 mass body and we have concluded that depending on the shape of the body, the pressure gradient has a strong role to play. So if the body has a particular curvature or maybe a bluff shape, then the pressure distribution on the body may give rise to interesting and important rack forces. Now regarding the pressure distribution it has to be remembered that there is a distinctive behavior if the boundary layer separation has occurred. But before the boundary layer separation has occurred, the pressure distribution is something which should follow as a consequence of what happens in the outer stream or the free stream. Because one of the important assumptions or one of the important conclusions that we got from the boundary layer theory is that the pressure gradient which acts on the boundary layer is same as the pressure gradient what is imposed on it by the free stream. And if the pressure gradient has a strong role to play in terms of dictating the rack force, we have to quantitatively understand that how the pressure gradient acts on the body of a given shape. And for that a clue may be that what is the pressure gradient imposed on the body from the outer stream or outside the boundary layer. This is valid so long as the boundary layer theory itself is valid that means so long as there is no boundary layer separation. But till there is boundary layer separation, till that limit one may use the boundary layer theory and provided other assumptions are valid that is Reynolds number is large and then we may evaluate the effect of the pressure gradient without referring to the boundary layer but just by referring to the free stream or far stream outside the boundary layer. And outside the boundary layer, the flow does not understand the effect of viscosity. So what is the typical characteristic of the flow outside the boundary layer? So if you have a boundary layer like this, outside the boundary layer you have some free stream. The free stream velocity is u infinity but as it goes past the body, this may change because of the curvature of the body and whatever is the important thing, we have to keep in mind that initially say if it was u infinity that means if you consider say a 2 dimensional flow for example, you have the x component of velocity as u infinity and y component of velocity as 0. Now originally is it an irrotational flow, this is an irrotational flow. Now an irrotational flow will remain irrotational if it is inviscid. So outside the boundary layer, the viscous effects are not felt and therefore whatever flow was originally irrotational will tend to remain as irrotational and when it remains irrotational always, we may always describe it through the gradient of a scalar potential. The velocity field, we may always express through the gradient of a scalar potential if the velocity field is irrotational and such a flow is known as potential flow. So potential flow therefore as good as inviscid and irrotational flow, the key here is that if it is irrotational initially as the free stream condition but there are viscous effects inside, the viscous effects will make the irrotational flow a rotational one. Therefore you must have an inviscid condition on the top of this one to make sure that it is irrotational forever for all conditions. So that for all conditions you may find out that for all conditions you have the velocity as the gradient of a scalar potential. It is important to look into the potential flow. See when we discuss about potential flow, the first bottleneck or the first mental block that comes to our mind is that therefore we are talking about a case when viscous effects are not present and when viscous effects are not present, it is an ideal fluid flow. Will such a flow exist? The question is not whether viscosity is 0 or not. The question is whether effects of viscosity are important or not. So we have seen that outside the boundary layer, the effect of viscosity is not directly important. It does not mean that the fluid has no viscosity, it simply means that the viscosity is not coming into the picture in terms of dictating the fluid dynamic characteristics and therefore it is as good as inviscid flow outside the boundary layer. So when it is as good as inviscid flow outside the boundary layer, it is also irrotational as the free stream condition, it will remain as irrotational for the entire region over the body till you come to a condition where these conclusions you may never draw, thus what are those conditions? Boundary layer separation. Because if boundary layer separation has occurred, there is no question of a boundary layer, no question of something outside the boundary layer, so all those things become irrelevant. So these considerations we may apply till the boundary layer is growing, till there is no boundary layer separation. But there is always a limit up to which the boundary layer is growing, at least up to that limit if we want to evaluate what is the effect of the pressure gradient, it is important that we calculate the pressure field outside the boundary layer because the same pressure field is imposed on the fluid in the boundary layer and the pressure field outside the boundary layer may under these conditions be calculated by the potential flow theory and that is why the potential flow is important even in the context when you have the effect of viscosity or boundary layer. The next question is that how should we approach for evaluating the velocity and the pressure fields in such a flow field? The potential flow cases that is where you neglect the viscous effects are really mathematicians paradise and that is how the entire subject