 I will be starting this class of, Professor Sharma is in the back, he will take over from Wednesday. So today and tomorrow I will be talking about essentially fluid mechanics as is required for CFD. So I will try to explain what that means in a minute and before we begin though I would like to thank all of you. It's been really enthusiastic support is what I have heard, of course 50 people are here so it proves it. Essentially let's just accept that we have taken a fairly serious job at our hand and what I will request is that while let the interaction be fairly informal throughout the five days. Let us be reasonably focused also so that whatever material we have outlined we can complete in the time. It's not going to be easy to be honest with you to complete it for various reasons as you will see as the course goes on. So I will request that let us be quite let's say professional about what we are doing here and let us try to stay focused. It's not going to be easy I know it's not going to be easy sitting here for close to six plus hours a day for five days in a row but like I said we have decided to do it so there is no option but to do it. We have all done this in some sense when we were students and now we have to do it as part of our duty. Let me also say that there is no real teacher student relation here it's a fairly informal business let us treat it that way it's a good interaction that we want to have to make sure that whatever we are talking about here make sense to you and also something that you can really incorporate in your class work that is the objective. Of course all people that have been mentioned here are known to use let me just go ahead. Before we get on with the class let me clearly point out what the objectives of the course are and the idea is that we want to introduce the fundamentals of CFD and associated with it the heat transfer part. It will be very clear to you what we mean by fundamental. The level of the material that we are going to present to you is essentially at the advanced undergraduate level which means that typically either third year or perhaps fourth year student having gone through a mechanics class and heat transfer class and some numerical methods background should be able to handle this. So our objective really is that you take this material package it whichever way you want and offer it as an advanced undergraduate course when you go back to your college and in particular I am hoping that the coordinators will definitely do it. So there is a little bit of more pressure on the coordinators in that sense that when you go back you please try to really see whether you can incorporate the material that we are going to present here in a typically advanced undergraduate level class in your college. I think that will be really the success of what we are up to. So let me point out the last point which is underlined right at the beginning. We are not going to introduce any specific CFD software we are not going to use a word of any of it. I will not use the word fluent CFX star CD nothing. This is the last time I am using these words let us say. The objective is not to introduce a CFD software but to introduce the methods that actually run the software. So that is the third point there that we want to introduce the methods and algorithms that are working behind the screen when you are actually using the software. Now this is a point of philosophy in the sense that we believe in this there is no reason for you to believe in this. In the sense that you may think that why do all this software is all available might as well use it. Of course I am not going to go and stop you from that you are free to do it. Only thing is that we hope that if you have gone through the material which goes into making the software there is a better chance that the software will be used slightly more intelligently rather than just like a black box where a student or even some of us we just sit and start clicking something here and there without really knowing how to use the software intelligently nor knowing how to interpret the results correctly nor knowing what sort of choices we are going to use when we run the software. You just heard about professor Darte being mentioned I think there is a good chance that many of you know him if not by my face but by name he is considered to be one of the leading authorities in CFD how many of you know professor Darte many of you of course good I remember this very small aside I was sitting with professor Darte on one of the MTech examinations where the student was presenting something that he had done with Fluent and you are just showing plots after plots and this and that and so on. So after a while I think professor Darte decided enough is enough he asked him so just tell me what analysis you are doing how you went about generating these things what choices have you made what discretization scheme you used and then he was blank absolutely and then you know professor Darte said that see CFD is not to be done as a mouse driven exercise please keep these words in mind if you start using some blindly by simply clicking on things never ever is going to be clear to you neither it is going to be clear to the students that you teach and an academic institution we do not believe honestly in in saying that let us use Fluent or let us use CFX eventually the student can go to an industrial outfit and depending on their let's say requirements and preferences they can use whatever they want but as an academic institute we cannot endorse a certain specific product because today I will say that I prefer Fluent and there is a philosophical problem here