 We will start today with the discussions on fluid kinematics. As you understand when we will be concentrating on fluid kinematics, we will be somewhat abstracted from whatever are the forces which are giving rise to the motion. But primarily, we will be now concentrating on the nature of motion. That means displacement, velocity, acceleration, when we say motion we also mean angular motion, deformation. We have earlier seen that deformation is one of the very important criteria for demarketing a fluid as compared to that of a solid. So we will try to see that how we can characterize the kinematic features of a fluid motion. To do that, we have to go through certain formalities because we need to first appreciate that there is a necessity of describing the fluid motion in a manner different from that of a solid. Usually whenever we are dealing with particle mechanics, what we are trying to do? Most of the times you have like a particle. You are trying to find the evolution or locus of the particle as it moves with time. So you have an identified particle which is tagged like this. This particle at different instants of time comes to different positions and the locus of the particle can be obtained by joining those positions successively. This type of approach is known as Lagrangian approach. So Lagrangian approach is nothing but tracking of individual particles. So you identify particles and you tag those particles and track the particles so to say as they evolve. This Lagrangian approach is good for particle mechanics but when you come to fluid mechanics you see that it has certain limitations. Not that it cannot be employed in principle but when it comes to practice because fluids have numerous particles well solids also have numerous particles but the difference is that fluid particles when they are in motion they are continuously deforming. So it is very difficult to keep track of individual particles and therefore this type of description of motion is not so convenient. If it was convenient how would we have described that motion? So let us say that we would have set up a coordinate axis like say x, y, z and let us say that this was the initial position of the particle which we try to designate by some position vector say r0. So the position vector at any other instant of time say the particle is at this location at an instant of time t and the position vector is r. So r at time t we can say in this approach is a function of what? It is a function of the initial position and the time that has elapsed. So when you describe the displacement in this way or the position vector rather then this is a Lagrangian description. As we have discussed that for fluid it is difficult because you have numerous particles which are continuously evolving over space and time. So for fluid there is a more convenient approach which is known as the Eulerian approach. So in the Eulerian approach what we are doing? You are not tracking individual particles. Instead of tracking individual particles what you are trying to do? You are trying to focus your attention not on a particle but on a position. So let us say that you are focusing your attention on this position. Now what is happening? There may be many particles which had come to this position and had left. You are not bothered about which particle has come from where and which particle is going where as if you were sitting with a camera and the camera is focusing on a position. You are bothered about what is the change that is taking place across this position. So what you are bothered about is that maybe what is the velocity at this point? So velocity at this point is nothing but the velocity of a fluid particle at that instant which is passing through that point. So what is a fluid particle? I mean there are involved concepts associated with it but in a very simple term it is just like an inert tracer particle moving with the flow. So it does not interact with the flow. It does not have any density difference with respect to the flow. So it is just a passive inert particle moving with the flow. So if such was a particle then at a particular instant the velocity of the particle which is seen by an observer who is focusing the attention here is known as that velocity is known to be a velocity that is obtained by using Eulerian approach. Again that velocity would be same as the velocity of a tracer particle that at that instant of time was passing through that position. So what you can say from here is that we can write the Eulerian description in the following way. So we can write just like in this way we could write the Lagrangian approach. The Eulerian approach we can say that the velocity at a particular position is a function of what? It is a function of the position r and the time t because it also varies with time. So the key difference is here you focus your attention on a specified r across which you are trying to see the changes in the fluid property. So here velocity is the fluid property that you are looking for as an example. Regarding the fluid particle one important concept is like this that all of us know that fluids are comprising of molecules at the end. So where from the concept of particle comes? Now if you try to extrapolate the concept of molecules to particles then it can be thought of in this way. So there may be a collection of molecules in the fluid which on a statistically average sense have a common group behavior and that collection may be conceived as a fluid particle. And the behavior of that is equivalent to the behavior of a inner tracer particle put in the fluid and it just moves with the flow. So