of fluid mechanics initially developed through the mathematicians and one of the important techniques by which this entire thing was entire mathematical development occurred for potential flows, one of the important mathematical techniques was by using the theory of complex variables. So we will be using the theory of complex variables very briefly we will discuss that why theory of complex variables may be important in such a context and how it may be utilized in this context but we will mainly look into the developments using that theory that how you generate different types of flow, different types of potential flows by using the calculus of complex variables. So we start with defining a function say capital F which is equal to phi plus i psi where what is phi and what is psi, phi is the velocity potential psi is the stream function. So what kind of flow we are talking about? We are talking about case where both phi and psi are defined. So 2 dimensional incompressible irrotational flow okay. So that is the example that we are taking. Now when you have these as F, see why we have chosen such an F? See when you choose a complex number say z equal to x plus iy, fundamentally it is ordered pair of real numbers x and y. These ordered pairs are chosen by some orthogonal basis x and y okay. Here phi and psi are also 2 orthogonal basis because we have seen lines of constant potential and the stream lines there is lines of constant psi are orthogonal to each other everywhere in the flow field except the stagnation point right. Therefore this is sort of a mapping from say a complex plane with x and y as coordinates to phi and psi as coordinates nothing more than that but the orthogonality is preserved. So because they are orthogonal we can choose such a complex function even preserving the general characteristics of what we have for such a complex number. Now the next thing is what is the important advantage of this? See if we write, so this is therefore a function of z, it is a sort of a transformation on z. So if we have such a function what is the advantage that we extract out of it? The advantage is that in straight away in one shot we are writing the velocity potential and stream function together through a single complex function. So if we write the complex function then we may find its real and imaginary parts one will give the stream function other will give the velocity potential that is one of the straight mathematical advantages. Next let us look into the differentiability of this function. We should remember that z is x plus i y. So let us try to find out what is df dz. The first question that we will like to answer is does df dz exist or not? So to test that we will soon see a criteria that what should be the corresponding test but just let us try to look into it in a simple way. Now the simple reason is z is a function of x and y. So what is this one? This is 1 right because z is x plus i y and what is this? 1 by i right. That means you multiply both numerator and denominator by i so this becomes minus i. So now next step is let us partially differentiate this with respect to x. So you have, so now what is the big test of the existence of this df dz for a complex function? One of the important tests is that the evaluation of the derivative should not be dependent on the direction in which it is calculated. See now the direction is important because you are approaching say you may approach a limit as delta g tends to 0 by using the definition of the limit that delta g tends to 0 you may approach from various directions because now it is not a single direction it is a plane of points. So from for example you may approach the limit as g tends to 0 by going in this way you may choose by going in this way you may choose by going in this way. So you may choose various directions in which you may approach towards the limit as g tends to 0. Now the existence of this derivative requires that the value of this df dz should not be dependent on the direction in which you approach in the limit to calculate the derivative. So that means that if you approach it along say x equal to constant line that is along y axis. So what you get? If you approach it along y equal to constant these are special directions and if we satisfy these directions any combination of these directions will satisfy because these are 2 mutually orthogonal coordinate directions. Interestingly you see that if you recall our basic fluid kinematics discussions this is what this is u right this is also u what is this v and what is this – v. So that means because we have u is equal to v is equal to this one you are having that no matter in whatever direction you calculate this it comes out to be u-iv and that means this df dz exists because we may calculate it to be the same irrespective of the direction in which you are calculating. The whole key is because these conditions are satisfied. So not in general but only for such cases when these relationships do exist these are known as Cauchy-Reimann conditions. So when this Cauchy-Reimann conditions are satisfied we can see that this df dz it exists that means it is called as complex differentiable that is it is differentiable in a complex sense. So that means it is complex differentiable and such a function which is complex differentiable is also known as an analytic function in complex theory that means f is analytic that is these are just terminologies it is important to know these terminologies. So you see that if f is analytic and yes from our basic fluid mechanics considerations we show that yes it is analytic if we frame f in this way then you see the next