someone will say why Fluent I like star CD more or someone will say why why these two I will want CFX or CFD or CFD plus plus and whatnot so there is no end to this and therefore let me be very very clear again that our objective is not to introduce the software but to introduce the methods that go into making the software obviously we are doing at a first level class clearly the methods that run the software are far more sophisticated but they are all based on what we are going to talk about here so the the ABC is in some sense of all those methods will be covered here if you get interested in it and if you understand this over a period of time you can do all that yourself getting into that one so that is the idea as they say you you have to teach someone to fish rather than giving the fish directly so then you can improvise and do many many things so in that sense what we have what we have done is that using that open source programming environment called Sylab we have developed a set of I should say design problems let's say which are essentially nothing but codes which have been developed that use all these methods and algorithms and the idea is that during the lab sessions here you will be provided laptops tomorrow not today but tomorrow onwards you will utilize those programs and run those codes to see how the methodology functions what sort of output is generated whether we can play with the parameters that go into the code and change the output and how to interpret that output and so on the idea is that at the end of the course you will be given a CD or a pen drive or whatever on which the Sylab software along with all these developed codes will be provided so that between now and the main workshop in June the idea is that you should be sufficiently comfortable with knowing what's in there so again think about it is a reasonable amount of responsibility in the sense that the course is not just finished in five days here when you go back you have to keep playing with this set of codes in Sylab making sure that you understand how the codes are working how to change parameters because when in in June you are going to be handling your remote center and ideally we are really hoping that you should be able to handle the 50 or whatever number of people will come here so it's it's very exciting in some sense it's just the fact that we have to be somewhat serious about it it's not like five days and done it's not like that so let's be prepared for it is what I'm any question on this because this is the philosophy and there will not be any more discussion on the software part is that is that fine I know that many of you will not agree with it and so be it I think there is there is nothing that we can do as I said you know you're free to do use of software the way you want once you go back but our job and our mandate is to make sure that the methods that work the software are informed to you in a proper manner is that fine thank you all right so first part of this course is going to be discussion of fluid mechanics ideas and again we are calling it essentials of fluid mechanics underlined essentials what that means is that we are not really going to cover a complete fluid mechanics course that is just not possible the main purpose of the course is CFD so the essential essentially implies that whatever is minimally required from fluid mechanics plus a little bit more let's say is what we will cover so that everyone is on the same ground when it begins to or when when we begin the CFD work now let me be very honest and say that there is I'm pretty sure there is a whole bunch of people here as the introductions were going on your thing that you have you have taught this before and that and so on so there is a good chance that to many of the people here this material in part one will be familiar and that's fine I mean you just treat it as a revised crash course of certain part of fluid mechanics which is useful for CFD on the other hand I'm all sure that some of the material here will not be familiar no matter what so it's a it's a good exercise to to make sure that you note it down and when you go back you can revise it as and when required again the the material here that I am planning to cover is essentially at the undergraduate level perhaps a little bit more here and there I will also say that whatever is going to be in the set of slides you may not find in one single book that is never the idea I don't have to tell you you're all teachers you know each book that you pick up clearly you have seen thousands or hundreds at least on fluid mechanics each book will have some advantages and some limitations each author thinks that this is the way to do it and so on so there is no single book that I can point out to you where I'll say that just read that and you'll be fine it's not going to be like that I am listing here only three books as part of references however informally we can talk about more it's not that you know I have read all these 20 or 30 definitely I have read the three that I am mentioning to a really good extent and I feel fairly confident that these three should really serve the purpose of taking care of the fluid mechanics obviously you will have your own favorite also that's also okay so let's just begin in the first part that is his fluid mechanics but we are what I'm planning to cover is just a brief introduction what we call an integral analysis and then get on with the so-called differential analysis which will be divided into kinematics and derivation of the governing equations after having derived the governing equations I will outline a few analytical solutions of the so-called