 previous lecture we did analysis for boundary layer over a flat plate. Now that gave us an estimation of the thickness of the boundary layer, the wall shear stress parameters like that but what if we exactly want to solve those equations without going for any order of magnitude analysis. So to understand that how that might be possible, let us rewrite the momentum equation that is we are still considering the flow over a flat plate, that is the momentum equation that we were dealing with. Our objective is to solve this equation, we keep in mind that it is not that you have just one equation and two unknowns because you also have the continuity equation to support the momentum equation. Now you can clearly see that even in such a simple form, this is a nonlinear partial differential equation. So the question is that solving this just by using this particular form may not be very very simple but there are certain techniques in which under certain circumstances the partial differential equations may be transformed into ordinary differential equations and one such important transformation is known as similarity transformation or stretching transformation. We will try to see that whether that means what we are trying to do, we are trying to investigate whether it is possible to convert it as a function of a single variable not 2 variables x and y but a single variable where the single variable will contain both the information of x and y and if we are successful in doing that what is the objective that we will achieve because of having function of a single variable the partial differential equation will be converted into ordinary differential equation. So to highlight whether that is possible let us just look into first a qualitative way in which historically this phenomena was understood. When this phenomenon was understood historically a lot of effort was given by a famous scientist and engineer known as Blasius basically a mathematician. So Blasius what he tried to do he saw one of the important behaviors that is if you look into the velocity profiles you see that these velocity profiles are not same as you go along x the velocity profiles change but the velocity profile looks as if it is a stretched version of what it was at an x before that. So this motivated him to make a plot of say u by u infinity versus y by delta because it sort of normalizes the stretch because this is always confined between 0 to 1 this is also always confined between 0 to 1 and then it was found that if you make such a plot then u by u infinity as a function of y by delta is such that this functional variation this is one this is one this variation is same at all sections that means behavior at all sections may all be combined and normalized in this particular functional form and this gives a very important physical insight that u by u infinity is a single valued function of y by delta and you see that we have just seen from the order of magnitude analysis that delta is some function of x right. In fact it scales with square root of x that is what we have seen. So that means see the dependence of both y and x these are there and you may introduce a new variable which is like sort of y by delta let us call it a new variable eta which as a single variable is dictating the behavior. So this physics is see how physics is related to mathematics that is what is very very interesting because we will rigorously derive and come up with the same conclusion which from a very little physical insight could be obtained that this gives a motivation that this velocity behavior is a function of a single equivalent variable where that equivalent variable carries the information of both y and x y explicitly x implicitly through delta and that means that this variable eta may be of the form of y into some function of x right because delta is some function of x and y is there so let us call eta or let us say introduce eta as y into gx okay. Now let us say that we write u or say u by u infinity as a function of eta because that is what we get from the normalized picture based on this one and we will remember that what is this eta? Eta is equal to y into gx. We will try to understand physically that what this transformation is trying to do. We will do that once we get an estimate of what is gx. We will see that mathematically that gx will scale with 1 by delta. Physically there is no physically we are seeing that this eta is a function of separable function of y and x. So this similarity transformation is a special case of method of separation of variables that you have learnt in mathematics course. So the variables are separated so you have effect of y and effect of x separated and because of a particular physics that is occurring this separation is possible and we will see this separation will become mathematically consist separation of variables whatever physics is governing this it will also become mathematically consistent that means this gx will indeed come out from the mathematics to be of the order of 1 by delta and that we will show just from pure mathematics without going into the physics and that will give us a sort of a equivalent between these 2. So the objective now is to use this similarity variable and make a transformation of this partial differential equation to ordinary differential equation. To do that let us say that we want what first del u del x then so different terms that we are looking for. So what is this just be using the chain rule right. So du d eta is what u infinity into f dash. So when