 Good morning, I welcome all of you to the workshop on computational fluid dynamics and heat transfer. In the yesterday's lecture I had talked about the computational heat reduction and then moved on to computational heat convection. And till now towards the end of the yesterday's lecture I had started a topic on finite volume method. So, I started with computational heat conduction then moved on to computational heat advection and then computational heat convection. So, note the word heat in all these topics. So, I had taken an examples of diffusion phenomena advection phenomena and the combination of the two convection taking an example of heat transfer. I would emphasize this yesterday also that in real world we have a pure diffusion phenomena which is a heat conduction. But in real world we do not have a pure diffusion phenomena as far as momentum transport is concerned because pure diffusion phenomena in momentum transport corresponds to the sum of the viscous forces acting in a fluid flow situation is equals to 0. And then I took pure advection situation in heat transfer although in heat transfer also it is in hypothetical situation where the flow rate is very large. But it is not if you take an example of a momentum transport pure advection phenomena it is difficult to get a feel of what is happening. Because I could take an example of if you are standing between ice and fire only for case of heat transfer where you have a better feel of this phenomena. But when I started the finite volume method yesterday there the objective is to show you that whatever approximations or whatever finite volume method which we have used for conduction advection and the combination of the two convection are indeed applicable for fluid flow phenomena momentum transport. So we did in earlier topics we basically took energy equation now we are taking momentum equations which is also transport equation and there is quite an analogy and that is what is shown through this topic of finite volume method. Towards the end of today's lecture I will initiate a new topic which is on solution of Navier-Stokes equation on a staggered grid. Let us start with today's lecture with an animation which gives the overall philosophy with which we are of the finite of the finite volume method which is being discussed here. So I will show you an animation okay. So in our you know what are the conservation laws in fluid mechanics in heat transfer law of conservation of mass, momentum and energy and in fluid mechanics as well as heat transfer from under graduate days the way we do analysis is that we apply this conservation law not in a Lagrangian fashion but we apply for Ilarian control volumes. So when you take an Ilarian you follow an Ilarian approach to the and apply the conservation laws then this is what is done in undergraduate heat transfer and fluid mechanics course that we apply this conservation law and divide by the volume of the control volume then we get differential terms and we get a differential equation which is called as the governing partial differential equation. Now this governing partial differential equation the problem is that this governing partial differential equation although it is applicable for most of the problems in fluid mechanics and heat transfer although for more complex situations certain additional terms need to be incorporated in this partial differential equation but for different problems you get different results because your shape and size of the domain and the boundary conditions dictate the results. Now this differential equation I would say that if you can obtain analytical solution of this differential equation for all the fluid mechanics in heat transfer problem it is like a video camera which has an infinite time and special resolution okay what I am trying to say is that if we are able to obtain the analytical solution what is analytical solution of a differential equation functional relationship and so in this case the analytical solution is velocities pressures and temperature as a function of special coordinates and temporal coordinates. So you can create a picture for any values of x y and z so it is like a picture with infinite resolution secondly you can create a movie because in this case it is a function of x y z as well as time and you can put any time step so you can create a movie of which has an infinite frame rate. So this is the wish list and this is one of the most important unresolved problems in science and engineering it is as big as problem that if you are able to obtain analytical solution for a general fluid flow problem that all this CFD software in fact this subject has to close down okay this is a as big as a problem like this. So this is what we know from our undergraduate days how to derive the partial differential equation. Now what is done is most of the CFD book is that the differential equation is conservation law per unit volume we take the volume integral of that differential equation which is per unit volume so basically on one side we had divided by volume in case while deriving the equation and on the side when we want to get algebraic equation using finite volume method now we are integrating it so once we are dividing or you can say differentiating and then we are doing then we are doing integration to get an algebraic equation which is basically coming from the conservation law so rather than following this path the finite volume method which is discussed in most of the CFD books this is what is being done in fact in almost all the CFD books however I had realized by teaching this course to various not only IITs I had taught to various places also I go for invited lectures in this subject so what I had realized is that when I follow this approach which is given in most of the CFD books I had found difficult to connect to the audience especially when you give CFD lectures to different colleges in our country where the level of