 Alright, so let us begin with the kinematics discussion. As I said kinematics is an auxiliary topic to differential analysis. Many of the results that we will derive here in kinematics will be eventually used in the derivation of the differential equations of motion. So kinematics on its own has usually two tasks associated with it. One of them is that we want to describe the motion of a fluid continuum and also to describe the associated fluid properties. And many of you will perhaps know and we had mentioned it yesterday also a little bit that there are two standard approaches within fluid mechanics with which you can do these or achieve these two tasks. One is what is called as the Lagrangian approach and the other one is called as the Eulerian approach. So the Lagrangian approach is as we said yesterday is exactly similar to exactly identical I should say to the solid body dynamics from mechanics that we have seen in our earlier studies. So what we say is that in a Lagrangian approach we identify a fluid particle. So what we mean by a fluid particle is that it is a infinitesimally small amount of the same fluid mass that we are following as it moves from one position to the next. So this is essentially exactly same as following let us say a projectile in case of solid body dynamics from one place to other. So that way conceptually there is no difference between what you know as solid body dynamics and the Lagrangian approach. So mathematically what we do is that we say that the fluid continuum is composed of a very very large number of these fluid particles which are essentially infinitesimally small masses of fluid matter each of these containing the same amount of mass as it moves from one location to the next and a very large collection of such particles is what constitutes the fluid continuum. So what will happen is that under the action of forces each of these fluid particles so to say will keep moving and therefore the motion of the continuum is described in terms of the motion of each of these particles. So if you want to describe the motion of the fluid continuum you have to come up with an aggregate of the motion of each of these fluid particles. Typically you can imagine that if you want to describe a fluid continuum you will need a very very large number of fluid particles to be incorporated in your analysis. The number could be as large as 10 to the power 10, 12 whatever. So even though conceptually the idea is very straight forward and directly comes from the solid body dynamics practically to follow each and every of these 10 to the power 10 or more particles in a real life situation either experimentally or computationally is essentially out of question. So you will see that usually the analysis that is done in fluid mechanics is not really using Lagrangian approach. Mind you there are some people who in fact do this Lagrangian approach for some special situations but for standard continuum flow situations Lagrangian approach is essentially not possible because it is cumbersome. It requires too much effort to track and figure out how each of these particles is moving and come up with some sort of an aggregate description for the fluid continuum. Nonetheless it is a conceptually useful tool and mathematically what we say is that each particle will be assigned its own reference coordinate at some reference time. Let us say that I have identified particle number 1 and that particle number 1 at some reference time let us say t equal to 0 is situated at some specific location within the fluid domain. So that location is assigned to that particular fluid particle as its reference starting location. In fact it is what is popularly called in mathematical terms as the material coordinate of that particular fluid particle which is what is written out here. And then as time progresses we essentially follow this fluid particle to wherever it goes and its subsequent positions denoted by the position vector r are always described in terms of its initial location and as a function of time. So mathematically speaking for each fluid particle we come up with a description such as this that at any instant later than the reference time a given particle is located at whichever location it is and that is described as a function of its initial location let us say at time equal to 0 if that was the reference time and the time itself. So this is as far as the motion of the fluid continuum is concerned. In general in the Lagrangian approach any property of interest is expressed in terms of its value at the reference coordinate which is also called the material coordinate as I said and as a function of time. So therefore what we are doing really is nothing but following each of these fluid particles in the same sense as a solid body dynamics and then trying to come up with an aggregate description of each of these particles to totally come up finally with the description for the fluid continuum. So as I said conceptually fine but in practice it is essentially impossible to carry out. So on the other hand what we do is what we employ here as an Eulerian approach. So in an Eulerian approach what we do is we express any property so I am denoting any property by this symbol eta this can be pressure this can be density this can be temperature this can be velocity can be whatever. So any of these properties is described at all positions within the domain of interest and for all times. So mathematically we say that eta is a function of r and t where this r actually is the position vector of all possible locations within the domain of interest and this t is essentially all times and that is what is written as the second sentence here specify eta at all locations for all times. And this is what we call as a field