we write f dash we what we are meaning is df d eta that is the shorthand notation we will use and then the partial derivative with respect to x y into dg dx right. So we will write it g dash so we will use again a shorthand notation g dash is dg dx. So both dash but the variables are different for f it is eta for g it is x what is partial derivative with respect to y. So u infinity okay first let us write the chain rule description and then we will write. So du d eta so du d eta is u infinity f dash then that will be g. We also require a second derivative with respect to y so let us just do that. Second derivative with respect to y is like first of all you have this f dash term so let us consider its second derivative. So you have df dash d eta that into g plus f dash is there dg d eta okay let us see g is explicitly a function of x right. If you want let us write and see whether it is consistent or inconsistent I do not mind let us just write if you feel that this type of chain rule is going to work let us just keep it as it is. So I mean before going into further let us investigate whether this sort of chain rule is going to work or not. See one important thing is you have to look for the description of the function in terms of explicit representation and implicit representation. See you have g as a function of x both explicitly and implicitly. So you have to think eta and x and 2 different variables 2 different sort of independent type of variables. And then if you look into it in this way see g does not understand what is x this is what is you are writing in terms of explicit. So g explicitly is a function of x only so it does not understand eta maybe there is an implicit inter linkage between eta g y whatever but it does not understand explicitly what is that. So this is clearly equal to 0 okay. So this one d eta dy is what g so this becomes g square yes and that is what you see you are not writing del g del eta you are not writing partial derivative that is what you have to understand okay you are just writing the ordinary derivative. So if you see this is where knowing too many things is bad say you have started with a very basic calculus what you know g as a function of x if you are not asked to find out a derivative of g with respect to anything else other than x that will be 0. So that is what we are doing. So you have g is equal to 2x you are asked to find out what is dg dy so what you will say so it is just like that so this is not a variable which is contained within g and ordinary derivative not partial that you have to be careful. So now let us try to write this expression see what should be our strategy see v we do not know so we will write v from this expression and then eliminate that from the continuity equation by using the continuity equation. So what will be v? v is equal to that is the first term nu u infinity f double dash g square see there is some lot of algebra I need so if I make any mistake please correct-in place of u it is u infinity into f and partial derivative of u with respect to x so you have u infinity another u infinity so u infinity square f f dash y g dash divided by u infinity f dash g. So let us write v then term by term so nu f double dash by f dash g right that is the first term second term-u infinity f y g dash by g if we want to eliminate v then basically you have to find out what is del v del y and then equate that with-del u del x in the continuity equation then v will be eliminated so we have to differentiate it once with respect to y. So when you want to differentiate it with respect to y g is a function of x that is like a constant for that partial derivative so nu g then basically you are dealing with this is a function of eta so dd eta of this one f double dash by f into del eta del y that is g so another g has come so this g into this g will make it g square then next term-u infinity g dash by g is like a constant for it so for it there are 2 variables one is f another is y so for f it is like df d eta-unit is I mean if there is some explanation has to be given then that is by what algebra so if I have made a mistake in algebra you let me know otherwise no explanation so u by u infinity is f so when you have substituted u that is u infinity into f that is how f has come okay so-u infinity g dash by g then f into y u differentiate so this df d eta into del eta del y is g plus f into the partial derivative of y with respect to y so that is 1 so nu g square dd eta of f double dash by f-u infinity g dash f dash df d eta is f dash g and g get cancelled out-u infinity f g dash by g which one d d eta yes this is f dash right okay. Now this dv del v del y is equal to-del u del x from the continuity equation so that is equal to-del u del x from the continuity and del u del x expression we already have so that is equal to-u infinity f dash g dash y here this term has y right yes g into y right yes okay yeah because it was a product of f into y so y has to be there right okay. Now if you look into these equations so we have this as in one side and this in the right hand side right so if you just compare these 2 you will see that first of all of course there is another term in the left hand side this one also okay so if you compare these 2 these 2 terms get cancelled out right so we are left with the form of the equation let us just write it so we write this as nu g square dd eta of f double dash by f dash is equal to u infinity there was no u infinity here okay that is fine u infinity f g dash by g. So we may just isolate the