 So, we are going to take a detour through fluid mechanics, but first I will just take a recap of what we had just studied yesterday in convection one. That is we said that convective heat transfer is can be classified as forced convection and pre convection and in forced convection essentially there is going to be a pump or a fan, but in natural convection the convection is essentially occurring because of density gradients created because of temperature gradients, but these density gradients alone would not create natural convection, but then we need acceleration due to gravity. So, in space natural convection would not be there because of the absence of acceleration due to gravity, so then we introduce what is called as Newton's law of cooling that is q convection equal to h a s into T s minus T infinity one of the professors had asked me the question yesterday, why is it called Newton's law of cooling can I call it as Newton's law of heating, but I do not see any problem in calling in Newton's law of heating as well, but historically it has been termed as Newton's law of cooling. So, one can call this as Newton's law of cooling slash heating, so it is given by q dot convection equal to h a s into T s minus T infinity, here q dot convection is the convective heat transfer in watts, h is the convective heat transfer coefficient in watts per meter square Kelvin, a s is the surface area or the bathing area over which the convection is taking place, T s is the temperature of the surface, T infinity is the temperature of the fluid sufficiently far from the surface, why this is said sufficiently far from the surface because we are not within the boundary layer, I had not mentioned this yesterday because I had not introduced the concept of boundary layer, but now that we have introduced the concept of boundary layer, so we want this to be sufficiently away from the boundary layer that means it has to take the free stream temperature essentially. So, then we introduce what is called as local heat flux, so if I have to understand local heat flux I need to get to this, so here before I get to this I had defined the heat transfer coefficient, before going to the definition of the heat transfer coefficient I need to pin point one point that here heat transfer coefficient is not dependent on heat flux or temperature difference, it is dependent on the flow distribution and the fluid which it is being handling, so we should not be under the impression that if I increase this heat transfer rate my heat transfer coefficient is going to increase or for a given heat transfer rate if I decrease the delta T that is T s minus T infinity my heat transfer coefficient is going to increase, no it cannot happen like that. H is essentially dependent on the thermal boundary layer temperature gradient, let us see how does that happen that is we said that there is q convection that is within thermal boundary layer before going to within thermal boundary layer, we said that in a fluid when a fluid is flowing on a stationary block, on a stationary block or a solid block which is not moving, we said that there is no slip boundary condition that is velocity just at the tip of the wall that is at y equal to 0, if this is my y direction that is at y equal to 0, we said that there is velocity is equal to 0, so that is what is called as no slip condition. Similarly, for temperature also there is no temperature jump that means the if I maintain the solid block as at a constant temperature let us say T s just at y equal to 0, the fluid temperature also is going to be T s, this is what is called as no temperature jump condition. Having understood this no slip and no temperature jump condition, so what is happening in the thermal boundary layer, what is happening is that the q conduction that is essentially within the thermal boundary layer conduct heat transfer is there, that conductive heat transfer is given by again Fourier's law. Whenever I say it is conduction, the quantifying law is the Fourier's law of conduction, so here actually q dot conduction all the I have written what I mean here is that it is heat flux, so q dot conduction equal to minus k of fluid, here you see it carefully this is thermal conductivity of the fluid not the solid plate, most of the times we get confused and people write this as k of the solid, no, so this is k of the fluid into del T by del y at y equal to 0, there is another question yesterday I need to know only del T by del y at y equal to 0, so I do not have to worry about the thermal boundary layer, it is not so only if I know the temperature distribution in the thermal boundary layer then I would be able to compute this slope that is del T by del y at y equal to 0. So if I get this conductive heat transfer that has to be equated to q convection which was given by Newton's law of cooling that is heat transfer coefficient into T s minus T infinity, so that is h equal to minus k fluid into del T by del y at y equal to 0 upon T s minus T infinity, what we understand by that by this here is that whatever the heat flux I apply, so my temperature gradient gets adjusted, so this temperature gradient along with the thermal conductivity of the fluid is going to decide my heat transfer coefficient, so again I re insist that heat transfer coefficient is not dependent on heat transfer rate or the temperature gradient that is T s minus T infinity, temperature difference not gradient temperature difference but it is dependent on the temperature gradient, so having understood this in fact let me re emphasize that this is the definition of heat transfer coefficient and Newton's law of cooling is not going to define the heat transfer coefficient that is a means of measuring the heat transfer coefficient, of course this also can be used for measuring the heat transfer coefficient that is if one uses laser inter-phenometry and gets the temperature gradient within the thermal boundary layer one can compute the heat transfer coefficient. So with this basics we can now realize that there is something called as local heat transfer coefficient, if I take flow over an aerofoil, if I take flow over aerofoil see here one would expect by just intuition that the heat transfer coefficient at this point is significantly higher than the heat transfer coefficient at this point, why because the flow is directly facing this aerofoil tip that is what is called as stagnation point, if you do not understand what is stagnation point now you do not worry we will deal with this when we go to flow around cylinders. So what I mean here is that the flow is directly hitting this tip, so it is going to get cooled significantly higher, so one would expect the temperature gradient quite large here and the heat transfer coefficient quite high, but here there will be lesser heat transfer coefficient, so I need to differentiate between the local heat transfer coefficient and the average heat transfer coefficient, so far