advantage is if you differentiate f with respect to z this is once you know that it is differentiable you write f as a function of z is differentiation is just ordinary differentiation. So you write f as a function of z find out df dz exactly in the same way as you could find out dy dx when y is a function of x. Then in one step you get another complex number if you equate its real and imaginary parts you get u and v the velocity components. So the strength of using this approach is if you frame a complex function then its derivative where it is phi plus i psi then its derivative directly will give you u-iv the velocity components u and v. Therefore it is a very convenient way of extracting the velocity components. Now this u-iv is in the context of description through the Cartesian coordinate systems but we may describe the same thing in terms of polar coordinate system. So let us try to see that how we describe it equivalently in terms of a polar coordinate system. So let us say that you have these are these are x y coordinates and let us say that you have this as r theta coordinates. Along the r and theta directions you have the velocity components as say vr and v theta. Along the x y components you have the velocity as u and v. Our objective is to write vr and v theta in terms of u and v. So what is vr or if we write u in terms of vr. So u equal to vr cos theta-v theta sin theta. What is v? vr sin theta-v theta cos theta. So our complex function df dz was u-iv. So let us just write u-iv. So vr cos theta-v theta sin theta-v r i sin theta-v theta i cos theta. So this you can write vr-iv theta-cos theta-i sin theta. That means by using the de Moivre's theorem you can write this as vr-iv theta-e to the power of-i theta. So we should remember that the alternative way of writing this in terms of the polar coordinates is u-iv is as good as vr-iv theta-e to the power of-i theta. Now the next thing is that with this mathematical background how we may utilize these to generate different flows. To understand that we will go through some examples. So the first example say our objective is to generate a uniform flow. Uniform flow. So you have to keep in mind that what is our objective? Our objective is to generate mathematically some physical flows by finding out the proper expressions of the capital F function. So all these flows are univc and irrotational type of flows. Let us say one is uniform flow. So uniform flow may be in different directions. Let us say that this is x and this is y and you have a uniform flow which is having a velocity v making an angle theta with the x axis or say alpha with the x axis. So you have such uniform flow. A free stream flow inclined at an angle alpha with the x. That is the flow we want to generate. So what is u here? u is v cos alpha. What is v? That is small v, v sin alpha. So u-iv is what? v cos alpha-i sin alpha. So v e to the power-i alpha. And this we want as dFdz. So f is v e to the power-i alpha z. May be plus something, plus some p-i q you may put. It is not always necessary to put that. But we will see that if you want to generate a body of a particular shape, then this constant will be useful. But just for a free stream flow you need not use this because then you need not generate a body of a particular shape. So this is not necessary. So you may choose a reference such that. So you have to keep in mind that this f is like, f is called as a complex potential. So the name of f is a complex potential. So it contains both the stream function and the velocity potential and you see those are not fundamental quantities. Those are just mathematical quantities. Fundamental quantities are the velocities for which gradients of those parameters are important. Absolute values are not. That is why if you add some extra constant it does not matter. When you calculate the gradient of the derivative this does not matter at all. So this is just a choice of reference. Depending on your choice of reference you may put this as 0 also. It makes no difference. It gives still the same velocity field. Now if you, so using this example let us say that you want to generate a uniform flow along x. You want to generate a uniform flow along x with a velocity say capital V equal to u infinity and alpha equal to 0. So what is f in that case u infinity into z. Now this is phi plus i psi right. z is x plus i y. So what is phi? Phi is u infinity x. Psi is u infinity y. So how do the streamlines look like? Constant psi lines, lines with y equal to constant right parallel to x axis. So these are the streamlines. These are psi equal to constant. Lines of phi equal to constant, x equal to constant that is lines parallel to y axis. So these are phi equal to constant and you can clearly see they form a net of orthogonal lines. This is a very simple example but this demonstrate the use of the method. From this we will go to more and more complex examples. The next example that we will talk about is something called as a source. What is a source? Source is something like this. Let us say that you have a point. This point is radially emitting some flow in the radial direction. So this is a point through which you have flow emerging radially from all directions. Just imagine like that so that the velocity component vr is inversely proportional to the radial location. That means at r equal to 0 this point if it is r equal to 0 then vr theoretically tends to infinity and therefore r equal to 0 is like a singularity point that you have to understand. Now let us say that vr is equal to c by r and v theta equal to 0. So this is a pure radial flow okay. Now how is the source defined or