 We continue with the discussing in the last lecture that there is a tank and through the bottom of the tank, the water is being drained out and the height of water in the tank is therefore changing with time. Our objective is to find out how the height changes with time. Now we were discussing about what is the significance or impact of the unsteady term that is retained or that should not be retained or should be retained is our doubt in the Bernoulli's equation. Now if you try to approximate it in some way, see in engineering we try to get a feel of the order of magnitude. So we may try to approximate it by a certain term which should be like a derivative of velocity with respect to time times some height. So let us say that if this dv dt was a constant, if it was a constant, not that it is a constant, if it was a constant it could have come out of the integral and then it would have been some equivalent constant dv dt times s2-s1. So s2-s1 may be roughly like the height, if you take a streamline which is coming I mean which is just along the axis then it is exactly equal to h but you cannot just write it as some equivalent dv dt into h because v is changing with time in an unknown way. So you do not have really an equivalent constant dv dt but you may make a kind of approximation. You can say that I approximate this dv dt with dv1 dt. If you see that except for very close to the outlet the streamlines are almost parallel to each other and when the streamlines are almost parallel to each other it represents a case when that v is not varying very much, see why v is varying, see there is a flow rate confined between these. So when the streamline the distance between these 2 streamlines remains the same you have say a1 v1 equal to a2 v2. So a1 is like this, a2 is like this both are like as if cross sectional areas with the streamlines as envelopes. In fact you can have a large number of streamlines their envelope may look like an imaginary pipe or a tube that is known as a stream tube. So it is a collection of streamlines making an imaginary tube within which the fluid is flowing. If you consider such a tube you can always see that the extent of that tube that remains almost the same till you come to the exit where it is really accelerating because now the area available to it is so small that it has to get adjusted to itself. So when the area is very small and it has to get adjusted to itself that is only a small portion in comparison to the tank extent. So if you approximate this dv dt with dv1 dt it is wrong but it will give us some picture or some idea of what is the effect of what is the impact of this term. So if you make an approximation that this is equal to dv1 dt times h. You have to remember that both are functions of time h is a function of time v1 is also a function of time. So if you write this equation in a bit different way you can write say v1 v2 square-v1 square by 2 plus or –g z1-v2 is equal to – of okay and z1-z2 is equal to h which is itself a function of time okay. So these are valid locally at each and every interval of time. At that time you have a dv dt and you have an h. Now if you try to compare different terms say we want to compare this term with this term. So if these 2 terms are compared then let us say this is term A and this is term B. So when can you neglect term B in comparison to term A when you have this mod of this divided by mod of gh when this is much-much less than 1 then B is much-much less than A. So if this is the condition well h is something which you do not consider locally because this is like h is always a constant I mean always a local constant that means whatever h is a function of time here in term A same h is there in term B. So only that means you have you are comparing dv dt with g. So the rate at which the change of velocity of the free surface it is there that is a sort of acceleration if it is comparable with the acceleration due to gravity then you cannot drop this term and then you should retain this term at least frame a differential equation it cannot be solved analytically. But if this is the case which is true for most of the practical cases then it is possible to drop this term. The second important point is irrespective of whether you drop this term or not A1 v1 to A2 v2 is what you are always using. The reason is straight forward the origin of this does not come from steady flow although this is valid for steady flow it does not mean that it cannot be used for cases when the flow is unsteady because the fundamental way in which it was derived from what from the continuity equation first by dropping the partial derivative of rho with respect to time equal to 0. So if rho is a constant partial derivative of rho with respect to time is 0 it may still be unsteady flow because the velocity may be function of time but rho not being a function of time was the first thing to drop the first term in the continuity equation that derivative with respect to time. For the other terms then what how we came up with this we integrated this that differential form of the continuity equation and then if they are say rho at the inlet and the exit sections are equal again if rho equal to constant that is valid then you have A1 v1 equal to A2 v2. So a very important thing is for A1 v1 equal to A2 v2 to be satisfied it is not necessary that it has to be a steady flow only thing rho should not change that is a very important thing that we have to keep in mind. So even when it is varying with time you can use that. Now let us say that this is the case so that we can drop the term B. So if we can drop the term B then you can write v2 square minus v1 square by 2 is equal to gh. Now what is v2 you can express v2 in terms of v1 so v2 is