never stock's equations the purpose of these analytical solutions is to point out some specific issues which of course will be pointed out tomorrow when we come to it and then before I hand over to professor Sharma for the CFD per se what I want to do is I want to just introduce you the methodology of a numerical solution through a technique called finite differencing again I'm very sure that many of you would know it but just to sort of transition from the basic fluid mechanics into the CFD part which will be formally done through the finite volume method with professor Sharma next Wednesday onward this is just a bridge so to say between fluid mechanics and getting on to the formal finite volume technique which will be discussed in detail just to introduce you know what happens in a numerical solution methodology again you know many of you would know it so just treat it as a revision if you know it if you don't know it it's useful to know to some extent all right so let's get on with the introduction part as we know what do we do include mechanics we study equilibrium and motion of fluids necessarily we are dealing with studying of whatever forces that exist in fluid whether it is in equilibrium or in motion what I'll do is I'll try to be very very brief and the reason is because of the time pressure we have to finish a certain material in in the certain time that we have however that does not mean that you should not stop me if you have any serious question or any question for that matter please stop me try I'll try to address it as as cleanly as I can in the time that is available and then we'll move on applications fluid mechanics as professor Gaitande was saying extremely widely applicable I have written here various branches of science and engineering is there anyone here who is not from engineering everyone is here from engineering right so I don't want to talk about engineering you know exactly what but do you know any branch of science that may utilize very quickly physics for example any yes biology yes chemistry where will you how do you use fluid mechanics in chemistry surface tension is just one aspect let us talk about the fluid flow phenomenon yes medicine so I'll say that biology and medicine go together that's fine how about anything more mainstream physics I'll code two things one is what we will call plasma physics and one what I'll call astrophysics you'll actually see that although in both plasma physics and astrophysics it's not just pure fluid that we are dealing with usually we are dealing with plasma which are which involve charge particle however you will see that if you if you look at the analysis of these two types of situations equations of fluid mechanics are inbuilt into it so anyone who is doing a plasma physics or an astrophysics study actually needs to know enough of fluid mechanics is very interesting natural phenomenon so it's not just related to technology and engineering and science we see fluid mechanics in natural phenomena day in and day out where yeah ocean waves that's one yeah hurricane, tornadoes, tsunamis all bad things everything is fluid mechanics anything else clouds yeah cloud formation very difficult from a fluid mechanics point of view but yeah it is a fluid mechanics problem yes that's fine salt okay that's interesting yeah how about the flight of bird that that's the easiest thing that I can think of motion of fish all these are natural phenomena and everything revolves whatever we have listed around fluid mechanics if you want to analyze the fish motion which people are trying to because it's one of the most efficient ways of propulsion in a fluid you need to know fluid mechanics well sports, swimming, cycling, cricket, all balls I like that good that's the answer that I was looking all balls so now let me ask you tennis is fluid mechanics in tennis yes so if I ask you how many of you watched French Open I hope you know what is French Open is played on clay course the the red clay and the person who is good at top spin usually scores well in French Open if you carefully follow the tennis so what is special about a top spin what sort of fluid mechanics involved in a top spin absolute Magnus effect where else do you see Magnus effect which other ball game okay what about baseball how many of you have seen baseball anywhere not much it's not the sport that is played outside the US much but yeah if you see baseball it's all about Magnus effect nothing more nothing else anything else how many of you play table tennis as anyone played you do so backspin top spin side spin I hope you all remember that nothing but fluid mechanics all that we are doing there is essentially a force balance which will tell you how the table tennis ball if hit in a certain manner will curve in a certain manner that is all because of what he correctly said the Magnus effect what about football so who is those famous for that the movie on it David so how many of you seen bend it like Beckham this movie no seriously someone has seen good you have seen so what is the word or what are the words bend it mean he bends the ball when he kicks it he's a free kick expert isn't it what does he do he can actually curve the ball from a very long distance supposedly and get into the goal post so that curving of the ball to the air is happening through what Magnus type force fluid mechanics nothing else so you can assume that even though Beckham has never studied fluid mechanics his brain is wired to know fluid mechanics perhaps I don't write so fluid mechanics by far anywhere you want you'll see actually I don't even want to talk about the technological applications because that