in the conduction problems whatever we have done we have always dealt with average heat transfer coefficient, we never bothered about the variation of the heat transfer coefficient with phase, but here we need to be worried about that, so here if I know the local heat transfer coefficient, if I integrate that I am going to get over area as I get the local heat I get the average heat transfer coefficient, so this is actually h bar, so with this we went ahead and solved a problem and we found how we can relate the average heat transfer coefficient with the local heat transfer coefficient. So with this of course we said that we will come back to Nusselt number little later we defined internal and external flow, laminar versus turbulent flow for pipe internal pipe flow and we defined what is called as Reynolds number to inertia to viscous forces that is rho V L by mu, so L is the characteristic length rho is the density of the fluid V is the velocity typically if it is flow over a flat plate free stream velocity, if it is flow in a pipe it is average velocity mu is the dynamic viscosity if it is it is also written as V L C by mu, mu being mu by rho that is the kinematic viscosity, so then that is where we had stopped in yesterday's class. So today we are going to get started with few definitions again and what we will do is we will get to fluid mechanics what after the few definition and then get back to heat transfer again, so first is 1, 2 and 3 dimensional flows, so that is we have velocity if the velocity is a function of x, y and z like temperature it is going to be 3 dimensional if the velocity is going to be only a function of x, y or y, z or x, z then it is going to be 2 dimensional if velocity is only a function of x or y or z then it is going to be 1 dimensional flow, so that is how we handle or we define 1 dimensional, 2 dimensional and 3 dimensional flows by and large we are going to handle 1 dimensional flows in our in our convective heat transfer because we need to get the closed form solution. So another important aspect of flow is how to differentiate between laminar and turbulent flow, we said that in laminar flow the flow goes as laminae and 1 laminae does not talk with the other laminae, but in turbulent flow it is not so 1 laminae is going to talk with the other laminae or many other laminae, so turbulent flow is usually visualized of course we will get to this in very much detail, but for now I want to touch and go we will come to the turbulent flows very much in detail in next to next hour. So for now all that I want to tell is that if I work to measure velocity in a turbulent flow somehow actually there are some measurement technique that is there is something called hot wire animometer or laser Doppler velocimetry or particle image velocimetry if I employ any of these technique I am going to get the velocity as a function of time it can be in a pi or it can be on a plate or it can be flow over a cylinder, but the flow is turbulent if it is turbulent the velocity you will see that it is going zigzag that is if I average this zigzag motion for a considerable period of time, so then I would be getting the average velocity, but on this average velocity if I super impose this fluctuating component you can see that velocity at any instant is average velocity plus u prime it need not be always plus this u prime need not be always plus here it is plus for example here it is negative. So any velocity at any instant of time is we are saying that it is going to be comprised of the average velocity and the fluctuating component. So this is how we characterize turbulent flow that means we have average and fluctuating this is true not only for velocity for also pressure and also temperature, so p bar plus p prime t bar plus t prime. Now how do I differentiate between laminar and turbulent let me use white board, so what is happening here is in case of laminar flow in case of laminar flow in case of laminar flow in case of laminar flow the shear stress the shear stress we saw yesterday is given by or I have told this much earlier and we all know from fluid mechanics that shear stress is equal to del u by del y. Today we are going to understand what is the meaning of del u by del y within few minutes, but for now I just want to touch and go this del u by del y, but for turbulent flow the shear stress is going to be largely dependent on what is called as rho u prime v prime bar that is what was u prime and v prime. So this is the fluctuating component in the x direction and fluctuating component in the y direction, so that is what is this turbulent shear stress, so in the if the boundary layer is turbulent then if the boundary layer is going to be turbulent I am going to have the shear stress characterized by minus rho u prime v prime and if it is laminar it is going to be given by mu del u by del y, so that is how I differentiate laminar and turbulent boundary layer. So now that is what I have written here that is turbulent shear stress is equal to minus rho u prime v prime bar. Similarly, for heat transfer if it is laminar if it is laminar heat flux is given by minus k del t by del y, but if it is turbulent we have seen that it is given by minus rho Cp v prime t prime that is q double dash or q double dash turbulent is equal to rho Cp v prime t prime bar that is this is the fluctuating component in the y direction velocity and this is the temperature fluctuation component. So if you are not understood this for now you do not have to worry I just want you to carry the message for now because I am going to deal this turbulent characteristics in great detail in today's lectures. So for now all that I want to tell you is for laminar flow the shear stress is characterized by velocity gradient and for laminar flow the heat flux is characterized by temperature gradient, but for turbulent flow the shear stress is characterized by fluctuating component and for turbulent flow the heat transfer is characterized by the fluctuating component. So fluctuating components come into picture in case of turbulent flow and in case of laminar flow we quantify the shear stress and the heat flux by either the velocity gradient or the temperature gradient. So that is the point I want to make now in the introductory lecture why because I just want to introduce you what is called as velocity boundary layer. So let us formally introduce ourselves what is called as velocity boundary layer. So here you see this is the flat plate so let me draw this so I have a flat plate please note is that this is a sharp edge this is a sharp edge so I have taken a flat plate ok this is made sharp because the boundary layer will start forming at the edge. So I have initially laminar boundary layer and my boundary layer transits between laminar and turbulent ok and later on my boundary layer become turbulent. So in this portion it is laminar and in this portion it is transition transitional boundary layer and