v1 into capital D square by small d square. It is v1 square capital D square by small d square minus 1 by 2 is equal to gh okay and the remaining work is very straight forward you can find out. So v1 is of the form some constant into root 2 gh okay where that constant is basically d square by d square minus 1 by that square root of that. See this gives a contradiction what is the contradiction when small d is very small says you consider the limit as small d by capital D tends to 0 that it is a very big tank of a large cross section area and there is a very small hole through which the water is coming down. Then how does this work yes how does this work c is almost 0 if c is almost 0 then v1 is almost 0 I mean practically it is true that if it is a tank of very large area and if there is a very small hole the velocity at which the free surface is coming down is not perceptible it is very small. So that is okay let us not bother about that too much let us just try to complete this one by writing this as minus dh dt is equal to c root 2 gh now if you integrate with respect to time you can find out how h varies with t this is a very simple work. Now try to relate this with a kind of again formula that you have used earlier in your studies. Let us think that this tank I mean this hole is not located here but located at the side okay that is this is a different example just I am drawing in the same figure to save the effort. So let us say that now this height is h which is changing with time so there is no hole here but there is some hole here there is a nozzle that is fitted and water is coming out okay. So when you are doing that the way in which most of you have done is like you have assumed the velocity that which the jet is coming out is root 2 gh this is known as Torricelli's formula. So how you have arrived at that equation you have used Bernoulli's equation between 1 and 2 at that time you are not very careful about whether they are along same streamline or not just out of pleasure you have applied between 2 points and then when you applied between 2 points you put v1 equal to 0 you put p1 equal to p2 the difference between the 2 heights h and so v2 will come root 2 gh right. So what are the assumptions under which that is valid that is not a very bad formula Torricelli derived it long back I mean in a historical perspective it is a great development because nowadays we can speak big words but the subject when it was fundamentally developed this itself was not a very trivial matter to resolve. So then when Torricelli found out this expression what are the assumptions in which this expression you expect to work still. So one of the thing was taken as v1 equal to 0 that means v1 equal to 0 when when capital D is much much greater than small d. So v1 is approximately tending to 0 the other approximations are that you are having a streamline like this with respect to which you have the points 1 and 2 and the unsteady term does not appear in that analysis and it is assumed to be an inviscid flow the greatest deviation from reality is because of the assumption of the inviscid flow. So that is one of the very important features that we have to keep in mind. So with that assumption this formula is not illogical but a very important thing is we must keep in mind that some of those assumptions are to be questioned one of the important assumption is like capital D is much much greater than small d which is true if it is a very large tank and from that there is a small hole through which water is coming out and the dropping of the unsteady term and we have discussed that I mean when how this unsteady term this particular term in what conditions it may be dropped or not. So this is a very simple problem but if you try to look into this problem very carefully it will give you a lot of insight on the use of Bernoulli's equation under different conditions and I would encourage you to think about it more deeply under what conditions different terms are important in different ways not just satisfied with finding h as a function of time but to write the differential equation of maybe say v1 as a function of time in a very simple case and in the most general case and try to compare them that is what are the terms that are making them to be different. We will consider another example in the unsteady Bernoulli's equation in the use of the unsteady Bernoulli's equation that is given by the next problem. Let us say that you have 2 plates these are circular plates we have solved problems with rectangular plates just for a change let us consider that it is a circular plate. So this is like this plate is coming down with a uniform velocity v okay this is a circular plate the radius of the plate is r and say that we are considering a coordinate system the local coordinate as small r. So small r is the local coordinate radius r. Now with this we are interested to see so the bottom plate is stationary there is some water with rho equal to constant and when this plate is coming down what is happening water is squeezed out of the place because whatever water was there say originally this was b0. So b equal to b0 at time equal to 0 but as this is coming down this b is changing b is decreasing. So where will that water go that water will be squeezed out radially to make sure that the continuity is maintained okay. So we are interested to find out how the pressure varies with r assume inviscid flow rho is constant that we have already defined or we have already assumed. So as we have seen that in all these cases it is important to get a feel of the velocity profile. So if it is an inviscid flow the velocity variation over the cross section over the section is not there so the velocity is uniform over each