becomes a very boring discussion you all know where all fluid mechanics goes as far as the technology part goes but these natural phenomena and sports are something that is worth talking about if you look at yacht racing those who follow ESPN and the star sports and such you'll see once in a while they show this competition called America's Cup I think it's called it's a yacht racing how many of you have seen a yacht with a Marathi we call it Sheed what is it called in English I don't even remember anyway it's all about utilizing the wind force most efficiently to propel the boat it's all about fluid mechanics and no one knows fluid mechanics there they just know through empirical evidence and their practice and such that okay if you do it this way you are going to propel it the most efficient so again fluid mechanics interesting stuff right so we all know how we define the the fluid at least the kinds of fluid that we are going to look at are such that they simply cannot resist a shear force when they are at rest no matter how small the shear force is that is the way we usually that's the textbook definition of fluid and I have placed a star on fluid because not all fluids behave this way the kinds of fluids that we are usually interested in mechanical aerospace typically not chemical but definitely mechanical and aerospace do behave this way so any other fluid that you know which doesn't behave this way chemical people will know rheological fluids yeah for example I think something like a toothpaste it doesn't behave this way unless you provide a certain non-zero I mean large enough shear stress it's not going to start flowing but for our purpose the fluid that we are talking about is something which will not remain in equilibrium static equilibrium that is under the action of a shear force no matter how small that shear force that's the definition that we will work we all know fluids both come in liquids and gases depending on their intermolecular forces some are liquid some are gases how do we study fluid mechanics usually you will see that there are the so-called two approaches molecular approach and a macroscopic approach so molecular as the name suggests you just realize that a fluid material is composed of a large number of molecules and ideally if you follow each molecule and come up with some sort of an aggregate you should really describe how the fluid is behaving that simple as that to describe in in practice it's incredibly troublesome to implement some people do it though some people actually prefer that this approach be followed however it's not followed as a follow each particle or follow each molecule sorry type of method it is actually followed in some sort of a statistical approach and those who have slightly more physics background perhaps know there is something called kinetic theory of gases there is something called statistical mechanics where these ideas are usually explained from standard engineering point of view we go with the next approach which is the so-called macroscopic approach where we are looking at overall or average behavior of the fluid using the so-called continuum assumption what is this continuum assumption it simply assumes that we are not going to bother about the distribution of fluid matter on molecular level we will simply say that disregard that molecular structure and we'll simply assume that the fluid matter and whatever associated properties that it has have a continuous distribution that is the baseline assumption of the continuum model so associated properties mean mean really what you can talk about pressure you can talk about density you can talk about temperature if you want these are all I will say are associated properties and a continuous distribution of those is what will be assumed in a continuum model the biggest advantage of using a continuum model is that since we are using then a continuous distribution we can use methods of differential calculus for the mathematical description of it and that is the idea so the last point here points that out that we assume essentially a point wise smooth distribution of fluid matter as well as the property and therefore you can use methods of differential calculus let me take one more minute on this and point out that mathematics is an integral part of fluid mechanics whether we like it or not we cannot really get away from it now there are some people here I'm very sure who are extremely comfortable with the mathematics part of it on the other hand I'm sure there are some more people here who perhaps are not as comfortable those who are comfortable I don't have to say anything to them but those who are not you must try and at least get to the basic level of mathematics that is required for a good description of these fluid flow phenomena the reason is because unless and until you really master that straightforward math we are not really getting into anything seriously heavy some ideas are not that clarified not to yourself not to the students so even if perhaps you are right now slightly uncomfortable is just a matter of putting some effort in it and I'm sure if you do it there is there is no problem with the reason I'm pointing this out right now is because of the last point where I'm pointing out that we will be using the methods of differential calculus all the time and I'm doing it on purpose I'm doing it on purpose so that you some of those you of you rather who don't want to touch mathematics will be forced to we have all gone through it it's not like you know people who will be standing here and teaching supposedly know this really well and such no it's not like that it's just a matter