mathematics specially for an undergraduate course is little low then there is a difficulty among the audience to get a feel and understanding of what is this divergence theorem and another problem with this approach is that finally you get an algebraic equation so when you are getting algebraic equation then you have coefficient so the numerical procedure talks in terms of the coefficients then the connection of the fluid mechanics and heat transfer is broken to the numerics so the approach so this has motivated me to come up with an approach which is easy to feel and connect to so this is the approach which I had been using since last maybe 3-4 years where the idea is let us start directly from a control volume apply the conservation law apply the approximations which is in fact the approximations which are being used here are nothing different than what is given in most of the CFD books or what is being mostly followed even if you start from a partial differential equation and do a volumetric the same approximations are used so as we are here using same approximation it is not surprising the that we end up with same algebraic equation so whether you start from a differential equation or whether you start directly from a control volume as we are using same approximation we end up with same algebraic equation but it is just that in this approach which I called as physics based approach and the traditional method as mathematics based approach I found that students get a better feel it is not only in the formulation I try to make it physics based even during the programming level or implementation details I try that even when you are programming I define terms like heat flux in the x direction heat flux in the y direction so rather than working coefficient based of an algebraic equations it is more of a physics based so programming also go hand in hand and it is more of a physics because when you are programming the different terms which you are having represents physical quantities I have you can understand and appreciate the objective okay now in the previous lecture there is a second animation which I would like to show the this is the conservation law the for momentum and energy equation so the x momentum equation the left hand side is the rate of change of term now that rate of change there are two terms as we are following an Eulerian approach rate of change of x momentum inside the control volume plus across the control volume and this rate of change as you know from Newton's second law of motion is equals to force when you take a fluid control volume most of the time most commonly encountered forces are the viscous forces pressure forces and the body forces this is the x momentum similarly we have a y momentum which is also a transport equation and energy equation which is also transport equation and this are the different terms unsteady term unsteady term advection term diffusion term and whatever remaining is a source term okay so this is so in this topic of finite volume method let us come back to the slide so in this finite volume method which is being discussed here we are using finite volume method from the conservation law directly applied to the control volume these are some of the things which are taught in the last lecture I will not go into it the last lecture before I finish the lecture I had shown you that discretization of law of conservation of mass this is different from diffusion advection and convection so this discretization is something new for as far as what has been discussed in the earlier topics so anyway I had shown you this description in the last lecture and here again I had mentioned that this right now I am showing you the mass conservation law for incompressible flow where the rate of change of mass although it has two component rate of change of mass inside the control volume plus across the control volume what I am showing you here is the rate of change of mass across the control volume which is basically taking into consideration the mass fluxes which is mass flow rate per unit area on the different surface of the control volume and I had shown you that the two level approximations are quite similar to what has been shown earlier because mass flux is mass flux is also a flux so the approximation involved is similar to what we had done for conduction flux and thalpy flux so the first level approximation is as the flux varies on a surface area so we are taking it by one point representation which is a good approximation as the surface area goes on reducing and anyway in our CFD computations we take the size of the control volume very small but when we say very small that small enough word depends upon what is the action of fluid flow and heat transfer in a particular problem each problem has its own story each problem has its own action so yesterday there are many question that how do we decide the to give you a simple example let us suppose if you want to capture explosion if you want to create let us say a movie of how explosion occurs and if you want to capture each in every detail you need a video camera which has very high temporal resolutions okay and on another hand if there is some phenomena which is slow you can have a camera which is a usual camera with a normal frame rate similarly in fluid mechanics just imagine that you have different so that is the bottom calling is action there are certain phenomena which are very changing rapidly in time there are certain fluid flow phenomena which are not that rapid so the grid so the answer to the question your question that what should be the grid size we should take that depends upon our that is guided by fluid mechanics and heat transfer that is guided by how much action is occurring in a particular fluid flow problem or a heat transfer problem okay so this is a philosophical understanding of how we decide the grid size so in a particular problem a priori many a times you do not know