119
00:14:07,589 --> 00:14:13,790
that x equal to 0, we are putting a thermometer
x equal to 0. And we are measuring temperature
120
00:14:13,790 --> 00:14:20,790
as a function of time, this is a time t. At
a later time t plus delta t, this point blue
121
00:14:24,690 --> 00:14:29,380
point would have moved. So, I am going to
draw it roughly at the same locations. So,
122
00:14:29,380 --> 00:14:33,930
the red point would have moved to blue point,
where the blue point was x equal to 0. And
123
00:14:33,930 --> 00:14:40,930
the blue point would have moved somewhere
else. So, this distance is still this is the
124
00:14:43,220 --> 00:14:50,220
location x equal to 0, let me call this, let
we write this in green ink the spatial location
125
00:14:52,600 --> 00:14:55,639
x equal to 0. Now, this becomes x equal to
plus delta x.
126
00:14:55,639 --> 00:15:02,639
So, let us call this spatial location x equal
to 0 x equal to minus delta x this x equal
127
00:15:03,000 --> 00:15:10,000
to 0, this is now this is time t plus delta
t. The thermometer is keep still at x equal
128
00:15:10,529 --> 00:15:17,529
to 0, this is a thermometer fluid is flowing.
So, fluid motion takes the point, blue point
129
00:15:19,500 --> 00:15:25,339
which was at x equal to 0 to x equal to delta
x. Since the velocity is constant v 0 delta
130
00:15:25,339 --> 00:15:32,339
x will be v 0 times delta t. delta x is uniform
motion velocity is constant at each and every
131
00:15:34,100 --> 00:15:40,420
point in space in time. So, it is a constant
velocity. So, velocity is the displacement
132
00:15:40,420 --> 00:15:47,420
delta x is v 0 delta t. Now, what is a thermometer
measuring at let us also, before I proceed
133
00:15:51,519 --> 00:15:56,620
further, let us also denote the Eulerian labels
of this particles. So, this particle is denoted
134
00:15:56,620 --> 00:16:03,129
by it is position at time t 0. Let us call
it to be consistent with previous lecture,
135
00:16:03,129 --> 00:16:08,899
let us call it t 0. So, all points are marked
by the positions at time t 0.
136
00:16:08,899 --> 00:16:15,759
So, this point, the Lagragian coordinate is
x equal to 0, we need not have the vector
137
00:16:15,759 --> 00:16:21,509
substitutes, because we are just worrying
about one direction here, this is x equal
138
00:16:21,509 --> 00:16:26,870
to minus delta x. Now, even at a later time
the Lagangian coordinate of this point is
139
00:16:26,870 --> 00:16:33,870
simply, still x equal to 0 or let us to be
specific, let us call it x naught equal to
140
00:16:34,029 --> 00:16:39,899
0, this is x naught. And here, the Lagrangian
position of this particle is still, x naught
141
00:16:39,899 --> 00:16:46,899
is minus delta x, delta x is 0 delta t that
is the magnitude of the displacement. So,
142
00:16:47,589 --> 00:16:54,589
the labels of these particles are simply the
blue particle, the Lagrangian labels. Lagrangian
143
00:16:58,879 --> 00:17:05,879
label of blue particles is x 0 equal to 0.
So, I am sorry, I do not need the vectors
144
00:17:06,949 --> 00:17:13,949
symbols and the Lagrangian label for the red
particle, is simply x 0 is minus delta x.
145
00:17:14,069 --> 00:17:20,560
But we also realize that at time t equal to
t naught, it also coincides the blue particle
146
00:17:20,560 --> 00:17:24,610
coincides with the spatial location, that
x equal to 0. The red particle coincides with
147
00:17:24,610 --> 00:17:29,909
the spatial location x equal to minus delta
x. Now, what is the substantial derivative
148
00:17:29,909 --> 00:17:36,909
of temperature? We mentioned in the last lecture,
that substantial derivative is detonated by
149
00:17:38,260 --> 00:17:45,260
a special symbol capital D D t, is simply
D D, the rate of change of temperature with
150
00:17:45,409 --> 00:17:52,409
time as we keep x naught constant. So, that
is the key difference. So, this is nothing
151
00:17:53,190 --> 00:18:00,190
but when we keep x naught constant T of x
0 is minus v 0 delta t minus delta x T naught
152
00:18:08,370 --> 00:18:15,370
plus delta t minus t of x naught is minus
delta x at t 0, divided by delta t limit delta
153
00:18:21,500 --> 00:18:25,460
t going to 0. This is by definition, what
is substantial derivative? The substantial
154
00:18:25,460 --> 00:18:30,400
derivative is the time derivative, as you
follow; let us say the red particle.
155
00:18:30,400 --> 00:18:36,970
The red particle is currently time T naught
plus delta t at a position x 0, x equal to
156
00:18:36,970 --> 00:18:43,970
0. But at the earlier time, it was a position
here. So, let me just draw the red particle
157
00:18:46,350 --> 00:18:53,350
figure it this is the earlier location of
the red particle x equals. Let me just write
158
00:18:54,789 --> 00:19:00,299
it in green color this spatial location, x
is minus delta x, this red particle has moved
159
00:19:00,299 --> 00:19:05,740
from here to here. As you follow the red particle,
how was the temperature changing, that is
160
00:19:05,740 --> 00:19:10,990
the definition of substantial derivative,
but the thermometer measures the completely
161
00:19:10,990 --> 00:19:12,039
different.
