 Let us start. What I want to do is this square flow I want to do it thoroughly. I like this I assume hopefully you will also like this because it is a complete very very what this can be done completely in the class starting from the beginning to the end by hand. But that is not the only thing. There are some very nice concepts which come out of it. And this is one of the very few as always we say one of the very few exact solutions of the Navier-Stokes equations. Navier-Stokes equations are very complex with certain assumptions definitely. But when we do this also the flow is very simple one of the simplest flows. Actually I would say this is the simplest problem in convection. It is a convection problem. It is a simplest in convection. You will see why it is so. You will actually do it along with me. I want you to do step by step starting from the beginning. So today and tomorrow we will completely allot this. I will ask you to do something at home later. Let us start. This is a parallel plate problem basically. Usually we divide the problems. It is a flat plate problem, pipe problem, square duct problem, curvature, what is it? Bluff bodies. This is a parallel plate problem point number one. We call it in literature quiet flow problem. C O U E W T E quiet flow problem. So that is what in particular we are talking about a quiet. This was a this is called quiet flow in literature. It was a I think it was a French person. His name has been given. We have tried what was his contribution. It is nothing great actually but it is referred to as a quiet flow. It has a certain practical application which may look very trivial but it gives you lot of information. That is why I like this and hopefully you like it. So I will also add a adjective to this. This is called plain or simple quiet problem. Now objective of this is as usual to obtain velocity and temperature profiles and getting the problems is not the end of the solution. We would like to estimate, calculate, rate of heat transfer. I will say in the system now. Then we will see what it is and then study this a little further. Study the results and see what comes out of it. This also involves interpretation. Just getting the profiles at heat transfer is not enough. Then you have to look at the consequences of that. Study a little bit. Analyze. When you do this for any problem then you know how to use it. So simply solving a mathematical model through a code you will give the nice profiles. That is that is not the end of it. You have to know the results, interpret them, study them and then say okay. Study them means you know what happens if I change this? What happens if I change that? Why is this happening? Where is this possibly used? How can I control the flow? How can I control the heat transfer? That should come from the results part of it. So a complete solution involves interpretation and study also always and there is no end to that. Now we will start. What is the very first step? We will do it in steps. When you want to study a problem. In any problem but here you are talking about convection. What is it? I have simply stated a overall title I have given parallel plate problem. You must have an idea what it is. We will write it down. And it is called a plane or a simple couette flow. When I give that objective you should immediately think oh there should be something other kind of a couette flow. More complex, not so plain, not so simple obviously. We will see. Now in any convection problem, conduction, radiation, whatever it is. What is the first when you want to study a problem? What is the first thing that you do in our scientific methodology of not the experimental part scientifically? Okay. I will give you a, we are actually going to model this let us say. So what is the first thing that you write? You should or? Pardon me. Hey come front here. Why are you sitting at the infinity? It is like y tending to infinity there. So y equal to 0 is vacant here. Tremendous resistances. Heavy shear stress here. Okay. You want to be free or all free stream people. Okay. Tell me now. That is all. I will not say anything more about it. What is the first thing that you do? You have to do. What is it? Geometry. Can you give me a slightly better term? Boundary conditions will come later. Exactly. That is why I want to clear it out here. How can you talk about boundary conditions without even talking about the equations? But before you talk about the equations something else has to be known clearly which we, for clarity we write down. Diagram. Diagram you said what? Geometry. Flow conditions. Can you combine all this and give one? Pardon me. What is it? Say that it is okay. Assumptions will come later. We will do that. I am now trying to identify and write down. All these things are correct. Geometry you said? Diagram. Flow. Description. Okay fine. In what? You want it to write in words or you want to? Formulation. Correct. Formulation. Okay. I will just want to use the word. A dash model first we should have. That is exactly the problem. It will come later. So now comes the diagram geometry you are talking about. We