how much action it is so you take a grid size you do a simulation and then you do a second simulation with a larger number of grid points and then when you compare the results between the two grid size if there is more action maybe probably your when you do on a larger grid point the things have changed as compared to the first simulation so this way you compare the result and this is what is called as grid independence so the first level of approximation is surface averaging and the second level of approximation is what we had done earlier in conduction and advection is basically the way we express the flux like heat flux it is a function of gradient of temperature the enthalpy flux it is a function of value of temperature now these fluxes are at the face center so this value or gradient comes at the face center and the second level of approximation is basically the discrete representation of the value which we had done what we called as advection scheme and gradient the discrete representation of normal gradient of temperature in case of conduction now here the flux is not conduction flux or enthalpy flux the flux is mass flux now the mass flux is expressed in terms of value of the velocity at the face center so the second level of approximation is the discrete representation of the value of the normal velocity in terms of the neighboring cell center right now I am showing you by using linear interpolation assuming that the face center is lying exactly in between the node capital P and capital E but later on the next lecture on staggered grid I would show you that if you use this interpolation it causes a wavy velocity field wavy velocity field which is obeying the continuity equation so people initially if you look into the history of development of computational fluid dynamics initially people were struggling in getting the solution from the new stoke equation and there was alternative approach which was quite popular in the early days of CFD which is called a stream function intensity formulation where you do not solve specifically mass conservation equation and you eliminate the pressure gradient term because later on I will show you that this linear interpolation you not only do for velocities but you need to also do for pressure and this is what creates lot of problem and I will come back to more issues more reasons why people were struggling in the next lecture now so with this two approximation we get a discrete or algebraic representation of the mass conservation law okay this is where I had stopped in the last lecture finite volume method for the so now we are moving on to we started with the energy equation the length such as conduction and the convection now we are moving we are you can say increasing the scope we started from energy now we are including the momentum also along with energy and what we are calling it as a transport equation so we are basically we now will discuss momentum along with the energy equation the discretization by physics based finite volume method so the way we handle the unsteady term is similar to what has been shown but it is just that instead of temperature I am showing you a general variable phi so earlier I had shown you that the first for unsteady term the basic issue is that the in conduction I had told you that the rate of change of temperature in a volume control volume varies point to point so we have but in case of let us say x momentum the u velocity temporal gradient the rate of change of u velocity point to point varies so we need an average value in a volume so you we do what we call as volume averaging and the second level of discretization is discrete representation of that rate of change of temperature for energy rate of change of u velocity for x momentum rate of change of v velocity for y momentum equation this is the final discrete or algebraic representation of the unsteady term now from unsteady let us go to the second term which represents rate of change of momentum or enthalpy across the control volume note that unsteady term was representing inside the control here now it is across the control now in the advection term the first level of approximation is again I am showing you this first level is in fact the same basically there are two situations which occur if it is a flux term then we have to do surface averaging and earlier I had shown this for conduction flux mass flux and here I am showing you for x momentum flux also and y momentum flux so the first level of approximation is this surface averaging of advection flux and what is advection flux or what is advection is shown in this figure calling it as a in the earlier slide I had shown a more in a more generic way and in the coming next three slide I will be showing pictorially specifically corresponding to x momentum y momentum and energy equation so if you take x momentum conservation now the way we have put actually the way we do in undergraduate courses that we write this as at x this as at x plus delta x this at y and this at y plus delta y and what is this this is mass flow rate per unit area multiplied by u velocity what is this this is x momentum flux so this is just x momentum flux which is expressed as small a here called as advection flux so this advection flux here which is x momentum flux multiplied by surface area gives you the rate at which x momentum is entering into the control volume from this phase here from south phase note that here in this case the mass flux which is a driver the mass flux always consists of normal velocity so on this phase the normal velocity is v so there is a u velocity and it is being carried away by the v velocity so that is why I had put the arrows like this okay so because many times you find it difficult to understand and appreciate that how this transport is occurring so what I am showing you here is that there is a u velocity due to the flow and there is some mass flow on this phase there is a v velocity and