162
00:19:12,039 --> 00:19:18,740
What the Eulerian time derivative as measured
by the thermometer, because it is sitting
163
00:19:18,740 --> 00:19:25,740
at a same spoice, same point on space and
so it simply, measuring at a given x, lets
164
00:19:26,400 --> 00:19:33,400
x equal to 0. What is this? This is, limit
t delta going to 0, T of x equal to 0, t naught
165
00:19:40,360 --> 00:19:47,360
plus delta t minus T of x equal, to 0 t divided
by delta t. So, somehow, we should get this
166
00:19:52,539 --> 00:19:59,539
information this is the substantial derivative
as we follow the same material particle from
167
00:19:59,760 --> 00:20:06,570
this information, from this description, the
spatial description. So, how do we do that
168
00:20:06,570 --> 00:20:12,360
very simple terms, we simply take this and
then add and subtract the following quantity?
169
00:20:12,360 --> 00:20:19,360
First, we realize that x equal to 0 t 0 plus
delta t is nothing but so we could also do
170
00:20:26,240 --> 00:20:33,230
it something similar instead of doing it at
x equal to 0, we can call it x not x that
171
00:20:33,230 --> 00:20:39,520
is so this is perfectly fine. So, I am going
to re label the Eulerian independent coordinates
172
00:20:39,520 --> 00:20:46,520
with Lagrangian independent coordinates T,
let us just go to the figure. So, in this
173
00:20:47,130 --> 00:20:54,130
figure at x equal to 0, at time t plus delta
t it is the blue particle that is let me just
174
00:20:59,150 --> 00:21:06,150
go here. At x equal to 0 at time t plus delta
t, it is the red particle that is present.
175
00:21:09,779 --> 00:21:16,529
So, I am going to change the label to the
Lagrangian label, limit x naught equal to
176
00:21:16,529 --> 00:21:23,529
minus delta x t naught plus delta t at x equal
to 0 at time T. It was the particle that was
177
00:21:26,919 --> 00:21:33,260
present at time t or t naught, that is at
x equal to 0 is the blue particle with identity
178
00:21:33,260 --> 00:21:38,690
the x naught equal to 0. So, I am going to
simply write this as, x naught equal to 0
179
00:21:38,690 --> 00:21:44,870
T, and then close the bracket divided by delta
t ok. So, this is the partial derivative.
180
00:21:44,870 --> 00:21:51,110
So, the partial; normal partial derivative
at x equal to 0 is given by limit delta t
181
00:21:51,110 --> 00:21:58,110
going to 0. Now, I am going to add subtract
a term, which is help us identity the relation
182
00:21:59,250 --> 00:22:06,250
delta t minus T x naught is minus delta x
t 0 plus T x naught is minus delta x t 0 minus.
183
00:22:14,980 --> 00:22:21,980
I am adding and subtracting this same quantity
minus T x naught is 0 t, t 0 divided by delta
184
00:22:26,470 --> 00:22:33,470
t. Now, we can identify the following, that
is at this, if you look at this combination
185
00:22:38,600 --> 00:22:45,600
here this is nothing but here you are identifying
the same material particle, the x naught is
186
00:22:46,830 --> 00:22:53,100
minus delta x, but the time is different.
So, this is nothing but this term divided
187
00:22:53,100 --> 00:22:58,640
by delta t as time delta t going to 0.
So, the left side remains normal partial derivative.
188
00:22:58,640 --> 00:23:02,630
So, one of the terms on the right side is
the substantial derivative that, we want to
189
00:23:02,630 --> 00:23:09,630
calculate this is nothing but partial T partial
t as x naught is kept constant, which is minus
190
00:23:09,650 --> 00:23:14,490
delta x here and that essentially, the substantial
derivative plus we have and additional term.
191
00:23:14,490 --> 00:23:21,490
Let me write it, which we will simplify now
T of x not is minus delta x t 0 hence, T of
192
00:23:24,049 --> 00:23:31,049
x naught is 0 t 0 divided by delta t. So,
sorry this is partial by partial t here. At
193
00:23:38,539 --> 00:23:45,539
x equal to 0 is the substantial derivative
plus instead of writing at as delta t, delta
194
00:23:46,909 --> 00:23:53,909
t v 0 delta t is delta x. So, instead of writing
it as delta t, I am going to write this as
195
00:23:58,960 --> 00:24:04,940
so it is delta t, I will write it as delta
x by v 0. So, I will get v 0 limit instead
196
00:24:04,940 --> 00:24:11,340
of delta t going to 0, I will get delta x
going to 0 T of now x naught equal to minus
197
00:24:11,340 --> 00:24:15,520
delta x.