of sitting down and working through it and again you know you may have to see a few books not just one that's about okay so when do we do this when can we utilize this continuum model sufficiently let's say or what is the validity of it the idea that if you have a large number of molecules that that you can think of in the flow situation that you are handling such that you can imagine a fairly small volume that you identify in the space fluid space that volume is such that it's such a diameter or some linear dimension of that volume is sufficiently large compared with the mean free path of the molecule so what do we understand by mean free path is the distance between successive collisions that molecules will cover so if the if the volume that I am identifying in the fluid space is sufficiently large than the mean free path and at any given instant of time if you find that there are very very large number of molecules available in that volume so that if I want to define some sort of a property at a point now this is not a mathematical point that we are talking about this is a point let us say which we can say is the center of this small volume wherein the average density for example over this small volume will be simply assigned at that point so unless you have large number of molecules in that small volume you will not get a statistically consistent number for that density unless you have that you cannot utilize a continuum model very simply if you have very large number of molecules available in a fairly small volume that you are dealing with you are fine with continuum model this is all fine in the sense that the description wise the standard applications that mechanical engineers aerospace engineers chemical that perfectly fine within the continuum assumption the only two or three special situations where we see that the continuum assumption is invalid is if you are looking at highly rarified flows such as what would happen if you are dealing with vacuum applications for example or if you are dealing with very high altitude space application so as you know if you go up in the atmosphere the density will keep on decreasing sufficiently high if you say are about 100 kilometers from the surface of the earth the atmosphere is very rarified so what does that mean you that means that if you identify a fairly small volume and keep monitoring the number of molecules inside that small volume at some instant there will be certain number of molecules in there but at some other instant they may not all be there so depending on a situation like that you may get if you have a probe which measures density for example at some instant it will show a non-zero value of density some other instant it will show a zero value of density because there is no molecule in that volume if we are dealing with a very high rarified so there you cannot really use continuum similarly nowadays there is this new field coming up MIMS application nano scale application so here again the volume that you are talking about the characteristic volume itself is so small that there is a chance that you may not have enough number of molecules in there in which case you cannot really use continuum so there is a parameter which exists called the Knudsen number which is simply defined as the ratio of the mean free path over the characteristic length of a problem let's not get into the characteristic length right now but roughly speaking let us say if you are talking about flow inside a pipe our standard mechanical application the diameter of the pipe usually we take as the characteristic length so imagine that that pipe you shrink to such a small diameter that that diameter itself becomes of the same order as the mean free path of the molecule then we are in trouble because in that sense there will not be too many molecules at any given instant in that cross section of the diameter and you may not really be utilizing the continuum assumption at all so roughly speaking those who know I think some of you perhaps are already working in this non continuum application that Knudsen number less than about 0.001 roughly continuum assumption is valid all that means is that there are a very large number of molecules available in a fairly small volume the linear dimension of which is much larger than the mean free path in this case what happens is then that you can talk about the fluid point as I was trying to explain to you the fluid point is not a mathematical point as I said it's simply a point that will be assigned at the center of that small volume where you will be assigning the average value of density pressure etc for that small volume as the point value that's about it and then we move on to the next point so if you can imagine a very large number of such points placed right next to each other then I have what we will call a fluid continuum where the material also very smoothly and the properties such as density pressure etc very smoothly from one point to the next and we will be always dealing with continuum model this is just essentially a let's say condition that we are going to work with continuum situations you are not going to work with non-continuous very quickly these are the properties that we will be utilizing on and off first is pressure pressure as a molecular origin so anyone wants to point out how pressure is explained from a molecular point of what do we say exactly that is absolutely right what we imagine is that imagine some sort of a surface in the in the fluid domain and on this surface there will be a very large number of molecules that will be impinging let's say and exchanging their momentum the aggregate effect of a