 So, welcome everyone to the second day here. What I will do briefly is go through some of the material that we covered yesterday especially the integral analysis. So, that is what is projected on the on the screen now. So, let me let me just point out the important parts from the integral analysis that we did yesterday. So, if you remember what we did was we started our analysis with a balance statement which is projected on the board right now. And the balance statement so far that we have seen was for mass balance and in some sense you can call the other principle a momentum balance. This is the primary equation if you want written in a principle form or in words and then what we ended up doing was we chose the mass as the quantity of interest in the beginning and then carried out an analysis to figure out what are the mathematical expressions for each of the terms in this balance statement. And thereby we came up with our integral form of the mass balance. So, here again just to recap the first term on the left hand side here signifies the rate of accumulation of mass within the control volume and the second term on the left hand side here signifies the rate of outflow of mass minus the rate of inflow of mass or in other word the second term here the area integral is essentially net rate of outflow of mass from the control volume. Then what we did was we employed that balance statement for a linear momentum quantity let us say and then initially we talked about the linear momentum in the x direction. So, in a manner similar to what we had done for mass balance we obtained an integral equation of linear momentum principle applied to control volume which is shown on the projected screen right now in the box. So, again here the first term on the left hand side is the rate of accumulation in this case of x linear momentum within the control volume and the second term on the right hand on the left hand side I am sorry is the net rate of outflow of x linear momentum from the control volume. It is essentially outflow minus inflow term and then on the right hand side f suffix x is the net force both surface as well as body if at all it is acting in the x direction on the material that is contained within the control volume. So, here we realized and noted that the control volume will contain a certain amount of material which will be essentially treated as a free body having been isolated from its surrounding and when you isolate this material because of the control volume you have to mark all sorts of forces that will come on to this material in an appropriate fashion. And then we have generalized this linear momentum principle in a vector form on the way we also wrote it for a y direction and then we generalized to a vector form. And then what we did was we went through a few examples five of them wherein in each of these examples we have step by step employed these integral analysis procedure. If you recall each of these examples involved first an integral mass balance equation and then the momentum equation in whichever direction that was considered appropriate for that particular problem. Let me not go through the examples obviously, but let me just point out that these were our guiding principles which have been projected right now on the board. When we want to perform integral analysis we try to follow these guiding principles as carefully as we can. Specifically we choose control volume such that the inlet and outlet velocities are normal to the corresponding areas then we do not have to bother calculating the dot products for the mass flow rates. They will be simply algebraic products the forces which will involve both surface forces and body forces. Now, here let me make a point that in the five examples that we chose to discuss yesterday none of the examples had a body force included, but that is fine at the moment the idea is not to completely get into the details of the integral analysis but at least give a flavor of what is involved in these. What I suggest is that I have suggested a few fluid mechanics books yesterday. Having discussed this material here yesterday I am very sure that if you go back to some of those books you will be able to follow many more examples outlined in the book as well as the problems that are given in the book some of those will involve inclusion of these body forces. Anyhow the second guiding principle was the forces acting on the control volume are essentially the forces acting on the material isolated by the control volume and we said that this material is to be treated as a free body just like what we do in solid mechanics or strength of material. And as far as the choice of control volume is concerned we choose it such that the force of interest that has been asked in the problem statement or at least an equal and opposite reaction to it should directly act on the control volume and that is how the control volumes are typically chosen. And finally, the linear momentum in and out of a control volume can be positive or negative. It turned out that all examples yesterday that we talked about had the linear momentum both in and out positive. The reason was it is determined by the inlet and outlet velocity whether it is positive or negative in the chosen coordinate system. So, mass flow rate as I said yesterday will always be positive but the momentum linear momentum that is that it carries with it can be positive or negative depending on the inlet and outlet velocity being positive or negative in the coordinate system. So, that is more or less what I want to talk about as a brief summary of what we discussed yesterday. These are the 5 examples which I am going to skip now. We have discussed these enough yesterday. So, let me carry on today with the remaining balance equation that we did not talk about yesterday and that is the total energy balance for a control volume. It is essentially the same sort of a balance statement that we wrote earlier in the general form and then specialized it for a mass balance and a linear momentum balance. Here what I am talking about is a rate of accumulation of total energy in the control volume on the left hand side here. E dot in and E dot out are essentially the terms containing total energies that are associated with the fluid flow