 So, the whole understanding is that surface with such a curvature we have seen in the previous lecture at least qualitatively that because of the curvature of the surface pressure gradient may be developed along it. Our objective is to have an assessment of the effect of pressure gradient. We have to keep in mind that pressure gradient for an external flow may be induced by such a curvature effect but for an internal flow when it is say flow in a pipe or a channel you automatically have a pressure gradient. So, there it is not a curvature induced pressure gradient but natural consequence of the confinement that is there and here it is the curvature that is creating the pressure gradient. Now our important objective first will be to see that what is the effect of or what is the demarcating effect of dp dx less than 0 and dp dx greater than 0 and we will look into these cases separately. So, when we say x remember we are talking about a boundary layer coordinate. So, this x need not be a rectilinear coordinate x and a rectilinear coordinate y. So, let us say that when we are talking about dp dx less than 0 maybe we are talking about any flow in this part and whenever we are talking about dp dx greater than 0 we are talking about the part in the right hand side. So, here you have dp dx less than 0 and here you have dp dx greater than 0. We will try to draw the following profiles say u versus x just the other way we will plot u versus y del u del y versus y and del 2 u del y 2 versus y. We will try to make a qualitative sketch of these. Let us first do it for dp dx less than 0. So, for dp dx less than 0 I mean we have already seen dp dx even dp dx equal to 0 we have seen for flow over a flat plate. So, the situation is quite similar even for dp dx less than 0 and what you will get is say this is how you will vary with y till you come to a condition where like when y tends to delta u will tend to u infinity something like that. Now if you want to plot du dy versus y. So, first of all at the wall what is du dy positive or negative at the wall what is du dy positive or negative positive right u is increasing with y. So, you have a positive du dy at the wall what is du dy at the first stream as y tends to infinity it tends to 0. So, maybe like this in between what should be the variation the variation is monotonic there is no separate physical behavior. What is d2u dy2 at y equal to 0 negative because du dy is decreasing with y. So, you have a negative d2u dy2 and eventually it will come down to 0 as y tends to delta. So, that is the qualitative picture and this kind of qualitative picture is general hallmark of a flow with favorable pressure gradient. So, this is favorable pressure gradient next let us consider the case of dp dx greater than 0 which we have seen is known as adverse pressure gradient. Let us try to make such plots velocity profile du dy versus y before making the plot let us try to look into the corresponding boundary layer equations. So, the boundary layer equation what you have with pressure gradient u del u del x plus v del u del y is equal to nu del 2 u del y 2 minus 1 by rho dp dx. Let us see that how we applied this for dp dx less than 0 see at the wall let us say we are interested to apply it at the wall. So, when it is at the wall u is 0 by no slip 0 by no penetration. So, basically at the wall what you have is 1 by rho dp dx. So, what we conclude out of this is that when dp dx is negative this is negative and that is what we got here. So, this has to be consistent with the governing equation this we did intuitively from our previous experience in dealing with such cases, but it agrees with the governing equation. When it comes to dp dx greater than 0 for dp dx greater than 0 you must have d 2 u dy 2 at the wall greater than 0 right. So, at the wall let us start with this diagram. So, d 2 u dy 2 is greater than 0 at the wall right. The next question is that what is d 2 u dy 2 at the far stream far far away if it has to be 0 right at the very far stream away from the wall there should be no sort of gradient like u is like u infinity constant. So, far away you must have like asymptotically d 2 u dy 2 is 0. The question is then would we have d 2 u dy 2 something like this maybe something like this or even could we have it cross the y axis somewhere there could be many possibilities that is this is one value at the wall this is the value at the far stream far far away. Think of a case could you have in between a point where u is a maximum you could in between you should have a point where u is a maximum right that means you just has attained the maximum and then maybe further it is change is very less typically just outside the boundary layer. So, when u is a maximum you have d u dy equal to 0 and what is d 2 u dy 2 negative. So, you have it is value positive somewhere it is value positive somewhere because at some point away from the wall you must have u as maximum and this is a continuous function. So, it must cross 0 somewhere right. So, the correct variations will not be these ones right try to understand this very carefully by logical reasoning that you have d 2 u dy 2 greater than 0 at the wall there must be some point away from the wall where u is a maximum d u dy is 0 and d 2 u dy 2 is negative. So, that means there is a variation from and d 2 u dy to the second derivative is a continuous function. So, it has a variation from positive to negative somewhere it must have cross the 0 right and that is why its variation should be like this. So, that means we have identified a point where you have d 2 u dy 2 maybe 0 and not the far stream but somewhere in between. See the far stream d 2 u dy 2 0 is obvious this is not something which is obvious this is