 If you look into the momentum equation I had already talked about rate of change of momentum is consist of two terms unsteady term when you talk of rate of change of momentum inside the control volume and advection term when we talked about momentum change across the control volume. So the rate of change of momentum is equals to force. Now when you talk of force there are two types of forces like if you consider yourself as a fluid control volume and what are the different types of fluid forces acting on the surface of your body. There are two type of forces which acts in your body one is consider yourself as a fluid control volume one is the pressure which is directly proportional to your surface area and one is the weight which is directly proportional to your volume if you consider yourself to be a fluid. So there are these are the different types of body forces and the surface forces I will not go into much detail of it as I have to quickly go to the CFD part. So this surface force and body force you can resolve into components I would here like to point out that the way we had done the derivation this is what is called as the conservative form of the Navier-Stokes equation and in finite volume method we use this form of the we do not take the mass flux outside the derivative and this way we ensure that the mass conservation occurs to a very large extent. When you talk of surface forces right now I am here showing you the stresses let me tell you I am showing you okay so rate of change of momentum is equals to force. Now this force here what I will show you mainly are the two types of forces two types of surface forces which are there in all the fluid flow problems which are like viscous forces and pressure forces. Gravitational force in many a problem it is found to be negligible there are other forces which comes like surface tension force if you have two phase flow if you have electric field magnetic field other forces comes into picture but the viscous force and the pressure force are there in all the problems. So here I will discuss only these two forces what happens if you see a surface of the control volume the force may act in any some arbitrary direction here I am talking about the viscous forces the force may act in any arbitrary direction but here what we do is that we resolve into components because how we are doing balance as you do in let us say engineering mechanics course you do a three body diagram the force may act in any arbitrary direction but you resolve into component and then only you balance in fluid mechanics also we do the same thing there it was solid here it is fluid control volume. So we resolve into x component and y component so this force is expressed in terms of fluxes as stresses. So these are vertical phases so the surface area is delta y these are horizontal phases the surface area is delta x stress is a second order tensor and the first subscript here represents the plane in which it is acting and the second subscript represents the direction in which it is acting. Again you this surface forces it is a flux multiplied by area when you divide by volume you can easily get a differential term. Okay so in conduction you get del by del x of q x del by del y of q y in mass conservation you get del by del x of m x del by del y of sorry del by del x of m x plus del by del y of m y here you are getting del by del x of sigma xx plus del by del x of sorry del by del y of sigma yx note that this is stress in the x direction the x plane this is the stress in the x direction the y plane this is normal stress this is so you get sigma term in conduction you got get del dot q term here you are getting del dot sigma term in mass conservation you get del dot m m mass flux is a vector and as I said there is a subsidiary law like Fourier law of heat conduction to convert q x into as a function of temperature there is an expression to calculate mass flux in terms of velocity here also there is an expression to express this stresses in terms of velocities in pressure which is that law Newton's law viscosity actually this derivation professor Puranic had discussed in the previous lecture where he has used flow kinematics and shown you that stress is directly proportional to strain rates and strain rates is expressed in terms of velocity gradients that way expression between the stresses in solid stress is directly proportional to strain but in fluid it is directly proportional to strain rate finally I will show you the expression this is the final expression for stress as a function of velocity gradients sigma xx is minus p plus 2 mu del u by del x sigma by minus p plus 2 mu del v by del y sigma x y is mu del v by del x plus del u by del y so if you substitute this stresses in terms of velocity gradients by using subsidiary law you get the gradients so note that in case of mass flux and in case of momentum flux or enthalpy flux you were having the value of the variable at the phase center in case of conduction heat flux and here in this case of viscous stresses you get gradient of in conduction it is gradient of temperature here it is gradient of velocity this whatever CFD I am teaching is mostly restricted to incompressible flow that is what as a mechanical engineer most of the time we encounter and most of you I think are mechanical engineers so I had limited my lecture to incompressible flow incompressible flow we can further simplify this equation let me do