 So, I think we will move on, what we will start talking about is now first part of the differential analysis which I am just going to call kinematics. So, as much detailed discussion as we want to do here will be useful. So, as I said this kinematics is basically an auxiliary part to the differential analysis. Formally what do you understand by kinematics anyone towards kinematics as you probably know from dynamics. Yes, please. Yeah, it is just the description of motion without really bothering to talk about the forces and such, correct. So, formally I will say that the tasks for this kinematics are now that we are dealing with fluid mechanics. Describe the motion of the fluid continuum and describe the associated fluid properties. That is all I can say. So, again now here we will come back to the discussion of the Lagrangian and Eulerian approach little more. We can do it both ways. Those two tasks that you want to achieve, you can do it both ways. If you want to employ the Lagrangian approach, we essentially write for each fluid particle so that each fluid particle is something that needs to be put in proper perspective. We are not talking about a single molecule here clearly when I say that it is a fluid particle. What we are talking about really is again a collection of a large number of molecules you can say which will something like gel with what we talked about earlier as that fluid point. Earlier in the morning we talked about a fluid point which is essentially let us say a small volume. The linear dimension of that volume is sufficiently large as compared to the mean free path. However, if you want to compare it with the characteristic length of your problem such as say the pipe diameter or whatever, that diameter is much much smaller. So, now you imagine the same situation. Imagine that diameter and the fluid material in that diameter and now you treat as if it is to be taken out and to be tracked as a separate material body with the same material, fluid material inside that volume. That is what I will call a fluid particle. If you want to look at it from a different point of view, you think about it as a small volume which is surrounded by an extremely thin membrane let us say. This is just for the conceptual understanding such that inside that volume surrounded by that extremely thin membrane it is the same fluid material that always exists which you are now tracking as it moves from one location to the next. As it moves it may change its shape or whatever that is fine but it is always the same fluid material that we are talking about. Now, this is not a very physically appealing description at all and I will agree to it. The reason we use it is because we essentially are going to use the laws of mechanics directly in fluid mechanics. We can do it in both two ways. We have done this control volume type analysis in the morning for the integral situation. We can do the same thing for a differential situation. There is absolutely nothing wrong with it. Just for the change of flavor what I am planning to do later is I will also introduce this so called Lagrangian way of representing fluid mechanics. So, when you are talking about Lagrangian approach you necessarily are talking about fluid material of fixed identity which means that is the same amount of matter, fluid matter that you are following inside a very small volume so to say. There are enough people who do not like this at all and I think that is perfectly fine because there is lot of problems with this description. Physically it does not make any sense whatsoever especially having understood what is the fluid continuum idea does not make any sense but that is fine. One way or the other everything finally boils down to a model. So, this way of doing modeling also results into something that works so it is fine. Anyway, if we accept that this is the way we are going to treat a fluid particle then mathematically the Lagrangian approach simply says that I will represent a fluid continuum using a very large number of fluid particles. The simplest way to imagine this is think about a flat tray just to get some visualization and think about a very large number of pebbles that you place in one layer right next to each other. So, those are basically I am treating as my fluid points and then under the action of forces all these pebbles will start moving together. They may have whatever velocity they may have but they will all move together and this is the way I can imagine the motion of a fluid continuum utilizing the concept of a fluid particle. So, then each pebble or each fluid particle now had at time equal to 0 let us say which is our some reference time a certain location. Let us say I am tracking 1 million such fluid particles. So, each of those 1 million fluid particles at time equal to 0 had some reference location. So, that location which is given by this position vector r naught here is what is popularly called as the material coordinate of a given particle. So, if you are tracking 1 million such particles you need to know 1 million such locations in your fluid domain and each will be different and then as a function of time each of these particles will move as they are moving. So, for each of those particles you say that its position vector at a later time is expressed in terms of the time and its original location which is given by that r naught vector. So, this is the idea of a Lagrangian approach. In principle it is exactly same as a solid body dynamics approach where we are saying that each fluid