effects of the variables eta and in this way the purpose of the way in which this was done is to show that now you are able to rearrange it in a way that the left hand side is a function of eta only right hand side is a function of x only these 2 variables do not explicitly know each other so this is a function of eta only this is a function of x only this implies that each has to be a constant let us say that the constant is k so long as this proportional relationship is satisfied it does not matter what constant we take that is what we have to understand because we are satisfied with this equality equal to what is not going to matter us a lot so we will choose this k in a way that just helps us in our algebraic simplification to get a clue of that what should be a good k for that let us consider this equation and maybe try to find out g as a function of x. So now if you integrate this this will give you g as a function of x so what is this one so you have dg dx 1 by g cube dg dx is equal to k into nu by u infinity that means g to the power – 3 dg is equal to k nu by u infinity dx so if you integrate it g to the power – 2 by – 2 is equal to k nu by u infinity x plus some constant of integration right. How do you know what is the constant of integration you must know g as at some point where you know x so at some point you must know the relationship between x and g so think of the flow of a flat plate this is the flat plate the boundary layer is there can you tell what is g at x equal to 0 yes try to remember the physical meaning of g is like scales with 1 by the boundary layer thickness so g tends to infinity as x tends to 0 one of the important things that you have to remember is the boundary layer theory singular at x equal to 0 that means you do not really have at x equal to 0 delta equal to 0 you only have at x tends to 0 plus delta tends to 0 plus that is all but exactly at x equal to 0 that is not always you see in books or whenever we say we loosely say at x equal to 0 delta equal to 0 obviously it is okay in an approximate sense but if you rigorously want to state what is the boundary condition then the important thing is at x tends to 0 you have that delta equal to 0 because at x equal to 0 it is a singular behaviour it does not have any definition of the boundary layer thickness so that means if you utilize that that at x tends to 0 plus g tends to infinity then c will be equal to 0 so from this what we conclude g square is equal to minus k by 2 nu sorry minus u infinity by 2k nu x one of the important physical restrictions is that k has to be negative because g is a positive quantity it is meaning the inverse of the boundary layer thickness so any negative k choice is fine but what we may choose may be a good number is k equal to minus half that somehow nullifies many of the bad numbers so let us say k equal to minus half this is just choose not a must not a ritual this is just like a convenient way of doing it algebraically so when you do that then g becomes square root of u infinity by nu x you see if you recall from the order of magnitude analysis we got delta scales with square root of x so this is this case as 1 by delta so from mathematics you are getting back the same physics that you got from order of magnitude analysis now with the same k important is with the same k because so long as you keep the same k does not matter what k you take by satisfying this condition you may solve for the f because f is what is important for you f gives your velocity profile u by u infinity is f so let us do that so you have dd eta of f double dash by f into 1 by f is equal to minus half then d of f double dash by f dash is equal to minus half f d eta so in principle it is an integrable form only thing is you do not know explicitly as a function of eta but the form is separable and integrable now to make it a bit more convenient let us say that we say that integral of f d eta is equal to capital F just because this is sort of an integral form we want to convert it to a new variable which will give us a convenient form where as if this integral is already there question is what could be physically this f so we can write df d eta is equal to small f see this f includes also constant of integration see because when you integrate this there will be a constant of integration but the constant when differentiated will give back the same thing so we will not explicitly use any constant of integration we will integrate this constant of integration is inbuilt with this capital F that we have to understand or keep in mind so if you see that f is velocity this is another special derivative of velocity so it is equivalently like a stream function so let us try to see let us recall the definition of stream function this is a 2 dimensional incompressible flow so stream function definition is valid so u equal to del psi del y right so you can write this as say if u is a function of eta only then stream function will also be an eta in that way or u by u infinity is a function of eta only then this is d psi d eta into this one so this is like dx but most important thing is that when you write u d eta see u is what u is u infinity into f so when you write f d eta by separating variables here then whatever you get as d of that that is the equivalent of a stream function with the of course of multiplying factor that means capital F has a significance of a stream function in a transformed manner but it has a significance of a stream function so basically as if we have eliminated v by using