not an obvious conclusion. So, now if you want to extrapolate this plot that is you want to get a d u dy variation. So, what will be the d u dy variation at the wall you have a d u dy something okay there we will see that there may be a limiting case when you may also have d u dy at the wall in the limiting sense 0 and then what happens we will see that what is that. But in a general sense d u dy at the wall is say positive then you know that at this point you will have d u dy as the maximum that is why d 2 u dy 2 is 0. So, you may have something like this. So, d u dy maximum therefore d 2 u dy 2 is 0 like that what happens to the y u versus y just look into u versus y just look into u versus y. So, now if you draw the velocity profile here at this point you have d 2 u dy 2 is 0 that means it is an inflection point that is there is no rate of change of d u dy with respect to y. So, you may have a curve may be something of this sort where the curvature changes. But what you have is the slope from by looking from 1 n is same as the so at this point if you consider the slope like this and the slope like this. So, it is may be just show it in a different way something like an essay. So, where this will come to a limit of course where it will not change any further. But this is the point where no matter we take slope from the bottom part of the curve or the top part of the curve is the same. So, that point we call as a point of inflection p i. So, the point of inflection in the velocity profile is one of the hallmarks of the dp dx greater than 0 which is not there for dp dx less than 0. This is the first thing that we are understanding. Now, next thing is that now what happens at the wall? See when you have a velocity profile of this particular shape it is highly possible that it comes to a limiting condition when d u dy is 0 at the wall just you look into this particular form. So, if you consider an extreme case of this will represent what? An extreme case of this may have a situation when like the velocity profile takes off almost in this way almost like a vertical take off. That means the with change in y u does not change. So, d u dy with such a shape of velocity profile it is possible with such a shape it is not possible. So, you may have a limiting case when d u dy with the adverse pressure gradient the d u dy at the wall may even become 0. See in a favourable case it started with greater than 0 when it is coming less and less favourable at the wall you may have d u dy equal to 0 and then if it continues in that way it may be less than 0 and less than 0 is such a case when actually u will be not in the positive direction but in the negative direction. So, reverse flow will occur to understand that let us try to draw a schematic say for flow over may be a circular shaped object or object with a curvature. So, let us say that we have an object like this. So, we are interested to draw velocity profiles at different locations at different radial locations. So, initially when the dp dx is less than 0 you see that it is a intuitive way in which the velocity profiles are there may be it maintains it like this in the process the boundary layer grows. Now when you come to the dp dx greater than 0 you will come with locations of inflection points in the velocity. So, now if you draw the velocity profile at some point you will come to a state let us say that at this point we come to a limiting condition when the velocity profile is something like this. We have just now seen that it is possible with dp dx greater than 0 that is what we learnt from this experience. So, this is a special case of course the velocity profile has an inflection point that is the first thing the second important thing is here you are having at the wall equal to 0 that is a limiting case when it is 0 at the wall and then the pressure gradient is going on getting more and more adverse then what is happening the flow may reverse in its direction. So, you might have sort of a reverse flow like this in this way you may get points. So, if you have a reverse flow we will see physically that why such a reverse flow is occurring, but if you come to a location let us say this point this point represents what this point represents a location where the velocity is 0. Similarly, you may draw such velocity profiles at all other points and you will find that there will be such points. So, if you join such points so get a locus of may be all the points where you have such cases with velocity as 0 and above that velocity is greater than 0 that means as if the wall which was the location of 0 velocity has got shifted to this new location because beyond which you are having the velocity as greater than 0 and it is maintaining its direction and here what is happening here what is happening is a local back flow that is a loss. So, to say because it does not contribute to the main energy of the flow. So, this point where this all initiates that is known as a point of separation and when this has occurred we say that the boundary layer has got separated. So, you no more have a monotonous growth of the boundary layer beyond this point and the name of this point is the point of separation. So, we call this as s this is separation point. You may have different stream lines passing through the separation point may be like this or whatever it I mean that is possible this is the stream line passing through the separation point just as an example I mean there it may be possible that you have locally recirculating flows. So, you may also have other stream lines like this because in some case you have a negative flow then you may have a positive flow like this. So, the question or the emphasis is not really that how should the stream lines