it does not do anything it does not interact with the flow it is so passive that wherever flow takes it it just goes there and it does not have any density difference. So it does not have a net buoyancy force also because of its existence on the flow and such a particle which has the density same as that of the fluid conceptually is known as a neutrally buoyant particle. That means a particle which does not have any resultant buoyancy effect because of its interaction with the flow. That means the densities of the fluid and densities of the particles are considered to be the same. Clearly the Eulerian approach is easier for implementing for fluids because when you are implementing that you are abstracted of individual particles. You are only concerned on a particular position and you see that some particle is coming and some particle is going you are not bothered about the identity of the particles which are coming there and leaving that place and that makes it more convenient for employment for analysis of fluid flows. So we will mostly be bothering about the Eulerian approach keep in mind that the basic laws of Newtonian mechanics were originally derived by using Lagrangian approach. So a major emphasis on fluid mechanics will be to convert those expressions into equivalent forms which are implementable using the Eulerian approach. So the expressions that we will be writing say equivalent to Newton's second law of motion in terms of Eulerian approach will not exactly look in the same form as we are familiar with for simple particle mechanics for solid mechanics say. But we will see at the end that these approaches are inter convertible and it is possible to derive one approach or expressions in perspective of one approach from the expressions of the other and that we will take up in a later chapter. But we will keep in mind that Eulerian approach is something that we will be trying to follow for most of the problems in fluid mechanics. Now regarding the use of the Eulerian approach we will try to understand or we will try to learn some basic terminologies. The basic terminologies are as follows first concept is steady and unsteady flow and do we say that a flow is steady whenever we are trying to focus our attention at a point say by following the Eulerian approach we are interested to see that the flow velocity or fluid properties at that particular point are changing with time or not. So if at a specified location the fluid properties or the flow velocity are changing with time then we call it an unsteady flow but if that is not changing with time then we call it a steady flow. What are the keywords here that we are having a fixed position and what we are trying to see is the fluid properties. So when we say fluid properties we have discussed many fluid properties maybe density maybe viscosity whatever fluid properties and flow velocities do not change with time. If that is the case at a fixed position then we say that it is a steady flow. If that is not satisfied as obvious that is unsteady flow. We will learn a related but not the same concept that is uniform and non-uniform flow. What is that? Just like steadiness discusses about change with respect to time at a fixed position. Uniformity discusses about change with respect to position at a fixed time that means let us say that you have a domain in which flow is taking place. In technical terminology sometimes we call it a flow field. What is a field? Field is a domain over which some influence is felt. So when you have a flow field it is basically a domain over which the influence of the flow is felt. So you are having some velocities at different locations. So in the flow field at different points let us identify different points. Let us say that these different points have different velocities. So let us try to mark these different velocities. Say that different velocities are like this. If they are such that at a given instant of time they are same at all points then we say that the flow field is uniform. So not only the velocities but also again the fluid properties. So fluid properties and flow velocities. So what we are doing now? We are fixing the time. So fixed time that is our concern. At a fixed time we are trying to see that fluid properties and flow velocities are invariant with position. Do not change with position. So if that is the case that at a fixed time we are finding that fluid properties and flow velocities do not change with r we call it a uniform flow field. So the distinction between steadiness and uniformity is quite clear. Let us ask ourselves certain questions related to this. If the flow is steady could it be non-uniform? That is very much possible. So if you have a domain like this with different points. So you have velocity at different points different but whatever those are those are not changing with time. So it is possible to have a steady but non-uniform flow. Is it possible to have a uniform but unsteady flow? So 50% of you are saying yes and 50% no. Let us look into this example. So let us say at some time instant these are the velocity vectors. Let us say in the next time instant you have the velocity vectors like this. Assume that they are all same. So with change in time at a given point if you focus your attention you can see that the velocity has changed. So it is unsteady but at a given instant of time it is same at all locations. So it is uniform. So it is possible that you may have unsteady but uniform flow just like it is possible to have steady and non-uniform flow. So the steadiness and uniformity should not be confused as equivalent concepts. They relate to 2 