the stream function so keeping that in mind let us complete the description of the equation so you have d of small f double dash means capital F triple dash by capital F double dash is equal to that means f triple dash 2 f triple dash plus f f double dash equal to 0 this equation is known as Blasius equation this is a ordinary differential equation but it is not a simple linear ordinary differential equation with which most of you are very very familiar so this does not have any analytical solution but this may be solved numerically by many ways and we will not go into those the numerical techniques of solving this equation this is not a course on numerical analysis but I mean there are straight forward ways for example it is possible to decompose this equation into 3 coupled ordinary differential equations of first order this is a third order you may decompose this into 3 coupled first order equations and then for each of those you may use a technique known as fourth order Runge-Kutta method or any Runge-Kutta method is fine so long as you can cast these in the form of an initial value problem you will see we will see that what are the boundary conditions first let us see and we will find out whether it is possible to cast it in an initial value problem so what are the boundary conditions this is a third order equation it will require 3 boundary conditions so boundary condition first at eta equal to 0 eta equal to 0 is what eta equal to 0 is the surface of the plate what is the value of f capital F now you remember that capital F is the having the significance of a stream function along a streamline the value of stream function is a constant and the surface of a solid boundary is always like a streamline why because there is no flow across it stream lines are imaginary lines in the flow field where there is no flow across it so the solid boundary here the flat plate is like a streamline question is when it is like a streamline what is the value of the stream function that is up to you because stream function is a relative quantity see you have u equal to del psi del y so with psi and psi plus c by both of these u is satisfied your basic variable is u stream function is just a mathematical way of looking into it so any constant value of stream function that you can choose at the solid boundary standard convention is we choose the constant as 0 obviously that is the most simple for calculations so we choose this equal to 0 fundamentally this is any constant but the choice is arbitrary because it does not depend on the difference also at the wall you have no slip boundary condition so no slip boundary condition means u equal to 0 that is small f equal to 0 that means d capital F d eta equal to 0 so at eta equal to 0 you have f dash equal to 0 capital F dash this is no slip this is wall is streamline and what else when you go at a distance theoretically infinity from the solid boundary what do you face you come up with a situation when there is no further gradient of u with respect to y that means that means small f is a constant that means d capital F d eta is a constant that means d 2 capital F d eta 2 is 0 that means as eta tends to infinity f double dash equal to 0 that is basically so it is a well posed mathematical problem and if you write if you introduce if you decompose this into 3 first order equations let us say you have you will have 3 variables say f 1 equal to f f 2 is f dash and f 3 is f double dash see this is like these are like initial value problems for f 1 and f 2 because at eta equal to 0 you know the values of the variables so it is like as if eta is like a time coordinate that is you have at time equal to 0 you march with time and get the solution instead of time it is the eta as the variables for the third one you see you do not know eta as eta equal to 0 but you know as eta tends to infinity so that is not a sort of initial value problem where you know the value at 0 you know at value at the other end. So what you may do you may guess with a value of eta equal to 0 at eta equal to 0 say you guess with what is the value of f double dash at eta equal to 0 and solve the 3 coupled initial value problems by some methods say Runge-Kutta method once you solve that you will see that at the other end when you go to infinity of course numerically you cannot treat infinity how will you get infinity you will just see that beyond some eta there is no further change that numerical value is like infinity for you. So infinity is what infinity is what that does not understand the effect of the plate for us infinity is anything greater than delta that does not understand the effect of the plate. So once you get the value of f double dash at that end you will see that you will not get this equal to 0 because you started with a guess but you will see that it will be somewhat deviated from 0 more accurate your guess was more close it will be to 0. So what you do is you have a guess you get something as non-zero say it is shifted to one side of 0 you make a different guess it will be shifted to the other side of 0 then interpolate with a new guess which is a sort of in between these 2 until or less your guess is such that it matches with the boundary condition at infinity this is known as shooting method that means you have a target and you are you are having a sort of a bullet which you want to you want to shoot the target the target is this boundary condition. So you are shooting with different initial conditions till you come up with an iterated solution where you