approach in fluid mechanics and this is the usually followed approach for most analysis. Now what you should remember is that if you look at any particular location within the fluid domain as time progresses that location will see different fluid particles due to the flow of the fluid. So at time equal to t1 there will be one particle at the given location at time equal to t2 that particle has moved out and some other particle has occupied that position. So in this Eulerian approach what you should realize that the eta at a given time instant at a given location of course is the eta associated with whichever fluid particle that happens to be present at the given location at the given time. So that way it is slightly different in the sense that we are not really tracking one particular particle we are essentially focusing our interest on one location and simply monitoring what is happening at that particular location as a function of time and you do this at all possible locations within the domain for all times and that is the way you monitor the motion of the fluid continuum. The good thing about Eulerian approach is that those who are performing experimental work and measurements will realize that this is precisely what you do in case of fluid mechanics measurements. Normally when you want to measure velocities for example in a fluid mechanics experiment what you do is you go to a certain location and at that particular location you install a velocity measuring probe and as a function of time the probe measures velocities at that particular location for various different times. So that way there is a good connection between what you do in an experiment and the Eulerian approach. Now let us try to come up with a connection between the Eulerian and the Lagrangian approaches and here what I have shown is a single given fluid particle. So what I am calling that as P here at time equal to T1 the particle of the fluid which is P is shown in this position where my highlighter is its coordinates are r1 vector which is essentially x1, y1, z1 if you want in addition and at time T1. This fluid particle then moves in a period of let us say delta T time period of delta T and it goes from the location r1 to a location r2. So this is another position vector this is obviously at a time T2. So the difference between T2 and T1 is delta T. So as the particle goes from location 1 to location 2 the properties associated with that particular particle whether that is density of that particle or pressure of that particle meaning what I am talking about is really particle is essentially not one molecule it is a collection of a large number of molecules contained in a really small infinitesimal volume. Temperature of the particle let us say going from location 1 to location 2 will change from whatever its value was at location 1 to whatever its value is at location 2. So that generally we are using property by the symbol eta and so I am saying that the eta associated with the particle P at location 1 is eta 1 by the time it shows up at location 2 that property has changed to eta 2 because of whatever reason we are not interested right now in knowing what the reason was or is because of which the property has changed. Let us simply say that it has changed. With this setting we immediately describe the Lagrangian rate of change which is defined as the rate of change observed by an observer that is moving with the fluid particle. In other words what we say is that let you be the observer you climb on to this fluid particle or you sit on this fluid particle with a sensor that is measuring whichever quantity of interest that you are interested in let us say temperature and as you move from location 1 to location 2 along with the fluid particle the sensor that you are holding in your hand will show that the quantity has changed from eta 1 to eta 2 over a time interval of delta t is that fine. And now with the delta t tending to 0 as usual in the limit we define the so called material rate of change or substantial rate of change which is essentially the rate of change in the Lagrangian approach as eta 2 minus eta 1 over delta t and this is given a special symbol of capital D by dt of eta and this is what we call a material rate of change or a substantial rate of change. The reason we provide this capital D dt as the symbol for this particular rate of change is to signify that we are always following the particle ok. So, keep this in mind that the capital D dt is to imply that this is a Lagrangian rate of change in the sense that it is the rate of change observed by the observer as the observer is moving with the fluid particle. So, that is what the substantial rate of change or material rate of change is. Let us go to the Eulerian description. We said that in the Eulerian description we describe a property at all times and at all possible locations. So, mathematically speaking we say that eta is equal to eta of r and t where r is a position vector and t is the time. In that sense if you go back in location number 1 the property was eta 1 the in location number 2 the property was eta 2. So, therefore, eta 1 is written as whatever that eta function is employed at r 1 comma t 1, r 1 is in terms of Cartesian coordinates x 1 y 1 z 1. So, therefore, eta 1 is expressed as a function of x 1 y 1 z 1 and t 1 similarly eta 2 is expressed as a function of x 2 y 2 z 2 and t 2. And now since delta t if you remember the time interval between the particle going from location 1 to location 2 is usually an incremental time interval meaning that we perform all these analysis as delta t tending to 0 we express eta 2 in a first order Taylor expansion about eta 1. This we can do because all quantities involved in particular the time interval is tending