different things. Of course there may be relationships between these 2 in very special cases. The other important remark is that whenever we are talking about the steady and unsteady flow we have to keep in mind that steadiness or unsteadiness is not something which is absolute. It depends on the choice of the reference frame with respect to which you are analyzing the flow. And this type of choice of reference frame is very important in mechanics. Fluid mechanics is of no exception. Let us try to understand this through an example again. Let us say that there is a river. On the top of the river there is a bridge and there is an observer standing on the bridge. The whole idea is to observe just qualitatively observe how the fluid flow is taking place below the bridge on the river. Now suddenly a boat comes. The boat approaches the bridge and it just crosses the bridge below it. So what happens to the water? The water was say earlier stagnant. Now the boat has come so it has disturbed the water and the boat has gone again after sometime the water will come back to its original state. So to the observer who is standing on the bridge how the flow will appear? It will appear to be changing with time. So initially so he is focusing his attention on a particular location. He is finding initially it is stagnant then velocity is changing because of disturbance created by the boat and then again the velocity is coming back to its original state. So it is a strongly unsteady flow. Let us take the same example in a different perspective. Say the boat is moving relative to the river at a constant velocity and a person sitting on the boat is trying to observe the flow. So when the person sitting on the boat is trying to observe the flow of the same river what is the interpretation? Because the velocity of the water relative to the boat is not changing and the person is only observing that relative velocity for that person the same flow of the river water is appearing to be steady not changing with time. So the same flow may appear to be steady or unsteady depending on the choice of the reference frame. So these 2 were 2 special reference frames. One is a fixed reference frame just like the bridge. Another is a reference frame that is moving with a boat like moving with something at moving with a reference at constant velocity. And you can see that with respect to the reference frame moving at a constant velocity it is possible to analyze a flow which is otherwise unsteady in terms of a steady concept. That means it is possible to convert the notion or transform the notion from an unsteady to a steady flow by such a transformation of reference. And this in mechanics is known as Galilean transformation. So we will try to maybe have a visual example of what type of flow takes place as a boat is moving. I cannot show you the bridge in this example but at least I can show the boat. So we will try to follow very clearly that what is happening across the boat as the boat is moving. It is an artificial tunnel. This tunnel is not very wide and just see that what is happening to the water. Water is not of course very clean but it will help you to visualize it better. See the boat was moving relative to the water at a uniform speed but what you can see is that after the boat has gone there is a change in the velocity pattern. We are now observers with respect to a fixed reference frame and we are observing it to be changing with time. See one important thing is we are not talking about inertial and non-inertial frames. Both are inertial frames because when something is moving with a uniform linear velocity that is still an inertial frame. So do not confuse this with the concept of inertial and non-inertial frames. So Galilean frame is still an inertial frame because it is moving with a uniform velocity. It is not an accelerating reference frame that we need to understand. So with this background what we will do next we will try to see that we have got a basic concept of what should be the description of a flow in terms of change in position, change in time and the description of the flow velocities. The flow velocities as you can see that the velocity vector v these may have their individual components like components along x, y and z. In fluid mechanics we give them certain common notations. The velocity component along velocity component along x we give a name commonly as u. Velocity component along y we give a common name v and velocity component along z as w. So these are just names. I mean just notations so to say commonly used in textbooks. In the index notation of course you can use ui where i equal to 1 will imply vx, i equal to 2 will imply vy and i equal to 3 will imply vz because it is a vector. So we have seen how to write vectors in index notations. Velocity is of no exception. Now whenever a flow field is having a velocity it is not so easy to map the velocity vectors in a flow. So there should be some mechanisms by which we are trying to develop a feel of how we visualize a flow and for that we have to consider certain imaginary lines in the flow. Just like you have lines of force in a magnetic field by which you try to visualize how the magnetic forces are acting. Similarly whenever you have a flow field you must develop some concept of imaginary lines. These lines are not exactly existing in the flow but as if these lines were there to aid you to visualize the flow both qualitatively as well as quantitatively. So we have earlier seen some examples of flow visualization. Maybe what we will do is we will try to see some more examples to see different methods of flow visualization. See