 Welcome to the next lecture on multi-phase flows. What I want to do is before I start just want to summarize what we have done so far in the last 4 lectures or so in this class and to tell you what is going to happen at least in the next 4 lectures okay and you will have some idea about what is in store for you. So what we have done is use the fact that all of you have some basic knowledge of fluid mechanics okay and we have used that and we did not bother to you know derive the Navier-Stokes equations or anything. We just said that you all know fluid mechanics and then we decided to analyze a specific 2 phase flow problem okay and the thing that we respected ourselves to is laminar 2 phase flow. Now I just want to emphasize that as far as the entire course is concerned we are going to mainly be looking at laminar flows. We are not going to be looking at turbulence at all because there is a completely different ball game. You need to analyze situations using computationally intensive methods. The focus here is not so much on computationally intensive methods but on getting a good physical understanding. So if you were interested in doing a turbulent flow analysis of a 2 phase flow problem what you would be doing is looking at doing direct numerical simulations, doing computational fluid dynamics okay. So we are not going to be doing all that. So the reason I am telling you this is tomorrow when you are reading papers you will see that there are people who are doing fluid mechanics who are using those kinds of principles. You will see people who are not using CFD but they are using the kind of methods that we are talking about. But the idea is for you to know the differences and see if we can learn from one technique, one approach and how we can apply it to another. Similarly you are doing turbulence how I can use ideas from that method to this. So that is something which you should be clear about and that is something which not only as far as this course is concerned for anything that you are doing. You are going to be looking at working on a particular problem. You are working on an area which is maybe reaction engineering or fluid mechanics whatever. There would be so many things in the same area and you should be able to know what the similarities are and dissimilarities are between what you are doing and what somebody else is doing. Because that is the only way you can become more clear in your concepts okay. So that is what I am trying to tell you now is that what we are doing is mainly non CFD approach okay. But then that gives you a lot of insight tomorrow when you want to do computational fluid dynamics. So that is the way you have to take this course okay. So what we did is we looked at stratified flow towards the end you have to do the core annular flow and we want you to basically work out couple of problems. The idea is normally what you would think when I am pumping 2 different liquids in a tube if I pump 2 liquids with let us say the flow rates of the 2 liquids is Q1 and Q2. The volume fraction which is going to be occupied by each of the liquids okay is not going to be given by the ratio of Q1 by Q2 or the volume ratio okay. So that is one of the things which we wanted you to understand by the 2 examples that we did the stratified flow example as well as the core annular flow example okay. So that is the reason why we wanted you to calculate the velocity profile. We wanted you to calculate the hold up. The hold up is the volume ratio. So let us say V1 and V2 are the volumes occupied inside the channel by each fluid. We want you to be clear that V1 by V2 is not equal to Q1 by Q2 and that would be something which is normally the thing. I am pumping in 2 liquids with the same flow rate. So you would expect that half of the tube is occupied by 1 fluid half by the other but those of you who have done the assignment will realize that this is going to be decided by the ratio of the viscosities okay. And the reason why the other parameters do not come into the picture is because we are assuming that we are in the lower Reynolds number regime so that the density does not show up okay. So that is one thing which I wanted you would take from the lectures that we have seen so far okay. Now what we want to do is we want to do things in a slightly more formal setting we want to extend. We have assumed that the interface is not changing in size in shape as far as the position is concerned. So we want to get to the position where we are able to incorporate the changes of the interface. So to make this entire course self-contained we are going to go back a little bit and I am going to derive in the next 3 or 4 lectures we will do some theory and then we will do some problems. So the theory is going to basically focus on deriving the Euler's acceleration formula which I have already seen which then we talk about the Reynolds transport theorem which is basically a conservation principle okay. And then we will see how we can generalize some of the boundary conditions that you saw earlier regarding the pressure jump across the curved interface. When you have a curved interface in a core annular flow the surface tension basically helps you evaluate what the change in the pressure is. You need to know the curvature you need to know the surface tension and you can estimate what the pressure difference