section but this uniform velocity is changing with radius. So how you can find out it you have to think that what is the rate at which this is pulling water downwards is the same rate at which it is being squeezed out. So if you consider a local radius r what is the rate at which this is coming down. So when you write a1 v1 equal to a2 v2 question is how do you write v1 v2 a1 and a2 what is v1 v1 is the rate at which so it is like an artificial flow imposed by the movement of the top plate. So that flow velocity is given by v1. So what is that a1 into so what is a1 if you consider only up to a local radius of small r so a1 is pi into small r square. So a1 is pi into small r square what is v1 v1 is v because it is a uniform rate this is uniform this is not a function of time this is constant is equal to what is a2 2 pi r into b, b is a function of time into v2 or v as a function of r. Let us write vr just to emphasize that it is v at a radius r. So you can write v at a radius r is equal to v divided by vr by 2b okay. Now so this is the velocity at a radius r next we are interested to find out the pressure. So if we are satisfied with inviscid flow and rho equal to constant we can consider a streamline that connects 2 points any 2 points a1 and 2. So the streamlines how the streamlines will look so the streamlines will virtually look like this. So the flow is being squeezed out in this way so that is how a streamline will look. So let us take any 2 points located on the streamline and write the Bernoulli's equation between those 2 points located on this identified streamline. But because it is an unsteady flow we need to retain the unsteady term in the Bernoulli's equation. So p1 by rho plus v1 square by 2 plus gz1 is equal to p2 by rho plus v2 square by 2 plus gz2 plus let us say that we apply that between 2 points 1 point is located at r equal to small r and another point 2 is located at r equal to capital R okay. So when you have such a case you are getting rid of many things one is between the points 1 and 2 there is no difference in height. So this of course if this gap be itself is narrow then even if there was a change in height because of taking the points 1 and 2 not exactly along the same line that term itself is not that large. But like if you take them along the same horizontal line they are identically the same. Then you are interested to find out p1 and p2 you know that p2 is the atmospheric pressure. So because it is at the exit plane. So you are interested to write p1- say p2 is p atmospheric p1-p atmospheric by rho is equal to now v2 square-v1 square by 2. So v2 square-v1 square by 2 is v square by 4b square into capital R square- small r square because v is having only this component. Then plus this term so by 2 will be there right so 8b square then plus let us calculate that third term. So what is the partial derivative of v with respect to t that is the partial derivative of vr with respect to t that is the only v component that is there which is the function of t here v is the function of t here. So this will be-vr by 2b square into db dt right and-db dt is equal to v. So-db dt is equal to v just like the previous tank problem that we were considering. So this term becomes v square r by 2b square. So that you can substitute here and ds will be dr because you have chosen your streamline in such a way that the change in s is like change in r. So this is from integration from small r to capital R v square r by 2b square dr very straight forward to complete it it becomes v square by 4b square into capital R square- small r square. So at a given instant you can see the pressure at the radius small r is varying with time because b is a function of time. So this only at a given instant you can say. So at different instants you have different values of b and you can find out what is the value of b at a given time how because you know that db dt is-v. So b equal to b0-vt. So if you are given a particular time so this will give you b is equal to b0-vt. So if you are given a particular time you can find out what is the value of b at that time then you may substitute the value of b at that particular time to get the pressure at a radius okay. So you can clearly see that the unsteady Bernoulli's equation how it can be utilized. Now the next topic that we are going to discuss in the context of this Bernoulli's equation is the use of such equations. See the Bernoulli's equation has been one of the very popular equations in fluid mechanics not just because of its simplicity but because of its applicability in an approximate sense in terms of quantifying the nature of or the principle of working of many engineering devices and we will look into such examples of applications of Bernoulli's equations. So some of the examples we will not detail very much but we will only get the essence. The details of most of these examples are uploaded in the course website through note on the applications of the Bernoulli's equation. So if you go through that in details you will get all the detail picture because we are going to divide we are going to discuss subsequently about certain devices. These devices have certain intricacies and we will only highlight the major or the important features but for the other detail features you should refer to those notes. Now before coming to any device of very great engineering application we may come up with a sort of a very primitive device which you have already heard of something called as a siphon. So if you have say water in a tank like this and you are having a bend tube which is a sort of sucking water and ejecting water to a different place from the tank. So this is called as a siphon. The apparent amazing feature of the siphon is out of