very large number of such collisions essentially results as a compressive force on that surface and that's what we call nothing but pressure obviously you know the pressure measurement devices that we have are usually some probes etc and no matter how small you want to make the probe is going to be still much larger than the intermolecular distance etc so we are still talking about a really large number of molecules hitting the surface of the probe if you want to measure it the pressure that is so what we get really is some sort of an average value again over that small volume and therefore we are again using the continuum idea of getting the pressure data out when we insert a probe in the fluid medium density as we know mass per unit volume specific weight is simply you multiply that density by acceleration due to gravity physically it is the weight per unit volume then we will use sometimes the specific volume which is simply the reciprocal of the density volume per unit mass and then there is nothing more to talk about with respect to this dynamic viscosity which I think most of you know is denoted as mu so this property then characterizes the resistance that a fluid offers when a shear force is applied to it so now I mentioned that the fluid cannot resist here for here I am mentioning that resistance offered by a fluid so is there a problem is there an inconsistency yeah what is that most exactly that though that's the key word when the fluid is in static equilibrium and then you apply small shear force it cannot resist so there cannot be a static equilibrium under the action of shear force however a fluid in motion can actually resist the shear force and the mechanism through which it resist is what is called as or it is characterized using the dynamic viscosity again the basic physical nature is of molecular type we are not going to get into the molecular discussion of how the viscosity comes about if you are really interested in that you must be reading then kinetic theory of gases where at least for gases it is explained how a molecular motion results into what is called as viscosity on a macroscopic basis it's very interesting but it's beyond the scope of what we are talking about okay fine so let's see if we can again use some basic mathematics to quantify this here stress and the viscosity and so on okay so what we have here is a simple shear flow so simple shear flow is nothing but you imagine two plates which are parallel to each other separated by a small distance let's say and the characteristics is that at least as far as thought process is concerned that the plane of the plates is very very large so we call these infinitely large plates obviously nothing in real life is infinitely large the idea is that if the linear dimension of a plate either the width or the length if you want if it is sufficiently large then the separation distance between these two we can roughly say that these are what we will call infinitely large plates fine there is a fluid in between those and let us say that the fluid motion is established somehow such that there is only one component of the velocity for the fluid which is in the x direction as it is shown and the x component of the velocity is what I'll be using the letter small u the characteristic of this simple shear flow is that the u velocity or the x component of velocity is only a function of y meaning that it will vary from one y location to the next but if you go from one x location to the next it will not vary and this is something that you keep in mind right now eventually we will when we talk about the solutions of the Navier-Stokes equations tomorrow we will essentially come back to this situation and try to explain where this simple shear flow actually you can think about and how you can actually show that the u velocity is only a function of y and so on fine so let's say this is the setting now imagine that there is a small vertical segment that pq that I have shown with a length delta y now this delta y has a mathematical notation it with it that that's fine but physically what we are understanding here hopefully is that that delta y sufficiently larger than the intermolecular collision distance okay there is a variation of properties from point p to point q over that the distance delta y so what I've shown there is that let the x velocity at the point p be u and then if you simply travel a distance of delta y the velocity x velocity that is at the point q is given by the first order Taylor expansion which is given by u plus du dy times delta y now somebody can question this they'll say why only first order I can add more terms then yeah you can there is absolutely no no problem the idea is that this is a model it's a model what do you understand by model a mathematical model what do you interpret by the words mathematical model yeah so you want yes you want to represent some phenomenon using some mathematical expressions and it's up to you really how accurately you want to do things since we are talking about a smooth variation of properties in the flow and in general these distances delta y later on we'll use delta x etc these are sufficiently small then they're not as small as obviously the mean free parts they are larger than those but they are sufficiently small so that usually what is found is that a first order Taylor expansion type situation is sufficient for description accurately keep this in mind people students ask this question all the time to me so why are we going only with a first order Taylor expansion and the answer is that this is a model there is no God sent let's say dictat that you have to use on the first order