this first example. So in this example it is like flow in an artery junction. So there are certain tracer particles. So how this flow is visualized? You have tracer particles. So these particles may be very small beads of may be nanometer size or few micron size and these beads are put in the flow and the flow is illuminated by say a laser source. So in that illuminated condition when the particles are moving it is possible to not track individual particles all particles as such but it is possible to statistically track them that means it is possible to get a statistical picture of the displacement and velocities of these particles and from that it is possible to have a post processing and map the velocity field also by a statistical operation on the motion of selected particles and that principle is followed in one of the devices which we commonly use for flow visualization in advanced research applications known as particle image velocity or PIV. Now we will not go into such analysis for this elementary course but if you just try to look into some other examples. See this is particle visualization. This is also particle based visualization but visualization through smoke emission. So you can clearly see basically a wing of aircraft has passed and when it has passed it has created a vortex see the rotations and by the smoke emission that is being visualized. So by illuminating with a smoke, smoke is a natural emission and it is possible to visualize the flow at least qualitatively by using that. The same type of thing that is possible to be observed in the chimney of a power plant and we have seen this example earlier when we were discussing about the introductory concepts and you can see that that also gives rise to a good visualization of the flow field. When you have these particles put in the flow and those particles are in forms of beads. So you can see that nice rotating structures or vortices can be observed with these particles. So these are called as tricks and we will see that why these are called as tricks. We will look into one or two more visualizations with the tricks and then we will try to formalize that how to mathematically write or express these concepts of visualization. This is a numerically or computationally simulated flow with the help of synthetic particles. So these are not really particles in the physical sense but this is just the entire flow is simulated in computer to have this kind of visualization. Now what we will do? We will try to see that what are the conceptual lines which are important for quantifying these visualizations and for that we will learn certain concepts. So the first concept that we will learn is something which you have heard of earlier and that is the concept of a streamline. So we are discussing about some conceptual paradigms which help us in visualizing fluid flow. So when we say streamlines, how do we define streamlines? Streamlines are imaginary lines in the flow field. These are not existing in reality. So imaginary lines, what type of imaginary lines? These are such lines that at an instant of time, tangent to the streamline at any point represents the velocity vector at that point. So if you have a streamline like this say, so when you have a streamline like this, you may have tangent to it at different points and these tangents are representatives of the velocity vector at these points. One important concept that we miss many times is that it is defined at a particular instant of time. That means at different times you may get different streamlines. Only when the flow field is not changing with time, as the steady flow you get same streamlines at all instants of time. Otherwise you may get different streamlines. Now to express the streamline in terms of some equation, so this is a line, this is a locus, so it should be expressible in terms of certain equations. Let us try to see that how we can express that. Towards that we will first recognize that let us say that there is a point P or P1 located on the streamline. The fluid particle at a particular time, the fluid particle is located also here. It is coincident with this point. When the fluid particle is coincident with this point, then after some time the fluid particle has come to a different point and so on. We are not bothering about the motion of the fluid particle. We are just bothered about say 2 points which are located on the streamline which are quite close to each other. Say P1 and P2, not that P2 represents the location of the fluid particle at a different time not like that. Just it is another point on the streamline which is very close to the point P1. The vector P1, P2 let us say we denote it by a change in position vector dr and let us say that v is the velocity at that particular point. When we give it a name dr, we have to keep in mind that it is very small and it is differentially small. It is as good as writing delta r as delta r tends to 0. So when delta r tends to 0, then what is the status of the points P1 and P2? They are almost coincident. When they are almost coincident that means P1, P2 then represents tangent to the streamline at the point P1. What is the tangent? Tangent is the limit of a chord in the limit as the gap, the distance between the 2 points becomes infinitesimal. So in the limit P1, P2 becomes tangent to the streamline at the point P1. So dr in that differential limit is tangent to the streamline at which point at P1. By definition v is also tangent to the streamline at P1. That is the definition of the streamline. That means these 2 are parallel vectors. If these 2 are parallel vectors, their cross product