coming in and going out of the control volume. In addition to these there was a source term if you remember in the balance equation and therefore this source term has been added E dot source. In terms of an energy balance equation this source can be typically some sort of a chemical reaction happening within the control volume. So, the chemical reaction can be either exothermic which means that it will give out energy in the form of heat or it can be endothermic which means that it will actually suck energy in the form of heat. Additionally when we talk about a control volume analysis you should realize that there will be two more energy terms that needs to be incorporated. If you want you can treat the last two terms on the right hand side as sort of part of source term as well or they can be separately considered as energy in and energy out terms which are somewhat of special nature in the sense that they are decoupled from this E dot in and E dot out which are essentially the energy brought in and energy taken out by the fluid that is flowing in and out of the control volume. So, this Q dot here is essentially the heat transferred to the CV and that is why it has been chosen as a positive sign here and this could be because of a process such as a conduction heat transfer from the surroundings into the control volume or it could be even some sort of a radiation heat transfer from the surroundings into the control volume. One way or the other it is the energy transferred in the form of heat that is not because of the flow going in and going out of the control volume. So, it is essentially the heat transferred due to either conduction or radiation if possible. And the last term here is the work done on the control volume and the work done on the control volume will essentially come because there will be forces acting on the control volume surface and wherever there is an inlet and outlet flow there will be a velocity of the fluid involved and the forces multiplied by appropriate components of the velocities will give rise to these work terms. And again in this particular situation a positive value of work done on the control volume has been chosen. So, I am talking about work transferred into the control volume from its surrounding as being positive and that is why the word on is underlined here. Similarly the heat transferred to the CV is also underlined because we are talking about heat transfer being positive when the heat is transferred from outside into the CV. So, typically the energy balance as we say here is not really required for a fluid mechanics analysis unless there is an explicit heat transfer occurring or if you are dealing with a compressible flow type situation where there are some additional effects because of the compressibility of the fluid that we need the energy equation also included. If you are talking about a constant density or incompressible fluid flow per say many times if there is no external heat transfer we do not really need this energy equation at all you will probably realize that this form of the energy equation is required when we talk about convection heat transfer. So, those people who have undergone the heat transfer workshop earlier will definitely recognize this as an equation of interest when convection was discussed in that workshop. What I am going to do here is I will simply bring about the standard integral forms for all these terms and I will leave this equation as it is you know after bringing the integral forms. When we come to the differential equations of motion tomorrow we will revisit this energy equation in more detail. Right now just the way we had brought those integral forms out for the mass balance and the linear momentum balance what I will do is in the next slide I will outline simply the integral forms for each of these terms and then we will leave this at that for the time being. The total energy that I am talking about here which is capital E is expressed as the mass content times the specific total energy which is given by small e right at the bottom of the slide and this specific total energy will be composed of specific internal energy which is due to the random motion of molecules or atoms that you have for the fluid which are contained in the control volume. Addition to that you have a specific kinetic energy which exist because of the fluid velocity being non-zero and then specific potential energy because typically we are talking about all these interactions happening in a force field such as due to gravity. So, there is going to be a specific potential energy involved. So, the total specific energy then is the addition of these three terms and this capital E that we are talking about in the main balance equation is the sum of these three types of energies in general. And now to bring about the integral forms if you go back to our discussion yesterday for the mass balance and momentum balance we will follow exactly the same procedure and immediately write down the expressions for the accumulation term and the rate of energy in minus rate of energy out term. So, if you go back a couple of slides let me go back to let us say the momentum balance right here. So, the top two equations here which were written for the accumulation of linear momentum in the x direction within the control volume have been written as time derivative of the linear momentum associated with an elemental mass within the control volume. So, dm is the elemental mass within the control volume multiply that by the x direction velocity and that gives you the elemental x direction linear momentum and then you integrate that over the entire control volume and you get the linear momentum contained within the control volume. In a similar manner what we had seen was the linear momentum in the x direction coming in minus the linear momentum going out in the x direction again was equal to minus area integral rho times v dot n dA is essentially the mass flow rate and you multiply the mass flow rate by x direction velocity when it comes to an outlet area this v dot n will provide a positive sign when it comes to an inlet area this v dot n will provide a negative sign and then that negative and this negative will take care of each other. So, therefore, the expression for linear