designated? It is defined or designated by the rate of flow from the source. So how do you find out the rate of flow from the source? If you consider an imaginary circle located at a radius r from the center then what is the rate of flow? This is a planar thing. So the rate of flow when you consider you have a circle like this you have flow perpendicular to the face of the circle. It is a planar thing. So when you consider that area the area just becomes sort of the perimeter over which the fluid is flowing for a planar 2 dimensional case. So say you consider unit length or a length of l it becomes 2pi r into l. Here it is a planar case so just 2pi r. So the flow let us say small q is vr into 2pi r. So that is c by r into 2pi r. That means c is q by 2pi. This q is called as strength of the source. Why it is a strength of the source? It physically indicates the rate of flow from the source. So if q is larger and larger you have the higher and higher rate of flow emitted by the source. So this is called as the strength of the flow. So what is the velocity field here? Let us just write the velocity field. You have vr is equal to q by 2pi r. v theta equal to 0. Remember df dz is vr-iv theta into e to the power-i theta. So this is q by 2pi r e to the power i theta. Now remember that r into e to the power i theta is just another way of representing the complex number z in the polar form. So z is r e to the power i theta. So this is q by 2pi z. So you look into this thing that although df dz exists it is an analytic function but that has a singularity at z equal to 0. So at z equal to 0 it ceases to be complex differentiable or analytic because of this division by 0. So these such points are known as points of singularity. Now what is the form of f that you can find out straight away as if by integrating this. See once you write it as a function of g or a single variable it is as good as the simple differentiation. So f becomes q by 2pi lnz. So this is the form of the complex potential function f and you can write it in terms of the phi and psi. So you can write for example f as q by 2pi lnr e to the power i theta. So that is q by 2pi lnr plus iq by 2pi theta. So it is just like phi plus i psi by definition of f. So what are the phi equal to constant lines? Phi equal to constant lines are r equal to constant lines that is the radial lines and psi equal to constant lines are the theta equal to constant lines. So what are theta equal to constant lines? So if you have if you consider a net like this just different lines like this. So what do these straight lines indicate? These are what? These are theta equal to constant. So theta equal to constant are basically the radially diverging lines and phi equal to constant are sorry r equal to constant are these lines. So r equal to constant lines are what? r equal to constant lines are the equipotential lines. So these are phi equal to constant lines and theta equal to constant lines are psi equal to constant lines. So you can see that physically it represents a sort of that is a source at the origin and there may be flow like this. So you can see that it the shape of stream lines gives you an intuitive idea of the nature of the flow. If the flow is in the opposite direction that means instead of radially diverging it is converging to the point the origin then we call it a sink just in place of a source. So if q is positive we call it a source till now we have implicitly presumed that q is positive but if q is negative we call it a sink. Now we have seen that what are the shapes of the stream lines and the equipotential lines for the source and the sink. Let us now consider a super position of the source and the sink. So we consider now an example like this that let us say that this is the real axis that the x axis then you have a point epsilon, 0 and you have a point – epsilon, 0. At the point epsilon, 0 you have a sink of strength q – epsilon, 0 you have a source of strength q. Our objective is to find out the velocity field and the stream function velocity potential all these things okay. So we can see that whatever we have learnt till now it is a combination of that effect but the question is it is it a linear combination or not. To understand that keep in mind that both the stream function and the velocity potential satisfy the Laplace equation. So you have Laplacian of phi equal to 0 Laplacian of psi equal to 0 that means Laplacian of phi plus i psi also equal to 0 right that means Laplacian of f equal to 0. This is a linear differential equation that is why if f equal to f1 is a solution and f equal to f2 is a solution then f1 plus f2 is also a solution right that means you can consider this problem as a linear superposition of the effect of the source at a point and a sink at a point. So you can just add those together that is the advantage of the linearity of the problem. So let us try to do that so we will now write try to write f. So now you tell what should be the f what should be the contribution of f say f1 for the source of strength q. So q by 2 pi ln of what? See now it is a shift of origins earlier we considered the source or sink at the origin. Now it is located at a different point. We will write it in terms of the z. So remember that f is q by 2 pi ln z. So we will write z we will make a transformation or a translation of z to something z plus something or z-something. What is that plus or minus epsilon. So as if you have used a new coordinate system capital Z with its origin located at the source right then for the sink-q by 2 pi ln of z-epsilon. Now it is located at epsilon. Let us consider that epsilon is small but how small