here it is turbulent. So now let me get back so that is what I have shown here. You see here very closely if you see there is within the laminar boundary layer there is no mixing the velocity earlier in one of the participants had asked me a question within the boundary layer is the fluid still no you are seeing within the boundary layer because there is velocity there is velocity ok. So there is velocity and the velocity the velocity is the nature of the flow within the velocity boundary layer is laminar but then I have in transition in the transition region for some time it is laminar and for some time it is turbulent ok. So it is always transiting between laminar and turbulent but in turbulent boundary layer it is thoroughly turbulent you can see that it is mixing it is mixing but in a turbulent boundary layer again it is having three regions. As you can see here if you are not able to see this in few in few centers there are three regions again here. So this is let us say we will call as a region 1 region 2 and region 3. Region 1 is laminar sub layer laminar sub layer and number 2 is buffer layer and number 3 is number 3 is turbulent boundary layer. That means even within turbulent boundary layer very much close to the wall it is laminar in nature and then subsequent to that it is buffer and then subsequent to that it is turbulent. But how do I differentiate between laminar sub layer, buffer layer and turbulent boundary layer? Laminar sub layer is in that region where in which my shear stress is given by velocity gradient it is given by velocity gradient. But in turbulent boundary layer as I had told it is given by rho u prime v prime bar with the negative sign ok. I will tell you little later why that negative sign is there ok. Because when we derive Reynolds averaging later on we will appreciate this negative sign quite easy for now you will have to take that on my face value. But here this is what is called as tau t and this is what is called as tau l. So within the laminar sub layer or viscous boundary layer there is no fluctuating component in the turbulent boundary layer there will be velocity gradient but the order of the velocity gradient will be very less compared to that of the fluctuating component. But in the buffer layer both velocity gradient and turbulent shear stress will be both will be of the same order. Please note this notation I am using tilde they are of the same order ok. That means both are important within the buffer layer. Let me restate this at the cost of being repetitive whenever because this is the figure which we see in almost all books but even as a student I could not understand when I read this on my own that is the reason I want to reemphasize this. First thing is I have a sharp edge why because I want the boundary layer to start from here. What will happen if I have a flat plate like this? My boundary layer will form from here and then it will form here. So that is my boundary layer will be formed in this portion also. So which I need to avoid that is the reason I have made my edge sharp. That is what it would be there in most of the text books they would show you sharp edge. They might have not told why they have made it sharp. So initially we have initially means for certain length that is for up to x critical for certain length we have the boundary layer as laminar. Boundary layer as laminar means that is there will be there is velocity no doubt about that and there is gradients of velocity within this laminar boundary layer and in this laminar boundary layer the shear stress is characterized by velocity gradients and I have transitional boundary layer that means there is intermittency between laminar and turbulent. That means for some time it is laminar and for some time it is turbulent. That means intermittently it is laminar and turbulent. So how long it is turbulent is defined by what is called as intermittency factor. Let us not worry about that. So in turbulent boundary layer there is thoroughly mixing. There is thoroughly mixing in all directions you can see the velocity. But there are again in the turbulent boundary layer it is classified or it is observed. These are all based on experimental observations. This theory is based on experimental observations. So people have measured that is why they are able to see that. How does one say that there is a fluctuating component? Because you can measure the fluctuating component. How do I measure that? As I said using either hot wire animometer or particle image velocity or laser Doppler velocity. So here we have in the turbulent boundary layer we have three regions that is laminar sub layer and buffer layer and turbulent boundary layer. So very much close to the wall is the laminar layer and immediately after that you have buffer layer and then you have turbulent boundary layer. So in the laminar sub layer velocity gradient decide the shear stress and in the turbulent boundary layer the fluctuating component decide the shear stress. But in the buffer layer both the laminar shear stress and the turbulent shear stress are of the same order. This is I would say this concept is very important. We are going to quantify this in a little different way a little later. But I need all of you to remember this otherwise it becomes quite difficult to follow subsequently whenever I say turbulent boundary layer and things like that. So this is what is being told in this transparency. That is we have velocity boundary layer. So we have laminar, transition and turbulent. I think that much is sufficient and of course when I say all of this the boundary layer thickness is defined as that region over which the velocity is reaching 0.99 times the free stream velocity. So that is the velocity boundary layer definition that is what is being shown here. You see delta is given by at that location where the velocity at that location is given by 0.99 times the free stream velocity. So with this basics we will move on to why do I need this? The main question always ask is why do I need this velocity gradient? Why am I worried about this velocity gradient? That is what we are going to do in subsequent lectures today. That is we are going to get velocity gradients through momentum equations which are called as Navier-Stokes equation and we are going to derive what is called as energy equation through which we will get the temperature distribution. Let me write that. Why am I going to do that? So what am I going to do today is that we are going to derive what is called as little later what is called as conservation of conservation of mass and Navier-Stokes equation that is conservation of momentum that is conservation of momentum and then conservation of energy. So this conservation of mass and Navier-Stokes equation if I solve I am going to get the velocity distribution u, v, w and p. But if I solve the energy equation I am going to get the temperature distribution. Now the question is why do I need