nothing it is pulling the water in the upward direction that is the apparent amazing feature but if you look into it a bit carefully it is not at all any amazing feature because eventually when it is discharged it is discharged at a level below. So the actual head difference which is working on it is this one which is a favorable one because effectively it is coming from this elevation to this elevation and this net elevation difference is actually giving it a velocity. So with that velocity the water is being sucked. So the fact that it is going up is nothing very special because eventually it comes down and it gets ejected from a height which is less than or below the level of the tank. But the good thing is that while doing it it can traverse a vertically upward distance. One is how much distance it can vertically traverse. So what should be this say if you call this as h then what is this h max? This is given by a practical consideration. Let us try to identify a streamline which connects the points say the streamlines will be bent like this but let us just consider a streamline which is confined between that points 1 and 2 which are almost like a located on a vertical line. So if we are interested to write the Bernoulli's equation we can write p1 by rho plus p1 square by 2. So every time whenever we are writing the Bernoulli's equation we are not repeating the assumptions but you should keep in mind that what are the assumptions on the basis of which we are writing it. So p1 by rho plus v1 square by 2 plus gz1 is equal to p2 by rho plus v2 square by 2 plus gz2. Now you can clearly see that at the level 1 you have pressure as the atmospheric pressure. So this is p atmospheric v1 is approximately equal to 0 just like the Torricelli's equation because the area here is so large that the velocity with respect to which this level is changing is very small as compared to the velocity here. v2 is same as the velocity at which the jet is ejected here if the area of cross section remains the same. So v1 is small because a1 is large as compared to the area available at 2. Then v2 is equal to vj that is the velocity at which the jet is coming out if the cross section is same and you can find it out that vj is nothing but approximately root 2g into this capital H by writing the Bernoulli's equation between 2 points on the same streamline whether if you continue with that streamline it goes like that and comes out. So the net elevation difference will remain this one if you write the Bernoulli's equation along the streamline between say points 1 and say a point j which is located here. Now when you write that one what you will get? You will get p2 by rho is equal to g into z1-z2. So g into z1-z2 is –gh-vj square by 2 okay. So you can clearly see that if you take the atmospheric pressure as 0 reference. So this is written by taking atmospheric pressure as 0 reference. So this is like a gauge pressure. So when you take the atmospheric pressure as 0 reference then p2 is negative because h is positive vj square is positive that means the pressure at this point is below atmospheric. So if it is below atmospheric it may come to a state when it comes to the local vapor pressure. So when the pressure falls below the local vapor pressure then what happens? Then vapor bubbles are formed. So when the vapor bubbles are formed it is nothing very special that vapor bubbles are formed but what is special is that when these vapor bubbles are transported or moved to a different place but the pressure is again higher they will collapse again to form a liquid. And once they collapse what happens? Basically then they were occupying a large volume but when they collapse again to be converted to liquid again there is a volume change. So it creates an unsteadiness in the flow and it can create a lot of vibration and noise and that is not so good for the flow and that type of phenomenon is known as cavitation. We will see in details what is cavitation when we will be discussing about the fluid machinery which will be our last chapter in this particular course. So we will not go into the details of like what is cavitation at this state but we have to keep in mind that it is better if we keep the pressure at 2 below the local vapor pressure that is below the vapor pressure which should be there at that corresponding temperature so that vapor is not formed. So that means we are keeping a restriction that P2 must be less than the vapor pressure at that local temperature of the fluid. So then you can see that you get 1 h max from that and that is the maximum h with respect to which you should design your system. So that you do not have a problem with formation of vapor. So the siphon in principle may be designed to be very like tall in height in terms of this bent tube but in practice one should not make it too tall because if you make it too tall it is possible that the pressure is so low that vapor are formed and that can create other disadvantages in terms of operation of the device. The next application when we will consider we will keep in mind that now whatever applications we are going to study our objective will be to have the Bernoulli's equation utilized in devices through which we are interested to measure the velocity or the flow rate in a say pipeline. So let us take an example. Let us say that you have a pipe like this horizontal pipe. Now water is flowing and you make certain holes in the pipeline. What holes you make? So first you make a hole like this. So when you make such a hole what will happen? The water will rise and it will come to a height. The height with respect to which the water rises will be an indicator of the local pressure at that location. Pressure at where? See we are interested