look there the big emphasis is that from the till the point s the boundary layer did grow as we expected and as we discussed may be like for flow over a flat plate. Flow over a flat plate is what is the special case what is the special case dp dx equal to 0. So, when dp dx equal to 0 for flow over a flat plate where lies the point of inflection look into the governing equation dp dx equal to mu at the wall. So, if dp dx is 0 that means this is equal to 0 that means point of inflection for flow over a flat plate is at the wall at y equal to 0. So, that case is the limiting case it is saved from such a phenomenon because if you do not have a point of inflection within the flow domain this type of limiting situation does not occur where you might have the velocity gradient 0 at the wall. So, when you have the velocity gradient 0 at the wall from the point s what we have come to a conclusion that the boundary layer separation has taken place point s onwards. That means because of this type of reverse flow the boundary layer is not monotonically growing as it was doing earlier and as if it has got detached from the wall as if the new boundary that now represents the growth of whatever apparent growth of boundary layer is this one is the new one. So, the very important conclusion we get out of this is that this line sort of represents a wall that has got detached from the main wall like that. So, it is therefore considered as a flow separation or boundary layer separation when you consider the case of a boundary layer theory. So, very important restriction is that beyond this point s you cannot apply the boundary layer theory. So, when we were discussing about the boundary layer theory and its assumptions one of the important assumptions that we made is Reynolds number is large that remains, but the other important assumption is that there is no boundary layer separation because if boundary layer separation occurs from the separation point onwards the boundary layer theory is no more valid and that we have to keep in mind. So, these are the two most important fundamental assumptions behind the boundary layer theory that we have developed. Now, what happens beyond the separation point actually there is a low pressure region that is formed beyond the separation point and because of this formation of the low pressure region this low pressure region behind the separation point it is called as a wake and what is important here is to note that because of the formation of that wake or the low pressure region or the separation the symmetry in the pressure distribution around the body is disturbed. So, if there was no such separation the pressure distribution with respect to the vertical axis would have been same for the left and the right, but because it has occurred now it gives rise to a low pressure region and the pressure distribution does not remain uniform and therefore there occurs a net resultant force because of the pressure distribution that acts on the body and the pressure distribution that acts on the body it is very critical because depending on that you may have a resultant force that acts on the body. We have seen that the resultant force that acts on the body is a combination of the effect of the shear as well as the pressure gradient. The pressure gradient is absent for flow over a flat plate, but for flow with such surfaces with such curvature that effect cannot be neglected that is one of the important conclusions that we should keep in mind. Now, before moving on to the consequence of this separation let us maybe look into one example to see that how the velocity profile gets affected because of it. Let us say that we want to have an estimation of the velocity profile at the point of separation and let us say that it is given in this form u y u infinity is equal to a0 plus a1 y by delta plus a2. This is an example of assume that this is velocity profile at the inception of separation. Our objective is to find out a0 a1 a2 a3 consistent with the physical conditions. So, it is just like estimating a velocity profile or fitting a velocity profile in a polynomial form that we did for flow over a flat plate, but the boundary conditions which should be consistent with this one the boundary conditions are somewhat altered. So, what are the boundary conditions that we need to consider for this? Again there are 4 constants. So, we should consider 4 boundary conditions to specify the velocity profile completely. So, what are the boundary conditions that we should consider? Number 1, y equal to 0 u equal to 0 fine. Then at y tends to infinity or at y equal to delta here which is like tends to infinity at y equal to delta u equal to u infinity at y equal to delta you have no further change in u and fourth one at y equal to 0 this is equal to 0. This is very very important because this is the hallmark of separation at y equal to 0 you cannot apply d2 u dy2 equal to 0 that was for flow over a flat plate because dp dx was 0. Here dp dx is not 0 because you are given that the velocity profile at the inception of separation when there is a chance of separation it must be dp dx greater than 0. Now the big question is that why such a separation will occur? See when the fluid is there within the boundary layer it has some energy but because of the reduction in the velocity that is being imposed by the shear effect within the boundary layer the boundary layer tries to or the fluid in the boundary layer has a tendency to get slowed down. Now when it has a tendency to get slowed down still it has to move forward then what is that which will try to make it move forward? One is the shear interaction with the outer layer that is what it tends to make it move forward. So it has sort of