that v velocity carries this u velocity inside the control volume and when it this v velocity carries this u velocity inside the control volume there is a x momentum transport from the bottom phase similarly from the top phase and this is the x momentum which is leaving from the right phase note that the arrows between use in derivation of partial differential equation the same direction is used here so on the negative phases it is inflow on the positive phases it is out so this when you do a balance finally what you get is the rate of change of x momentum of the fluid across the control volume I would like to mention that why this is a first level of approximation because this values mass flux velocity is this vary from point to point on the surface area we are expressing by a one point value which is west note that in this control volume there is a small red line that small red line here is representing the centroid of the surface area denoted by small letters small e small w small n and small s okay so that way we are doing surface averaging and that is what is first level of approximation for momentum flux here same thing we do for maybe if I can show you through an animation probably you get a better feel I will show you the same thing in animation in this window so this is the x momentum transport in the x direction from the positive and negative phases this is the y momentum transport in the vertical direction this is the sorry everything here is the x momentum transport here this is an y momentum transport in the x direction now here what I am showing you is that now on the vertical phases now the v velocity now v velocity is like a passenger and what is the driver the mass flow rate the mass flow rate consists of a normal velocity u so the v velocity is carried into the control volume by the driver which is the u velocity which is the normal velocity on the surface area that is why the arrow here is shown like this this is the way y momentum is transported from all the surfaces of the control fortunately temperature does not have any direction so the direction of arrow is not here zigzag it straight arrows however there is a constant which is coming into this expressions which is specific it so this represents the first level of approximation for advection term of energy equation which finally represents rate of change of enthalpy of fluid across the control volume so when you balance I am shown here all the three slides as a single figure in a generic form in all the three slides earlier three slide there was a mass flux this constant was CP in energy equation and one for momentum equation so I had given a general variable what is this phi this phi is a passenger or transported variable which is u velocity in x momentum v velocity in y momentum and temperature in energy equation so this is a generic representation because if there is certain commonality generality we develop program test of a program and then convert it into a module so we develop advection module and this is used three times it is used in x momentum equation so we develop a subroutine for this advection and in that subroutine when we want to calculate the total rate of change of x momentum across the control volume we send to that subroutine that phi in that subroutine is u velocity c is 1 okay once you send this two information this is a generalized subroutine so it follows a procedure and give us the total rate of change of x momentum across flowing across the control volume for y momentum phi is v c is 1 for energy equation phi is temperature c is equals to CP okay so this slide shows the general generic representation where phi is the advected variable and the product of the mass flux c and phi f is the advection flux and this is the total balance we do as this is an outflow minus inflow if you if you take an energy equation this is the total enthalpy which is leaving this is the rate at which enthalpy is entering so that is out flux minus influx so this is total heat lost by advection in energy sorry yeah total heat lost by total enthalpy or heat energy lost due to advection phenomena for x momentum this is the total rate at which x momentum is lost when the fluid flow across the control volume similarly for y momentum and this is what I am showing separately many times you will see that I will show the same slide but actually these are the important concepts many times repetition I know that when you write a research paper or when you teach you want to avoid repetition but these are concepts which take time to for you to digest so that is why expecting that most of as far as the CFD development is concerned formulation is concerned expecting that the most of the audience is looking at not gone through it in more detail so I show this things repeatedly this is just to emphasize the concept so advection flux as I said is a generic form and this are the specific for x y momentum and energy equation now so that was the first level approximation now the second level approximation is basically advection scheme which we call as advection scheme earlier in the my earlier lecture I had shown you what is advection scheme I had given you how we derive the expression for advection scheme so this is basically an extrapolation or interpolation in procedure I had discussed that this expressions which we derive is guided by the fluid mechanics and heat transfer principles I had mentioned that in this mathematical procedure we need to take into account the direction of mass flux and based because based on the direction we come to know which is the upstream neighbor which is the downstream neighbor which is the upstream of upstream neighbor and based on that the expressions are derived for advection scheme so note that everything we do in any mathematical procedure we adopt in CFD is guided by the fluid mechanics and heat transfer principle this slide I had shown you earlier also so it is basically just to