particle is essentially equivalent to a solid body that we deal with in dynamics is just that since it is a continuum situation what we have is that all these 1 million or whatever particles that you want to follow they have their own material coordinate at time equal to 0 with respect to which its motion will be written or expressed and in general any property of interest which I am just calling using the letter eta can be represented in this fashion let us say it could be a temperature of a fluid particle. So, I am a fluid particle and I am moving around at time equal to 0 I was here and my temperature at this location was even let us say as time progresses I am going through this fluid domain something is happening perhaps there is some external heat transfer or whatever because of which I keep on going from location to location and if I am carrying with it a temperature sensor I keep looking at it and I see that I go from here to here and my temperature changes. So, that is what I am calling as a eta which is essentially expressed as function of time, but it is always with respect to my original location. So, you have to do this for 1 million particles. So, if you want to do this practically you can imagine how troublesome it is. There are some people who actually do this for description of fluid motion they prefer this so called Lagrangian way, but to do this first of all you will have to do this computationally because otherwise you cannot really track this 1 million molecule and to do this also it is very cumbersome the larger the number of particles the more accurate is going to be your solution and so on. So, it is a very cumbersome approach even though in principle some people will say that this is exactly like what I knew from my solid body dynamic. So, I do not have to think any different. So, that idea of a fluid particle is something that is inherent in a Lagrangian approach. On the other hand if you want to do this from the Eulerian approach what we do is that we express the eta whatever the eta is as a function of all possible locations in my domain for all times that is all simply the Eulerian approach and this is what is called as the field approach. So, then computationally this makes sense when when professor Sharma starts his CFD you will realize that this is what we do in the sense that you have a domain over which you want to solve for the fluid flow you essentially identify a whole bunch of locations which are monitoring for all times and all these locations and for all times at all these locations you will basically monitor whatever property that you are looking at whether it is the pressure whether it is the velocity whether it is the temperature. So, inherently you know Eulerian approach is something that is more intuitive especially when it comes to the continuum fluid mechanics. Note though that if I am looking at this location it is not the same fluid particle that will always be here this particle will move something else will come. However, I am monitoring the temperature let us say at this point as a function of time. So, whatever I keep monitoring is not of one particle, but whatever particle that happens to be there at whatever instant of time I am recording. So, that is what I have mentioned last point is basically to emphasize that this is what we do in experiments if you are doing a lab experiment in fluid mechanics you cannot really identify some millions of particles and start following those. What you do is you measure velocity at some location for example, many locations for example not just one and that is why the Eulerian approach makes sense because really speaking experimental this is what you can do that you come up with a whole bunch of locations in the fluid flow and monitor whatever property that you want to monitor for all times at all those locations and this is what we will call a field approach or the Eulerian approach. So, now how do we connect these two guys? Everything here now occurs on a rate of change basis fluid mechanics heat transfer everything is on a rate of change basis. So, we want to now connect the rate of change in a Lagrangian situation to a rate of change in a Eulerian situation. So, to do that we actually say that let us have a fluid particle again which is at location r1, t1 the fluid particle is p and at this location at this time the property that we are interested in knowing for that fluid particle could be temperature could be density b eta suffix 1 and then over a period of delta t this particle moves from r1, t1 to r2, t2 just for the sake of showing that it may change its shape I have shown some other shape it is the same material that is included in the particle but it has deformed as it moved and when it came to this other location at the later time it is the same particle but its property has changed from eta1 to eta2 could be some external heat transfer was going therefore the temperature change from t1 to t2 fine. So, the Lagrangian rate of change is what we call normally the so called material rate of change or substantial rate of change as was pointed out in the morning is basically the rate of change of any property of interest observed by an observer moving with the fluid particle. So, imagine that you are actually the observer with a probe measuring probe moving with the fluid particle and you are constantly seeing what the probe shows if it is a temperature probe and if it keeps on showing different values of temperature as you are moving with the fluid you call that the rate of