to 0. Therefore, in principle what we will say is that typically location number 2 and location number 1 are close to each other. So, this close is to be taken in the calculus sense that the distances delta x delta y delta z between location number 1 and location number 2 are all tending to 0 because typically delta t is tending to 0. Let me define the velocity vector with its components u in the x direction v in the y direction and w in the z direction. On the first line on this slide we have expressed eta 2 as a function of eta 1 with the first order Taylor series expansion. So, because it is a first order Taylor series expansion we are keeping only the first derivative terms. Remember that eta is a function of more than one variable it is a function of x y z and time. So, I have to talk about three first derivatives of the space and one first derivative with respect to time and formally we are expanding eta about eta equal to 1 or eta 1 or location 1. Therefore, formally speaking all these derivatives should be evaluated at location number 1 and that is why you see subscripts 1 on all these later on we are going to drop this to make it general. x 2 minus x 1 is simply the x separation between the two locations. So, it is what we are going to call delta x as here right here y 2 minus y 1 accordingly then is delta y the y separation between the two locations similarly delta z and t 2 minus t 1 is delta t. The third step involves bringing this eta 1 onto the left hand side and dividing the entire equation by this delta t. So, therefore, we get just look at the left hand side right now we have eta 2 minus eta 1 over delta t clearly as delta t tends to 0 which just a few minutes back we had argued to be the substantial rate of change or the material rate of change that is why the left hand side then is immediately written as the substantial rate of change of eta equal to capital D Dt of eta. On the right hand side we are left with four terms and now we are dropping these subscripts 1 and 2 etcetera because we are now talking about two general locations we do not have to always keep talking about 1 and 2. Since we are following the fluid particle delta x over delta t is simply the change in the x location divided by the time which is nothing but the x direction velocity of the fluid particle. So, therefore, dx over Dt is then replaced in the last equation as the u velocity or the x component of the velocity. Similarly delta y over delta t is then the y direction velocity which is v in here delta z over delta t is accordingly the z direction velocity w and partial derivative of eta with respect to time remains as the fourth term and this is really the connection that you are talking about between the substantial rate of change and on the right hand side what you see is now written in a vector form as a generalized expression that the substantial rate of change of eta is equal to the partial derivative with respect to time of eta plus all these first three terms u times partial derivative of eta with respect to x plus v times partial eta with partial y w times partial derivative of eta with partial z can be compactly represented using the vector representation v dotted with grad eta. So, if you just put in your Cartesian components u v w for v and the gradient operator as we had discussed yesterday operating on the eta you will actually obtain exactly this expression on the right hand side. So, what you see in the box is the relation between what we call the substantial rate of change which is the inherent rate of change while we are talking about Lagrangian approach and on the right hand side what we have is the rates of change when we are talking about an Eulerian approach of description. So, now rather than writing this operating on some quantity eta we write the above equation in the form of what we call an operator notation. So, now simply say that capital D by dt is an operator type of rate of change which will operate on any quantity eta. Therefore, in terms of the operator notation what we say is that substantial rate of change on the left hand side here is equal to the local rate of change given by the partial derivative with respect to time plus what we call as a convective rate of change. So, let us just talk about these two little in little more detail. So, the local rate of change is essentially the rate of change at a given spatial location with respect to time. What this means is that let us say we go back to our example of velocity monitoring at one particular location within the fluid. So, let us say you are monitoring velocity at one particular location of the fluid using some sensor and as a function of time the sensor is showing different values of velocity at the same point. So, therefore, you are in a position to compute a partial derivative with respect to time of the velocity at the same location and that is what we will be called as the local rate of change. The convective rate of change comes in because at the instant of interest the property eta is varying from one location to the other location and therefore, as a particle of fluid if you are travelling with the particle of fluid, when you go from one location to the next because eta is changing from location to location, when you are following the particle you will realize when you are holding that sensor in your hand measuring eta that you are measuring different values of eta as you go from one location to the next. So, therefore, this convective rate of change which is given by the velocity dotted with the del operator is what we call a convective or sometimes people will also say advective rate of change. Physically, purely physically the convective rate of change occurs because at the given instant of interest there is a