this velocity distribution and why do I need this temperature distribution? So if I get the velocity distribution so I can get what is called as let me go back to the document. You can see here as I said till now I have been telling that shear stress is given by shear stress. Shear stress is given by viscosity into del u by del y at y equal to 0. This is shear stress. Why do I need this shear stress? This shear stress is defined or non dimensionalized what is called as skin friction factor that is C f equal to tau all upon half rho v square half rho v square. So I get shear stress from this shear stress I get the skin friction coefficient. So this skin friction coefficient is going to tell me the resistance offer that is how much pumping power is required. So this if I go back to whiteboard and what am I done? What have I done? So if I solve conservation of mass and Navier-Stokes equation I am going to get the velocity profile and pressure. So using this I can get the shear stress and this shear stress gives me the skin friction coefficient that will tell me the pumping power that is from velocity profile I can get friction factor or skin friction factor from I will tell little later we will define what is the difference between skin friction factor and friction factor for now let us take it as skin friction factor. From that we will get what is called as pumping power that is what is the capacity of the fan or a pump I have to put I have to put. So that I can I can fix the rating of my pump is given by pumping power. So if I solve the energy equation I am going to get the temperatures I am going to get the temperature using this temperature distribution I can get the heat transfer coefficient and from this heat transfer coefficient I can compute either heat flux or if heat flux is known we can compute what is the maximum temperature my plate can take. This is why we are interested in setting up the equations for conservation of mass, momentum and energy. It is this engineering necessity to fix the pumping power and fix the heat transfer rate which drives us to solve these conservation of mass, momentum and energy to solve and get the velocity and the temperature distribution. So that is that is what is stated here as surface shear stress skin friction coefficient and of course from this we can compute the friction force which is given by in friction coefficient into surface area into rho v squared by 2 that is what is the frictional force. So I think now I will give this to Dr. Arun all the fundamentals related to the classification of flows I think has been completed and we have essentially told you why convection heat transfer problems convection heat transfer problems require what is a coupled solution of the conservation of mass, momentum and energy equation of course all the three of them would be derived later today and the end result of this kind of computation or this kind of a solution will give us velocities u v and pressure and the temperature distribution. So from fluid mechanics we have seen that mu d u by d y represents the shear stress skin friction coefficient is there and then total frictional force pumping power as I rightly pointed out. So these quantities are engineering quantities which have to be calculated computed. So for an engineer he really probably does not care too much about nitty-gritty details of say velocity distribution overall percent if you go and ask a plumber that you know give me tell him I need this much mass flow rate this much discharge is needed he will be able to do it. But if you tell him I need a velocity profile which is given by u by u max is equal to 1 minus r by r square or something like that he will say what are you talking about. So we as academicians or persons dealing with fundamentals we need to understand the conservation laws the physics that goes behind these kind of concepts of shear stress power and temperature distribution heat flux etcetera those are the fundamentals which we as students and teachers have to understand. But when we finally deal with engineering problems we say a pipe of diameter 1 inch carries water at the rate of so and so velocities 2 meter per second. So we give an average velocity there might be whatever profile laminar turbulent inside, but as a person who is going to use it we give average or bulk quantities. So there is a difference between what is done in the classroom and what is done at a in a engineering world, but fundamentals always have to be kept in mind. So with this idea let us see the concept of thermal boundary layer. Now all of us are familiar ensure every every college every university in fluid mechanics features as about hydrodynamic or velocity boundary layer velocity boundary layer or hydrodynamic boundary layer refers to some kind of a region let us I will call it region essentially because I do not like to call it a boundary layer when when we define something we should not be using that word. So if I have a flat plate which is stationary there is a fluid coming in at u infinity u infinity t infinity the plate is stationary fluid mechanics with all the concepts of viscosity wall shear stress so on and so forth has told me something that there will be a region which is formed like this and this we will call as velocity or hydrodynamic please introduce these words also when you each student do not just give velocity boundary layer they should be able to relate to another more scientific term also hydrodynamic boundary layer and let us say this is given by the bracket x. So we said this boundary layer has been formed y when the plate is moving you will get the velocity at this location at the y equal to 0 location if I draw the coordinate axis this is y and this is x at the wall at y equal to 0 what we call as no slip condition tells me that the fluid velocity would be the same as the plate this is called no slip condition and no slip condition dictates that the fluid attains 0 velocity in case of a stationary plate or the velocity of the plate at y equal to 0. So I have a very common example which all of us give probably in class is if you are trying to catch a running bus you have to run at the speed of the bus to be able to enter it obviously if you are going to run slowly you are not going to catch it if you are going to run fast yes you will catch it but you will probably go and hit yourself somewhere because you have to slow down and reach the velocity of the bus. So essentially what this no slip condition is saying that when the fluid is coming on the stationary plate it has to stop ok. So the fluid has a velocity u infinity which we call as free stream velocity and one layer of the fluid remember these fluid are like sheets of paper ream of paper. So I have this set of paper here which is like a fluid this fluid is flowing and for some reason and it is flowing I have to curtail the movement of the water most sheet of paper and all these are talking to each other