said parallel plates, right? Parallel plates. Two parallel plates. Now we will define this plate. These are large plates. In our written, there are some words which give you some ideas which you should use. Large. Now what do you mean by plate being large? Plate is large. What meaning do you make out of it? So very good. Very long. Long in one particular direction. The other one is very, long is length. The other one is okay. This is okay. Now I can also add this is thin. Why do we say it is a thin plate? We always, we should actually, whenever we talk about solid, you know, usually we say thin plate because there could be thick plates also. What would be the difference between these two in terms of what you should consider in heat transfer? Then you have to consider the conduction in the thick plate also. That becomes a conjugate problem kind of thing. Thin means the entire, that thickness is at one temperature. You can take it like that. Now very long and very wide. So let us now have the coordinate system X, Y. For me, there is always, you know, in convection, X and Y, these are the directions. Even if I go to free convection, X along the surface or in the direction of the flow, Y perpendicular. X, Y. Now I will also write this. Z. This is for the flow. Now if you take that in terms of the plate, that means this X is very long. So this is very long. Very long means it is infinity. These are all very simple things. I am trying to make it clear to myself and to you. So how you can write it down? X is infinity. Now we will not make Y infinity. We will make Z infinity. Z is perpendicular to the plane of the board. So the example I usually give is, this is the top plate. The roof is the bottom plate. And they are extending in two directions, infinity. This is Y. This is a parallel plate problem. Actually this is a duct enclosure problem. If you take this room, it is an enclosure problem. Four walls and top and bottom. But remove these four walls, you get the parallel plate problem. And therefore in the Z direction also it is. Now this is why I have to obviously say what is this. Let us say it is, they are displaced. There is a gap of L. We will not specify at this point how much it is. We will just put it as L. Now where is the flow? He said we have to define the flow, you know. Where is the flow? This is a very peculiar problem. Usually what you would do, you put two parallel plates and then say flow is coming from this side. No. Now I will say. First of all, what is there in between these two plates? So now let us say there is some fluid. We would not even say air or water. The moment I say fluid, so what is the number we are specifying? So we say there is a fluid number, there is a fluid. It has a certain contour number. Meaning it has certain properties. Pressure and temperature come into the picture. Mu C p by K, mu function of temperature, C p function of pressure, temperature probably K function of temperature. So pressure and temperature is taken care of. That gives you the fluid. So if it is point, if you say it is 1, it is a gas. If it is 4 or 5, it is water up to 10, 20. If it is 100s and 1000s, it is oil, 40, 50,000 is heavy oil, 0.01010 is a liquid metal. That is fixed. Now, so there are two parallel plates. Now we will define a little bit further what are these plates? So now I am not saying there is a flow here. How would then a flow exist? Now I say this top plate is moving. Bottom plate, you can, you know in convection you can have any kind of geometries and situations and you can try to solve them. I will say the bottom is fixed. This hatching for me is either it is a fixed plate or it is insulated. In this particular case I am saying it is fixed plate. There are two plates parallel to each other displaced at a distance L. So we call this the bottom plate. This is the top plate. So the bottom plate is fixed. The top plate is moving. Then I have to prescribe something. It is moving with a velocity u1. 1 refers to the top plate, 0, okay, this refers to the bottom plate subscripts later. Now anything else to be defined? I am now completing the physical model basically. You have the diagram. You have the flow. Flow means now I am saying in steady state by the way, okay. We will define it again. The top plate is moving. So it is not fixed. It is not insulated. Bottom plate is fixed at this point in time. For me to have heat transfer, one would think there should be a temperature difference. So I will say this is at a temperature T1. This is at temperature T0. So now you say we have prescribed the velocities and the temperatures. Anything else required for us to continue? This physical model is complete now. Two plates, large x infinity, z infinity, displaced at y. Thin plates. There is no imposed flow. Bottom plate is fixed. Top plate is moving. One would think therefore there is a flow is induced. Induced is different from imposed for me. Imposed would be I put a fan here or a compressor. That would be inducing the flow. But as long as you understand, you can use whatever word you want. Bottom plate fixed. Top plate moving. Moving in the x