or how large we will see that is what we need to consider. Before that we will just do a bit of a manipulation on this. The whole objective of this manipulation is we will write the terms in the form of ln of 1 plus something so that we can use the logarithmic series expansion okay. So q by 2 pi. So let us just try to expand this. Log 1 plus this one we assume that epsilon by z is small. So log 1 plus x what is the expansion? x-x square by 2 and so on. I mean we need not write all the terms-q by 2 pi then this is-epsilon by z then-square terms remain the same and so on right. In the limit as epsilon by z is small then like you may write it approximately as q by 2 pi into 2 epsilon by z right. In the limit as epsilon by z is small. So you have q epsilon by pi z. So this we just write in a shorter notation as some m by z where m is q epsilon by pi. Such a case when epsilon is tending to 0 this is known as a doublet. So that is the name of this one when you have this as epsilon by z or epsilon tends to 0. But important thing is although epsilon tends to 0 we have assumed that q epsilon by pi is finite. But q epsilon by pi is finite that is very important because you have to understand that if epsilon tends to 0 you always have a chance of q epsilon by pi also tending to 0. But maybe q is so large so it is a product of 2 quantities that is important. q is very large epsilon is very small but q into epsilon is finite that is what we are thinking about and this is known as a doublet that is the name. So let us try to make a sketch of or try to identify that what should be the corresponding nature of the stream lines and the equipotential lines for a doublet. So what is f? f is m by z that is m by or m r e to the power m by r sorry m by r e to the power – i theta that is m by r cos theta – i sin theta. This is phi plus i psi. So let us try to identify what are psi equal to constant lines. So what is the expression for psi then? – m sin theta by r right. We may easily convert it into Cartesian coordinates by multiplying both numerator and denominator by r. r sin theta will become y and r square will become x square plus y square. So if you want to identify what are the stream lines? Stream lines are psi equal to constant lines. So you can easily do that by noting that x square plus y square plus m y by psi that is equal to 0. So you can write this as x square plus y square plus 2 m by 2 psi y plus right. So x square plus y plus m by 2 psi whole square. So it represents a family of circles. x – a whole square plus y – b whole square equal to r square that form. So it represents a family of circles. The values of course depend on m and psi. So let us try to make a sketch without going into the values. So what is the center of this circle? 0 – m by 2 psi and the radius m by 2 psi. So let us try to make a sketch. Assume m as positive and psi as positive. So this is x axis, this is y axis 0, – m by 2 psi and the radius m by 2 psi right. So you have whatever is this distance that is same as the radial distance. Even if you have a smaller radius it is like this. Similarly you may also depending on the value of m and all this you may also have it a part of it at the top right. So you may also have exactly the same way. So as if you have a source which is located very close to the origin and you have a sink located very close to the origin on the other side and there is a flow from the source to the sink. That is what it is sort of qualitatively representing. Now phi equal to constant lines they will just be similar and orthogonal that you can easily calculate by looking into that phi equal to m by r cos theta again multiply both numerator and denominator by r. So it will be mx by x square plus y square and that will represent sort of family of these types of lines. And these sets of lines are orthogonal to each other and you can see that just if you look into plain mathematics it is a very interesting way of generating orthogonal lines. This is just a way of thinking a bit differently from just the fluid mechanics perspective but for us the fluid mechanics perspective is the important concern that we are talking about. Now we will consider another example uniform flow along x plus a doublet okay. So it is a superposition of uniform flow along x axis plus a doublet say of strength m. So what will be the resultant complex potential? First due to uniform flow along x that is u infinity into z plus m by z right. So this is as good as u infinity into r e to the power i theta plus m by r e to the power minus i theta right. So u infinity into r cos theta plus i sin theta plus m by r cos theta minus i sin theta okay. So u infinity r plus m by r cos theta plus u infinity r minus m by r i sin theta. So this is we should remember that phi plus i psi. Let us say that we want to generate a body of a particular shape over which the flow is flowing by using this. We are going just one step forward. When you want to generate a body of a particular shape we always have to keep in mind that the contour of the body is a streamline because there cannot be any flow across it right. So if you want to generate a contour of a body it should represent the psi equal to constant by convention that constant is taken as 0 that is the reference. So if you want to generate the shape of a body the shape of the body should come out from the condition of psi equal to a reference which is 0. So the psi which is here you can see that if you now generate the contour of a body this should be 0. You can see that this psi solely depends on the radius r right. So you can fix up a radius such that on that radius the psi is 0 and what