because it is a fluid and not a not sheets of paper in essentially we are saying that the behavior is like different sheets of paper but they are all going to talk to each other. So once this fluid is brought to rest the next layer of fluid cannot by default go at u infinity it will be restrained because of whatever we call as viscosity shear stress so on and so forth and therefore we will end up getting what we call as a velocity distribution I will I will redraw this because it looks as if the distribution is come from somewhere in the middle and this is the boundary layer. So velocity is equal to 0 at the wall and then this is the this are the velocity vectors and this boundary layer represents if I put across here it represents the locus of points at which u is equal to 0.99 times u infinity. So at this x location this velocity will reach almost u infinity at this height further and further if I move along the length of the flat plate the it takes a longer while for the fluid to reach the free stream velocity what it means is that before that the boundary layer essentially is a region where viscous effects are important. So viscous effects are present here viscous effects are present inside the boundary layer outside the boundary layer viscous effects are absent that is not the correct way of saying we are not saying viscous viscosity goes to 0 no what it means is that in comparison with other forces viscous forces are small therefore they can be neglected in the analysis it is not that viscosity goes to 0 that is the same fluid which is flowing. So for the same fluid viscosity will not go to 0 at one location so on and so forth. So this boundary layer separates the fluid flow field into two parts the part which is inside the boundary layer which is between the flat plate and the so called boundary layer is the region where viscous effects are important that means the fluid carries the effect what is this boundary layer if there was no flat plate when air is going in this room you do not have a boundary layer formed in air when there is no solid surface when there is a solid surface which is stationary which is coming in the path of the flow what is happening is the flow has to adjust itself to take care of the boundary condition of 0 velocity and that taking care happens because of the fluid property called viscosity and the effect any change that you impose somebody beats you you probably move a little forward because of the impact and then you will stop. So that effect is felt for some distance for some time and that effect is what is this velocity distribution inside the boundary layer. So beyond this delta at y greater than delta what we are saying is that the fluid if I am a fluid particle which is standing outside of the boundary layer somewhere here I do not know that this fluid is encountered a stationary flat plate that is what I mean but if I am a fluid particle at position a I do not know at if I am a fluid particle at position b I know that look I was travelling at u infinity velocity and now I have been supposed to come to a lower velocity because of this viscosity effects. So this viscous effects propagates this way and the boundary layer represents the demarcation between what we call as the viscous region and the inviscid part of the flow. In fact inviscid or potential flow that you that is taught in fluid mechanics I do not know how many places it is taught but I think most undergraduate curriculum have this source sink doublet rank in half body etcetera these are all potential flow solution. What it means is you are taking viscosity viscous effects negligible. So that is essentially that you do not even you do not even start the concept of boundary layer at that point. So that is the inviscid fluid mechanics this part and this is the viscous part. Now why am I saying all this we are saying all this because we want to emphasis something very important just as see how fluid mechanics in heat transfer are married it is like you know many of you would have seen your mother or grandmother wear a ear ring with some kind of a gold string which goes around and keeps the ear ring in place they are always together it cannot go one without the other. So fluid mechanics in heat transfer are so deeply coupled that you cannot throw away fluid mechanics why I am saying is this is now when I have when I go to heat transfer exactly the same plate same free stream velocity a surface which is at temperature T s greater than T infinity the flow is coming in at u infinity T infinity this is my y direction this is my x direction. Let us say for now velocity boundary layer let us for example call we have not defined Prandtl number let us say the fluid is such that there is what we call as Prandtl number some property is equal to 1 we will come to that in a few minutes. What happens in fluid when we studied hydrodynamics we did not include the concept of temperature now I am saying logically all of us know if I have a hot plate a tawa or whatever a pan on which you make a dosai to heat the pan there is going to be air which is going to rise because of the change in temperature near the heated surface. If you blow air and try to cool it what happens that the air which is in contact with the heated surface which is in contact with y equal to 0 this part definitely will get slightly hotter and further and further away it is going to get cold. So if this is the if this is the hot plate I should not show my hand if this is the hot plate which is being heated and I keep my hand here I feel greater amount of heat if I take it away I feel lesser amount of heat essentially it is the same air but this air closer to the hot plate has a higher temperature closer further away has a lower temperature all this is common observation there is no equations involved. So when there is a flow which is happening say from this way this direction the air which is coming in contact with this hot surface because of no temperature jump condition will have to reach instantaneously that first layer we talked about the layer of fluid right. So let us say this is the first layer of fluid this first layer of fluid will have to reach the temperature of the solid surface which is T s every subsequent layer of fluid let me call these I have kept it stacked like this some 5 or 6 layers of fluids are there. So each layer of fluid will have a slightly different temperature lower than what is the temperature of the solid surface why is that because if this is the scale represents T s T infinity is smaller than T s the first layer is going to have a temperature T s second layer is going to have a temperature T 1 T 1 is lesser than T s third layer of fluid I am deliberately drawing it far apart T 2 is lesser than T 1 T 3 the third layer third layer of fluid so T 1 T 2 T 3 T 3 lesser than T 2 so on and so forth. So what am I trying to say when I have a temperature difference between the free stream temperature