is. So now what we want to do is we want to generalize that to the case where there is flow. So that is what we are going to do and then we start working out some problems okay. So next 3-4 lectures is going to be basically on trying to establish the thing on a firm theoretical footing. Some of it may be repetitive to what you already seen but it is always good to you know revise and revisit some things and then go forward. So let us look at the theoretical basis of let us say fluid mechanics okay not being very careful with the English here but it is okay. So normally 2 approaches you are already familiar with okay which you have seen in the past and the 2 approaches used in tackling flow problems. The first one is the Eulerian approach and the second one is the Lagrangian approach okay. So what is the difference between these 2 approaches? What we do in the Eulerian approaches? We are focusing our attention on a fixed region in space okay and we are trying to understand how things parameters like velocity, temperature, etc. change in that fixed region in space. So now this region in space can be very infinitesimal, it can be very small, it could be just a point and supposing you actually were to measure temperature at a particular point in a keep a thermo at a point, it is going to measure change in temperature. So you will get a temperature as a function of time but that is at a fixed position. So what you are doing is you are basically using an Eulerian approach whereas the Lagrangian approach is one where what we are doing is we are not looking at a fixed region in space but we are looking at a fixed particle or a fixed collection of molecules or a fixed molecule because it is a fluid flow problem the molecule is going to move. So in order for you to be able to track the change in the temperature you need to be able to have a probe which is tagged on to the molecule okay. If you have a probe, if you have a thermocouple which is able to sense the temperature of the molecule as the molecule moves you will be able to actually measure the temperature. So then you are actually doing a Lagrangian approach okay but now what is happening is this probe is going to occupy different regions in space but it is tracking the same molecule okay or same particle. So there is a difference between these two approaches. Now the reason why we need to be clear about this is because most of your fundamental laws that you have come across earlier like your conservation of mass, conservation of energy you have are used to those laws in the framework of the Lagrangian approach okay whereas because when you talk about mass cannot be created, mass cannot be destroyed you are talking about you know the stock piece same sort of molecules is going to remain. So you are talking about fixed set of particles fixed you know object whereas what you are talking about here is fluid mechanics where things are going to be flowing okay it does not really and what you are interested in is how is the velocity at a particular point. So you need to understand the difference between these two approaches because what we want to do is we want to see how we can extend the fundamental laws like Newton's law of motion where you say force is equals rate of change of momentum okay we want to see how because that is for a fixed particle we want to see how I can extend that to a fixed region in space because that is what we are interested in we want to find out rate of change of momentum and how that depends upon the forces etc in a fixed region in space okay. In fluid mechanics since you have a fluid flow problem we are so normally what happens here is in a fluid flow problem you know I just put a diverging channel because I am going to use this later on you will have you know let us say fluid moving from left to right. If you focus at a particular point here this particular point is going to be occupied by different molecules at different times because there is continuous flow. So if you want to measure the temperature here you will be measuring the temperature of different molecules but at the fixed point okay. So if we have a probe at let us say A okay this measures the property different molecules but at one point. So because the probe is here my thermocouple is right here okay because of the fluid flow it measures the average temperature of all these particles there. Now this is what approach is this? This is the Eulerian approach and this is what as an experimentalist you would be doing you would be measuring a fixed point in your reactor or your heat exchanger or whatever it is whereas in if the probe moves with the fluid and is attached to a molecule okay or a particle then the probe measures changes of the property of the particle okay which moves around in the fluid and this is your Lagrangian approach and the fact is what we are really not interested in what is going to happen to the temperature of the particle. So if you want to have a probe here stuck to this particle maybe some kind of a radioactive dye or something and you want to try to find out how the concentration is changing it moves. We really want to find out what is happening inside my system. Once it goes outside my system it may not be of interest to us. So there is no point in trying to talk about changes of the material particle because you are really interested