show what is the upstream neighbor and downstream neighbor connecting to the example which I had discussed earlier that ice and fire so if the flow is in this direction flow is from the ice side and if the it is the first order of unit then I had mentioned that the phase center value of the temperature now here this is equally applicable for u velocity in x momentum equation v velocity in the y momentum and these are the different advection scheme which I had discussed earlier constant extrapolation first order of one scheme involving only one upstream neighbor linear interpolation which is a central difference scheme involving one upstream and one downstream neighbor second order of wind scheme which is a linear extrapolation involving two upstream neighbors quick scheme which involves one downstream and two upstream neighbors in this figure I am showing expressions for one case when the flow is in the positive x direction there will be a similar equation for flowing the negative x direction more over I am showing you this on one of the phase center east phase but in a 2D control volume there are four phase center so this thing has to be applied on all the four phase centers that is west phase center north phase center and south phase however a phase is common to two control volumes so because this east phase becomes west phase for this control volume so if you do if you do a proper tagging and in your implementation if you are good enough you can avoid double computation and you compute only on the east phase and north phase for this control volume when you go to this control volume you calculate on the east phase and north phase because this east is already being calculated from here that way you can avoid that is why I had shown you that those green squares and red inverted triangle just to emphasize that how you can avoid double computation at the phase centers and actually at the phase centers you have to do many things mass fluxes enthalpy fluxes momentum fluxes so conduction fluxes viscous stresses so if you do double computation and there are many fluxes you unnecessarily waste time and here I am showing you when you do a balance here I am again showing you off flow minus inflow you get total rate at which net momentum across the control volume for x direction and y direction and this is the rate at which net and internal energy and enthalpy across the control volume is lost due to advection heat transfer this is a total balance okay so we started with unsteady moved on to advection then we are going to diffusion or diffusion discretization we had already discussed for conduction heat transfer the new thing here is that we will show I will show you that this is indeed applicable for the same approximations the same procedure the same finite volume method with a difference in the instead of temperature here you will have u velocity for x momentum v velocity for y momentum but the expressions are quite similar instead of conductivity here you will have a dynamic viscosity so first level approximation is same what with what I discussed for conduction fluxes that we do surface averaging this is there for any flux to the first level approximation is surface averaging of diffusion flux to calculate the total diffusion and the second level so I am here showing you what is diffusion flux for x momentum equation the diffusion flux for x momentum equation is the viscous forces in the x direction note that pressure I will take separately here I am showing you only viscous forces and I had shown you in my first lecture that for an incompressible flow although the generic expression for sigma x is 2 mu del u by del x the general expression for sigma y axis mu del u by del y plus del v by del x but for incompressible flow I had shown you in my previous lecture a topic number 1 you can refer to the slides that we can simplify and instead of 2 mu del u by del x if we take 1 mu del u by del x and instead of 1 mu del u by del x if you take from here and you take mu del v by del x then by applying continuity equation it can be shown that it becomes 0 so that way we simplify the expression of viscous stress in the x direction is a function of velocity gradient and the purpose of this simplification is to demonstrate that the diffusion flux is expressed in terms of normal gradient of a diffused variable now that diffused variable is u velocity for x momentum v velocity for y momentum and temperature for energy equation now that diffusion coefficient because this viscous viscous stresses or conduction fluxes are directly proportional to the gradient but there is a constant of proportionality and that constant of proportionality varies from fluid to fluid material to material and what is that constant of proportionality dynamic viscosity for fluid flow and kinetic viscosity for conduction internal so if you take copper versus let us say iron or if you take 2 fluid let us say water versus air or oil you can see that the diffusion phenomena will be different so with this I am showing you that this diffusion fluxes which is a viscous stress in the x direction are expressed at the centroid of the surface area that is what is first level of approximation this is same thing is shown for viscous stresses in the y direction note that in the previous slide this wherever now there is a u in the previous slide it was sorry in the previous slide this was u now here is v and you can see that where this sigma x y is acting the sigma x y is acting on the vertical surfaces what is the normal to this vertical surface x and that we are taking the gradient in the x direction so this is in general a viscous stress is directly proportional to normal gradient and that is what happened in conduction it fluxes also note that the arrows direction of arrows are different like in x momentum all arrows are horizontal all arrows in case of viscous stress acting in the y direction they are vertical but in conduction it is