this is called as free stream velocity this is called as free stream temperature when I have this difference the no temperature jump boundary condition requires that the fluid attains the surface temperature at y equal to 0 and beyond that the temperature progress progressively decreases till where does it decrease it decreases until the effect of the flat plate is completely lost. Let us go back to your tawa if I have the pan where you are making dosa if I person sitting on the first floor of your building does not know that you are making dosa because there is no effect of the heat that you are doing but if you keep your hand this far probably there is an effect if you bring it closer and closer there is more effect essentially what we are saying is beyond a certain point in the flow the presence or presence of a change in the boundary condition is completely lost just as you had in fluid mechanics this wall cause the velocity to become 0 at this point beyond that if you are a point sitting here you do not know that the fluid encounter a wall same thing the fluid particle sitting here does not know that the fluid actually has seen a heated surface at the bottom. So, that location where the effect of the wall is lost is called as the thermal boundary layer there is a mathematical criterion will come to that but this thermal boundary layer essentially is formed because the same reason that there is a temperature gradient between the fluid which is in contact with the heated surface and the next layer and this effects propagates in the y direction in the vertical upward direction and the location the locus of points remember one thing in fluid mechanics we have local velocity and u infinity now we have in heat transfer three temperatures T surface T infinity and local temperature. So, in fluid mechanics I could write u by u infinity is 0.99. So, in fluid mechanics I could write u by u infinity is 0.99 basically because I had only this is the local velocity this is u infinity which is the free stream velocity I had only two variables whereas, in heat transfer I have T local x comma y T surface and T infinity. So, I cannot write T by T infinity is 0.99 that does not make any sense. So, what is the criteria it is the we have a condition it is already written here let me just go to that T minus T s is equal to 0.99 times T infinity minus T s. So, I will write it as T minus T s divided by T infinity minus T s is equal to 0.99 what is this has come from the fact that the denominator represents the maximum possible temperature difference. So, this is the maximum temperature difference this is the local temperature difference between fluid and surface. So, when this difference reaches 99 percent of the maximum possible temperature difference we will say that is where the effect of this is the solid surface. So, if I recast this I will get T local x comma y is equal to T s plus 0.99 times T infinity minus T infinity minus T infinity minus T s it is come out badly sorry T of x comma y is equal to T s plus 0.99 times T infinity minus T s. I am just rearranging this equation I am rearranging this equation to get the local temperature. So, I will calculate the local temperature and say that because of this presence of the solid wall this is the value that it has obtained and locus of all these points at various location is what we call as the thermal boundary layer. This thermal boundary layer is given by the symbol delta subscript T and again just as you had in hydrodynamics and fluid mechanics this is also going to increase with respect to x direction why is it going to increase with x direction. Because the flow is going to attain I will have the effect of just as you had as the as the fluid went over the flat plate when more and more portion of the flat plate is covered by the fluid more and more portion is going to see the effect of the no slip condition. So, the it is going to increase with respect to x same thing here this boundary layer will increase with respect to x and what is happening just as in fluid mechanics we said there was in viscous region and an inviscid core or inviscid portion of the flow field. We had written viscous and inviscid was outside here was the viscous part here also we are going to say something similar we do not have something called viscosity, but we are going to segregate the flow field into two parts what is that we will see. So, with this background on why a thermal boundary layer is formed we will go and develop the concept of phantom number and thermal diffusivity further. So, this shape of the temperature profile in the thermal boundary layer dictates the convective heat transfer between the solid and the fluid what does that mean. So, yesterday we had told from definition here let us just go back to the definition of the heat transfer coefficient very elegantly it was told what heat transfer coefficient is it is nothing, but minus k of the fluid d t by d y at y equal to 0 divided by T s minus t infinity what is d t by d y d t by d y at y equal to 0. Just go back I will write it here d t by d y at y equal to 0 represents the temperature gradient at the wall minus k fluid d t by d y is the heat removed by the fluid the layer of the fluid by conduction why is it conduction it is conduction because that fluid is forced to become stationary at that location. So, because of no slip condition no temperature jump condition it is forced to become have a 0 velocity. So, essentially for the first layer of fluid you have conduction heat transfer. So, minus k fluid d t by d y represents the gradient. Now, will the gradient change for different flow situations yes just as velocity gradient change we can also say that the temperature gradient will change for different flow problems what does that mean let me just draw this you have a boundary layer thermal boundary layer let us say let us say d x is same as d t x both the boundary layers are of same thickness we will see when that is possible. And if this is the velocity distribution this slope represents the velocity gradient at the wall similarly thermal T s greater than T infinity the profile will look something like this. This is T s maximum temperature and progressively the fluid is getting heated you will have a temperature profile like this just as you had a velocity profile like this. This is u profile x comma y this is temperature profile T of x comma y T of x comma y. So, this represents the slope. So, tangent to the temperature profile at y equal to 0 d t by d y at y equal to 0 represents the gradient of the temperature at the wall that times multiplied by k f and Fourier's law is there. So, it has a minus sign this represents the heat flux conducted conduction and we say this h therefore, was minus k fluid d t by d y at wall or y equal to 0 divided by T s minus T infinity larger the temperature gradient at the wall larger the temperature gradient at the wall greater is the heat transfer