So, let us go to the figure x naught is minus
198
00:24:15,520 --> 00:24:22,520
delta x is x equal to 0 and x naught is minus
delta x at time t 0 x naught is minus delta
199
00:24:24,880 --> 00:24:28,260
x at time t 0 corresponds to the Eulerian
location x equal to minus delta x. So, I am
200
00:24:28,260 --> 00:24:35,260
going to write this as, x equal to minus delta
x
at time t 0 minus x naught equal to 0 at time
201
00:24:44,330 --> 00:24:51,330
t 0 is T, sorry, we remove the bracket here,
this is T and x equal to 0 time is 0. So,
202
00:24:58,309 --> 00:25:02,940
we are converting this Eulerian labels to
Lagrangian labels here to Eulerian labels,
203
00:25:02,940 --> 00:25:09,770
just by looking, where this point x naught
was this is x naught, the x not equal to minus
204
00:25:09,770 --> 00:25:14,210
delta x was exactly at time t equal to 0 at
the spatial location, x equal to minus delta
205
00:25:14,210 --> 00:25:20,380
x, x naught equal to 0. The point, the Eulerian
label, the Lagrangian label corresponds to
206
00:25:20,380 --> 00:25:26,080
the location at time t equal to 0 x equal
to 0 divided by delta x.
207
00:25:26,080 --> 00:25:33,080
This is nothing but so let us forget this
term is nothing but minus partial T by partial
208
00:25:37,409 --> 00:25:44,409
x. So, partial temperature by partial time
x equal to 0, is the substantial derivative.
209
00:25:48,820 --> 00:25:55,820
As you follow I am sorry, as you follow a
material particle, which was at x equal to
210
00:25:56,809 --> 00:26:03,779
0 at time t naught minus v 0 partial T partial
x r.
211
00:26:03,779 --> 00:26:10,779
In other words, the substantial derivative
of a particle, that was at time t equal to
212
00:26:12,279 --> 00:26:16,320
0 at x equal to 0 substantial derivative of
temperature as you follow. The particle which
213
00:26:16,320 --> 00:26:23,110
was at time t equal to t 0, x equal to 0 as
this particle is moving, it is rate of change
214
00:26:23,110 --> 00:26:27,890
of temperature is because of the local rate
of change of temperature at the fixed location
215
00:26:27,890 --> 00:26:34,440
in space at the x equal to 0 plus. A convicted
rate of change of temperature, which happens
216
00:26:34,440 --> 00:26:39,470
because of fact at this particle is moving,
probably moving presumably moving from a region
217
00:26:39,470 --> 00:26:43,080
of lower to higher temperature. So, it is
temperature will change not just, because
218
00:26:43,080 --> 00:26:47,320
of the inherent rate of change at a given
point also, because of gradient. In gradient
219
00:26:47,320 --> 00:26:51,049
space and temperature as you follow the same
point.
220
00:26:51,049 --> 00:26:56,940
So, that is the substantial derivative, this
is the very important result. So, we can generalize
221
00:26:56,940 --> 00:27:03,940
this for any arbitrary velocity, instead of
having V 0, we can generally write this as
222
00:27:05,539 --> 00:27:12,539
partial T and constant x plus V x is a direction
of velocity in the direction plus the gradient
223
00:27:13,899 --> 00:27:20,899
of temperature in the x direction. So, this
is called, the local, this is the substantial
224
00:27:22,010 --> 00:27:29,010
time derivative of temperature, this is the
local rate of change of temperature, this
225
00:27:42,490 --> 00:27:49,490
is the convicted rate of change of temperature.
226
00:27:50,580 --> 00:27:57,580
So, we can also generalize this to 3 dimensions,
when velocity you have velocity vector in
227
00:28:04,330 --> 00:28:10,640
all the 3 direction. So, fluid is flowing
in arbitrary 3 D, this is called 3 D motion,
228
00:28:10,640 --> 00:28:17,640
where V x, V y, V z are not equal to 0, in
which case you can analyze this.
229
00:28:19,000 --> 00:28:24,769
So, the substantial derivative in 3 dimension
is given by substantial derivative of scalar
230
00:28:24,769 --> 00:28:31,769
filed like temperature is given by the local
partial derivative plus v x just by generalization
231
00:28:32,029 --> 00:28:39,029
partial T partial x, plus v y, partial T partial
y, plus v Z, partial T partial z, we can also
232
00:28:42,570 --> 00:28:49,570
use, the symbol from vector calculus this
is nothing but partial rate of change of temperature
233
00:28:50,679 --> 00:28:57,679
at a fixed location plus v dot the gradient
of temperature. Grad t is of course, i partial
234
00:28:59,570 --> 00:29:06,570
T partial x plus j partial T partial y plus
k partial T partial Z this is the gradient
235
00:29:08,510 --> 00:29:14,070
of the temperature. Familiar from gradient
of any scalar field is familiar from vector
236
00:29:14,070 --> 00:29:19,640
calculus like this. So, this is the substantial
derivative of temperature.
237
00:29:19,640 --> 00:29:25,110
Again what this means in the Eulerian context
is the suppose, you have a frame of reference
238
00:29:25,110 --> 00:29:31,470
Eulerian frame of reference in the lab. And
let say fluid is flowing in some orbit limit
239
00:29:31,470 --> 00:29:36,130
and you are sitting at a point in the fluid
aerodynamic, you are putting a thermometer
240
00:29:36,130 --> 00:29:42,899
and measuring temperature. And you get this
information as a function of time, temperature
241
00:29:42,899 --> 00:29:47,970
as function x y z you have to keep many thermometers
and get this information. Suppose, you are
242
00:29:47,970 --> 00:29:54,970
interested in point, you are fixing x and
you are asking. Suppose, I have a particle,
243
00:29:55,690 --> 00:30:02,690
that was here a time t 0, as I follow this
particle by time t 0 what is the rate of change
244
00:30:02,909 --> 00:30:06,019
of temperature? As I follow the particle that
is the meaning of substantial derivative,
245
00:30:06,019 --> 00:30:10,559
this is gradient.