change of temperature which will be given as the substantial rate or the material rate of change of temperature and here what is it then it is basically for the same fluid particle p eta2 minus eta1 over the period delta t as simple as that with delta t tending to 0 and by definition this is what we call material rate of change substantial rate of change and just to specify that this is indeed a rate of change observed by an observer moving with the particle we use that capital DDT for the designation of it. So, that capital DDT of eta is then called the substantial rate of change or material rate of change. Formally it is simply what you know from your solid body dynamics when you write for example f equal to ma for a rigid particle from your solid body dynamics that a is what acceleration is the rate of change of velocity that rate of change of velocity when you write it in that Newton's law of motion is actually this because it is even though it is never explained to you perhaps it is really the rate of change of velocity in the Lagrangian sense that you are mentioning when you employ f equal to ma to a rigid body particle. So, it is the same idea since the Lagrangian approach is exactly same as the rigid body dynamics we say that the rate of change in the Lagrangian approach is nothing but the capital D by DDT as before. In the Eulerian description it is slightly different in the sense that what we just remarked earlier that the Eulerian description says any property expressed as a location and a time correct. So, eta 1 if I want to now write I will write it as the value of eta at r1, t1 r1 is essentially a position vector which I can write as x1, y1, z1 and t1 similarly r2 I will write as x2, y2, z2, t2 and now I can do a little bit of mathematics. In particular what I am doing here is I will simply express that eta 2 in a first order Taylor expansion about eta 1 and here let the fluid velocity be ui plus vj plus wk just for the generality is three dimensional. So, let us see what happens is something that many of you would have done anyway. So, right at the top you write the Taylor expansion using the only the first order terms. So, since the Eulerian description I have eta as a function of x1, x, y, z and t I bring about all partial derivatives. So, eta 2 I write as eta 1 plus partial derivative of eta with respect to x technically evaluated at 1 because we are expanding E2 eta 2 above eta 1 times essentially it is a delta x if you remember, but it is in this case it is x2 minus x1 plus the same thing for y same thing for z and also for time because eta is a function of x, y, z and t 4 variables. And now you simply do a bit of rearrangement first you realize that x2 minus x1, y2 minus y1, z2 minus z1 are all delta x delta y delta z etc. Similarly, t2 minus t1 is delta t and then you bring about this eta 1 on the left hand side divide entire equation through delta t. But what is this? This is we just argued two minutes back that this is nothing but the substantial rate of change or the material rate of change. So, we write that as capital DDT of eta and now knowing that we are anyway talking about some specific point I am going to simply drop the subscripts 1 I do not want to talk about that and I simply say that delta x over delta t because we are following a fluid particle is going to be the velocity in the x direction. Similarly, that in the y direction similarly that in the z direction will come as u, v, w multiplying these partial derivatives with respect to x, y and z and this is something that we get in a Cartesian form. So, if you simply rewrite this in a vector form we write that as the substantial derivative of eta equal to now instead of writing the partial derivative of eta with respect to time at the end I will write it here and whatever is left out this guy this is scalar. So, I can write it as u times d eta dx plus v times d eta dy plus w times d eta dz etc which then I can write as v vector dotted with gradient of eta. So, that gradient expression is something that we must know in the at least minimally the Cartesian coordinates. So, this is essentially an operator as you can see it is a vector operator and in the Cartesian system that gradient has an expression given by that and because it is an operator this entire boxed equation can be written in the form of an operator form where the left hand side is written as a material rate of change and now what it is operating on is simply vanished it can operate on whatever that is equal to partial derivative with respect to time plus this v dot grad technically or v dot del as some people will say this is one of the most important expressions in differential fluid mechanics capital d by dt as we said is the material rate of change or substantial rate of change partial d dt if you see the way it was brought about is basically at one. So, we were talking about point number or location number one and location number two and with time tending to 0 delta t tending to 0 we are basically bringing those two arbitrarily close to each other. So, what we are talking about is the rate of change of this eta at the location of interest which is say one plus what we call this an advective or a convective rate of change that v dot grad that v dot grad if you simply write it as a Cartesian form this will come out as this and this is what is called as the convective rate of change basically that is happening because there is a spatial variation in eta at the given instant of time that is why that rate is coming about local