in the in the viscous forces the forces of vector there that way you show it as an arrow but here the way we are showing actually this is a total rate of heat transfer here we are showing it is an inflow and outflow so from the negative surface we take inflow and from the positive surface we take as an outflow so this is the total diffusion where this eta represents the surface normal direction the coordinate corresponding to the surface normal and this gamma phi is dynamic viscosity for fluid flow and conductivity for heat transfer now the this is the diffusion flux shown for fluid flow it is a viscous stress for momentum transport and for heat transfer it is a conduction heat flow and the discretization as is what I had been discussing earlier for conduction the same thing is shown here the second level of discretization is piecewise linear approximation this I had shown earlier for heat conduction the same thing is shown here in a generic way and this is that when you balance the total diffusion outflow minus inflow this represents net viscous forces acting in the x direction I would like to mention actually there is I am showing you here as outflow minus inflow but in the previous slide I would like you to note that this diffusion coefficient I am not showing it as minus k I am showing it as a plus k intentionally to have a generic expression because what happens is that for forces I have to take forces acting on the positive faces minus forces acting on the negative faces but when I go to the heat transfer if I take positive faces minus negative faces it becomes conduction out flux minus conduction heat flux and you remember that in my conduction topic I had talked about influx minus out flux and I always used to say heat gain conduction heat gain rate of change of internal energy is total heat gain by conduction. So to have a same physical interpretation here gamma I am taking as plus k but when you take here positive faces minus negative faces and when it is a minus k now this that minus k finally when you talk of heat in terms of heat transfer although it seems to be out flux minus influx but as I had instead of minus k I had taken plus k this finally gives us total heat gain by conduction heat transfer this is just for a generic representation because in a program this is the way we calculate the total viscous force in the x direction viscous force in the y direction but when we are calculating the conduction fluxes we want to have a physical interpretation not as conduction heat lost by the control volume but to have a physical interpretation that it is heat gain by the conduction heat transfer I had used that. So this completes the unsteady term advection term diffusion term all this I had already discussed in case of heat transfer in separate topics computational heat conduction for diffusion computational heat advection for advection phenomena and the combination I had discussed in convection and here just I had demonstrated that for fluid flow also there is an advection phenomena there is an advection term there is an advection flux there is a diffusion flux so it is just a repetition with a certain generality and note that the level of discretization which was used earlier in heat transfer the same discretization the same approximations are being used here to get an algebraic equation. Now we are left with the source term now the source term is the pressure force and the volumetric heat transfer pressure force for momentum transport so how do we represents pressure force pressure force is always acting normal to the surface area now pressure is also per unit area term it is a pressure is a force per unit area it is just a different types of force different from the viscous force. Now pressure is also a flux because it is a per unit area term so the first level of approximation is same is that what we use for all the fluxes surface averaging to represent phase on a surface on the surface of a control volume that is surface averaging which is shown here and the second level of approximation is expression to calculate pressure at the phase center in terms of neighboring cell center this we had done for mass flux also for the normal velocity note that if you look into the discretization unsteady adhesion diffusion I had shown you earlier there are two discretization which I am showing you new in this topic one is that mass flux and second is this pressure force because in our heat transfer we do not have an analogous term and second level of approximation so this is the first level of approximation this is second level of approximation now in this slide I am showing you the first level of approximation to calculate the pressure force in the x direction note that although pressure is acting on all the force surfaces but just to demonstrate the pressure force acting in the x direction I am just showing you this pressure force and this pressure force this I will be showing you for pressure force in the y direction in a discrete form how you do this is that you this is the pressure force in the positive x direction this is the pressure force in the negative x direction you take the difference of positive force and the negative force and divide by the volume surface area divide by volume gives you delta y in the denominator and you get px plus delta x sorry px minus px plus delta x this will become your px this will become your px plus delta x so px minus px plus delta x area is delta y here when you divide by volume you get delta x delta y delta y cancels down and when you apply the limit you get minus del p by del x here in this case by first level approximation you have expressed the pressure at the centroid of the control volume and by second level approximation you can write p small e is equals to p capital p plus p capital e divided by 2 p small w is equals to p capital p plus p capital w divided by 2 and then you can obtain algebraic representation of this pressure