So, the answer is you have a local rate of
246
00:30:10,559 --> 00:30:16,450
change plus a convicted rate of change as
you follow the particle. The particle may
247
00:30:16,450 --> 00:30:21,019
go from regions of 1 temperature to higher
temperature or lower temperature that will
248
00:30:21,019 --> 00:30:25,460
cause a gradient of temperature. And when
you dot that the velocity vector that will
249
00:30:25,460 --> 00:30:29,779
give you the convicted rate of change while,
this is the local rate of change.
250
00:30:29,779 --> 00:30:36,779
So, this is the notion of substantial derivative,
where in you can actually get information
251
00:30:37,149 --> 00:30:42,669
from Eulerian description, this is a Eulerian
description. You can get on this Eulerian
252
00:30:42,669 --> 00:30:47,980
description of temperature as the function
of 3 coordinate directions in time. And how
253
00:30:47,980 --> 00:30:53,519
temperature will change as you follow particle,
which is at a given spatial location at the
254
00:30:53,519 --> 00:30:59,880
given time, this is the meaning of the substantial
derivative. Now, we can also generalize these
255
00:30:59,880 --> 00:31:06,880
2 more complicated objects, such as velocity
temperature was a skill. So, that was much
256
00:31:07,720 --> 00:31:14,720
simpler. So, Dt, Dt was partial t partial
t and constant x, plus v dot del t. So, what
257
00:31:16,799 --> 00:31:23,799
is the substantial, if I have the velocity
Eulerian velocity filed, this is the Eulerian
258
00:31:25,010 --> 00:31:32,010
velocity filed.
Suppose, I have the Eulerian velocity filed,
259
00:31:32,510 --> 00:31:39,510
how do I complete the substantial derivative
of the velocity, what is the meaning, physical
260
00:31:39,630 --> 00:31:45,940
meaning of this? If I have a particle at x
at time t 0, this particle would in general
261
00:31:45,940 --> 00:31:52,940
move to x plus delta x at time t 0 plus delta
t. So, what is the rate of change of velocity
262
00:31:54,049 --> 00:32:00,309
as I follow the particle. And that physically
is the acceleration, in the Eulerian description,
263
00:32:00,309 --> 00:32:07,309
the acceleration of a fluid particle is obtained
by the substantial, this is the acceleration
264
00:32:07,820 --> 00:32:14,820
of a fluid particle
is equal to the substantial derivative of
the Eulerian velocity filed.
265
00:32:24,809 --> 00:32:31,809
Now, the idea is very similar, a is Dv Dt,
just by looking at the previous formula that
266
00:32:45,179 --> 00:32:50,899
was written here, we can write by partial
v, the local rate of change of velocity at
267
00:32:50,899 --> 00:32:57,899
a given spatial location plus v dot grad instead
of temperature here, which was a scalar and
268
00:32:58,279 --> 00:33:05,279
here we wrote v dot grad t well we wrote grad
t. So, this is the Eulerian acceleration field
269
00:33:06,240 --> 00:33:11,539
obtained from the Eulerian velocity field,
nearly by taking the substantial derivative
270
00:33:11,539 --> 00:33:17,700
of the velocity. Now, acceleration is a vector,
so we will have to take individual components.
271
00:33:17,700 --> 00:33:24,700
So, for example, the x component is given
by D vx the substantial derivative of the
272
00:33:25,309 --> 00:33:31,110
x component of the fluid velocity.
So, this is partial v x by partial t plus
273
00:33:31,110 --> 00:33:36,880
v dot grad. So, remember that v dot grad is
the scalar operator, while velocity is the
274
00:33:36,880 --> 00:33:41,240
vector and grad in the vector there is a dot
product. So, this makes it a scalar operator
275
00:33:41,240 --> 00:33:46,000
v dot grad. So, v dot grad will remain as
such and since, we are taking the x component
276
00:33:46,000 --> 00:33:53,000
of this vector a x here. So, you will have
v x here. There is no any need for taking
277
00:33:53,980 --> 00:34:00,630
component here, because v dot grad is already
a scalar operator. Because both v and dell
278
00:34:00,630 --> 00:34:05,980
or vector operators and if you take a dot
product you will get a scalar operator. So,
279
00:34:05,980 --> 00:34:12,980
ax is partial, vx partial t D vx by Dt. This
ax plus vx partial vx by partial x plus vy
280
00:34:19,950 --> 00:34:26,950
partial vx by partial y plus v z partial v
x by partial v z. So, this is how accelerations
281
00:34:30,200 --> 00:34:37,040
can be computed whenever, you have Eulerian
field velocity information.
282
00:34:37,040 --> 00:34:44,040
Now, the next topic that we are going to discuss
is the following. Suppose, how to describe
283
00:34:44,160 --> 00:34:51,160
the notion of flow further? So, we are going
to in this course, in fluid mechanics and
284
00:34:51,430 --> 00:34:58,430
in most fluid mechanics courses are researched
one normally, uses the Eulerian frame of reference.