it's just that we are talking about points which are sufficiently close to each other in a smoothly varying field and it is found that a first order Taylor expansion is sufficient to describe things correctly now what is meant by is found to sufficiently correctly described is something that I will talk about again tomorrow right now you noted down okay so if that's the case then we say that the x velocity at q would be simply u plus d u dy times delta y so now we come up with a what is called as a rate of shear strain first we will talk about shear strain shear strain how many of you have done solid mechanics all of you so here we are going to talk about the time rate of it so first let us look at the shear strain itself with respect to that segment p q so what is going to happen in general the velocity u and the velocity u plus d u dy delta y in general they are going to be different so p will move a certain distance q will move a certain distance which is not going to be the same as what is moved by right and that is what is pointed out in the expression here this guy here u plus d u dy delta y times delta t is simply the distance moved by q in the time delta t minus u delta t is simply the distance moved by p in the same interval so therefore what is going to happen is this fellow will go like this and build tilted like this that's all and then we from solid mechanics what we define as a shear strain is simply this distance which is the difference between the distances divided by the original distance between those two for shear that's it so that's been divided by delta y so whatever is inside the square brackets is nothing but shear strain and then we take a time rate of it which is simply dividing by the time interval over which all this action is happening and let that time interval 10 to 0 mathematically speaking taking the limit as delta t tending to 0 here for the only time I think I have written this specifically that we will be taking limits as delta t is will tend to 0 from now onwards I will not be mostly writing it again and again but it will be understood that we are always talking about these time limits being taken as the interval going to 0 so that's it if you just simplify it u delta t u delta t d delta y delta y etc will cancel many of these things happen you can work out the algebra which I by the way want you to do this is very simple but there will be more troublesome algebra situations later when we derive the governing equations and unless you are ready to derive those once at least yourself I personally think that you have no right to ask the students to do it this is my take you can you can vary you have to do it once otherwise if someone tomorrow some student comes I say how do I do this now what will you tell him or her you have to be able to say that okay this is the way to go about it so once you do it right so then finally it simply comes out to be this d u dy which is the gradient of the you velocity in the y direction and the way to connect then the the shear stresses now I will call as the stresses set up in the fluid to oppose the imposed shear stress usually you will see that those stresses are proportional to that rate of strain raised to some power if that power happens to be one we have what we call a Newtonian fluid and then we have tau equal to that dynamic viscosity times d u dy okay so this is in some sense the formal way of showing how you can bring about that tau equal to mu times d u dy expression as the so-called Newton's law of viscosity using a simple parallel shear flow where I have only one component of the velocity in the x direction but it is a function of later on when we derive the full governing equations we will actually generalize this to what we will call a generalized Newtonian law of viscosity where there will be more than one component of velocity there will be u as well as v and in principle a w also I am not going to work out the entire algebra for three component I will work out for two but then generalizing it to three is just more accounting so those who are good at bookkeeping can do it well also to note later that there will be more than one component of the velocity and each component will be a function of more than one special variable so u then will be a function of both x and y and z also and time also if you want similarly v which will be the y velocity will also be a function of x y z t and so on so we will see how to generalize that later but right now just to introduce the idea of viscosity where it comes about we are saying that we come up with a rate of strain expression and in general the stresses that are set up in the fluid which will oppose the imposed shear stress are going to be proportional to some power in general of the rate of strain expression we will be dealing with the Newtonian fluid where that n equal to so apologies to the chemical engineers how many of you are here chemical no one from chemical I thought there were a couple so good then you don't have to bother it's just Newtonian fluid that we are working with chemical engineers are the ones who work with non-Newtonian fluid where I think as he was mentioning you have to look at rheological fluids all sorts of weird things in principle the methods of analysis are not different but the way the stress and strain rates behave there is different you have to take that into account appropriately that's it that's my introduction let me take a minute on this as again these are my in the sense that when I was a student I have used Fox the most let's say now that I am on the other side of the the table let's say I'm clearly using more and the one at the top