direction. This is very important also. For us, there is a Cartesian and we have selected the Cartesian coordinate system, okay. Now, what kind of a fluid here? Now we will go one step ahead. What kind of a fluid we would like to have here? We cannot of course. I will prescribe that. You will prescribe. We will say this is a Newtonian fluid. Many a time we take all this for granted. I am trying to make it a little bit more. So I do not think it is so simple. It is known. Sometimes you make mistakes here also. It is a Newtonian fluid. Meaning it follows the Newton's law of viscosity, tau equal to mu del u by del y. It is a Newtonian fluid. This is the fluid, number one. Two, what kind of a flow we will now assume? We will now say it is incompressible. Is a fluid, do you refer, incompressible to the fluid or a fluid flow? You call it incompressible fluid or incompressible fluid flow. Compressible fluid, compressible fluid flow. Think about it. We will now say it is incompressible. I am prescribing. This is not an assumption at this point. I am saying I want to study this incompressible. What else in terms of flow? What else I have to say? Study. That is enough? Is that enough? A point from the coordinate system. What is it that I have to write? 1, 2, 3. So, you have to say. Let us write down. Then what else? Very good. Something from the point of view of properties. Oh, it means we do not use. No, that term we do not use, but properties. Either the properties are x or they are y. You decide at the initial point. All your equations, when you write, this point is very important because the type of equations will change according to this. What is that? So, the properties for our problem are considered as, I want the answers from you. Therefore, I want to have this interaction. It does not matter if it takes time. The properties are either actually there is a very important assumption. That is how you write your u, del u. All equations you write only then you can write those equations in that form. I will give you a del u. I will come to that continuum. In this particular case, constant we mean with respect to the temperature and pressure, not with respect to time. That is one way of doing it. With respect to time, the properties will change only if the pressure temperature changes with time. We will leave it as that. This constant property is not a very nice terminology from the English point of view, but that is used in literature. Constant with respect to temperature and pressure. So, properties won't change is what we have. Now, you see this is the beauty and complexity of a convection. There are so many things that you have to specify. When you do this and then get the mathematical equation and then put the boundary condition, for every one boundary condition and this combination you will have one solution. You change one boundary condition, you have a different solution. Keep the boundary condition same, change one term in the mathematical equation, solution is different. Those equations depend upon this. So, if it is not incompressible, you have a whole set of new equations where the properties as the row especially goes into the derivative. Steady and steady obviously you know. Laminar disturbance obviously you know. 3D, 2D, 1D. Single phase, two phase. So, when you look at flat plate, no salt equal to 0.332, Reynolds, Hoffprandt, one third, all this must immediately come to your two dimensional laminar incompressible constant property, Newtonian fluid of course air. All these give you that solution. You change any one of these definitions of the problem, your 0.332 does not hold good. So, you cannot simply as a flat plate put 0.332 Reynolds Hoffprandt. You should know what is behind that particular solution. This is the, yeah. So, I would like to take these out. This is a very important condition which we assume at the beginning. There is a difference between assuming and defining the problem. Constant property could be an assumption, but I am now trying to define. But this is, I am defining the problem here. Then assumptions will come in about the situation, about the conditions. Then you simplify the equations. Now what I want you to do is the, for me this is the full physical model now. This is, I am considering this problem. Diagram is one of the aspects. All these things you have to be very clear if you are, it is better to write down always, so that you can proceed to the next step. So, there is a physical model. So, from physical model you graduate to, what should you now get? Now the physical model is ready. So, now we have to write down the math model. Math model consists of two things. What are these? Bond conditions will come later, first before the governing equations. Only boundary conditions when you do this, the mathematical model is complete. So, do not think, when I say do not think out of experience I am saying, mathematical model simply people write the equation and leave the boundary conditions to me. Only when you write the boundary conditions looking at the proper equations, how many boundary conditions, how many boundaries, how many