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.