should be a null vector. So you have dr cross v is a null vector. So we can write dr and v in terms of components. You have r as how do you write r, xi, yj and zk where i, j, k are the unit vectors along x, y, z. So you can write dr as dxi plus dyj plus dzk. So when you write this cross product, it is possible to write it in a determinant form. So let us write that i, j, k, then components of dr, dx, dy, dz and components of the velocity vector uvw. That is equal to null vector. We can easily see that it boils down to 3 scalar equations for each for the x, y and z components. So what are these scalar equations? wdy minus vdz equal to 0, then wdx minus udz equal to 0 and vdx minus udy equal to 0. So if you combine these 3 together, you can get a compact expression dx by u equal to dy by v equal to dz by w which is nothing but the equation of the streamline. This is the locus that we are looking for. You can easily obtain the locus by keeping in mind that uvw are functions of position like x, y, z and also time but when you are considering a streamline, you are freezing the time. So at a given instant of time, so that does not become a variable in this case. So x, y, z are the variable. So if these are substituted as functions of x, y, z, you may integrate these to find out the locus. That is very straightforward. Later on we will work out some examples to illustrate that how we can do that. So this is the concept of a streamline. Now related to this concept, there are certain other terminologies again. Sometimes they are confusing because streamline is more commonly used. Those are not very commonly used but those are sometimes more fundamental and more relevant than the streamline. So we will see the next example that is called as a streak line. So what is a streak line? Let us say you want to visualize a flow. We will try to identify the concept from where it has come. So when you want to visualize a flow, say this is a flow field, you want to visualize a flow. A very common technique is what? You take an injection syringe. In that injection syringe, you take some color dye, say a blue color dye. A common name of a blue color dye is called as thymol blue. So you have taken a thymol blue. It looks like the ink. So when you have taken that blue color dye and say that you are trying to put that blue color dye, inject that blue color dye through this point P. So the blue color dye is coming here through an injection syringe. So now you are going on injecting the dye here. So what is happening? Whatever fluid molecules or fluid particles which are passing through this point at different instance of time, they are illuminated by the dye. And so wherever they go, that tag of illumination remains. So when you get an illuminated line, it is at a particular time, then what does it represent? It essentially represent locus of all fluid particles which at some earlier instance of time pass through this point of dye injection. That is how they were colored by the dye. So when you see a colorful line in the flow field, it represents that locus and that locus is called as a streak line. So what is the streak line? Streak line is the locus of all fluid particles which at some prior instance of time, all of which had passed through a common point. Mathematically we stop here, but physically we understand that common point is the point of dye injection. So that is called as a streak line. Let us try to conceptually draw a streak line. Let us say that this is a point at which dye is injected. So when a fluid particle passes through this point, say at time equal to t0. So when at time equal to t0, the fluid particle passes through this point. The fluid particle then undergoes a locus. So what is this locus? We introduce a third line which is called as a path line. Path line is the simplest and the most trivial concept to understand. It is the description of a flow from a Lagrangian view point. So it is a locus of an identified fluid particle. So if you identify a fluid particle, how it moves? The path traced by that is called as a path line. That is very simple and trivial, requires no explanation. So when we want to draw different path lines, see at say time t equal to t0, you have one fluid particle which pass through this point of dye injection. That fluid particle at subsequent instance of time, it is passing through different points. So this is the path line 1. Say what is that path line 1? Path line 1 is the path line of a fluid particle which pass through the point of dye injection at time equal to t0. Let us say there is another fluid particle which has pass through this at time equal to t1. And let us say that this red line represents its locus. So this is something which was injected at t equal to t1. The path may be different because it may be an unsteady flow. So it is possible that the velocity field has changed with time. So when it has changed with time, the particle may be forced to move along a different locus. So this is path line 2. Let us consider a third path line maybe for completeness. Let us say that we have a third path line like this which corresponds to the injection here at time equal to t3. And again the path line is different because the velocity might have changed at different points with time. So this is path line 3. Now say we are bothered about at time equal to t, say now. So at time equal to t, that particle which passed earlier through this point at t0, say now it is here, that particle which passed through this point at t1 is now at here and that particle which passed through this point at say t3 is here. Right now at time equal to t, say there