conditions. So, what is the order of the derivative? Then only the problem is now complete. Hopefully it is now ready for the solution. Now, so we will now get the mathematical, you will get the mathematical model for this. What I want you to do is, write down three dimensional, for this situation the Navier-Stokes equations, complete Navier-Stokes equations. We have not talked about boundary layers here. We have not talked about boundary layer. Take a difference. Write down the complete Navier-Stokes equations. Continuity, X component, Y component, Z component, energy. No species. Start. As and when you finish, you please tell me, I will write it down. Continuity equation. Three dimensional, laminar, incompressible. Incompressible is very important here. Constant property. So, give me the three dimensional equation. What is it that you say? You are somehow jumping ahead. May be I am a little slow. Give me the first three dimensional continuity equation for these conditions. What is it? If it were to be unsteady, what would it be? Doe, if it is incompressible. So, this is also for incompressible. It is a unsteady state also this is the equation. For unsteady, you get only del rho by del t in that term. You do not get u v w. That is only the property variation. If it is incompressible, if it is compressible flow only that will come in as a continuity equation. In incompressible flow, del rho by del t is 0. So, del u by del x plus del v by del y plus del w by del z equal to 0 is the continuity equation for incompressible flow, steady or unsteady. Now, X component. Now, we will write momentum. We will start giving numbers please. Momentum equation please. X component. Doe u. We will write it because although I have said here unsteady, steady at this point we will say unsteady and then make it steady. Good. Give me this please quickly. u doe u by doe x. So, yeah equal to what? Minus plus, you have to write this. Plus, you are okay. I will say the x component of the body force but I have divided every way by rho. Write this for the y and z component. Please, completely write. Convective terms or inertia terms, pressure, body force and the viscous. Write for w also. Three components you have written. Now, write the energy equation now. Can you tell me one by one terms? Energy. Doe t by plus. This is very easy to write. u del t by del x, v del t by del y plus w del t by del z. You all pronounce it as doe. I always pronounce it as del. You understand. Is equal to what now here? Alpha is the grad square t actually. Anything else? Write the complete energy equation for incompressible. Then we will make assumptions. Sometimes we write phi, sometimes mu phi. If you do not write mu phi and write phi, in the definition of phi you would have included mu. Here, that mu you can take out, take it away from that. Okay. Thank you. Is that all? Anything else? There are actually two more terms. One is a compressibility term but we have said incompressible. Otherwise, you should write the compressibility term. Beta comes into picture. Expansion coefficient. So write down q triple prime by, so this is 2, 3, no, no, no, no, 2, 3, 4. No species equation. Now actually you have to solve this set of equations. Three dimensional. This is the complete Navier-Stokes equation now what we have written for this particular situation. Now we make, yeah this can be done with a, after writing the boundary conditions of course for u, v, w, t, p and everything. Only on a big computer you can do this. And not analytically. Even your approximate, the integral method may or may not work here. Almost definitely you have to go to a, basically what we call as a DNS, you know, direct numerical simulation. But we will see, we will avoid it. So tell me, we will make some simplifications now. This is step number, okay, 2a, simplification. We have not completed the mathematical model. Now I will give you, now this is a parallel plate example I am giving. Now if I want to consider the flow somewhere in the middle of this plate, in this portion. So physically what would you, how would you interpret, how do I ask this question? I want the flow, although I have talked about 2 plates, I want to talk about the flow only in this region. What am I implying actually when I say that? I am, what is it? No, no, no, no. I did not communicate the question properly. I am interested only in the flow between these 2 plates but only here means, now tell me what I am trying to neglect, okay, let me put it the other way, give me a hint. I want only in this portion here, instead of taking the entire plate. That is why we have given the example, we have used the word large actually. Now I am coming to that. When you say large plate, in the majority of the plate zone, what is it that you can neglect? Entry effects. Entry effects and once you, correct, once you, actually there is no entry here, there is no entry but let us take it that way. Entry, when you say entry, another thing is exit on only on one side, on both sides. But there is no entry here, no exit. So, we would, very correct. Now we would say in the so called x direction, we