rate of change which is the partial derivative of eta with respect to time is basically at the given location you are monitoring what is happening with a probe. So, you fix the location and you simply keep on monitoring what is happening whatever is happening at that location is given as the local rate of change if nothing is changing at a given location with respect to time we call it a steady flow. So, this is the equivalent of that no accumulation term that we were talking about on a integral basis in the differential basis if the local rate of change is 0 then we call this a steady flow. So, let me try to explain this with a physical situation or two the convective and the local rate of change. So, here is a situation what I have is a air conditioned house let us say which is maintained at 25 degrees and it is a hot ambient outside at 40 degrees and let us say a person is now walking from this hot ambient with certain velocity into this room or into this house. So, which is the fluid particle here the person good fine the person is the fluid particle let us say that the person is carrying with him or her a temperature probe. So, that every time he goes or he goes somewhere it shows that this is the temperature fine good. Let us say that as he walks or she walks in with a 2 meter per second because there exists a spatial difference in the temperatures from 40 degrees to 25 degrees as the person enters the room enters the door there will be a sudden rate of change of temperature that will be felt what will that be which one of those two will load. Now, I have already written here but there will be a convective rate of change why because there existed a spatial difference in temperatures which gave rise to that v dot v dot grad the convective rate. Now, I am adding something to it let us say someone is sitting on the roof and just as that person is about to enter the room or the house to pour ice cold water on that. So, in addition to that convective rate of change there will be a local rate of change that will be felt. So, the total together is what will be recorded by the probe that the person is carrying as the substantial rate of change. Let us talk about another example which I do not have a picture of but I can draw it here quickly. So, this is a liquid flow let us say steady flow which means what if I am monitoring the velocity in this section 1 it will always show v 1 as a function of time there will be no change. Similarly, if I am looking at the section 2 it will always show v 2 right. So, there is no change of these velocities with respect to time at a given location fine. Is there an acceleration here there is there is a convective acceleration if you imagine that now there is a fluid particle which is now traveling when it comes to section 1 it will have velocity v 1 by the time it goes to section 2 it will have velocity v 2 since we are now dealing with liquid flow with area converging we expect that the velocity will increase and so there exists a acceleration it will be the acceleration of the particle but it is coming purely because of a convective acceleration in this case is that fine. Now, if I add something to it roughly some sort of a gate wall in this section and then I start closing the section clearly there is going to be an added acceleration in this section 1 itself as I keep closing the wall and that is what basically will be local acceleration in addition to that there obviously can be a convective acceleration. So, if a fluid particle now shows up at this time instant when this fellow is closing it will experience not only a convective acceleration because there is a change of velocity from section 1 to 2 but because something is happening in section number 1 it will also feel that the velocity will change here itself in section 1 which will be the local acceleration. So, this is the way you can imagine what is a local acceleration or a local rate of change and a convective rate of change and the total rate of change alright. So, that is about the rates of change the substantial and the local and the convective. Now, let us talk about something which I think most of the times we talk about in an undergraduate class these I will call as graphical descriptors what to say the first one is what is called as a path line which is simply a line that describes the trajectory of a fluid particle again the trajectory as you know from solid body dynamics is exactly the same as what this is there is no difference whatsoever. So, graphically you simply track one given fluid particle and just see how it moves in the fluid flow. So, it is the same fluid particle that has been shown at different instance of time it may be doing all sorts of things rotation this and that is fine but it goes from this location here at T1 to this location here at T2 and so on if you simply join these successive locations of one given particle you will get what is called as a path line. So, it gives you a history of the time history of the flow time history of the flow it basically tells you how that fluid particle has traveled over a certain period of time from one location to the next and mathematically speaking if you want to obtain the equation of a path line all you need to do is you integrate that dx over dt equal to u and dy over dt equal to v with initial conditions because now we are tracking one given particle. So, we are talking about a Lagrangian description when you integrate that you need to provide the initial conditions for it which are given in terms of its