spatial variation that is point wise variation let us say in the property eta and because of which this convective rate of change shows up. So, immediately there is a terminology that I would like to introduce here is that if this local rate of change meaning if the time rate of change of any quantity at a given location is equal to 0 then the flow situation is what is called as a steady flow situation ok. So, if this partial derivative with respect to time turns out to be 0 we call the flow situation steady flow otherwise it is unsteady flow. So, let us try to look at a couple of physical examples which will hopefully try to explain the ideas associated with a convective rate of change. So, what I have shown here is a house which is air conditioned let us say at 25 degree Celsius. So, that is maintained temperature here whereas, the ambient is at 40 degree Celsius which is which is hot outside and let us consider a person walking with certain velocity 2 meters per second in the x direction. The idea is that this person is representing a fluid particle ok. So, as this person walks from the hot ambient of 40 degree Celsius into a cooler ambient or cooler house I should say which is maintained at 25 degree Celsius. The moment the person walks into the house through the door he or she will realize that there is a convective rate of change associated why? Because there exists a spatial variation of the temperature as the person goes from outside to inside with a certain velocity. So, since we are talking about the person walking only in the x direction here we can say that u velocity is equal to this 2 meters per second say V and W are 0. However, the temperature gradient exists from 40 to 25 over certain characteristic length if you want which is giving rise to a convective rate of change. Again the convective rate of change is existing because at the instant of interest which could be the instant when the person crosses the door and walks into the house at this instant there exists a spatial variation in temperature which is giving rise to this convective rate of change and that will be the rate of change experienced by this particle which is this person in this case when he walks from a hot ambient to a cooler ambient. Now, if you want to add a local rate of change to this situation you simply imagine that someone is sitting on the roof here and at the instant when this person just walks into the door the person who is sitting on the top of the roof pours some really ice cold water on this fellow who is walking inside. So, what is happening then is that because of the pouring of this ice cold water at this given location at the instant when this fellow tries to walk in in addition to this convective rate of change the ice cold water is also going to introduce a local rate of change that will also be experienced by the person when he or she is traveling in at the door. I hope the idea is clear and that is precisely what I have written down at the bottom of the slide. Let us take one more example here is an example where let us say we are talking about a steady flow to begin with in a converging passage and let us say that we are talking about a liquid flow. So, most of you would imagine that when we go from a larger cross sectional area at the inlet to a smaller cross sectional area at the outlet if it is a liquid flow the velocity will increase and that is precisely what is happening as it is shown in the figure. So, V1 which is the velocity at the inlet is smaller than V2 which is velocity at the outlet and let us say that it is a steady flow what we have is a valve at the at the entry which is kept at a fixed location when we are starting this discussion. So, therefore this is a steady flow in the sense that if you take a velocity probe and mount it in the inlet section take another velocity probe and mount it in the exit section. The first probe will always show value of V1 with time second probe will always show value of V2 with time and that is precisely what we will mean by steady flow that there is no local rate of change. However, if you are a particle of fluid let us say which is now shown by my highlighter. Let us say this particle shows up in the inlet it will experience a velocity of V1 it will travel to the outlet and by the time it comes to the outlet it will experience a velocity of V2 which is greater than V1. So, therefore a particle going from inlet to outlet has experienced an acceleration, but this acceleration is a purely convective acceleration because there is a spatial rate of change of velocity from inlet to outlet. On the other hand if now we start operating the valve meaning that if we start either closing it or opening it as a function of time your probe mounted in the inlet section will start showing different values of velocities. So, V1 will not remain a constant with respect to time, but it will start changing similarly V2 will also be start changing with time and then in addition to being a convective acceleration situation there will be an additional local acceleration as well. This will occur as long as you are opening or closing the valve because that action will keep changing the value of V1 and V2 with respect to time at the inlet and outlet respectively. So, this is what we mean by local rates of change and convective rates of change. Going back again local rate of change is rate of change with respect to time at a given location convective rate of change appears or arises because there is a spatial variation with respect to spatial coordinates of the property eta at the given instant of time. So, let us try to work out a numerical example immediately to see how we can utilize these expressions. So, what has been shown here is that some temperature field is given