call this the ends of the plate, end. And in the other direction, we call it the edge. So, there is an end, two ends and two edges. Now we will say, if I want to consider the flow only here, the effects at the entry and exit, I may neglect. How do you neglect? Now mathematically, how do you neglect entry and exit effect on both sides? What would you, what would you say in terms of mathematical terms? I will write here, you came up with the right answer physically, no end effect means what? I will give, I have to give you a hint. What kind of derivative terms you can neglect? You neglect this. Then, so I will say neglect. I will say no edge effects. What would you neglect? Huh, go by? That means ladies and gentlemen, no, no, no ladies here. The flow is really three dimensional, in this box for example, you know where in those corners. That is a three dimensional corner. This is a two dimensional corner. Here you have the two flows, if there are flows, interacting to give you those two dimensional three dimensional effects, especially three dimensional effects. But if you take somewhere here, a simple flat plate, now you can take this as a flat plate, flat plate need not be horizontal, flat plate can be vertical and then the flow can come here. In the majority of this area, if you leave out the leading edge as you have known and the trailing edge, you say those effects you can, but in a boundary layer flow you cannot do that. In fact, that flow at y, x equal to 0, that is a leading edge thing that creates lot of problems actually and especially in your numerical, if you are going to solve numerically, what is the condition that you are going to put at x equal to 0 is a big problem. y equal to 0 is no problem, x equal to 0, what? That is a singularity situation there. No end effect, no edge effect. Now I will say one more thing, only g effect, only g as a body force, you know what this leads to. What are the other body forces? Yeah, then, then in actual that is good, but in our applications in mechanical engineering for example, body force acts on every molecule of the volume. Surface forces are act only on the surface, so pressure and viscous. The body force acts on every molecule, one is g. No, viscous is a surface. Pardon me. Normal reactions. No, that is also a surface, normal stresses. Normal and viscous is act only on the surface. I will give you a hint, a turbine, go further ahead. The earth is rotating on its own axis and rotating around the sun. What forces come into picture? Coriolis forces. So electromagnetic forces, Coriolis forces, centrifugal, centrifugal, all of them act on every particle. Therefore, they are called body forces, the body, the entire body otherwise they are surface forces. So here, we will take the simplest g and therefore, it is a natural force g. Now, as and when we go further, we will make other simplifications. With this, can you now simplify this? We are now still doing the mathematical model. Please simplify this. See what equations do you get. Now, get me four equations with all these assumptions that we have made. Take the, take the first. Take the, by the way, you are right, you are right. Thank you. We also take study. Very nice. Start from the continuity equation. Simplify it. No end effect. So, what is it? That can be neglected. No edge effect. There is no assumption on this. So, the continuity equation now is del v by del y equal to 0. Please write down. Right? Go to the x component please. So, I will say del v by del y equal to 0. Then, x component, del u by del x, del u by del z. Now, this is a problem. We will come back. Del square by del x square, del z square. This is not there. Now, we will say del p by del x is 0, edge effect because we are not imposing the flow with a pressure difference from outside. The inertia terms and viscous terms are taking care of each other. That is what is keeping the flow. So, and also we are saying no edge effect. So, all del by del x is cancelled. So, steady state. Now write down the expression now. What is the x component momentum? It is not a boundary layer thing. There are no boundary layer assumptions. It is Navier-Stokes equations with certain geometry, flow and some assumptions we have made like no end effect, no edge effect. So, what is this? Del square, give me the entire expression that you got here. Del u by del y is equal to, now, before going further, we will make a small thing. Now, here look at the continuity equation. Del v by del y equal to 0. Okay, we will do it later. Leave it, leave it. We will come back later. So, this is 5. We will come back later. We will further simplify it. Then, write down the y component. This is very interesting. Anything here? Del v by del x will go. Del v by del z will go. We will invoke a boundary condition and do it. I wanted to write it later but we will do that. Tell me the boundary conditions on v. At y equal to 0, please write down. At y equal to 0, what is v? Why? No penetration. No penetration? No penetration. No slip like this. v equal to 0. Slip doesn't come under v. For v, v equal to 0 is perpendicular to the plate. So, for