285
00:35:01,970 --> 00:35:06,940
That is the most convenient from an experimental
point of view and that is the most useful
286
00:35:06,940 --> 00:35:13,350
from a practical view point also. So, Eulerian
frame of reference will be what is used in
287
00:35:13,350 --> 00:35:19,830
this course as well as most applications,
in fluid mechanics will encounter. But the
288
00:35:19,830 --> 00:35:24,000
connection between Eulerian and Lagrangian
are always important, because thats what helps
289
00:35:24,000 --> 00:35:27,120
us to arrive at the notion of the substantial
derivative.
290
00:35:27,120 --> 00:35:31,520
Because even, if you want to work with in
the Eulerian description, it is always useful
291
00:35:31,520 --> 00:35:36,190
to have the notion of substantial derivative,
because only then we can compute quantities
292
00:35:36,190 --> 00:35:43,190
such as acceleration. So, in fluid mechanics,
we will stick to Eulerian frame of reference
293
00:35:43,700 --> 00:35:48,960
and within the Eulerian frame of reference,
the velocity is the vector is denoted as the
294
00:35:48,960 --> 00:35:55,960
function of three spatial coordinates and
time. In fluid mechanics, we will use the
295
00:35:56,200 --> 00:36:02,170
Eulerian frame of reference and the Eulerian
velocity field is given by v, the velocity
296
00:36:02,170 --> 00:36:08,560
vector. Remember that the velocity is a vector.
In general in fluid mechanics all the three
297
00:36:08,560 --> 00:36:15,560
components of velocity will be present. So,
we need to preserve this, v as a vector and
298
00:36:16,070 --> 00:36:20,990
it is a function of three spatial locations,
that you chose to work with and it is also
299
00:36:20,990 --> 00:36:26,030
function of time.
So, imagine you have the situation, where
300
00:36:26,030 --> 00:36:33,030
you have a channel like this. And let us say
that at a give point in space, the velocity
301
00:36:34,900 --> 00:36:41,900
is independent of time. So, at various locations
in this fluid is flowing like this. This is
302
00:36:45,220 --> 00:36:52,220
the channel valve and fluid is flowing like
this at various locations. Let us assume that
303
00:36:53,750 --> 00:37:00,750
for an experimental realization, that velocity
is independent of time. So, at a given location,
304
00:37:01,790 --> 00:37:08,790
velocity, partial derivative of velocity at
any given location is 0.
305
00:37:14,040 --> 00:37:17,670
Then the Eulerian; this is on within the Eulerian
description. This is the Eulerian description
306
00:37:17,670 --> 00:37:24,670
of velocity field, this is the Eulerian description
where velocity is given as function of spatial
307
00:37:26,430 --> 00:37:33,430
positions and time. If at a given location
the partial derivative of velocity with respect
308
00:37:34,670 --> 00:37:41,670
to time 0, that means the velocity of independent
time any location x such flows are denoted
309
00:37:41,890 --> 00:37:48,890
as steady flows. So, in a steady flow in the
Eulerian sense, if you look at various points
310
00:37:51,620 --> 00:37:55,640
and space measure the velocity at each point
of the velocity will be independent of time.
311
00:37:55,640 --> 00:38:01,870
So, take the partial derivative of the velocity
it will be 0 at each point. You fix a point
312
00:38:01,870 --> 00:38:06,500
and then measure the velocity and take it
is time derivative, it will not change. But
313
00:38:06,500 --> 00:38:12,950
this does not mean, study in the Eulerian
and it does not mean, that the velocity of
314
00:38:12,950 --> 00:38:17,590
the given fluid particle is not changing.
For example, if you follow this green particle
315
00:38:17,590 --> 00:38:22,100
from here to here, it is going from the region
of higher cross sectional area to lower cross
316
00:38:22,100 --> 00:38:29,100
sectional area, if mass, if the fluid is incompressible,
then the fluid is incompressible, then the
317
00:38:29,560 --> 00:38:33,970
amount of fluid is flowing here must be the
same amount of flowing in here. As the cross
318
00:38:33,970 --> 00:38:38,930
section area as more here, compare to here,
this fluid particle will get accelerated as
319
00:38:38,930 --> 00:38:44,770
it goes from here to here. So, even though
at given fixed location, the velocity does
320
00:38:44,770 --> 00:38:50,620
not change the time, as you flow fluid particle
it will it can accelerate in general. So,
321
00:38:50,620 --> 00:38:57,620
this is what we mean by the convicted contribution
to the substantial derivative. Even if there
322
00:38:57,780 --> 00:39:04,390
are local even, if it locally studied, that
is study in Eulerian scenes.
323
00:39:04,390 --> 00:39:08,270
Given fluid particle can acquire acceleration
or deceleration by the virtue of moving from
324
00:39:08,270 --> 00:39:13,140
regions of higher velocity to lower velocity
to higher velocity or higher velocity to lower
325
00:39:13,140 --> 00:39:20,140
velocity. So, but in fluid mechanics by steady
flow, we mean that the Eulerian velocity field
326
00:39:20,620 --> 00:39:27,620
is independent of time. Otherwise it is called
unsteady, if not flows are called unsteady,
327
00:39:28,350 --> 00:39:34,490
that is velocity is indeed a function of time
at various points of space, then the flow
328
00:39:34,490 --> 00:39:36,920
is unsteady.