it turns out that the temperature field is only a function of y coordinate and time and it is given as 20 multiplied by 1 minus y squared multiplied by cosine of pi times t over 100. A velocity field is given in the x direction and again it is shown that it is only a function of y u as a function of y 2 multiplied by 1 minus y squared and what is being asked is determine the rate of change of the temperature experienced by a fluid particle when it is at y equal to 0 and at t equal to 20. So, remember that at y equal to 0 location at different times different fluid particles are going to be present. So, we are actually going to calculate the rate of change of temperature experienced by that specific fluid particle which happens to be at y equal to 0 at time equal to 20. So, please keep that in mind. So, because the question asks the rate of change of temperature experienced by a fluid particle we are essentially calculating the substantial rate of change that is capital D D t of temperature and we will evaluate this at y equal to 0 and t equal to 20. If you go back this is our general expression. So, instead of eta we will simply substitute capital T temperature. So, then there will be a local rate of change of temperature and a convective rate of change. So, explicitly I have written the local rate of change and the convective rates of change in their Cartesian forms and let us see what I have done here. So, I have local rate of change as partial derivative with respect to time of temperature. The convective rate of change when explicitly written in Cartesian coordinates is u multiplied by partial derivative of x partial derivative of t with respect to x plus v multiplied by partial derivative of t with respect to y plus w multiplied by partial derivative of t with respect to z. All this is to be evaluated at y equal to 0 and t equal to 20. So, now given these two fields we realize that t is not a function that is temperature is not a function of x at all. So, this D t dx is identically equal to 0, v and w are identically equal to 0. So, the latter three terms are completely 0 in this case. So, it turns out that at t equal to 20 and at y equal to 0 the fluid particle is not going to experience any convective rate of change of temperature. So, the only rate of change of temperature that the fluid particle will experience at y equal to 0 and at t equal to 20 is going to be the local rate of change and that local rate of change is simply that you have to differentiate this with respect to time and that is about it and substitute t equal to 20. So, this is a fairly straightforward example where the expression for the substantial rate of change has been explicitly evaluated for a given function of temperature and velocities. So, let us move on and talk about a few quantities of interest which are kinematical quantities, but as far as CFD analysis is concerned many of these kinematical quantities will be of interest because as part of your post processing of the CFD results many times you actually generate these kinds of plots which will show how the flow field is behaving either as a function of time or at a given time instead. So, there are these three standard graphical descriptors that we normally utilize. The first one is what is called as a path line which is nothing but simply the trajectory of a given fluid particle. So, what I have shown at the bottom here is how one given fluid particle which is denoted by p is travelling from one location to the next to the next to the next and so on as time progresses. So, p is at this given location where my highlighter is at time t1 then it goes here at t2 then it goes here at t3 t4 t5 etc. So, if you simply join the line that this particle has been in these places at various times of instance you will essentially get the path line which describes how a particle of fluid has travelled over a period of time starting from t1 to t5 in this particular case. And since this is essentially a trajectory of the fluid particle it is no different from the trajectory that you know from solid body dynamics is exactly the same. So, therefore, in this particular case we can actually fairly easily derive an equation for the path line of a fluid particle. All we need to realize is that the u velocity will be given by dx over dt for the fluid particle v will be given by dy over dt if it is a two dimensional flow in x and y coordinates. Corresponding to that the initial conditions for the particle would be that at time equal to t0 which is some sort of a reference time the x location of that given particle was x0. Similarly, at the reference time t0 the y location of the given particle was y0. So, all you need to do is we integrate these two differential equations which are of first order differential equations subject to these two initial conditions. And we obtain if you want a parametric set of equations which will describe the path line of the particle. So, the parametric set will come in the form of x as a function of x0 and time and y as a function of y0 and time. Again x0 and y0 are the x and y locations of the particle at the reference time t0. So, this is what is called as a path line which is as I said again simply a trajectory of the fluid particle over a period of time. Then what we have is what is called as a streamline which is essentially a line drawn in the flow field such that the tangent at every point on it points in the local direction of the velocity vector. And this streamline provides an instantaneous picture of the flow field. What I mean by this is that if you are dealing with a truly unsteady flow field the velocity at each point will keep changing as a function of time. And therefore, the streamline pattern will also keep changing with respect to