it takes some energy from that and tries to retain its energy and make its forward movement and if you have dp dx less than 0 that on the top of it helps it. If dp dx is equal to 0 then also it is okay there is no extra effect from dp dx as a help but there is no extra effect also in terms of resistance. But when dp dx is greater than 0 then the pressure gradient resists the forward movement. So there is a poor fluid element which is located within the boundary layer has a tendency to get slowed down and the pressure gradient externally on the top of it tends to slow it down further and further. So it has a tendency of continuous reduction in momentum and that may give rise to shift in the direction of the velocity. So that is how you can get a back flow or a reverse flow. So the momentum in the boundary layer is not sufficient and external pressure gradient which is acting adverse to it if dp dx is greater than 0 may be just sufficient to make it flow in the reverse direction. But the question is if dp dx is greater than 0 is it necessary that it has to flow in the reverse direction. The answer is no it may not because still it might have sufficient momentum to overcome that adverse pressure gradient and somehow maintain the flow but if the pressure gradient is so adverse that it does not have enough momentum to overcome that then only it will be a reverse flow. So existence of an adverse pressure gradient is a necessary condition for boundary layer separation but it is not a sufficient condition. So you cannot say that there is an adverse pressure gradient and because of that there will be boundary layer separation. But you can definitely say that if there is no adverse pressure gradient there will be no boundary layer separation. So adverse pressure gradient is a necessary condition for boundary layer separation but not sufficient. The sufficiency will come from what is the strength of that adverse pressure gradient is it good enough to overcome the forward momentum and create a reverse flow that is the important concept that we need to learn. So what we have understood is that if you have a flow if you have a solid body in a fluid the solid body in a fluid may be subjected to different forces and these forces may come from one is the skin friction drag which we have identified for flow over a flat plate even without pressure gradient that is because of the shear plus there may be a force due to the pressure gradient. So that force due to pressure gradient may give rise to a net effect which may also be a drag force but to understand that what is the net effect is it just a drag force or something more than that we have to formally define that what are the forces which may act on a body which is immersed in a fluid. So let us do that. So we have now at least got the origin of the forces that is the pressure gradient and the shear but how we combine these to get the net force. So the resultant force on a body immersed in a fluid under motion. So do not confuse it with fluid statics it is not fluid at rest. So we are not talking about forces like buoyancy forces like that. So this is solely because of the relative motion between the fluid and the solid whatever is the force that force we want to identify. So generally the convention is that let us say you have an arbitrary shaped body let us consider a 2 dimensional flow. So what are the forces which are acting on the body? One is there is a pressure distribution around the body. The pressure distribution is not uniform. Why it is not uniform? Because for example there may be flow separation because of the flow separation after the flow separation has occurred it will be a low pressure region and that means there is asymmetric pressure distribution that is created because of flow separation. So the net effect is that the pressure distribution will be non-uniform there will be a force due to pressure distribution plus there will be a force due to the viscous shear effect. So you will have some shear effect also. Let us say that there will be some force due to shear. So you can see that even mathematically whatever we considered as tangential forces and normal components of forces those are automatically coming physically through the effect of pressure as the normal force and the effect of shear through the tangential force. The resultant force whatever if you integrate the net effect over the body you may resolve it into 2 components. What are of course you may resolve it into 2 components in many ways but what is the important convention? Let us say that this is the free stream direction. Let us say this is the u infinity direction. So the force on the body what is usually done is it is resolved into 2 components. One is in the direction of the free stream and another is perpendicular to the direction of the free stream and let us say that this is the resultant force. So this force which is along the direction of the free stream is known as the drag force and which is perpendicular to the direction of the free stream is known as the lift force. This is where we are talking about a 2 dimensional flow but if it is a 3 dimensional flow there also may be a side wise force in the third direction. If there is also a component perpendicular to the plane where we are drawing the figure but otherwise these 2 components must be there because the flow at least must be a planar flow and these are very important because this drag force is something which will come in the form of frictional resistance and the lift force is something which is may give an upward motion to the body and this is very important for wings of aircraft. So if you have a aerofoil section like which represents the wings of aircraft then because of the lift force