rate what does that mean larger the temperature gradient what do you mean by larger temperature gradient larger temperature gradient means sudden change the larger change in the temperature between the surface and the fluid for example, if the let us go back to the example of you of a pan on which you are making dosa if if that if the pan is only at 40 degree centigrade and the air around is at 30 degree centigrade you will not feel that much warm. But, if the pan has been heated you turn the gas to high and you are talking on the phone you have forgotten that it is on and then you suddenly put your hand on top and see it is really very hot that is what we are seeing larger is the temperature gradient what are you seeing larger heat transfer you are the feeling of warmth that you get is what is it is the manifestation of the heat transfer. So, larger the heat transfer larger is the temperature gradient at the wall the cause of the larger heat transfer is the larger temperature gradient and what is the larger temperature gradient due to it is due to the sudden or high value of T s in this case T infinity is the same the kitchen air is still at 30 degree centigrade T s has become so hot therefore, you know you will say oh this is too hot I cannot make dosa. So, you will sprinkle some water and cool the pan. So, this is essentially representation or manifestation temperature gradient manifests itself in the increased amount of heat transfer coefficient or consequently the heat removal rate or heat flux. So, next point was let us go back quickly these are all concepts. So, we are going quite slow on this, but nevertheless you have to be understood very carefully. So, inflow over heated or cool whatever we have told for T s greater than T infinity is going to be valid conversely for T s less than T infinity there might be questions on that. So, I will quickly draw the profile of how it would be when T s is less than T infinity that is straight forward T s less than T infinity I will have a thermal boundary layer which is still formed the temperature here is T s T infinity. So, I would have a profile which is like this unlike your velocity distribution where this was 0 this is not a 0 value if you are plotting the difference yes this would become a 0 value if you are plotting difference. So, be careful what is being plotted if you are plotting local temperature minus surface temperature if you are plotting T minus T s then at the wall T minus T s is equal to 0 you will get a profile which looks like the velocity distribution otherwise this constant T s this constant value of T s would be added always that is why you are having a non zero here it does not mean no heat no temperature jump condition is violated no temperature jump condition is obviously satisfied because the first layer of fluid has reached the solid surface temperature noting that fluid velocity will have a strong influence on a temperature profile why we have made a statement fluid velocity will have a strong influence on the temperature profile the development of the velocity boundary layer and the thermal boundary layer will have a strong effect on convective heat transfer very profound statement why do we say that well this is a statement we have not justified it all of all of us know the answer for it how does fluid velocity affect the temperature profile just take your example of your pan which is getting hot you do not know what to do so you have turned on the fan and try to cool it very quickly what is happening when the when you turn on the fan you have locally increase the velocity ok. So, there is a fan which is sitting here and the pan is here if you force the air to flow over the flat plate what is happening you can look at it this way also when of course heat transfer coefficient will increase therefore, what will happen is the ability of the fluid to carry away more heat is going to be there that is heat transfer conceptually also let us think the when there is very less velocity associated with the fluid the fluid will just be around that place only it will have the ability to stay there and take away the heat that means it is going to be a very slow diffusion of thermal energy from the hot plate to the air. So, it will it will have its own pace, but when I increase the velocity the fluid is not going to have the luxury of sitting there and it is like you know in your your gulab jamun or rasgula if you put it in a sugar syrup and take it out immediately after one minute the effect of the syrup is not felt inside, but the more it soaks in the effect is felt the same thing the velocity has a similar influence if the fluid is able to sit and collect everything and go away it is going to have a different characteristic different temperature profile and what are we the manifestation of all this heat transfer comes from this temperature distribution. So, the temperature distribution will be different when I have a different velocity if the same fluid is made to be the same air in the room it will it will the temperature profile will be very different as opposed to when you have a fan which where the velocity of the air is at some 2 meters or 3 meters per second. So, fluid mechanics therefore comes in there and this temperature gradient and temperature profile are inherently coupled. So, how this velocity and temperature distribution are coupled that we say there is a fluid property called as Prandtl number all of us use this all of us know what it is in the first quiz we have asked this question 70 80 percent of you have written the definition also. Prandtl number we have made a statement the relative thickness of velocity in the thermal boundary layer is described by the dimensionless parameter which is the Prandtl number why is this definition given like this and then we say we here we talked about relative thickness and then we are writing some molecular diffusivity of momentum molecular diffusivity of heat. What is happening is this is the physical phenomena that is happening what is diffusivity of momentum all this while when we said a boundary layer is formed the fluid here is 0 velocity slightly higher velocity more velocity so on and so forth. What is happening is that that fluid particle is transferring momentum from one particle to another. So, inside the let me let me just go back we are all dealing with steady state situation there is no transient effect involved, but this definition is inherently valid for say if I am saying I am trying to say carry 1 kg of something versus 5 kg of something. So, if I am going to carry 5 kg is from a starting point to end of a line say 100 meters away I am going to take some amount of time if I have to carry only 1 kg of that thing I am going to go much faster. Essentially the time scale is always there in the background so for that same time if I have only 1 minute 1 kg I am going to carry over a longer distance 5 kg I am going to carry over a shorter