how we can let me show how I can simplify so let me do one thing I will take out of this two I will take one of the del u by del x is that clear and out of this two term I will take one of the term which term I will take I will take this term and then I will show you that if you do a balance it becomes 0 how I will in the next slide out of the two if I take one and do a balance and divide by volume so if I take one means mu del u by del x x plus del x minus mu del u by del x x into delta y and out of this two term I take this term del v by del x at y plus delta y del v by del x at y if I do this what happens you get a term like this this becomes del square u by del x square this becomes del square v by del x del y if you take del by del x common what you get del u by del x plus del v by del y so finally I can express the stresses as this so instead of two time two gradient terms or two times of a gradient I get only one gradient for incompressible. Now I would say that this equation is very much similar to Fourier Laffitt conduction heat flux acts in which direction on any surface normal gradient which is involved in the Fourier Laffitt conduction is in which direction if you talk of surface normal direction in conduction q x is del t by del x where q x x in vertical surfaces. So in conduction heat flux is expressed in terms of normal gradient this is which plane x plane here is this which plane y plane plane is x and gradient here is expressed this gradient is referring to you will see that this gradient is referring to plane and what is the direction of the plane normal so here again you will see that the viscous stresses are expressed in terms of normal gradient of u velocity for x momentum equation normal gradient of similarly I can do for y momentum also I can take one of this two and let us say del u by del y and I can show that it cancels down to 0 and finally I can show that the viscous stresses in the y direction are expressed in terms of normal gradient of v velocity conduction heat flux normal gradient of temperature viscous stresses in the x direction normal gradient of u velocity viscous stresses in the y direction is the function of normal gradient of v velocity. So if you are developing code you will not write separate subroutine for conduction flux viscous stresses you will write one subroutine because for all of them you have to obtain normal gradient so you need to realize and appreciate the commonality in the derivation that is why I am doing the derivation fluid mechanics I am not yet started CFD in true sense these are some of the things which you should know understand and appreciate as far as fluid mechanics is concerned. You need to understand that there are fluxes the fluxes are at the phases this fluxes are expressed in terms of certain dependent variables some of this dependent variables are expressed some of this fluxes are dependent in terms of value of the dependent variables at the phases like mass flux momentum flux they are expressed in terms of velocities value not gradient there are certain other fluxes like conduction heat flux viscous stresses which are expressed in terms of normal gradient. This realization later on help us to understand and appreciate the discretization finite volume method in a better way. In conduction this is the way you do derivation I am showing you in one slide first you get this I called as first part of the derivation you are calculating total heat gain by conduction per unit volume this I called as the first stage of derivation then when you apply Fourier law of heat conduction which I show in the next slide this is the Fourier law of heat conduction then you can finally get the double derivative but here I again point out very specifically my slides are little detailed because I am not giving you lecture notes but I had written the important points wherever possible. So I have emphasized this normal gradient normal gradient of view velocity gives viscous stresses in the x direction normal gradient of view velocity gives viscous stress in y direction and normal gradient of temperature gives conduction heat fluxes. This diffusion again I can write all this viscous stresses and conduction heat fluxes this are basically what is called as diffusion phenomena. If you go to the transport mechanism they are basically occurring from random motion of the molecules. So physically the mechanism which gives rise to conduction heat transfer or the viscous stresses are due to random motion of the molecules. So this are called as diffusion term so this D is general diffusion term and this gamma phi is mu for dynamic viscosity for fluid flow and minus k for heat transfer problem and this eta is the surface normal. I am not showing you the normal direction the direction here why I am not showing you the direction here the reason is because this control volume I want to use for conduction when I use for conduction which direction the arrow points out surface normal q x q x plus d x q y q y plus they are always surface normal. When I want to use this control volume for viscous forces in the x direction then what happens on two surface it is normal vertical surface it is normal on horizontal surface it is tangential. So that is why because in a general proposition I am not showing you the arrows but the expression remains same. So the diffusion flux is viscous stresses this is viscous stress in x direction this is the viscous stress in