are really shooting the target you are getting the condition satisfied at infinity. So this is known as shooting method so once you use a shooting method the advantage is by this method you may convert a boundary value problem into an equivalent initial value problem. So you will get 3 coupled initial value problems from these 3 and then you may use the well known solvers of initial value problems there are many standard solvers of initial value problems and like you may also write your own simple code and solve this. So if this is solved numerically how it looks so if you find out capital F you may differentiate it once with respect to eta to get small f that is your velocity profile. So you will get say you are plotting u by u infinity as a function of say other way let us plot say u by u infinity as a function of say eta. So if you plot this the general plot general behavior is like that it becomes equal to 1 beyond this of course it does not change further and we may therefore conclude that whatever is this value of eta this corresponds to the location of the boundary layer right because the velocity has become almost the free stream velocity. So this is f so this f versus eta plot and this value is roughly 5 if you do it numerically. So what we may conclude out of this what was this gx so or what was this eta? eta is y into gx so what was gx? What was gx? 1 by no no no in terms of x square root of u infinity nu x by u infinity u infinity by nu x because you have to keep in mind it is scaling with 1 by delta. So whatever it is 1 by square root of x should be there. So this is one thing like for all exams you tend to mug up there is nothing wrong with it but you get a lot of pleasure in mugging the problem is there is nothing wrong if you intelligently mug up. So when you intelligently mug up you see I mean you are just chanting it like a mantra that gx is this gx is this keep in mind that it scales with 1 by delta and then you need not mug up and then just a simple physical 1 or 2 line of work will give you what could be the scaling variation of gx. So that means when eta is equal to 5 say y is equal to delta roughly so 5 is approximately delta into root over u infinity by nu x. If you want to find out that what is delta by x you have to what you have to do so delta is approximately 5 into square root of nu x by u infinity. So delta by x is 5 square root of nu by u infinity x that means 5 into Reynolds number to the power – 1.5. See by order of magnitude we estimated up to Reynolds number to the power – 1 only this 5 multiplier is something what we get from this solution. So with a lot of toil lot of chance of doing bad algebra and all this we came up with something which is important but more qualitative interesting and important things we could get from the order of magnitude in a really very simple way. Now of course if you know what is this delta as a function of x you can calculate the other things. Next what we will do is see this is a method of solution which was never liked by primitive engineers because engineers never wanted to look into non-linear differential equations and they found that well in practice it will make our life much more tough. Of course like we should not say that engineers are dull sets of people all of us are engineers but at the end what happened is that people look for some approximate solution the whole idea of the approximate solution was like this that I do not care how delta varies with x. For me as an engineer what is the most important conclusion is how wall shear stress varies with x because that comes directly to my design. As a designer I do not want to understand delta of course it is a very bad approach but the thing is that it may be a working approach for an engineer who does not want to go into mathematics and sometimes in practice that is okay. So when that is the objective the objective is somehow could we come up with an approximate analysis where we avoid solving such equations but come up with a reasonably good estimation of wall shear stress. But the basic of that method is still mathematically very rigorous the application is somewhat approximate and that rigorous method we will look into in a very careful manner that is known as momentum integral method. For boundary layer analysis we will take the example of flow over a flat plate. Let us understand what is the essence of the method. Essence of the method is again like we start with the boundary layer equations. So let us say we start with the momentum equation. What is the method? The key word is the name of the method momentum integral. Momentum equation integral means it has to be integrated. So what we are basically trying to do in this method is we are integrating the momentum equation over something. What is that something? That something is across the boundary layer. So we are integrating this with respect to y all the terms with a range of y from y equal to 0 to y equal to delta. Now when you do that let us see that what simplification we get out of this. First of all let us consider this term. So let us call this as term 1, term 2 and term 3. Let us write separately terms 1, 2 and 3. So term 1, this one, right. Term 2, for term 2 let us integrate by parts because you have already a du dy type of thing when it is integrated it will give you u. So we integrate it by parts by considering these as the first function and these as the second function, okay. So the term 2 will become first first function into integral of the second-integral of derivative of