and the one in the middle is something that I personally like a lot tomorrow I'll bring all these right now I don't have their copies with me tomorrow I'll bring it how many of you know either one of these I am assuming that Fox and McDonald is something that many one many of you will know yeah how many of you are using any of these for the class good very good okay now here is a personal request unless you read good books you cannot really understand any of it and as I said there is no one single book that I can point out at the undergraduate level where more or less we are working at right now the material the level of it I feel that these are fairly good anyone has seen even Gupta and Gupta you have has anyone tried to use it for a class no what may I ask why or why not I'll tell you why not it's not an easy book for someone to just use one needs a serious outlook to use that anyone knows who these two guys are by nature both are or one is now somewhere else one is both were from IIT Kanpur one from aerospace one from chemical I really think it's a phenomenal book it's just that people like it because it appears difficult that's all I can say however if you have the attitude to really sit and not use it as a storybook but really as a proper engineering textbook you will see that there are so many things which are written so excellently in that book that you will be very happy again this is my take on Potter and Wigert I don't think anyone here probably knows who knows I'll be very surprised if you know this it's it's a relatively unknown book I think I made a mistake there the title is not mechanics of fluid that the original US title the Indian edition is simply called fluid mechanics and I think it's a 2010 not 2011 it's by publisher called Cengage learning it's available I'll bring it tomorrow available for something like 3 400 whatever I don't remember in fact I I actually asked the Cengage people to publish it as an Indian edition book the reason is because it's it's a very very well written very compact book if you want to look at it nicely and briskly explain things that's the way I look at it so whatever I've talked about so far you will find in chapter one of first two and in the in the Fox book you will see it in chapters one and any questions so far there is a good chance that some of you may have a good alternative take on some of the things that I'm talking about there's a good chance that you may disagree with something that I have said which is also fine in fact how many of your teaching fluid mechanics for the post graduates I think some people the raise there are you prescribing any book okay those were slightly let us say adventurous I'll write it here I want you to at least look at it the reason I'll point out why use kundu and I reckon this I think some of you seem to know this it's a proper advanced fluid mechanics book and somewhere if I remember correctly they mentioned that I think it's in here if it's not here I'm sorry but somewhere I have read it the purpose of a course really is to make the student more confused can you can you say why then you will say you are defeating the purpose of a course honestly I think it's written here I am correct the idea is that unless you make someone confused that person is not going to be going back and be restless and try to do something and only restless people will survive that that I can tell you if you think that everything is going fine there's no need to do anything then nothing will happen unless you make someone uncomfortable they are not going to do anything so your job is really to push hard these guys that is your student but to do that obviously you have to do the effort first and that is what we try to do here as much as possible and believe me it's it's a troublesome business we get hammered in the class every time we go and teach this basic fluid mechanics in the class by the time I come out I'm drenched in sweat because the students are really going at it they will not let you settle down even for a second why is this why not this yeah I suck you can I suck you need here you can take unless you are ready to accept those questions as first of all being reasonable and be ready to think and get back to them nothing is going to happen then very fast students will lose interest if you are not ready to address those questions in any manner that they have very small attention span let me that as teachers we are expected to have larger attention span but students will not have a large attention and they want to know things very fast and quick and to do that you have to go through a lot of background material you have to really push yourself hard Elsevier it's an Indian edition I think it's again about 500 or so I'll bring it tomorrow see there are too many I'm just pointing out some things that I use and find useful so the idea is that you have to make yourself uncomfortable you have to make everyone uncomfortable and that's the only way that people will do something otherwise as professor that is common you remember things will be all mouse driven and nobody will know really know what is happening so please I mean this is my my request that try to push yourself very hard and try to ask questions all the time why is it this way why is it not this way what makes it do it this way and this is what I tried to or we try here to tell our students that when you read a book any of these or whatever don't just read it as a bedtime story book you have to really question each and every line there and in doing so there will be obviously far more questions than answers but eventually there's a good chance that that will help someone