about the centre line. So if we are interested about a point in the centre line what we are doing? We are sacrificing one thing. We are not able to exactly probe at the centre line. At the same axial location we are probing at a point which is different from the centre line. And we know that it is very much possible that the pressure at the centre line should be different in general from pressure at this. When they are different? When you have a curvature of the stream line? We have just in the previous lecture seen that if you have the gradient of pressure in the direction of n you have only when the stream lines have a radius of curvature which is non-infinity. But here if we consider that the stream lines are parallel to each other then you do not have that effect of the stream line curvature in terms of the pressure gradient. So whatever is the pressure here should be the same as the pressure here. So then this is an indicator of the local pressure. Now say we are interested to have an indication of the velocity. So for that what we can do? We can have another tube where we make a penetration in the wall but before that we have the tube directly confronting with the flow. So this tube and this tube is different. This is not directly interfering with the flow but this is directly interfering with the flow. When it is directly interfering with the flow it is bringing the flow to a standstill or a dead stop. So it is creating like a stagnation point where the flow comes to a dead stop. It cannot go further. So whatever water was coming here it comes to a dead stop what it will do? It will enter, it will rise through the tube and the question is will the rise will be greater than this one or less than this one. See this rise was the function of the pressure. Now what so the entire energy which was there in the flow if we assume that assumptions on the Bernoulli's equation those are valid. Now we have made the kinetic energy to 0. So the entire energy now contribution of pressure term plus the kinetic energy term will be successful to make it go further up because that where will that energy go? You have made the fluid to a dead stop. You are assuming it is a frictionless flow then where will that energy go? It will obviously make the fluid rise to a greater height and the difference between these 2 heights is if these points are very close to each other the pressures are almost the same. The difference between these 2 heights is just v square by 2g. So from this principle v is the velocity of flow at this point. So from this principle it is possible to make an estimation of the velocity and if you know the estimation of the velocity and if you assume it to be uniform then you can also have an estimate of the flow rate. Now if it is not uniform you can keep it at different radial locations and you can even find out how velocity varies radially because these 2 you can put at different radial locations. So this is put at r equal to 0 at the center line but you can also keep it away from the center line. So at different radius if you put it will give you a picture of velocity at different radius. So it is possible even to get a velocity profile if this is quite accurate. Of course there are many doubts about the accuracy of such a simple arrangement but it gives us a conceptual understanding. So the device which is based on this conceptual understanding is known as a pitot tube. So the last t is silent so it is pronounced as pitot tube. So this of course is to honor the name of the inventor of this device and it is a very simple device and the working principle of this device is based on 2 important definitions which we will tell now. One is known as static pressure. So what is the static pressure? Static pressure is the pressure which is there because of the intermolecular collisions. So that means if one is moving with the flow then what is the pressure felt because of just moving with the flow is the static pressure. So this is the pressure experienced in moving with the flow. So this is the result of the intermolecular collisions and this is the pressure that we fundamentally define. Now we are also going to define something called as stagnation pressure. So what is the stagnation pressure? Stagnation pressure is the pressure that is there at a point if the fluid is subjected to 0 velocity at that point in a reversible and adiabatic manner. So pressure at a point at which fluid is subjected to rest in a reversible and adiabatic. We will not go into the details of the reversible and adiabatic processes because these we will learn more in details in the thermodynamics course that we will have subsequently. But important understanding in our context is that one of the important requirements of this is it is a frictionless flow. So that means when the fluid is subjected to rest at a point you have to make sure that it is subjected to rest in a frictionless manner. So whatever is the pressure that this tube is getting is the stagnation pressure. So this is also known as a stagnation tube because it is reading gives an indication of the stagnation pressure and this is known as a static tube. So you can if you want to write the Bernoulli's equation between 2.1 and 2 which are located in such a closed manner. So at 0.1 if you have pressure as p, p s or say p 1 by rho plus v 1 square by 2 we are not writing the g z 1 and g z 2 they are so close that the difference in height is negligible is equal to p 2 by rho plus v 2 square by 2 plus g z 2 we are not writing again. So what is v 2? v 2 is 0 because it is a stagnation point. So the definition of the stagnation point is velocity is 0. So you can see that you write p 2 which is the stagnation pressure as p 1 which is the static pressure