is one more fluid particle that is injected just here if it is a continuous process. So the locus of all these which at some earlier instance of time pass through the point at t1 is of die injection that locus is now the streak line at time t. So we can clearly demarcate between streak line and path line. Let us take the example of a very special case but a very interesting case. What is that special case? That special case is for a steady flow. So when you have a steady flow then let us try to draw these path lines. So first let us draw the path line of that particle which passed through this point of injection at t equal to t0. So let us say that that path line is this one. Now another particle which passed through this at t equal to t1, that will also follow this line because with time the velocity field has not changed. So it will be constrained to follow the same line. So it will follow this line. Similarly the third one that is the one injected at t equal to t3 that will also be constrained to follow this line. And what is the specialty of this line? This line is the locus of the fluid particle. So that means at some time tangent to this line depresents the direction of the velocity vector. So we can understand that this is also a stream line. This is also the path line and again this is also the streak line because whatever are the locations of fluid particles those are always constrained within this line. So we can say that for steady flow stream line, streak line and path line are identical. That is one very important concept that we should remember that in this case you have stream line, streak line and path line are identical. They are identical. So whenever we visualize a flow let us look into some example maybe some images through stream lines or streak lines. So let us say that we see first this example. You can see these are streak lines. So, die streaks which are injected at the tip. You can see that now at different instance of time they are forming different colored images. And if you track individual one then it is like a path line. Now if you see let us look into a path line example. See this is like a lawn sprinkler. So many times it is used to sprinkle water in a garden. You can see that if you track the water droplets you can clearly see that the path what they are following. So it is something like a visual representation of a path line. Let us look into a stream line example. So these are 3 cylinders in a steady flow and you can see that this green color die is giving an appearance of a stream line. Fundamentally this is actually a streak line. But because it is a steady flow stream line, streak line, path line these are all identical. So these represent different streak lines or different stream lines or different path lines whatever you say if it is a steady flow. If it is an unsteady flow these will represent streak lines rather than stream lines. So you can clearly see that these visualizations of fluid flow that we have seen as concepts these may also be visualized in experiments. Many times we are interested to construct the velocity vector. So see the example of the boat that we saw earlier and you may construct such velocity vector. So it is not a direct visualization but you may do it in 2 ways by post processing the visualization of the particles which are injected into the fluid or by doing a computer simulation. And sometimes these may be equivalently compared. Computer simulation of course is an idealization because you are using certain boundary conditions, certain properties which might not exactly prevail in reality. But sometimes it gives a very important idea of how the fluid flow is taking place and it is used for advanced designs also. So this is known as computational fluid dynamics or CFD. So that is a separate area of research altogether where the whole idea is to computationally solve the equations of fluid flow to get a picture of the velocity field. With this understanding we will try to quickly work out an example to illustrate the concept of these lines, streamline, streak line and path line. So an example we will consider a 2 dimensional velocity field as an example. So let us say that a velocity field has these types of components, u is given by this A x into 1 plus 2t, v is given by By and w is 0. So it is a 2 dimensional field. Usually whenever you have a velocity field we call it 1 dimensional, 2 dimensional or 3 dimensional depending on the number of independent velocity components that you are having. So if you are having 2 independent velocity components it is a 2 dimensional velocity field. A and B are some parameters and x, y are the coordinates, t is the time. And so A and B have certain dimensions with adjust this so that you get the dimensions of velocities at the end. So these are dimensional parameters but constants. Let us say we are interested to find out the equation of streamline at a time say t equal to t1. We are interested to find out streamline and streak line at time equal to t1 that is one objective. The other objective is to find out equation of path line. So to find out equation of streamline that is the easiest part. Let us do it first. So you have dx by u equal to dy by v, dz by w is not relevant because it is a 2 dimensional flow. So dx, so we are talking about first streamline. So dx by Ax into 1 plus 2t is equal to dy by By. At what time we are focusing our attention? At time equal to t1. So you replace this t with t1. So when you are considering a streamline you are freezing the time at the instant that you are considering. It is clearly an unsteady flow. So at different times you