time. So, at every instant of time you get a different streamline pattern. So, what we say is that the streamlines provide an instantaneous picture of the flow field. Clearly if the flow field is steady nothing is changing with respect to time. And in that case the streamline pattern does not change with time, but if the flow field is unsteady the streamline pattern will keep changing with time. So, here also it is fairly straightforward to derive an equation for the streamline. To do that what we say is that we consider an elementor vector element along the streamline. And let me call that dr as the elemental vector element sorry elemental vector along the streamline. And because the definition of the streamline is that tangent at every point is in the local velocity vector by definition velocity vector v and this elemental vector at any point on the streamline dr will be parallel to each other. And if we are dealing with two parallel vectors their cross product will be equal to 0. So, therefore by definition of the streamline vector v which is the velocity vector and the vector dr which is the incremental vector element along the streamline crossed with each other will provide 0. And if you substitute your Cartesian components for the velocity u v w and dx dy dz if you want you work out this cross product. And what you will end up with is the equation of the streamline which will be in the form of dx over u equal to dy over v equal to dz over w if you are talking about a three dimensional flow. If you are talking about only a 2D flow in the x and y coordinate system we will have only dx over u equal to dy over v. And if you want to actually form an equation for the streamline which will be y as a function of x you integrate this governing equation this is a differential equation with a constant time. Remember that it is an instantaneous picture of the flow field which means that we freeze time at some constant value and having frozen that time at some constant value then you integrate the equation the differential equation here to find the equation of a streamline. So, path line gives you how a fluid particle a given fluid particle has travelled over a period of time streamline provides you an instantaneous picture of the flow field is something else which is commonly used especially in an experimental fluid mechanics work and that is what is called as a streak line. So, streak line is an instantaneous locus of all fluid particles that have passed through a single given reference location at some earlier time. So, let me just try to explain it with this sketch here. So, here I have shown where my highlighter is standing right now is the reference location which is the location through which all these different fluid particles are going to pass. Now, let us say we take the first fluid particle p 1 which is at the given instant of time sitting where my highlighter is, but at some earlier time instant it has passed through the reference location and it has travelled along its own path line which my highlighter is tracing and then it has come to this particular location at the given instant of time. Similarly, p 2 is another fluid particle different than p 1 which at some previous time instant passed through our location which is the reference location and may be travelled something like this and landed up here at the given time. So, on for p 3, p 4 and p 5 is the fluid particle which at the given instant of time is just about appearing in the reference location as time progresses it will also start travelling wherever it wants. So, the streak line then is essentially the line joining p 1, p 2, p 3, p 4 and p 5 in this particular case at the given instant of time. So, it is again providing you an instantaneous picture of the flow field. However, in this case it is joining different fluid particles which have passed through the same reference location at some earlier instant of time. One of the most standard examples of a streak line is the smoke plume that you see if you see a smoke plume coming out of a tall chimney of either a house or a power plant or something of that sort. What you can imagine is that at the instant of time you have taken a picture of the plume which is the smoke plume which is coming out of the chimney and the smoke plume can be considered to be composed of different fluid particles which have passed through the opening of the chimney at some previous time instant and at the given time instant you are basically joining all those different particles through a single line or using a single line. The streak line is something that is routinely used in flow visualization experiments. So, those guys who are involved in experimental fluid mechanics may perhaps relate to this streak line. So, these are the three main graphical descriptors that we utilize in fluid mechanics and as I said as part of your CFD post processing many times you actually generate these kinds of lines. If you are dealing with a steady flow situation you will more or less always generate this stream line pattern for unsteady flows also at different instants of time you end up generating stream line patterns. So, that way these three quantities are of practical importance as well from the CFD point of view and not just as fluid mechanics entities. Let us look at another example where we take a velocity field given by the x velocity being equal to x coordinate itself. So, u is equal to x and v is equal to minus y and here we are determining the equation of the path line for that particle which was situated at 1 comma 2 at t equal to 0. So, t equal to 0 is our reference time and the reference coordinates for the particle that we are looking at are 1 comma 2 at time equal to 0 and for this we have to come up with the