the y direction note that here I had written two normal two shear on two surface it is normal on two surface it is shear and the conduction it is always normal conduction it fluxes are always normal here again I am not showing the arrow. So the total diffusion term if you do a balance it can be expressed as gamma phi del square phi this is k del square t in case of conduction in case of fluid flow this becomes mu del square u for x momentum for y momentum it becomes mu del square v. So that was all about although diffusion mechanism give rise not only to the viscous stresses but also causes pressure I am showing pressure separately for clarity. Now we talk about conduction heat transfer viscous forces and we talked about direction. Now let us talk about pressure where pressure act on a surface in which direction it acts non. So if I want to obtain pressure force acting in each direction on which of this surface I should draw pressure on the surface where the pressure acts in the horizontal direction because that will contribute the pressure force acting in the x direction. So I will only show it in the vertical phases I am not showing in the horizontal phase it is acting but just for derivation for clarity I am showing it only in the x because here I want my purpose is only to show force acting in the x direction again what is the flux term here what is small f here pressure and when I divide by volume note that here I get minus del p by del x because pressure is compressive and when I want to obtain net pressure force in the positive x direction it becomes p x minus p x plus del x become minus del p by del x this is the reason pressure you get as minus del p by del x. In fact in fluid mechanics class one of you were asking how it becomes minus del p by del x this is the derivation for that. Similarly if you this can be written in a vector form del p by del x as del p dot i where i is a unit vector in the x direction this is the pressure force in the y direction. So this completes the derivation this is the rate of change of x momentum per unit volume this is the rate of x momentum of the fluid inside the control volume note that word inside this is the rate of change of momentum flowing across the control volume rate of change of x momentum of the fluid flowing across the control volume this is due to Eulerian approach this is a temporal and this is an advective component this is equals to viscous force in x direction. Note that all the forces are in x direction viscous force in the x I am using subscript v here here it is not necessary because this does not consist contains any pressure term the normal stresses consist of pressure terms that is why I have used a subscript v to demarcate that this is viscous component. So this viscous this is the diffusion term this is the advection term this is the non-study term advection term diffusion term and then whatever remains in C of D language we call that as source term here source term is pressure gradient this is the x momentum equation similarly question is this is for incompressible or general right now it is for incompressible because I am using I had already used continuity equation as del u by del x plus del v by del y is equals to 0 when I was showing that when you balance the stresses out of the two term one cancels down. So there I had used an approximation of an incompressible flow so that way this is for incompressible this is for one momentum un-study advection diffusion and the source term similarly this is the energy equation rate of change of internal energy of the fluid inside the control volume rate of change of enthalpy of the fluid across the control volume total heat gain by conduction plus total heat gain by volumetric heat generation. There is a mistake here I am saying mathematically it does not come here it is a dot dot dot del dot is asking what is this there is a dot here so with this I come to the end of not this lecture session but the derivation because I have to complete this topic we will have 25 minutes before the T session but may be I will pause if you have some question I will be happy to answer till now I had shown you the derivation with the approach different from what force of pranic had shown you had shown you a mathematical approach I have shown you from control volume how to obtain why I am showing is that this understanding will help you to understand and absorb later on finite volume method which I will be discussing any question okay so final equation is this for incompressibility continuity x momentum y momentum as I said there is lot of commonality you can give to the student fill in the box type of problem this is what I commonly give I draw some boxes and tell them write down for x momentum y momentum equation there are two types of boxes one small box second a bigger box what is coming in the small box advected and diffused variable u velocity is coming in small box in x momentum v velocity is coming in small box and what is coming in big box minus del p by del x this small box finally when we talk of generality I will denote it as 5 and this big box I will denote as a source term s of y for energy equation actually this small box is not only temperature it is specific it multiplied by temperature and the big box is volumetric heat generation so this sorry so this is your final unsteady term advection term diffusion term and a source term from here real CFD start because from fluid