is that geometry which is of a fixed radius that is a circle right. So let us say that you have at small r equal to capital R psi equal to 0. You can see that from this super position that this in this example it is solely a function of the radius not a function of theta okay. So that means in that case you have u infinity into r is equal to m by r that means m is equal to u infinity into capital R square. What is the implication? Implication is very interesting. If you choose the strength of your doublet as u infinity into some radius square then you represent a particular flow. What is the flow? The flow in the plane is like past a circular body that means in a 3 dimensional sense it is flow past a circular cylinder. So see starting from very simple mathematics we have generated the flow past a circular cylinder by combining a uniform flow with a doublet. So this represents flow past a circular cylinder radius r. So what is the key? The key is you have chosen your strength of the doublet in a particular way that is equal to u infinity into r square. What is the velocity field? So let us calculate the velocity field. The velocity field will be calculated by differentiating f with respect to g. So dfdg is u infinity-m by g square. So this is u infinity-m by r square into e to the power-2i theta. Just writing g as r e to the power i theta. m is u infinity into capital R square that we will just keep in mind. Now we know that this we may write as vr-iv theta into e to the power of-i theta. So what we do? We write this as u infinity into e to the power i theta-m by r square e to the power-i theta. The whole thing multiplied with e to the power-i theta. So here it is multiplying by e to the power i theta into e to the power-i theta to get just the same form of this one. So it is u infinity into cos theta-i sin theta-m by r square cos theta-i sin theta into e to the power-i theta. So u infinity-m is u infinity capital R square by small r square cos theta-i u infinity sin theta-u infinity sin theta-u infinity-m by r square sin theta where m is u infinity into capital R square. So it is just like this has become vr and this has become v theta. So when you have, when you consider flow past a circular cylinder, so we have a circular cylinder of a particular radius capital R. We are interested to know that what is the velocity at the surface of the cylinder. Remember this is not a viscous flow analysis, this is a potential flow analysis therefore no slip boundary condition will not be satisfied, okay. So what is vr at small r equal to capital R? This has nothing to do with the no slip. This should be 0 at small r equal to capital R, no penetration boundary condition. So there is a circular cylinder on which fluid is flowing. So fluid should not penetrate the circular cylinder. So you can clearly see that at small r equal to capital R, vr is 0. So this is no penetration. So no penetration is satisfied. What about v theta at small r equal to capital R that is-2u infinity sin theta, right. So you can see that there is a tangential component of velocity of fluid on the circular cylinder surface. This is a so called a slip type of velocity and that is there because the potential flow does not have a consideration of no slip boundary condition. Now how do you find out the pressure distribution? See we have found out the velocity field. So what is the situation? You have a uniform flow coming with a velocity u infinity. Now the fluid is flowing past the circular cylinder and we are interested to find out what is the pressure distribution and the velocity field. So to find out the pressure at some point, let us say we are interested to find out the pressure at this point P which is given by the location say some r, theta. Since it is like the theta is even 180-theta if it is replaced it is the same. So it does not matter whether you choose theta in the proper polar coordinate sense or you choose theta from the left hand side sense all the same. So see because it is an irrotational flow and all other conditions are for the Bernoulli's equation that is incompressibility all those things are there. So you may use the Bernoulli's equation between any 2 points. So you may use it from a point at infinity to a point which is marked on the cylinder. So you have P infinity plus 1.5 rho u infinity square is equal to P square sorry P plus 1.5 rho v square at this point v square is v r square plus v theta square at small r equal to capital R. v r is equal to 0 and v theta is minus 2 infinity sin theta. So you can get a pressure distribution P-P infinity in a non-dimensional form by 1.5 rho u infinity square is equal to 1-4 sin square theta because 2 sin theta square will be 4 sin square theta. So this is known as coefficient of pressure. So non-dimensional pressure distribution Cp. So that means see we have got a pressure distribution and we may use it even for a boundary layer theory till there is boundary layer separation because whatever is the pressure imposed by the free stream from outside the boundary layer which is like a potential flow, same pressure is acting within the boundary layer. But of course this is the pressure at a point on the cylinder and this pressure the what you calculate even in the presence of a boundary layer will not be too much different from this one till you come closer and closer to the apex point and you encounter adverse pressure gradients. So till the pressure gradient is very very favourable, the actual pressure distribution and this pressure distribution will almost remain the same. We stop here in this lecture and we will continue again in the next lecture. Thank you.