v equal to 0, what should be the assumption that we have to make or the condition that we have to specify? There are no perforations as I have. There are no perforations. So, there is no v component means it is a solid plate. It is a impermeable plate, please. That we should have written here. Actually, I could have written right there. Just so there is no v component. But is that right? Just because it is a impermeable plate, you think there is no v? There is another condition where for a impermeable plate, there could be v. You are saying v, if it is, there are holes and then you put in air or whatever, the fluid. Suction or blowing is not there. But I would like to state that v could be there even for my impermeable plate under certain conditions. You know this. You know the answer. I want you to give me. I will give you two minutes. Give me a physical situation. Natural conditions also have a impermeable plate. There is no v component there. Perfumicular to the impermeable plate, there is no v component. That v is at the wall. It is there in the boundary layer. At the wall, we are talking about. At y equal to 0, we are talking about. You have a water body, our reservoirs, Madras reservoirs which do not have water, you know water. Then there is a flow over there. There is a wind. What happens? That is a solid surface actually. Impermeable surface rather. Water surface. But now there is a v component. That v component is actually evaporation that is occurring. Now can you go further and say something else to maybe at least definitely one other situation, two situations may be one situation. Evaporation is one. Related to that something else. You see it all the time in your refrigerators. There is a condensation. That is the boundary condition you have to use. There is a v there. So it is impermeable plate v equal to 0 because there is no blowing end section but also there is no v equal to 0 because there is no condensation or evaporation even with a solid plate, okay. So now we will do that impermeable plate. On the other hand, can I use some other term for us to say v equal to 0, some other term? Not evaporation, no boiling, no condensation. Any other generally accepted term I can say no this thing, okay. We will do that. So v equal to 0. We have not formulated the problem but it is okay. Now take that v equal to 0 at the wall, both walls. Take this continuity equation you have written del v by del y equal to 0. So what is the consequence of this? If you apply the boundary conditions on v for this, what is the consequence of that? Just look at that. v equal to 0 at the bottom plate, y equal to 0, please write down. y equal to l also v equal to 0. dv by dy equal to 0. Therefore what is the conclusion out of this? v equal to 0, where? Everywhere. v equal to 0 everywhere. There is no v component at all because at y equal to 0, v equal to 0, actually y equal to 0 I should not write v 0, we should say v w equal to 0. And y tending, y equal to l also v w equal to 0 and this equation. Together this component continuity equation and the boundary conditions tell you v is 0 everywhere. This is a very important physical interpretation. v equal to 0 everywhere. Now use this for the y component. See what happens to the equation? Here, what happens? Sorry, use it even for the x component first, v equal to 0. So what is the x component equation now? Please. Doe square, that is it. There is momentum equation. What have you done in one shot? All the convective terms are gone. Viscosity is there, but because of our, this nu del square u by del y square equal to 0 actually, but now it has come down to this square u by, actually del square u by del y square, del v by del y, I want you to do another thing. It is only in the y direction that every equation is coming along now. This is del v by del y, del square u by del y square. You can as well write this as, change the doe to, that is it. So now it is coming into an ordinary differential equation. Here, see how we have simplified that entire momentum x component. Now go to the y component. Go to the y component. I had to do that v equal to 0 everywhere so that you can simplify this. But what remains here in the y component? There is something still remaining. Pressure and, now, so write down that expression and tell me a solution for this. Just write down the y component and the solution is hidden there. F y is, only body force is g. So what is that F y? Y is the y component of the body force. What is the force there? Yeah, give me the variables there, in terms of variables. So rho g and in what direction is rho g? No, no, y direction. Upward or downward. So it should be rho g prefixed with a, so now get the equation and automatically you will get the solution. Find out what it is. 1 by rho d p by d y equal to rho g. So what is the solution for that? Delta p equal to? What is that equation? That is it. That is the solution. y component is delta p. Delta p means within the boundary layer, within the fluid layer, p minus p not or whatever you do. It is equal to minus rho g h, rho g y. There is no more solution required. That