mechanics we first propose that there is lot of commonality between momentum and energy transport we call that is the transport equation and once we realize that commonality then you develop one subroutine for this one subroutine for this and then use this for momentum as well as energy equation then this energy equation yeah I guess you assume that k is a constant yes yes his question is here I have assumed that k is constant in most of the cases indeed this is true however there are cases where heat transfer become non-linear when I will be taking conduction theory as well as lab we have different separate lab session for that I will discuss it separately but here this is the first course in CFD so that is why I am trying to make it simple as well as most of the commonly encountered fluid for heat transfer where k is constant more complicated situations we can take it later on we can discuss it separately okay so as I said that this is the just what I explained this is just like a notes you can see later on but before I end in CFD you should also understand and appreciate not only the different terms but the fluxes which is encountered well again emphasize in different slides what are the fluxes which you encounter because this is what I call as small f okay because this comes at the phase centers and then we need to use approximation in CFD so what is the flux which comes when you apply law of conservation of mass mass flux what is mass flux made up of normal velocity and the surface area and where this mass flux comes into the expression it not only comes in the continuity equation this mass flux also comes in the x momentum y momentum and energy equation where does it come because it acts like a driver where it acts in the advection term okay so note that this not only comes in the mass conservation but it also comes in the x momentum y momentum and energy equation to advect u velocity v velocity and temperature respectively the second flux in general is what I call as advection flux and what is this advection flux advection flux is made up of two variables one is called as advecting variable which is like a driver I said only mass flux and the second is advected variable which are velocities and temperature depending upon the equation so mass flux advection flux and then the third flux is diffusion flux okay so this fluxes comes at the phases and we need to use approximation to calculate this some of this fluxes are expressed in terms of value and some of them are expressed in terms of gradients so we need appropriate approximation whenever we do a CFD study especially as an let us say you are in academic institute and if you do a research or let us suppose you are doing a PhD normally it is suggested that you should follow standardize procedure is that you do a non-dimensional study okay later on when we will be giving you codes in the lab session we had made it non-dimensional we will be doing problem not only for isothermal flow but force convection mix convection natural convection so non-dimensionalization is an important concept in fluid mechanics so we do non-dimensionalize because in any problem we have certain non-dimensional variable like for flow in a pipe diameter of the pipe is taken as the non-dimensional length scale so when you do non-dimensionalization finally this is the dimensional form of the equation when you use a characteristic length scale and the characteristic velocity scale and you non-dimensional temperature for constant wall temperature in this manner and for constant heat flux in this manner finally you get a non-dimensional equation continuity x momentum and energy equation this I have written in a vector form I would like to mention that if it is a force convection problem then you do not have temperature in the momentum equation but if it is a mixed or natural convection problem you have a source term which is made up of temperature so in force convection there is a one-way coupling the flow does not depend upon temperature but in mixed and natural there is a two-way coupling in case of force convection you only have Reynolds number but in case of mixed convection you have Grasschef number in case of natural convection you have Rayleigh number as a non-dimensional governing parameter when you talk of properties there are two types of properties one is called as the fluid property and the second is called as the flow property fluid property is basically thermo physical property and the flow properties are the properties by which let us suppose you create movie in CFD which are velocities pressures temperature note that not only the movie of velocity pressures and temperature are good enough to understand fluid mechanics because fluid mechanics is very complicated phenomena even if you watch a movie of fluid mechanics you cannot understand it completely you need to create different types of movies you need to be clever enough to create different types of movies so there is vorticity plots also vorticity contours which we draw to understand the flow in a better manner there are different types of flow analysis tools which we use in CFD for to get a complete and although most of the problem let me tell you the flow are so complex that with certainty we find difficult to say that this is the reason for happening of something most of the time we feel because it is a highly very complicated so most of the time we need to write in research paper that