reference location at the reference time which is provided as x at t0 which is the reference time equal to x0 similarly y at t0 which is the reference time equal to y0. So, subjected to these initial conditions if you integrate dx over dt equal to u and dy over dt equal to v you will obtain essentially the parametric equations if you want in terms of time for the x and y. So, those will be the x and y locations of the same fluid particle as it is traveling from one location to the next as time progresses. So, we will work out one or two examples of this later but this is the idea. The next graphical descriptor is what is called as a streamline which again more or less everyone will know it is the line drawn in the flow field such that the tangent at every point on it is in the direction of the local velocity vector as it is defined and note this gives you a instantaneous picture of the flow field. So, if there is a truly unsteady flow the streamlines will keep changing from instant to instant. To obtain the equation of a streamline what is done is that you realize that the velocity is essentially tangential to the streamline. So, you identify a vector element along the streamline somewhere and simply form a cross product of that vector element with the velocity because they are in the same direction velocity being tangential the cross product must be 0 and using then our standard notation you can bring about the equation of a streamline as dx over u equal to dy over v into d. Note that if you want to actually obtain an equation for the streamline you have to integrate this with t as some constant value because it is an instantaneous picture that you are getting. So, in case there is a time dependence involved in this differential equation you actually have to choose different time instants to obtain the equations of the streamline at different times. If it is a steady flow then the time dependence normally does not come in and you are fine. Is that fine the idea how you get the this is something that I think most of you probably would know all that I am saying is that you realize that do this let us say this is my streamline. So, I am basically looking at this vector element and the velocity is also in the same direction by definition. So, that in the cross product of these two guys will be 0 and that is what brings about the equation of the streamline. One more descriptor is called a streak line. So, what is this? So, this is basically an instantaneous local as it is written of all fluid particles that have passed through a common point of reference at a previous time instant. So, here we are talking about instantaneous locus of different fluid particles and the easiest way to imagine this is to look at a plume that is coming out of the chimney and that is nothing but a streak line. So, do you see why? So, just imagine that the smoke line that comes out of a chimney if you see is composed of different fluid particles. So, the very first location on that smoke line will be the particle fluid particle that has passed through the opening of the chimney at some previous time. Then you look at the next particle that has passed through the same injection point which is the opening of the chimney at some other time and so on. So, you just look at all these different particles that have passed through the same common point at some previous time and simply join their line that is what will be called as a streak line. And what I have tried to show this here is that here is the reference location let us say at some previous time this particle P1 came out of here and at the present instant it is here. Similarly, the next fluid particle which came at some later instant from here it is sitting here at the instant of interest and so on. So, you basically join the line of all such fluid particles at whatever instant of interest now let us say that at that instant of interest there could be one particle that is just about emerging from that location. So, that is what I have shown by P5. If you look at the path line of P1 it can be whatever it may have just come out of here at a previous time and wandered off here and here and now it is here it does not matter is that fine. So, that is the idea of a streak line and as I have commented here it is quite commonly used in flow visualization experiments in fluid mechanics. In fact, most of the times what you end up seeing as flow visualization data is many times streak lines you have seen perhaps the smoke visualization or hydrogen bubble type visualization most of the times those are actually streak lines that you end up seeing. So, these three things are useful. As I said earlier you know many people do not like the idea of a path line because or even this streak line because they will say that what is the understanding of a fluid particle it does not have any meaning and it is a valid criticism. It is just that it helps in visualizing certain flow patterns let us say what I will say that why this business is useful is when you do the CFD now the results that you get out of CFD are many times plotted using these ideas streamlines or in fact some of these commercial software is actually do it path lines etcetera as well. So, this is where I think the utility of these ideas is it helps in visualizing the flow field from a computational persons point of view that you can draw these lines using the simulation data and you can try to see how the flow field is behaving and that is why it is useful. So, that is all about the graphical description.