329
00:39:36,920 --> 00:39:43,920
People also use the classification; 1-D in
kinematics, 2-D and 3-D flows and this is
330
00:39:47,560 --> 00:39:53,570
in the following sense, if you have only 1
velocity field that is the fluid is flowing
331
00:39:53,570 --> 00:40:00,570
in only 1 direction. So, imagine you have
a channel and the fluid is flowing only 1
332
00:40:03,220 --> 00:40:09,340
direction, this is x and this is y and the
flow is in the x direction. So, let us say
333
00:40:09,340 --> 00:40:14,860
only v velocity is none 0 and this vx can
in general be a function of the normal direction
334
00:40:14,860 --> 00:40:21,310
y, but the flow is only in 1 direction. So,
we can call this 1-D flow, because there is
335
00:40:21,310 --> 00:40:28,080
only 1 velocity component that is not 0.
And, if the flow is there in 2 direction for
336
00:40:28,080 --> 00:40:35,080
example, if you have a flow like in a square
like cavity and it is moving this is x y,
337
00:40:40,400 --> 00:40:47,010
the top plate is moving with some velocity
v naught the bottom side plates are stationary,
338
00:40:47,010 --> 00:40:54,010
then the fluid will go like this. So, both
vx and vy are not 0 and we can call it 2-D
339
00:40:54,940 --> 00:41:00,830
flow and if all 3 velocities are there here,
sometimes, we can call it as 3-D flow, but
340
00:41:00,830 --> 00:41:07,680
there is not unanimity among various text
books and the nomenclature of such things.
341
00:41:07,680 --> 00:41:14,680
Because in certain conventions, even if vx
is a function of y as in this case, even though
342
00:41:16,010 --> 00:41:21,350
in this 0 velocity, they will call it 2D flow,
because you need 2 directions x velocity and
343
00:41:21,350 --> 00:41:27,820
the y direction to describe flow. But this
is commonly used, but you may find some text
344
00:41:27,820 --> 00:41:34,790
books use this or some conventions use this.
But it useful to just think of single velocity
345
00:41:34,790 --> 00:41:40,850
field being a function of, if you 1 velocity
that is none 0 that we call it 1D flow and
346
00:41:40,850 --> 00:41:47,850
so on. Although, it is clear from the context
what we mean. So, the next thing is how to
347
00:41:48,120 --> 00:41:55,120
visualize fluid motion, how to visualize flow?
In kinematics, we are worried about we are
348
00:42:05,210 --> 00:42:12,210
interested in, how to describe flow, how to
measure various quantities? So, one of the
349
00:42:12,220 --> 00:42:19,220
fundamental descriptions of motion is what
is called the path line. Imagine you have
350
00:42:21,670 --> 00:42:28,360
a liquid and at you can mark a point in a
liquid with a colored dye and let us assume
351
00:42:28,360 --> 00:42:35,360
that, the dye does not diffuse then at time
at some time t is 0. And then this particle
352
00:42:36,730 --> 00:42:43,510
will in general will move to some other location
at a later time t, this is called the particle
353
00:42:43,510 --> 00:42:50,510
path or the path line. This is called the
path line. So, how do we do this experimentally,
354
00:42:51,530 --> 00:42:58,200
well we imagine putting a dye at a point in
space. And then just look at the motion of
355
00:42:58,200 --> 00:43:00,730
the point have been function of time that
is the path line.
356
00:43:00,730 --> 00:43:07,730
So, this is what, we formally wrote as p x
of t as x p as a function of x p naught and
357
00:43:07,830 --> 00:43:14,830
t. This essential what are the Lagrangian
description, can be obtained from experiments
358
00:43:15,270 --> 00:43:22,270
by putting a dot of dye or you can introduce
a puff of smoke in a gas. And you can use
359
00:43:24,150 --> 00:43:30,060
the smoke to visualize the flow and the puff
of smoke will serve as the identity of the
360
00:43:30,060 --> 00:43:34,650
particle, which was at location at time t
equal to 0. And assuming, that the smoke does
361
00:43:34,650 --> 00:43:41,430
not defuse too much within the time scales
of interest, then you can identify, you can
362
00:43:41,430 --> 00:43:48,430
visualize this motion of particle a fluid
particle from time t equal to time t this
363
00:43:49,080 --> 00:43:53,720
is called the path line.
This is inherently a Lagrangian motion, this
364
00:43:53,720 --> 00:44:00,720
has Lagrangian information, the path line
has Lagrangian information. Because you are
365
00:44:01,300 --> 00:44:06,180
following a point identified by it is visual
location through by means of for colored dye
366
00:44:06,180 --> 00:44:10,270
puff of smoke. Then you are viewing it is
evolution as the function of time, special
367
00:44:10,270 --> 00:44:17,270
evolution it is function of time. Now, another
useful notion is called as streak line.
368
00:44:19,160 --> 00:44:26,160
A Strike line is a, what you will get, if
you continuously inject a colored dye at a
369
00:44:41,040 --> 00:44:48,040
point. And you try to
the motion of the dye, I will give an example
after, I am finished with the definitions.