probably this is the reason we make a statement that this is the reason reviewer mail reviewer will ask you that how you can most of the time although you may show some pictures and movies for confirmation but many times it is not confirmatory in a sense you need to be very conservative in your statement so when I talk of from the properties let us go to the parameters when we talk of parameters there are two types of parameters one is the governing parameter and second is what we call as the engineering parameter what is governing parameter when you do a non-dimensional study your results change if you change the governing parameter okay in CFD why if you look into the fluid mechanics problem why if you how the results change results change if you look into flow in a pipe or let us say flow across a car why the results are changing because the domains are different the shape of the domain is also different okay in a non-dimensional form when you do a non-dimensionalization these are the common governing parameter what I mean is that like for flow across a aeroplane you can study for when you vary the Reynolds number the results change this has the governing parameter these are the parameters which you vary which you play with and see the results so this is like an input parameter these are user input parameter and this is an output parameter okay this is an input parameter you vary this and then this results change this is called as engineering parameter non-dimensional form this is non-dimensional form of wall shear stress this is non-dimensional form of pressure drop this is non-dimensional form of drag force this is non-dimensional form of lift force this is non-dimensional form of frequency of unsteady flow this is non-dimensional form for heat transfer. Now, after non-dimensional the next thing would I would like to discuss is the initial and the boundary condition. Mathematicians have classified the boundary conditions into three types that is what I will discuss here. If you this is what type of equation and two dimension and steady state heat conduction with polymetric heat generation. So, there is one derivative with respect to time. So, you need one condition two derivatives in next direction two derivatives in y direction. So, two boundary conditions in x direction two boundary conditions in y direction. This is our governing equation in Cartesian coordinate system. So, we have to take a problem in Cartesian coordinate system. So, let us suppose there is a plate which is taken from a furnace. Let us say the plate is very long in the third direction. So, we are using it as a dimension. The length of the plate is l 1 in x direction l 2 in x direction. Now, to talk about the boundary conditions the initial condition will be let me take a general in the heat transfer we have three types of boundary condition. Let us say constant wall temperature insulated constant heat flux and convective boundary condition. Now, mathematicians have classified initial condition is let us suppose you took this plate from a furnace at temperature T 0 and then this is subjected to this boundary condition. Mathematicians have classified the boundary conditions into three types. One called as Dirichlet where the temperature is prescribed. Second Neumann where the gradient is prescribed. Third Robin or mixed where you have a linear combination of the temperature and the gradient. So, which one is Dirichlet of this four wall temperature boundary condition? Left wall is Dirichlet temperature specific. Which one is Neumann? Gradient is prescribed. Gradient is normal gradient of temperature is 0 here dT by dy is 0 and dT by dx is minus qw by k. So, these two are Neumann and at the top wall when you apply q conduction is used to q convection you get a linear combination of the temperature and its normal gradient. This is called as Robin or mixed type of boundary condition. Now, that was for a conduction problem. Now, let I had I show you one more boundary condition or initial conditions for external fluid mechanics problem. Let us suppose you have a cylinder. Cylinder is a common shape which you which we teach in let us say fluid mechanics or heat transfer. This is just for simplicity we take, but this can be an aeroplane or a car also. So, let us suppose we want to study free stream flow across a cylinder. So, you have you need one boundary inlet. As you know in fluid mechanics there are two types of problem. In general we classify out of two types external flow and internal flow. What is the size of the domain in case of external flow? Theoretically when you call external flow what do you mean by the size of the domain? Infinite domain size. But when you do simulation or when you want to create a pictures or movie of fluid mechanics you need some fixed region. The first question is what should be the where should this left wall should come? What should be the distance of this left wall of the boundary from the cylinder? This is very important question. So, you have to decide the position of the domain. Where should be? So, this is called as inlet. Why this is called as inlet? Because the flow is coming from this side. Why this is called as outlet? The flow is coming out from this side. This I call as inlet boundary. This I call as outlet boundary. This is bottom boundary. This is stop boundary. In CFD many times people use software like flow across a car and they take this boundary very close to the