is the solution, minus rho g y. So component has given its own, the momentum equation has given its own solution. You do not have to solve it anymore. So delta p equal to minus rho g, actually you will get it as y. So it simply says very correctly also, minus actually. So it is reducing. p 2 minus p 1 is reducing means p 2, as you move away from the surface the pressure is reducing. Now look at the z component very quickly. Yeah, you have it. Give me the z component equation, 10, 50. You have to go. z component, del w by del tau 0, all del by del x, all del by del z. So what is it? What is the equation? You have an equation still. What is that? dou square, right? Now invoke again the boundary condition. What is w at the wall? What is bottom wall? What is w at the top? At y equal to 0, w equal to 0. At y equal to n, w is 0. So w is 0, complete the sentence everywhere. Quickly do the energy equation, can you do it? It is 10, 51. I have written down here, so we will do this. Del t by del tau is 0, del t by del x is 0, v is 0 here, del t by del z is 0, w is 0, del square t by del x square is 0, del square t by del z square is 0. And now we will put one more condition here, no internal heat generation we will now add. So that goes off, phi. Phi is full of derivatives. Can you write down? You must have it somewhere at the back. Del u by del x whole square plus del v by del y whole square plus del w by del z whole square multiplied by 2. That is single terms all squares. Then two terms come into picture. You have it, please write down. And then invoke this no edge effect, no side effect. Simply I will give that to you as a small exercise. Do that and obtain the energy equation. All that we have written here as simply del d square t by dy square plus mu by k du by du whole square. Where is this du by du whole square coming from? From the viscous dissipation. Everything else cancels out there. I have already put d, I should not have done it but I have already mentioned. You will see that all the derivatives are only function of y. Therefore instead of dou you can simply make them into od's. So you have d square dv by du equal to 0, d square u by dy square equal to 0. This is the solution already. This is d square w by dy square equal to 0, d square t by dy square plus mu by k du by du whole square equal to 0. We never made an assumption, any other assumption with viscous dissipation I have maintained that term. That is very important from that you will get a lot. Now what is it therefore you have to solve? This is already taken care of. This is solution v equal to 0 everywhere. This is v equal to 0 everywhere. Sorry w equal to 0 everywhere. This is already solved. So finally ladies and gentlemen the only two equations you have to solve is d square u by dy square equal to 0, d square t by dy square plus mu by k du by du whole square equal to 0. Please put the boundary conditions at y equal to 0 for u, v and t. At y equal to 0, t is t naught. Please, okay. Now this is completed the equations. The mathematical model is not complete till you write the boundary conditions. Please write the boundary conditions at y equal to 0, u is 0, v is 0. We have given the reason why it is so and w is 0, t equal to t naught, y equal to l, u equal to u1. Now what is this condition called? I did not mention. You know that. When I put y equal to 0, u equal to 0 and uy equal to l, u equal to u1, what is the condition I am invoking? Physical condition. What is it called? No slip condition. This is where all the problem starts for us. If there were to be slip it would have been wonderful. Then that would have been ideal fluid. Diamonds principle would have worked. The real fluid, the problem is this. This is creating the entire boundary layer problems later. That one condition. Then what about the temperature? y equal to 0, t equal to t naught, y equal to l, t equal to t1. What is this condition called? The other one was no slip condition. What is this condition called? No, we have taken it to be isothermal. Not isothermal plates. Yeah, I did not mention. Please write down isothermal plates. That is a one boundary condition that we have given. I am talking about writing down y equal to 0, t equal to t0 and y equal to l, t equal to t1 corresponding to no slip condition. There is a name for it. There is no slip means it is attached. Otherwise it was slipped. Now temperature in a very similar way we use another word. Okay, I will tell you. Perfect contact is very fine. Perfect contact condition, you can call it very nice. So we also call it as no temperature jump condition. Perfect contact is very fine, no problem. Now when you write all these boundary conditions, these two equations, ladies and gentlemen, you have now completed the mathematical model. We have to go for the step 4 which is the solution. I will do it tomorrow. But I want you to solve the momentum equation. There is no problem. Try to, from the momentum equation solution, try to solve the energy equation. We will do it tomorrow. We will finish it and I will, I want to calculate the heat transfers tomorrow. Thank you very much.