in what is happening inside your heat exchanger distillation column reactor whatever it is okay. So we need to therefore go to using only the Eulerian approach okay but what is so special about the Eulerian approach the fact that to repeat most of the laws conservation laws are in the framework of the Lagrangian approach and we want to know how to relate changes in the Lagrangian approach to changes in the Eulerian approach that is the idea okay and once we know once we are able to do that then we can go back to solving the problems. In fact we have already seen this but maybe in a different context or in a different framework so we will just try to relate that. So just to summarize most fundamental laws are in the Lagrangian framework okay and we want to apply these laws in the Eulerian framework. Now I will go back to this problem of flow in a diverging channel okay to basically point out what the difference is between these two approaches okay just to clarify things. So let us go back to this problem in a diverging channel. Let us look at steady flow to make my life simple what I am going to do is I am going to assume that the flow is uniform across the cross section okay. So this is a rectangular channel which is extending to infinity outside the plane of my board and just to illustrate the idea okay. So I am going to say and I am going to put slightly smaller arrows here. So the length of the arrow basically is reflecting the magnitude of the velocity okay and what I am trying to tell you here is that the magnitude of the velocity is uniform here. So the velocity is not changing in the direction perpendicular to the flow. So if this is my flow direction is x and this is y direction does not change in the y direction the velocity does not change in the y direction everywhere. So since the channel is diverging one of the things we do expect is that the velocity is going to decrease as you go along the flow okay. So you have a uniform velocity here you have uniform here but the value of the velocity here is going to be different from the value of the velocity here. So if you want to call this point this section 1 and this section 2 and if you are talking about liquids then we all know from the macroscopic continuity equation that rho 1 A1 V1 equals rho 2 A2 V2 okay and we are talking about liquids the densities do not change you will have A1 by A2 equals V2 by V1. So if A1 is lower than A2 V2 has to be lower than V1 and that is the basically is going to decelerate. So now I am going to ask you the following question supposing I were to have a probe which is going to measure the velocity at this point okay you keep your pitot tube or you have some device by which you are trying to measure the velocity and if you were to measure the velocity here what would be the value of the velocity which your probe is going to measure? It is going to be V1 and this V1 is not going to change with time right because I am basically assuming that my flow is steady. Steady means at a fixed point in space the V1 does not change with time okay. So as far as my fixed probe is concerned so if we have a fixed probe V1 measured does not change with time and this is my Eulerian approach that is what it will tell us right. This is in the framework but now what is happening to the actual liquid particle which is moving from here to here. Liquid particle supposing you were to actually track a liquid molecule it has a high velocity here it has a lower velocity here. So now the molecule is actually decelerating okay. So the particle per se is having a change in the velocity. So if you were to actually sit on a particular particle you will find that you are slowing down clearly because V2 is lower than V1 okay. So in the Lagrangian framework the velocity of the particle will slow down okay since yeah we have a diverging channel okay. So it may appear to you like there is some kind of an inconsistency you are looking at a particular point things are steady things are not changing and whereas if you were to actually go but the idea is that you are using 2 different frameworks 2 different reference frames for doing your analysis. So depending upon the frame of analysis that you are working in you will have 2 different observations. So what we want to do is we want to see if we can actually relate these 2 things okay and that is basically our thing and we are going to extend this to a macroscopic region and that is what Reynolds transport theorem is. So first we will do it for a very simple case which is for an infinite simul. So let us relate the observations using these approaches for infinite simul region in space basically what I mean is point in space occupied by an infinitesimally small object okay. So basically I am not talking about I am talking about points I am just trying to make relate changes in points to changes in infinitesimally small particles that is the idea okay. Now just to try to make things clear let us talk about a scalar quantity like temperature. Now because temperature is something which you can measure using a thermocouple okay. So now let us focus on the temperature in the system. So now if clearly in the most general situation the temperature in that particular system is going to vary with the spatial coordinates x, y, z okay and also time if you have an unsteady state situation okay. So now