370
00:45:02,730 --> 00:45:09,730
So, what the streak line does is, you are
fixing point and space and you are continuously
371
00:45:09,770 --> 00:45:16,600
introduce dye of that point. And the streak
line is the instantaneous locus of all the
372
00:45:16,600 --> 00:45:21,870
fluid particles, that have been ejected at
the same point at some earlier a time. So,
373
00:45:21,870 --> 00:45:27,060
you are continuously ejecting fluid particles
from time t equal to 0 at the same location
374
00:45:27,060 --> 00:45:33,150
and you are trying to see, what are the various
locations of particles of that are being introduce
375
00:45:33,150 --> 00:45:39,440
from time t equal to 0 at a later time.
So, again simple example is suppose you have
376
00:45:39,440 --> 00:45:45,560
a smoke chimney, from which smoke is coming
you are consciously injecting smoke and let
377
00:45:45,560 --> 00:45:52,560
say air is moving. So, the path that is taken
by the smoke is an example of a streak line.
378
00:45:52,690 --> 00:45:57,460
Because you are continuously injecting a black
or brownish colored smoke from a chimney and
379
00:45:57,460 --> 00:46:04,340
you are trying to locate and you are trying
to visualize motion of this colored, you know
380
00:46:04,340 --> 00:46:11,340
particle in air. Another very useful notion,
this is again this has Eulerian information,
381
00:46:11,630 --> 00:46:18,630
because you are not you are basically, worrying
about what stuff being introduced at a point,
382
00:46:18,700 --> 00:46:25,700
but various particles will come and by occupying
at that point. So, this as in some sense Eulerian
383
00:46:26,960 --> 00:46:33,960
information, finally, we have stream line,
stream line is a mathematical idea. It is
384
00:46:40,830 --> 00:46:45,840
we will have to see how it is visualize experimentally,
but it is concept.
385
00:46:45,840 --> 00:46:52,840
The concept is that suppose, you imagine in
the Eulerian description, you have the velocity
386
00:46:53,520 --> 00:46:59,880
vector as a function of three special coordinates
in time. Let us look at a given time at a
387
00:46:59,880 --> 00:47:05,180
given time t at various spatial locations,
you can plot how the velocity vector is going
388
00:47:05,180 --> 00:47:11,220
to look like. And you can plot the magnitude
by showing a larger arrow and the direction
389
00:47:11,220 --> 00:47:16,710
by the direction of the arrow. And if the
particle, the various points of the velocity
390
00:47:16,710 --> 00:47:23,340
field has different values, you can show it
by the both direction as well as the magnitude.
391
00:47:23,340 --> 00:47:30,340
Stream line is a line that is instantaneously,
tangential to the local fluid velocity vectors.
392
00:47:31,020 --> 00:47:38,020
So, let me try it, try to draw this as tangential
as possible as.
393
00:47:39,050 --> 00:47:46,050
So, stream line is a line, that is locally
tangential to each, to the velocity vector
394
00:47:55,270 --> 00:48:02,270
at each and every point on the at a given
instant of time, each point in the fluid at
395
00:48:12,080 --> 00:48:19,080
an instant of time at the given time. So,
stream line is basically an idea, but it comes
396
00:48:23,320 --> 00:48:28,260
to we have to understand, how it is we have
to prescribe, how this is measured experimentally
397
00:48:28,260 --> 00:48:34,700
or how it is visualized experimentally. So,
we will illustrate this through an example.
398
00:48:34,700 --> 00:48:35,870
So, imagine.
399
00:48:35,870 --> 00:48:42,870
So, I am going to illustrate, the notion of
path line, stream line and streak lines through
400
00:48:42,920 --> 00:48:49,920
an example. Imagine you have, North four directions,
East, south and west. Let us say wind is blowing
401
00:48:55,100 --> 00:49:02,100
from west to east, let us say air is blowing
or being blown from west to east. So, it is
402
00:49:02,970 --> 00:49:09,970
completely parallel to the east direction.
So, if you look at the stream lines, they
403
00:49:12,720 --> 00:49:19,720
look completely parallel and assume that the
flow is that steady. It is each and every
404
00:49:20,290 --> 00:49:25,100
point that the velocity vector is not changing
with respect to time. So, flow is steady in
405
00:49:25,100 --> 00:49:29,740
the stream lines will look like this. Let
us look at path lines, path lines you take
406
00:49:29,740 --> 00:49:36,610
any point and you inject a dye at that point,
time t equal to 0 and local and watch it is
407
00:49:36,610 --> 00:49:40,980
motion is at later time.
So, this point would have moved at a later
408
00:49:40,980 --> 00:49:47,980
time, but it would also be line that is parallel
to the stream line. Now, so it will be identical
409
00:49:49,370 --> 00:49:54,220
to the stream line, because you can inject
a particle here, it will move exactly parallel
410
00:49:54,220 --> 00:50:01,220
on that stream line itself. So, in the steady
flow, in the Eulerian sense, the path line
411
00:50:02,540 --> 00:50:06,500
and steam lines are the same and the streak
line will also be the same. Because if you
412
00:50:06,500 --> 00:50:13,500
inject continuously inject smoke or a point
then of course, this will if you continuously
413
00:50:18,500 --> 00:50:23,810
keep injecting smoke at this point this will
keep move. So, simple realization is that,
414
00:50:23,810 --> 00:50:29,820
you have a chimney, let me rewrite this. So,
you have this stream line, so they are parallel
415
00:50:29,820 --> 00:50:36,820
air is blowing from west to east. Imagine
that you have a chimney at a location, x naught
416
00:50:39,450 --> 00:50:46,450
y naught at point P.