generated it may go from a lower elevation to a higher elevation. So both the lift force and the drag force are very important and it is very important to characterize those. So when we characterize the lift and drag force we have seen that the drag force we may characterize by a coefficient of drag Cd. So the coefficient of drag is the drag force by half rho u infinity square into some reference area. We will discuss that what should be the reference area. Similarly you have a coefficient of lift. This is the lift force divided by some half rho u infinity square. So the coefficient of the lift and the coefficient of the drag are just non-dimensional ways of representing the lift and the drag force. So when you have this drag force, this drag force has 2 contributors because see the resultant force along the direction of the flow is a resultant force due to shear having a component along the direction of flow plus resultant force due to pressure having a component along the direction of flow. So when we have the drag force, its physical origin is divided into 2 parts. What are these parts? One is known as skin friction drag which originates because of the shear and another is known as pressure drag or form drag. Then from the name pressure drag or form drag it is clear. First of all why then from the name pressure drag it is clear that it is because of the pressure distribution on the body. So just like the skin friction drag is because of the shear force distribution on the body, the pressure drag is due to the pressure distribution on the body. Why it is also called as form drag? Because it is a strong function of the geometric form of the body because the geometry of the body is what induces the pressure gradient. So that is why it is called as a form drag. In engineering one of the important objectives may be to minimize the drag force between a fluid and a solid. And if one wants to do that then one has to see that which one is the stronger contributor the skin friction drag or the form drag and it all depends on whether the flow is laminar or turbulent. So we will now try to look into the issue that first of all if the flow is if we consider the skin friction drag is it greater for a laminar flow or turbulent flow? If we consider a form drag or pressure drag is it greater for turbulent flow or laminar flow? So let us look into that. So if we consider say a skin friction drag, clean friction drag depends on what? It depends on the wall shear stress distribution. Wall shear stress depends on what? One is the viscosity of the fluid that does not change if the flow becomes turbulent from laminar. What changes is the velocity gradient at the wall. So now you tell that if you consider the skin friction drag for a flow which is laminar and for a flow which is turbulent then which should offer more skin friction drag. Obviously that will have more skin friction drag for which du dy is more. If you consider mu du dy the wall shear stress. So which should have more du dy? Think of a situation. Consider that you have flow in a channel. If you have a laminar flow what is the velocity typical if you have fully developed laminar flow velocity profile like parabolic distribution. This is what we have derived when we were solving this by using Navier-Stokes equation. But if it is a turbulent flow of course when it is a turbulent flow we are drawing only the average component of the velocity. How it will look? See for the turbulent flow the important thing is that there is a great level of mixing in the turbulent flow because of the interaction between eddies of different scales. So because of very efficient mixing between the in the turbulent flow the velocity so mixing what it tries to do? It tries to homogenize it. So it will try to have a uniform velocity profile almost throughout but at the wall it should satisfy the no slip boundary condition. So it should abruptly come down to 0 velocity at the wall. So the turbulent flow average velocity profile will be something like this. So if you now consider that where du dy is more at the wall for the turbulent flow because here there is a high change in u within a very short y distance close to the wall. So this is the turbulent and this is the laminar. So we can see that since du dy at the wall for turbulent is greater than du dy for the wall for the laminar flow considering all other conditions unaltered and when we say turbulent du dy you have to keep in mind this is the mean velocity that we are talking about. So that means you have the wall shear stress for the turbulent is greater than that for the laminar. That means you have the skin friction drag is greater for the turbulent flow. Now let us come to the form drag or pressure drag. When we come to the form drag, the form drag we are bothered about the geometry of the body. So the form drag all depends on the pressure distribution around the body. So form drag occurs because of what? Because of a boundary layer separation. Boundary layer separation will occur when the fluid in the boundary layer has insufficient momentum so that adverse pressure gradient strongly overcomes that and creates a reverse flow. So now in which case within the boundary layer you have more momentum for a turbulent boundary layer or a laminar boundary layer. You expect a greater momentum in the boundary layer in the turbulent boundary layer and since you have more momentum within the boundary layer for the turbulent boundary layer you have a greater chance of resisting the adverse pressure gradient. Therefore flow separation may be delayed by virtue of having a turbulent boundary layer. So if the flow separation is delayed what will happen? The separation point will shift to further downstream and the