surface of the car. Why? Because they feel that most of the action happens near to the surface of the car. Let me take the boundary very close so that the number of grid points are less and I get quick solution. But is it correct? Because you should realize that the boundary conditions which we are using like this outlet boundary conditions which we are using. What is this boundary condition means? del u by del x equals to 0 means what? Flow has become fully developed. If you take this boundary very close just behind the surface of the car let us say whether the flow has become fully developed. But you may ask that how can I know because if the car is going with different velocity this distance may be different. Yes, you are right. What you have to do? That is why I say that go back to that Einstein statement. You have to show that u2 cull and you place this boundary here one simulation. Place this boundary here second simulation. Place this boundary further downstream third simulation. And if you show that this middle one as compared to first there is lot of difference in the when you see a result there are two types of result. Qualitative result like velocity pressure distribution. Quantitative results like in this problem lift force and drag force. Let us suppose the result changes a lot between this location and this location as compared to the downstream. Then this is what is called as domain length independent study or domain size independent study. So, you have to take this rectangular region in such a way that with further increase in the size of this rectangular region there is negligible change in the qualitative as well as quantitative results. On top and bottom wall although you may feel that on the top and bottom wall it should be u is equals to infinity v is equals to 0. But this boundary condition is better because even if your height is small this can give you an accurate result. This is called as a free slip boundary. Free slip is normal velocity is equals to 0 normal gradient of temperature velocity is equals to 0. This is a horizontal wall. This is a normal gradient of tangential velocity. Now, when I talked about the derivations the Navier-Stokes equation in a general form the non-dimensional flow properties fluid properties. Sir, excuse me sir. Sir, I got a doubt in initial and boundary conditions. Yeah. When I teach this. Thanks for correcting me. When I teach computational fluid dynamics to my students. Yeah. Place it on. If you talk about boundary condition initial condition you take a heat transfer conduction equation from a mathematical point of view I am able to tell for example if it is one dimensional unsteady. Please use my. Yeah. If it is one dimensional unsteady from mathematical governing equation I am able to tell that I need two boundary conditions one initial condition. Yeah. Like that Navier-Stokes equation will be able to generalize. Your question is. Boundary conditions for Navier-Stokes equation. Yeah. Yeah. From a mathematical point of view. We can generalize. Yeah. If you throw some light on that it will be better sir. Okay. Okay. When it at appropriate slides I will highlight that. Okay. Thank you sir. His question is basically I showed the boundary condition for temperature but the same thing can be done for Navier-Stokes equation also. Okay. Now, before I end this I would like to mention that once you get the results software what they do is that they call as post-processing because this results first you obtain in terms because first you obtain in terms of velocity, pressure, and temperature but if you show this movies let us say movie of flow or temperature to let us say industry they may not be that much interested. What they may be interested let us say car industry may be interested that how much petrol needs to be spent on let us say drag port which is exerted so that the car has to propel that drag flow. So they need parameters which we call as engineering parameters. Those engineering parameters are calculated using certain integral equations like drag force is calculated by taking the cyclic integral. There is a mistake here this is a cyclic integral. Cyclic integral if you take at the surface this is the surface normal this is the stress tensor and this is dot i. What is stress? This sigma is a second order stress tensor the dot product of a vector and a tensor is a vector and that vector if you take the ith component it gives you drag force and if you take the jth component it gives you lift force. So we do the surface integral and this is for total heat run. This all these integrals are done at the surface. You can also obtain this using an integral approach where you take a much larger size of the domain but here I am showing you surface integrals. Like if you have an inclined plate whose normal unit vector is expressed in terms of ai plus vj and this is a second order stress tensor. If you take the product of 2 you get a vector and if you take it is ith component you get the drag force and if you take the jth component lift force and this is the conduction if you take a flat plate and if the flow is on the both the sides then you can obtain the viscous stresses shear normal stresses and then finally calculate the drag force and lift force like this. I am not going into the detail actually the expression previous slide you can work it out. This is quite straightforward. So with this I have come to the end of the first lecture.