So, the path lines are the trajectory of point
417
00:50:46,720 --> 00:50:51,320
that was released at time t equal to 0 at
this location. So, that will also be parallel
418
00:50:51,320 --> 00:50:55,540
in a steady flow, this you just keep going.
Streak line will also be just identical to
419
00:50:55,540 --> 00:50:59,530
path line, because the flow steady it will
continue to move in the same direction. And
420
00:50:59,530 --> 00:51:05,620
all this will be identical to stream lines,
you can introduce instead of imagining here,
421
00:51:05,620 --> 00:51:09,270
you can draw another streak line here also.
Berceuse streak lines are completely parallel
422
00:51:09,270 --> 00:51:14,290
in this simple example, because the velocity
vectors are completely parallel to each other.
423
00:51:14,290 --> 00:51:21,290
So, in a steady flow, the path lines, streak
lines and stream lines will merge, will be
424
00:51:32,510 --> 00:51:39,510
identical, what happens if the flow becomes
unsteady.
425
00:51:42,190 --> 00:51:49,190
We will illustrate with the same example,
imagine that at this is north, east, south
426
00:51:55,100 --> 00:52:02,100
and west. So, imagine that at some time initially,
the flow is from East to West. Up to time
427
00:52:06,110 --> 00:52:13,110
t naught at some time t naught the flow changes,
from North-West to South East. So, at some
428
00:52:15,780 --> 00:52:22,300
later time, the flow instead of it is being
like this. And are continuously, injecting
429
00:52:22,300 --> 00:52:28,610
smoke from a chimney, that is what we would
imagine. And we are looking at a later time
430
00:52:28,610 --> 00:52:35,610
t, greater than t not what is the status of
the path line and stream line and streak line.
431
00:52:36,760 --> 00:52:41,110
Well stream lines at a later time t greater
than t naught. Steam lines are instantaneous
432
00:52:41,110 --> 00:52:46,710
descriptions of lines that are parallel to
fluid velocity vectors. If the velocity vectors
433
00:52:46,710 --> 00:52:52,680
are all parallel to the North-West to South-East
direction, you will simply see that the stream
434
00:52:52,680 --> 00:52:59,680
lines will be at an angle to the ok. There
will be at an angle like this.
435
00:53:04,280 --> 00:53:11,280
So, these are the stream lines. So, stream
lines are in red. What about the path line?
436
00:53:13,180 --> 00:53:20,180
The path line, I am going to show it in green.
So, you take a point P in which you have introduced,
437
00:53:21,440 --> 00:53:28,440
we have introduced a point at time t equal
to 0. So, this point will be moving from in
438
00:53:30,370 --> 00:53:36,210
from the North, West to East direction and
then at time t is 0, you are changing that
439
00:53:36,210 --> 00:53:42,280
the air is change in the direction from West
to East to North-West to South-East. So, this
440
00:53:42,280 --> 00:53:49,280
trajectory at this is t less than 0, t greater
than t 0, it will come here. So, the green
441
00:53:51,170 --> 00:53:57,530
lines are path lines, the red lines are stream
lines.
442
00:53:57,530 --> 00:54:04,530
Now, what about streak lines, which I am going
to plot in blue. You are continuously injecting
443
00:54:05,430 --> 00:54:11,260
material or dye or smoke from at this point,
the dye that was introduced at t equal to
444
00:54:11,260 --> 00:54:15,920
0, it would have a trajectory that identical
to path line. So, I am going to draw the motion
445
00:54:15,920 --> 00:54:20,620
of streak lines here, because it can be confusing
this. So, I am going to so you are introducing
446
00:54:20,620 --> 00:54:27,620
continuously and for reference to plot this
path line. The path lines are clear and time
447
00:54:28,010 --> 00:54:33,980
t equal to 0 and injecting something, it will
travel up to t 0 here and then t 0 to t, it
448
00:54:33,980 --> 00:54:40,700
will go in the North-West to South-East direction.
What about streak lines? Streak lines, will
449
00:54:40,700 --> 00:54:45,100
be slightly different, I will plot in blue,
that point which was introduced at t equal
450
00:54:45,100 --> 00:54:52,100
to 0. It will go all the way here, and it
will reach here, let me use blue color ok.
451
00:54:54,520 --> 00:55:00,850
But the one, it is introduce at time delta
t greater than t equal to 0, It will not have
452
00:55:00,850 --> 00:55:05,710
reached up to here, it would have reached
up here and it would have changed, it is direction,
453
00:55:05,710 --> 00:55:11,040
because of the change in direction even it
would have reached here. And like wise things
454
00:55:11,040 --> 00:55:18,040
that are introduced before t 0, they would
go up to here and they would and the stuff
455
00:55:18,900 --> 00:55:24,000
that is introduced just before t 0 will be
here and it will reach here. The stuff that
456
00:55:24,000 --> 00:55:31,000
was introduced after t 0 would directly follow
this line. So, this is the streak line, while
457
00:55:32,470 --> 00:55:38,870
this is the path line. So, the path line and
streak lines and stream lines will not agree
458
00:55:38,870 --> 00:55:44,680
for the unsteady flows. So, we will stop here
and we will continue from here in the next
459
00:55:44,680 --> 00:55:51,680
lecture and we will see you in the next lecture.