this is same as p 1 this is p static plus half rho v 1 square that means stagnation pressure is a sort of property of the flow if you know the velocity of flow but you have to keep in mind that this equation is derived by considering a frictionless condition and frictionless condition is valid when you are subjecting the flow to rest in a reversible and adiabatic process. So the definition of the stagnation pressure is to be kept in mind stagnation pressure is not just pressure at a stagnation point. What is a stagnation point? The stagnation point is a point where you have 0 velocity but it does not mean that pressure at that point is a stagnation point. Pressure at that point will be a stagnation pressure only if the flow is subjected to rest in a frictionless manner because the stagnation pressure is defined in that way it is not just sufficient it is necessary that you must have the velocity to be 0 at that point so that the pressure measured is a stagnation pressure but at the same time it is not velocity subjected to 0 in any way but it is subjected to 0 in a frictionless way. The second important thing is since these 2 points are very close to each other and you can just say stagnation point as stagnation pressure at a point just as a property which is dependent on the local velocity. So stagnation pressure did not always be measured through a stagnation point. So if you want to say find out stagnation pressure at a point you can simply say that it is the static pressure which is the regular of the normal pressure plus half flow v square that is the definition. So the stagnation pressure does not mean that you have to bring the fluid to rest at that point to get a pressure it is like how you physically conceive that pressure not that so it should not give you a false idea that whenever the velocity is non-zero stagnation pressure is not defined it is definitely defined it is just a physical way of looking into its interpretation. Now the next we will discuss 1 or 2 important flow measuring devices and the first device that we will discuss is known as a venturimeter. So what is a venturimeter? Say you have a pipeline and you are interested to measure flow through a pipeline. So what you are trying to do? Say you have a pipeline like you want to measure what is the rate of flow through the pipe. So how will you do it? There are many ways in which it can be done it. One of the ways is by utilizing a device called as venturimeter. So what is done? A part of the pipe is like replaced with a device. What is that device? The device is like this. So you have accelerating section by having a converging cone. Then you have a zone of uniform cross section and then you again come back to the pipe dimension. So this is known as diffuser. This is known as a throat and this is the converging section. So what is the objective? The objective is see by this way you are reducing the cross sectional area. So to maintain the continuity in a steady state you are what you are doing. So if you consider now the points, let us say that you consider points 1 and 2. The point 1 was having the velocity as same as that of the velocity of flow in the pipe. Now at the point 2 the velocity will be more or less. It will be more because the area of cross section has reduced. So since the velocity is more, now if you write the Bernoulli's equation assume that it is a frictionless flow then p by rho plus gz that term will be what? That term will be changing and if we can find out a measure of that change then it is possible to find out the velocity through the Bernoulli's equation. How we do that? Now let us say that you make a tapping of a manometer. That means let us say that you consider a hole in the pipeline and a hole here and connecting that with a manometer. So when you are connecting that with a manometer, see we have not taken the point 1 at the inlet of the converging section but at some location which is sufficiently away from that because here the streamline curvature effect will tend to become more and more dominant. So you want to take it away from such a place where the streamlines are almost parallel to each other. So pressure at this point and maybe pressure at this point should not be very different because of the streamline curvature effect. So we are having a manometer in which we have a fluid. Now in which limb the fluid height will be more or in which limb it will be less? Let us write the equation, the Bernoulli's equation along a streamline between the points, connecting the points 1 and 2. So let us say you have a streamline that connects 1 and 2. So you can write p1 by rho plus v1 square by 2 plus gz1 is equal to p2 by rho plus v2 square by 2 plus gz2. At the point 1 if you have this as the height of the limb and at the point 2 if you have this as the height of the limb. Now I have drawn it in this way. Do you accept that it should be like this? Let us say a fluid a mercury is there as a manometric fluid here. We call it rho m, the density of the mercury. Now is it an acceptable sketch in this case? The remaining is filled up with water. So if water is flowing through this tube let us say this is filled with water, the same fluid with which it is flowing here. So is this acceptable? By this you are expecting that pressure at 1 is greater than pressure at 2. We will see that that may not be correct also. Let us see. But this figure is correct. How that is possible? Let us see. So let us write, let us say that this is the difference in height that we measure. Say that is equal to delta h. So when you measure this height delta h then from that delta h it is possible to write the equation of the manometric principle. That is you can write that if you have 2 points a and b at the