first into integral of the second. In terms of the first c it is v at delta into u at delta. So let us say v at delta is v infinity that is what symbol we have used. Into u at delta is u infinity- v at 0 into u at 0. Both are 0, v at 0 because of 0 because of no penetration, u at 0, 0 because of no slip- now you may use the continuity equation and write this as- of del u del x by continuity. So this becomes this plus integral of 0 to delta. So this is term 1. So the left hand side becomes v infinity u infinity- 2 into term 1. So u infinity into v infinity- what is that we do not know in this equation we do not know what is v infinity. To know that let us use the continuity equation because continuity equation will relate v with u. So next what we do is we write the continuity equation and integrate it across the boundary layer just as we did for the momentum equation. So let us write the integrals of the continuity equation. This term we do not disturb. We just write it as this term what we do? This will be v at delta- v at 0. So this is v infinity. So we have got an expression for v infinity. That will substitute here. So this will be u infinity- into v infinity that is u infinity. So at least we have eliminated v but another important objective which you want to satisfy and before that we ask ourselves a question that can we take the partial derivative with respect to X outside the integral. We have to see then the general rule by which you You may flexibly take the differentiation from outside the integral to inside the integral or inside the integral to outside the integral and for that the rule of mathematics is the Liefnitz rule. So let us consider that Liefnitz rule differentiation under integral sign. What is this? If you have a function say f, so say ddx of integral f xy dy where the limits of integration are one function of x to another function of x, this is the Liefnitz rule okay. So you can clearly see that if the limits of integration are not functions of x, then you may easily keep the derivative inside and outside without any problem but here what are the limits of integration? See there is a limit of integration delta which is itself a function of x. Therefore, let us try to apply this say in one case our function of concern is u square another case is u. So let us take an example, example 1 say f is equal to u. So you have ddx of f equal to u, ax equal to 0 and bx equal to delta okay. So you have ddx of integral 0 to delta u dy is equal to 0 to delta plus fx delta okay. So f is u, u at x and delta what is that? So the function is substituted at x equal to x and y equal to delta. So what does it become? u infinity, u infinity into d delta dx, minus it is 0 because a is 0, so that is fine. If you take a second example as f equal to u square then similarly you will get this as ddx of 0 to delta u square dy, u infinity square d delta dx. So let us substitute that here. So if you substitute that here this is minus u infinity into this one is ddx of 0 to delta u dy minus u infinity d delta dx. That is for the first term, for the second term minus u infinity square d delta dx right. Then you can clearly see that this u infinity square d delta dx I mean this one combined and this term they get cancelled. So the left hand side becomes ddx of u square minus u into u infinity dy. That is the left hand side. What is the right hand side? Right hand side is just one line, let us just write the right hand side. So the right hand side is that is a term 3 that is integral of nu, nu is a constant that we have assumed these properties are constant so this is from 0 to delta. What is du dy at delta 0 because u does not vary with y. So this becomes minus nu del u del y at y equal to 0. So the left hand side equal to the right hand side that means we can write ddx of say integral of 0 to delta u into u infinity minus u square dy is equal to nu del u del y at y equal to 0. This equation was first derived by von Karman and that is why the name of this is known as von Karman's momentum integral equation. So this is basically an integral form of the momentum equation when integrated within the boundary layer or across the boundary layer. What is the advantage that it is giving? Let us just look into it briefly. So you know that what is the objective of us to calculate the wall shear stress. What is wall shear stress? That is mu del u del y at y equal to 0. That means the right hand side we may write as nu into tau wall by mu. So tau wall by rho. If you non-dimensionalize tau wall with rho u infinity square because that is what we commonly do then just let us write one more step for that equation. So tau wall by rho infinity square is equal to ddx of integral 0 to delta u by u infinity into 1 minus u by u infinity dy. So this is a very useful engineering expression or mathematical expression with no approximation. See where will approximation come? This is perfect. Approximation will come when you substitute any approximate velocity profile. So if you do not know a velocity profile you may still make a guess of u by u infinity. Do this integration and that will sort of give you a relationship between tau wall and delta. So if you know how delta varies with x you will get variation of tau wall with x. So that depends on the substitution of an approximate velocity profile. So how to do that? That means how to make the use of the momentum integral equation to get an expression estimation for tau wall and say even delta approximately that we will take up in the next class. Thank you.