pressure distribution which is there around the body that asymmetry will be less and less. Therefore the drag force because of the difference in pressure between the points which are located upstream and downstream of the separation. Since the separation point shift towards the end towards the downstream then what happens? Then obviously you have a much smaller wake. Wake is the low pressure region beyond the separation point. So beyond the separation point the region if it is very small then that is in a way better because it will give rise to less form drag or less pressure drag. So if you have a turbulent boundary layer it has greater momentum it will delay the separation and in the process it will have a less form drag. So the turbulent boundary layer has higher momentum so it delays the separation and that means it has less form drag or pressure drag. See the engineering design has therefore a sort of conflicting things. So if you have a laminar boundary layer you have less skin friction drag but a greater pressure drag or form drag whereas if you have a turbulent boundary layer it is the other way. Your question is not to minimize this individually but the total drag force you have to minimize. So when you need to minimize the total drag force you have to see which one is dominating and when you see that which one is dominating it depends on what is the shape of the body. If the shape of the body is such that it induces a separation much earlier then the form drag is what dominates. If the shape of the body is such that it does not induce the separation so early then the skin friction drag is what dominates. Let us look into some movies where we will try to figure out that what are the important characteristics of flow behind different bodies. So if you look into this flow this is flow past a cylinder okay. So if you have if you consider this that circular thing as a cylinder where the length is perpendicular to the plane of the figure. If you play it once more you see that after the separation so the separation has occurred somewhere like maybe at somewhat in the middle section beyond that you have the region in the back where you have these rotating structures and all those are low pressure regions. So but if you consider a different shape body you may also see a very similar thing. This is flow past a building. So if you so it is basically a building simulated and if you see that there is a flow separation there is a low pressure region in the back. So whatever is the region which is towards the right after the boundary layer separation has occurred that is a low pressure region and it strongly depends on the shape. So let us look into the case if you change the shape what happens. Let us have a shape like this. So if you see now this is such a shape where the boundary layer separation is not there it is almost avoided it of course depends on that what is the direction of the relative velocity between the solid and the fluid and that we will see but just you can see that because of the section the section is such that it is not blunt that you can see the previous bodies were sort of blunt and these those are technically called as bluff bodies. This type of a shape is called as a streamline body. So a streamline body will have a much less tendency of boundary layer separation and that let us try to see an example where we have a combination of streamline and bluff body. So let us so the sort of board that is moving it is like a streamline shape and it is sailing fine in the water but it you see now closely the flow around the stirrer you see that this is this is quite bluff and the flow around that is something which is having the effect of flow past a bluff body. So I mean you may have combinations of bluff and streamline body together in the same effect and if you want to see that what is the consequence in engineering design we will of course we will try to understand this later that what is the consequence in engineering design but just to see that if you do not consider any boundary layer separation effect boundary layer separation effect comes if you consider boundary layer effect. Now if you do not consider boundary layer effect it is like an inviscid flow which is a potential flow you see that case at the bottom it does not show the boundary layer separation at the top you see that you can get a boundary layer separation and you have a low pressure region but if you design the car the backside like a streamline shape then you may delay that separation and that will reduce the drag. So you will see that many of the racing cars are designed in that way so to avoid or to minimize the drag force we will come into these examples one after the other when we will look into more details of flow past different shape bodies but our important conclusion is that if you have bodies of different shapes depending on that different drag forces may dominate. So what are the important measures that you may take to delay the boundary layer separation because boundary layer separation is an important issue that gives rise to a form drag or a pressure drag. So what could be the measures? There are several measures which are possible one of the measures is known as boundary layer suction. So let us just note this down that what are the measures possible measures to delay boundary layer separation. Number one is boundary layer suction. Boundary layer suction is what? Let us say that you have a solid boundary and there is a boundary layer that is formed of course not for such a straight surface the form drag will be important. Let us say that you have a curved boundary. Boundary layer is formed close to it and somehow what you do is you suck away fluid from the boundary layer