will get different streamlines. So you can integrate this and what you will get is 1 by a into 1 plus 2t1 ln of x is equal to 1 by b ln of y plus say some constant of integration. See how do we evaluate the constant of integration? You must be given a point on the streamline. Say the streamline passes through some point. So when you are given that the streamline passes through that point from that you can find out c. That is if you know that at time equal to t1 whatever streamline you are drawing there is one point on it. So that point when substituted x say x a1 and y equal to y1 will give the value of c. So that will give the equation of the streamline if you arrange it properly and in a compact form. Let us consider the streak line. To consider the streak line we have to remember one thing that this is the velocity that means if a fluid particle is injected at a point it will also represent its rate of change of position. That means you will have dx dt is equal to a x into 1 plus 2t where x represent the x component of displacement of a fluid particle which is subjected to this velocity field. Remember fluid particle is inert to the velocity field. So whatever the velocity field is imposing on it to do it will do that. So this is what it is imposing. Now it is possible to find out how x changes with time. It is straightforward but conceptually not that straightforward. To understand why it is conceptually a bit more involved we will parallely write the equation of the path line. So let us write the path line. This streak line is not yet complete but we will draw a parallel analogy with the path line and see where is the difference. So for the path line again you see path line is what? It is the locus of the fluid particle. So for path line also there is no need to believe that it should be something different than this one. Now the difference in approach comes in the concept by which we integrate these 2. So when we integrate this one say we integrate the equation of the streak line. So how we do it? We write dx by x is equal to a into 1 plus 2t dt. Similarly here also we write the same thing dx by x is equal to a into 1 plus 2t dt. So in the left there are going to be limits of x and the right limits of t. Here also same. Now what limits we will put? Let us say that you are injecting the die at some point x0, y0. This is the point of die injection. The die injection starts at t0 and the die injection ends at tf the final time. So this is the interval over which the die injection takes place. And the time that we are bothering about t1 is something in between t0 and tf. So when we write the integration for the streak line what we will do? We will integrate at some time say ti x equal to x0 and say at some time t equal to whatever say t1 x is equal to x. When you are considering the streak line you have to keep in mind one important thing that the importance will be clear when you write the integration here. So when you write the integration here at time equal to now here the time equal to t0 because you are finding the locus of the particle when you are talking about the path line. So when at time equal to t0 x equal to x0 at time equal to tx equal to x. What is the difference between these two? See look carefully into the limits of t. So when you say this is t1 x is t0 this is a fixed t0 that means you are finding the locus of a particle which at time equal to t0 passed through the point of die injection. Here you are dealing with a variable t0 because you are dealing with locus of all particles which at different instance of time pass through the point of die injection. So ti is a variable which may be anything between t0 and tf. So this is a variable limit. So this actually needs to be eliminated. We do not know what is this. Only what we know is that ti has to be between t0 and tf. But it is not a specified time. This is a fixed specified time and that is how you are going to find the path line. So similarly when you so when you write this in terms of x so you have ln x and x by x0 is equal to a into t-ti plus t-ti square where ti is a variable. Fortunately you will also get another equation involving y. So you can write similarly that dy dt is equal to by. So dy by y is equal to b dt. Again you integrate with respect to the same limits x0 to x and ti to t, t square-ti square. Very good. So this is t square-ti square. So when you do this, this will be not x but y. These limits are y0 to y. So that will give you what ln of y by y0 is equal to b into t-ti. So you have an expression here which involves ti and you can eliminate ti from these 2 to find out the relationship between x and y. Here that is not necessary because here you can straight forward write this. So you can write ln x by x0 is equal to a into t-t0 plus t-t0. T square-t0 square and ln y by y0 is equal to b into t-t0. Here what is the variable? Here actually t is a variable parameter because at different time it will have different position. To get that locus it may be convenient. Even you may write it in a parametric form but conceptually you may eliminate t to write the locus y as a function of x. Whereas to write this you have to eliminate ti. So here ti is a variable whereas here t is a variable. So conceptually it looks very similar but there is a subtle difference and that subtle difference needs to be appreciated in the context of streak line and path line. Stream line it becomes more or less straight forward. So I hope you get the distinctive concepts between stream line, streak line and path line and how to find out the equations given a flow field. We stop here today. Thank you very much.