temperature is can be written as temperature of x, y, z and t okay. This is the most general variation of temperature that you can see spatially as well as temporally okay. You suppose have your probe suppose you can think of an ingenious probe which is tagged on to a particular particle okay and it is moving around the fluid. This particular probe is going to change its position is going to move around right. So what this means is the spatial coordinates x, y and z are not fixed anymore. They are also going to be changing because now if you want to use a Lagrangian approach if you were to actually try to change to find out how the temperature is changing of this particle you are going to measure the temperature change not only as a function of time but also as a function of position okay. So in the Lagrangian approach if the probe is tagged to a particle the temperature that is measured is going to be depending on x, y and z which change with time because the particle keeps moving and it also has retains this explicit dependency on time okay. So what we want to do is we want to basically talk in terms of rates of change of the temperature that you are going to observe from your reading okay. So if you have a temperature probe which is giving you data signals you will be getting temperature as a function of time and position okay but for the same particle. So this particular change of temperature with time the rate of change of temperature with time is going to be what is called a material derivative because I am focusing on a single particle okay. So dt by dt is nothing but the total derivative. So this is rate of change of temperature with time of a fixed particle okay and I am using this particular symbol capital D you will hear this kind of nomenclature when you are reading books some people call this material derivative some people call it substantial derivative. This is basically taking into account the changes of temperature with respect to both position and time okay. So in this framework dt by dt is nothing but dt by dt which is the total derivative. In mathematics you would have come across you know partial derivative total derivative okay. So this is the total derivative. So I am just going to go back to calculus and establish this relationship between Eulerian and Lagrangian frameworks okay without really complicating the issue too much. This tells me how in the for the particle the temperature changes time and now I need to only differentiate that expression of this functional dependency okay and keep in mind that the spatial position also are functions of time. So now when I want to differentiate this thing find the total derivative I will get dt by dt as partial derivative of t with respect to x times dx by dt plus dt by dy times dy by dt plus dt by dz multiplied by dz by dt plus dt by dt okay. So all I have done is I have just differentiated that expression and I am differentiating that with respect to time and I am telling you that I am incorporating the dependency of x on time explicitly to take into account for the fact that the spatial position is changing with time and when you are actually using the Eulerian frame of reference what are you doing? You are fixing yourself at a point and you are trying to measure the rate of change of temperature at a fixed point okay and that is the partial derivative with respect to time of the object that you are interested in which in this case is temperature okay. So what I am trying to tell you is that this last term here represents the Eulerian derivative and that is how that is the rate of change of temperature with the fixed probe would measure. This is the rate of change of temperature which are variable probe which is moving with a particular particle is going to measure and basically this equation which is nothing but which has come from calculus okay which is the relationship between total derivative and partial derivative is basically the relationship between my Lagrangian framework and my Eulerian framework okay. So what I am trying to tell you here is that this total derivative is what I measured in the Lagrangian approach this is what I measure in the Eulerian approach and all I have done is just use calculus nothing else okay. So now what I like to do is write this in a slightly more general form which is this velocity vector is nothing but dx by dt dy by dt and dz by dt right the 3 components of velocity I can write gradient of temperature as dt by dx this is also a vector dt by dy dt by dz. I am going to write these first 3 terms on the right hand side in a compact way as v dot del t okay and what that relationship gives me now is dt by dt equals v dot del t plus dt by dt okay. So this is the Lagrangian derivative or the substantial derivative or the material derivative how does temperature of a particular particle of a particular material change with time this is how does temperature at a fixed point changes with time okay and the relationship between these 2 is actually given by this okay and I can write this for any quantity that I want. In fact I can write this for velocity this becomes dv by dt equals v dot del v plus dx by dt. I did do this for velocity at the beginning because velocity is a vector and then at some of you may not have been comfortable with gradient of a vector. So I just did it for temperature and