force for this control volume similarly you can calculate the pressure force in the y direction this slide shows you the total heat gain by volumetric heat generation in most of the heat transfer problem volumetric heat generation is expressed as q bar expressed here as q bar is known to us and in case this volumetric heat generation is varying point to point then here the approximation the first level approximation is that we are representing the point by point by point variation of volumetric heat generation as a value at the centroid of the control volume however in most of the cases this volumetric heat generation does not vary point to point so we can take it as a constant and here it is taken at the centroid okay so with this I had come towards the end of this topic but before that I will show you the slides showing the all the discretization so let us let me show through an animation so this is the term representing the rate of change of x momentum of the fluid inside the control node the word inside here this is rate of change of x momentum of the fluid flowing across the control volume node the word across this is equals to viscous force acting in the x direction which represents diffusion phenomena this is the source term pressure force acting in the x direction similarly when you go to the y momentum this is the rate of change of y momentum of the fluid inside the control volume this is an addiction term this represents rate of change of y momentum across the control volume this is the viscous forces acting in the y direction this is the pressure force acting in the y direction for energy equation this is the rate of change of internal energy or enthalpy of the fluid or solid inside the control volume this is the rate of change of enthalpy or heat energy of the fluid across the control volume this is the total heat gained by conduction and this is the total heat gained by volumetric heat generation so with this I had completed the topic on finite volume method thank you for your attention what I will do is that I will take one or two questions before moving on to the next topic. In the topic 6 slide number 35 while explaining the net conduction heat transfer for energy conservation it is explained that the rate of change of internal energy represents the total heat gained by conduction is it rate of change of internal energy or enthalpy. So, the question is in the left hand side of law of conservation of energy let me just see if I can open that I use this word internal energy enthalpy quite alternatively or what I actually mean is that here we are talking of incompressible fluids where both the term can be interchangeably used so that is why I had used this word, but if you go for compressible flows then this definition will not be correct so for an incompressible fluid this I can either use an enthalpy or a internal energy so that is what I had restricted to so whatever I am teaching here is related with an incompressible fluid where this definition is okay thank you. Sir another question is while explaining these animations of the x momentum equation you have used the diffusion term can you show the corresponding governing equation or the momentum equation. So, the question is while showing the control volumes for the diffusion term in the x momentum I had shown you the actually what happens so the question is can I show the differential form let me do one thing let me show in the wide board how from the let me go to the control volume and then I can probably do in a better way so the question is basically to show the viscous forces which are shown at the faces of the control volume in terms of a differential form am I right yes sir okay so let me go to the slide where I had shown that control volume okay let us take this and what I will do is that on the wide board I will show you that how we get a differential form of it that is what the question is. So, the control volume is this control volume now when you right now had shown you as sigma in the surface if I want to draw the viscous force in the x direction this is a normal stress so this will be sigma xx on this surface again it will be sigma xx on this surface note that now the plane is y so the first subscript will be y and the direction sorry the direction which we are taking is x so the second subscript it will be x on the bottom surface here again it will be sigma yx now in a finite volume I had expressed this as at the east face this I had expressed at west face this at north face center this at south face center where these are the centroid of the control volume sorry centroid of not the control volume but the surface areas however when you want to do differential representation we do not use this approximation so instead of the small e this will be at x plus dx the small w will be at x this small n will be y plus dy this small s will be y and then when you do a balance let us say we do a balance for the faces in the vertical forces in the vertical direction it will become sigma xx x plus dx this is in the positive x direction this is in the negative x direction so it will become minus of sigma xx at x we are talking of force so this has to be multiplied by surface area and what is the surface area delta y and then this is when we take the horizontal surface this will become sigma yx at x plus sorry y plus dy now here the surface area is delta x then we apply what we do if we want a differential term so right now this is the total force acting in the x direction to convert into a differential term we have to obtain force per unit volume so when you divide by the volume which is delta x delta y here delta y cancels down in this case delta x cancels down and then you use limit here you use limit delta x tends to 0 here you use limit delta y tends to 0 and you obtain del by del x of sigma x sigma xx plus del by del y of sigma yx which is del dot sigma second order tensor we will stop the interaction.