into the solid. In the process what you do you are basically increasing the momentum of the fluid within the boundary layer. So basically you are sucking the fluid from the boundary layer. So in the process what you are doing you are enhancing the momentum of the fluid within the boundary layer. We have seen that one of the reasons of boundary layer separation is that the fluid in the boundary layer does not have sufficient momentum to overcome the adverse pressure gradient. So in this way so we have looked into one example earlier in the previous class say where you have some fluid which you put through the holes from the top and you suck that from the bottom. So this is one of the ways in which you can create greater momentum in the boundary layer. You could also do it in the other way. You might inject fluid into the boundary layer. So increasing the momentum is not so much dependent on this whether it is upwards or downwards. So you might have boundary layer injection. Then of course changing the shape of the body that is streamlining the body. So if you have a bluff body you may redesign it to make it of a streamline shape okay. Then may be by artificially roughening the surface. If you roughen the surface what happens? You may create sufficient perturbations to trigger the onset of turbulence. So roughening the surface artificially. These are some of the very practical things and later on in one of our lectures we will see that sometimes the cricket ball is roughened in one side and what effect does it have on creating the swing in the cricket ball in a particular direction. And I mean these are some of the basic scientific tools that go behind. So we will just work out a problem quickly on the calculation of the resultant drag force on a body. So let us say that a circular cylinder with p infinity u infinity as the free stream conditions measure an angle theta from this direction and it is given that boundary layer separation occurs. We will see that when boundary layer separation occurs in such a case in detail later on. But say for this problem it is given that boundary layer separation occurs at theta is equal to 108.8 degree. This is from experiments. The wall shear stress is given by tau wall is equal to 1.5 rho u infinity square into Reynolds number to the power – 1.5 into a function of theta where the function of theta is equal to 0 for a range of theta between 108.8 degree to 180 degree and integral 0 to 108.8 degree f theta sin theta is equal to 5.93. This is also from experiments. Also it is given the pressure distribution in a normalized way is given by p – p infinity by half rho u infinity square is equal to g theta such that integral 0 to 180 degree g theta cos theta d theta is equal to 1.17. This is also from experiment. So experimental pressure distribution and shear distribution is given as a function of theta. From that you find out what is the total, what is the net coefficient of drag. So you can clearly see that after the boundary layer has got separated say 108.8 degree beyond that you do not have a wall shear stress effect because as if the effect of the wall has got detached from the boundary. So now how do you find out the resultant force? So if you just consider a small element located on the surface let us say that you consider a small element d theta located at an angle theta. So the length of this one is what? Length of this one is r d theta. If you have a total force on this you have one as the shear tau wall and another as the pressure. So what is the force due to the wall shear stress df due to the shear? So that is tau wall into r d theta into the length of the cylinder that is number 1 but what you want is the drag force coefficient of drag. So you want the force component of this in the direction of the flow. So what will be that? So basically this into sin theta. So you will have df drag due to shear is equal to tau wall into r d theta into l into sin theta. So you have the total drag force due to shear is equal to tau wall into r d theta l sin theta integrate 2 into 0 to 180 degree. Tau wall you know as a function of theta and so you substitute that. So tau wall is equal to this into f theta. f theta sin theta d theta integral is already given to you. So you can substitute that here. It becomes a very simple calculation. Similarly what will be f drag due to pressure? In place of tau wall the effect will be because of p and sin theta will become cos theta. So 2 integral p r d theta l cos theta from 0 to 180 degree. So the total drag is the sum of these 2 that is the total drag and what is the CD? CD is the total drag force divided by half rho u infinity square into reference area. For this kind of a case so the circular cylinder is like this and flow is taking place cross it. So the reference area is taken as the projected area perpendicular to the flow. So what is the projected area perpendicular to the flow? L into 2r. So this a reference that is called as a frontal area or projected area. Projected or frontal area that is equal to 2r into l. So if you do that let me give you the final answer which you may check. The final answer is 1.17 plus 5.93 divided by square root of Reynolds number. So what we can see here is that the drag coefficient at very high Reynolds number may be virtually independent of Reynolds number. And what are the important things that we see? This 1.17 is from where? This is from the form drag. This is from the skin friction drag. So very high Reynolds number flow may have the form drag is the sole determining factor for very high Reynolds number flow. So you can clearly see the effect of the form drag and the skin friction drag. So let us stop here today. We will continue again in the next class. Thank you.