having done it for temperature I have just written it for velocity okay. We will see how gradients of vectors are defined later on in the course and this is something which you are all familiar with and this is called the Euler's acceleration formula. Going back this problem of diverging channel where we spoke about the fact that the velocity is actually going to decrease as the particle moves one of the things we want to do is we want to see how we can apply the acceleration formula here to that particular system okay and our flow was steady. So our flow is steady what that means is the partial derivative with respect to time is going to be 0 okay. So this is 0 so dv by dt is 0 since flow is steady and what about dv by dt? dv by dt is the rate of change of velocity of a particle and we know that it is not 0 because it is decelerating. The particle is actually slowing down so this is not 0 but what is the value of this dv by dt it is v dot del v and if you for a very simple system that we have where we have velocity only in one direction in the flow direction vx which is actually changing in x direction if you about to evaluate this expression this is going to be given by vx dvx by dx. So that tells you how the acceleration is for the particle okay and you can find out if you know how the area is changing you can actually calculate how this velocity is changing with x and you can actually calculate this. So the deceleration of the particle so I just wanted to illustrate clearly that this is the relationship between the Lagrangian approach and the Eulerian approach. This is the time derivative which you measure at a fixed point which is Eulerian framework. This is the time derivative you measure when you are tracking a particle and this is how they are related okay. What I have done here is for a small at a particular point I am just assuming the temperature is a function of x, y, z I am trying to relate changes of temperature at a point to changes of temperature occupied by a molecule at that point. An infinitesimal mass, infinitesimal volume our objective next is to generalize this to a macroscopic system okay. So the same conservation so when I do this when I actually try to relate the Eulerian and the Lagrangian frameworks for a macroscopic region I get what is called the Reynolds transport theorem okay. So extending this from an infinitesimal volume to a finite volume in space P obtain the Reynolds transport theorem. What I will do is tomorrow and the next class is talk about the derivation of the Reynolds transport theorem. Try and show to you that you have already seen the Reynolds transport theorem in a different form. It is just that we are having a slightly different way of looking at the problem okay. So you have all seen Reynolds transport theorem in some other avatar. You are going to see this in this new avatar now. Then we will apply it to deriving the continuity equation the Navier-Stokes equations that you are all familiar with which you have been using. So once that is established then we go back to formulating the boundary conditions and stuff like that. But let me since I do seem to have a little bit of time let me just add a few more things so that I can do things in a more relaxed way tomorrow. So just to I am just going to call this RTT Reynolds transport theorem. I want to just introduce 3 concepts which are very simple and so it makes my life easy tomorrow. Control volume is nothing but a fixed region in space okay. It can be of any arbitrary step but it is not changing it is a fixed region. So it is analogous to my infinitesimal point okay. This is space. The boundary of this control volume which is going to actually demarcate the volume from the environment is what I call the control surface okay. Surface is the boundary of the control volume okay. And then we will talk about this other thing which is control mass. Control mass is a collection of particles, collection of molecules which have the same mass okay. So this is a collection of material particles. So this has a fixed mass but it can move around in space okay. So just to tell you what the analogs are what we have just seen this is at a fixed point okay that is your when you are looking at a fixed volume you are doing a Eulerian approach when you are looking at a control mass which is moving around you are actually looking at the Lagrangian approach okay. So the mass can change shape okay it can change size but then it is not in the fixed region in space it is moving around with the flow. So what we want to do is we want to try and extend this particular relationship that we have got for an infinitesimal particle okay occupying an infinitesimal region in space to a finite size control volume and that is going to be the topic tomorrow and that would basically be the basis. In fact if you understand Reynolds transport theorem if you can actually apply Reynolds what you would be doing as an engineer is you would be applying Reynolds transport theorem to different systems and depending on the system the final form of the Reynolds transport theorem is going to be different. So what we will do is just give you the general form of the Reynolds transport theorem and then depending on the particular system you are going to be analyzing you will be looking at different final versions of the Reynolds transport theorem.