momentum in minus linear momentum out on a rate basis was something like this. So, now if you look at these two expressions I will analogously immediately generate the energy expressions for the accumulation term and the energy in minus energy out term. So, all that I am going to do there is instead of this u velocity I will simply replace this u by small e which is the specific total energy and that is about it. And let me go there and as you can see here the top equation which is the expression for the rate of accumulation of total energy within the control volume if you see I am simply having this specific total energy in here multiplying by the elemental mass. The elemental mass is further written as the density times the elemental volume which is exactly how it was written earlier and the e dot in minus e dot out which are the total energy in minus total energy out due to the fluid flow remember that fluid flow is going to carry some energy in and will take some energy out. In exactly the same manner as what we had written for a linear momentum will be minus area integral which is the area surrounding the control volume. The specific total energy multiplied by this mass flow rate term again v dot n will be positive for out flow areas v dot n will be negative for the inflow areas. So, that is all that I will really want to mention here and put it together as the energy balance equation. So, bringing this e dot in minus e dot out term on the left hand side I write this as the accumulation of total energy within the control volume plus net rate of out flow of energy total energy from the control volume. Remember that this plus integral over the control surface or the area surrounding the control volume small e times rho v dot n d a represents the net out flow of energy total energy from the control volume. The remaining 3 terms which were in the original equation that is the source term due to some sort of a chemical type reaction let us say if it exists and the heat transferred into the CV due to either conduction or radiation and the work done on the CV these are the terms I am leaving exactly the way they are without really doing any more analysis on them. And the boxed equation on the slide that is on the screen right now is essentially the total energy equation on an integral basis as it is applied to a control volume. Now in the next few slides that is this is slide number 21, 22 and 23 what I have actually done is I have taken this general form of the integral energy equation and I have obtained a simplified form assuming certain things and that certain things have been listed in the next few slides. At the moment I am not going to discuss this part the reason is because this part is actually usually discussed in thermodynamics. So, the simplified form that has been outlined in the next two slides is essentially derivation of what is called popularly in thermodynamics as steady flow one dimensional energy equation. And what I wanted to show was that you start with a general integral form such as what is outlined right now in the box here and then employ certain simplifying assumptions such as steady flow, uniform flow over the inlet and outlet etcetera. And you can systematically reduce this general form to what is called as this one dimensional steady energy equation from thermodynamics. So, what I suggest is that at the moment I am not going to discuss this in detail the reason is because it is not really relevant from our course point of view I just wanted it in here. So, that those who are familiar with thermodynamics can relate to it that you start from a general form and under the set of some simplifications you can systematically reduce the general form to the well known one dimensional steady flow energy equation of thermodynamics. Those who want to go ahead after the day today you can please read through the material on slides 22 and 23. It is fairly self-explanatory in the sense that if you simply follow the steps one by one and understand how the simplifications have been done in each of these terms in the box equation which is the general energy equation you will reach the final 1D steady form. So, what I will do is I will stop here right now as far as the integral analysis is concerned what we will do is we will revisit this energy equation tomorrow when we talk about the differential equations of motion which will include derivation of a differential energy equation as well. At that time we will revisit this equation and we will do more manipulations in order to convert this integral form to a differential form. So, the next couple of slides in this integral analysis are for self-study in some sense where you will see how the general form is reduced to the well known one dimensional steady flow energy equation. So, this is more or less the end of the integral analysis that I wanted to discuss. First to summarize again remember that even though our course is focused on computational fluid dynamics and from that point of view the integral analysis is not necessarily a requirement to be known. However, as I mentioned yesterday a couple of times there are some CFD books which follow especially the finite volume methodology which will utilize the integral forms of the equations of motion on a routinely basis. So, it is good to know what these integral forms are. So, that in case you are reading those books you should not be stuck as to what these forms are and how they have come about it is always good to know little more and that is the reason we wanted to do this integral analysis. The second part or second objective why we wanted to introduce this integral analysis is that as we saw yesterday by its own or on its own I should say it is a good analysis tool for several fluid mechanics problems. And also perhaps the third objective is that in order to obtain the governing differential equations we can utilize the integral forms and convert those integral forms into differential forms. In fact, tomorrow when we derive the differential forms we will in fact use these integral forms and convert them into differential forms. So, it is a very good idea to understand these integral balances and then be in a position to use them. So, with this I will end the discussion on integral analysis.