same horizontal level you have P a equal to P b. So when you write P a equal to P b, let us say that you are writing. Say this is your reference for measuring z1 and z2 in the Bernoulli's equation. So this is your z1 and this is your z2. You can use any datum but this is a convenient datum. So you can write P1 plus rho g z1 that is equal to pressure at a. Where rho is the density of the water that is flowing through the pipe is equal to P2 plus rho g z2 minus delta h plus rho mg delta h. So when you are finding out the difference in P1 and P2, P1 minus P2 you see that you can clean up the expression by noting that it is not just P1 minus P2 that is important. You have P1 plus rho g z1 minus P2 plus rho g z2 that is what is going to be important. So if you write P1 plus rho g z1 minus P2 plus rho g z2 then that is rho m minus rho into g into delta h. So in this figure you are expecting that delta h is positive. This is just the dimension. Rho m say this is mercury so we know that it is much heavier than water so rho m minus rho is positive that means we are expecting this to be positive. So what this reading gives us? This reading gives us not the difference in P1 and P2 but the difference in sum of P1 plus rho g z1 and P2 plus rho g z2. So it is not giving us the pressure difference so what it is giving us. So let us write this in a bit more explicit way so let us write it as P1 by rho g plus g1 so we are dividing it by rho g so P2 by rho g because we know that in this process we will get something called as head which we use as a terminology for this calculation. So this is rho m by rho minus 1 into delta h. This delta h is very important because this is what experimentally you can read okay. So when you read experimentally delta h you see that it is an indicator of not just the difference in pressure but the difference in pressure head plus the elevation. So when it is flowing from 1 to 2 it is possible that P1 is less than P2 but P1 by rho g plus z1 is greater than P2 by rho g plus z2 okay. So this flow is taking place from a higher value of this collected term to a lower value of this collected term. This collected term which is given by P by rho g plus z is known as a piezometric head. So P by rho g plus z this is called as a piezometric head. Why it is called as a piezometric head? The reason is that if you say have a pipe and if you puncture the pipe or if you penetrate the pipe say and if you have a tube through which the water goes up it is just like that static tube that we considered in the previous example. Then the elevation that it assumes here is the elevation because of its vertical location plus because of the static pressure at that point and this tube is commonly known as a piezometer tube. So that is why the name piezometric head. So in the manometer in this kind of an example we do not measure the pressure difference but we measure piezometric pressure or piezometric head difference. So in terms of head it is called as piezometric head. If you express in terms of pressure unit it is called as piezometric pressure. So always keep in mind in this case manometer is not measuring pressure difference. It is measuring piezometric pressure difference. These are very very fundamental mistakes that people make. See as I told in the very introductory class that we are born with certain intuitions that it will flow from high pressure to low pressure and you can clearly see that with a very simple example where it is not actually a practical example because we have considered a frictionless flow but even that it gives a very important insight that it need not be from a high pressure to low pressure. It is basically from a high piezometric pressure to a low piezometric pressure. Now fortunately what is important for this equation is only the piezometric pressure because if you see like if you if you write it in this form you will get p1 by rho g plus v1 square by 2 g sorry p1 by rho g plus g1 minus p2 by rho g plus g1 minus p2 by rho g plus g2 that is equal to v2 square minus v1 square by 2 g okay. So this is something which is a very simple term for us now because from the manometer we have got an explicit expression for that that is rho m by rho minus 1 into delta h. So this we can write as rho m by rho minus 1 into delta h and this is equal to now you can express v2 and v1 in terms of the volume flow rate. So if q is the volume flow rate then you can write as q equal to a1 v1 equal to a2 v2. What are the assumptions rho is constant and it is a uniform velocity profile over the section that is inviscid flow viscous effects are not there. So you can write v1 as q by a1 and v2 as q by a2. So if you substitute that in in in this expression it is possible to express v2 square minus v1 square as q square by 2 g into no this g is there. So no because of division by g v square becomes v square by 2 g. So q square by 2 g into 1 by a2 square minus 1 by a1 square is equal to rho m by rho minus 1 into delta h. So from here you can solve for what is q. Remember it is very theoretical. Why it is theoretical? Because it has considered many idealizations which do not actually occur in practice. So we will keep this in mind and in the next class we will try to identify that what are the idealizations which were here which need to be rectified and what are the important design considerations that should go with this device matching with the non-idealizations that we will discuss in the next class. But if it was ideal just by getting the delta h reading you could get what is the flow rate through the pipe because a1 and a2 you know are the areas of cross sections of 1 and 2 which are given geometrical parameters. So we stop here today. We will continue with that in the next class.