path line equation of the path line and also the equation of the stream line passing through the same point. So, the point is still 1 comma 2 and we want to find the equation of the stream line. So, for the path line if you want to go back couple of slides remember this we said that we will integrate dx over dt equal to u and dy over dt equal to v with the given initial conditions which is what we are doing. So, dx over dt which is equal to u in this particular situation the u velocity field is equal to x. So, dx over dt equal to x is one differential equation dy over dt equal to minus y is another differential equation fairly straight forward. So, you get x equal to c 1 times e raise to t y equal to c 2 times e raise to minus t we want to find out the constants of integration c 1 and c 2 for which you utilize the initial condition that at time equal to 0 x naught and y naught are equal to 1 and 2. If you substitute in here you will get c 1 equal to 1 and c 2 equal to 2 and therefore, the parametric equation of the path line is x given by e raise to t y given by 2 times e to the power minus t the parameter being time here. In this particular case is a very very simple example in the sense that you can eliminate the parameter very easily and obtain a close form relation in terms of only x and y you can do that if you want, but let me try to go to the next part which is determination of the streamline. Now, here the flow field is two dimensional only in x and y. So, what we need to do for the streamline determination is that we need to integrate dy over dx equal to v over u from here. So, dy over dx will be equal to v over u this dz and w does not show up here at all and v velocity in this particular case is minus y u is x again very straight forward integrate this use the coordinates of the given point which were 1 comma 2 here to find out the integration constant and what you will see is that the equation of the streamline will come out as x multiplied by y equal to that constant which happens to be 2. So, x y equal to 2 is the equation of the streamline and x equal to e raise to t and y equal to 2 times e to the minus t are the parametric equations of the path line. If you eliminate time from the equations of the path line you will actually see that the equation of the path line is exactly same as this equation of the streamline. If you eliminate t from here you will get that the equation of the path line is also x multiplied by y equal to 2 which is same as the equation of the streamline and this is a special situation where the path line and the streamline are identical where does this occur many of you would know it if the flow is steady we will see that the path lines and the stream lines and in fact the streak lines also even though we have not considered them here they will be identical. So, going back to our original velocity field you realize that there is no time dependence on for the velocity field. So, it is time independent velocity field which means that it is a steady velocity field. So, for steady velocity field what you will see is that the path line and the streamline will actually turn out to be the same. Let me mention here that the equation of a streak line can also be generated it is a little more involved process and for the present course we are not going to get into that detail. If you are to obtain an equation of the streak line as well in this particular case which is a steady flow situation you will see that it is identical to this x y equal to 2. So, this was about path lines, stream lines and streak lines what they mean and given such simple velocity fields we are in a position to evaluate the equations in a closed form. Obviously, when you do a CFD analysis for a complicated flow pattern the velocity field will not be as simple as this such as what we have shown here. So, there in order to generate the streamline patterns and so on you will have to do some numerical manipulation. In fact, it turns out that many of the plotting softwares that people use today some of the popular ones are tech plot or MATLAB even they may be able to directly give you the streamline patterns once you provide the raw velocity field data. One way or the other the point to be noted here is that stream lines, streak lines and path lines are important quantities not only from a fluid mechanics point of view but from a CFD point of view as well you will end up generating streamline patterns more often than not whenever you are doing a CFD analysis. This was first part in the discussion of kinematics. What we have talked about here is so far the two different approaches of description of fluid motion. One was Eulerian approach which is the more common approach, the other one is the Lagrangian approach which is although simple to appreciate practically it is very very difficult to follow. We have come up with the connection between the rates of change that are inherent in Lagrangian approach and Eulerian approach. So, in the Lagrangian approach what we have is the substantial rate of change. In the Eulerian approach what we have are the convective rate of change and the local rate of change and we saw that the substantial rate of change is equal to the local rate of change plus the convective rate of change. We saw that if the flow field is steady the local rate of change is 0 it is non-existent and toward the end we saw that there are three graphical descriptors which we call path line, stream line and streak line which are important not only from fluid mechanics but from a CFD point of view as well. There is a good chance that next week when the second part of this workshop is going on you will see several of these stream line type patterns for the fluid mechanics problems that will be discussed.