distance. So, what this Prandtl number tells me is that this fluid particle which is there the time remember time is always there in the background. So, if I say if I do one analysis for one hour and other for 5 hours then this we cannot compare this what we are saying is for the same time involved depending on the nature of the fluid the fluid will be able to transfer diffuse. Diffuse means what transfer momentum and heat momentum transfer is related to the velocity aspect. So, the fluid will be able to transfer momentum and how is the transfer of momentum happening in the boundary layer it is happening because of viscosity. So, it will be able to transfer momentum from this point to this point let us say in a given amount of time and heat the same fluid may not necessarily transfer it to the same location because each one has its own favourites. If you give a child you know potato chips he will eat it much faster than if you give him bitter god or something like that. So, same thing the favourite then quality of the fluid to carry either heat or momentum differently is inherent. If it carries them both very nicely in the same manner very good it is non partial, but most fluids will have a different behaviour aspect for each of these things. So, a particular fluid will carry momentum faster over a larger distance in a given amount of time as compared to heat. So, the say if I say 5 minutes later how the momentum has diffused from what is momentum diffusion the effect of the presence of this solid surface or this change in the boundary condition is felt to a larger distance within the same amount of time as compared to the heat transfer aspect. That means, if I am a fluid particle in say honey or oil I will have this presence of the wall felt at this location whereas, I will have the presence of yeah if I have the flat plate here if I am oil I will have the effect of this change in the velocity felt up to this point whereas, after the same amount of time heat would have diffused only up to this point that is what we are going to say this oil is partial to transfer of momentum from the solid surface to this that is what we are calling as diffusion. So, the thermal boundary layer will develop only to this height this portion whereas, the velocity boundary layer the presence of the wall is felt presence of 0 velocity is felt up to a greater height whereas, the presence of a higher temperature compared to free stream temperature is felt only up to a smaller height that is what we are calling as diffusion of heat diffusion of heat happens only to a small extent heat diffusion all of us can relate to because it is directly related to heat transfer momentum diffusion is what is a little bit confusing. So, the fluid is able to transfer momentum from this to a greater distance. Now, if you take a fluid for which it is non partial for example, where we are having that after a given amount of time the effect of the high temperature that is T is greater than T infinity is felt up to this location probably around the same location itself is where the effect of the 0 velocity is felt both the boundary layer are roughly of the same thickness. And for conversely for liquid metals which are very very high thermal conductivity heat will be transferred very very quickly. So, high thermal conductivity means heat is going to be transferred very quickly that means, in a given amount of time heat will diffuse to a greater height whereas, that fluid will not be able to take away momentum to the same location it will be a very very small region up to which the momentum has diffused. The hydrodynamic boundary layer in this case of liquid metal will be very small in thickness what is this boundary layers boundary layers are regions where up to which the effect of the solid surface is felt let us keep that in mind. So, momentum gets diffused only to a small distance heat is diffused to a larger distance because of a higher thermal conductivity of the fluid. These two quantities which are related to the diffusivity of momentum are put together in a non-dimensional form which is called as the Prandtl number. And this Prandtl number we say is a fluid property mu by alpha, alpha if you substitute it will come out to be a mu is equal to mu by rho, alpha is equal to k by rho C p the rho is cancelled you get mu C p by k. Essentially this Prandtl number will come back to that is a measure or representation of the relative thickness of the boundary layers. So, Prandtl number d by delta by delta t hydrodynamic or velocity boundary layer thickness divided by the Prandtl divided by the thermal boundary layer thickness is of the order of Prandtl number raise to the power n where n is a positive quantity. For Prandtl number of the order of 1 delta equal to delta t that means momentum and heat diffused to the same extent in that fluid. Of course, time is there at the background all the time you cannot compare for different times d delta greater than delta t is for a case where Prandtl number is much greater than 1 that is momentum will diffuse to a greater extent. The presence of the solid wall would be felt to a greater extent in terms of the momentum transfer as opposed to the heat transfer and delta lower than delta t is for Prandtl number less than 1 typical Prandtl number less values are given here liquid metals very very low Prandtl number liquid sodium is used for cooling in certain types of reactors. So, it is for good heat removal rate very low Prandtl number thermal boundary layer is much thicker than the hydrodynamic boundary layer gases are of the order 0.721 water has a wide range of thermal conduct Prandtl number light organic fluid oils are highly viscous. So, let us keep these things in mind and almost all universities will ask this question either in an exam or a quiz. So, please plot for Prandtl Prandtl number greater than 1 draw the thermal and the hydrodynamic velocity and the thermal boundary layer on the same scale. So, for Prandtl number greater than 1 the student should be able to write delta over delta t is of the order of Prandtl number raise to n n is a positive quantity this is the number greater than 1 for this case. So, delta is going to be above delta t and why are we studying all this. So, what what a boundary layer is there. So, on and so what is the big deal big deal comes from a very nice fact that see velocity distribution in this boundary layer is going to be like this I am going to draw a big diagrams this is the velocity distribution temperature distribution for this fluid is going to be like this and in this part the fluid is virtually at t infinity this is t infinity why because the effect of this high temperature t is greater than t infinity is felt only up to the thermal boundary layer thickness beyond that the fluid has a free stream temperature u infinity t infinity which it came with it the fluid particle here does not know that it is flowing over a hot plate. Whereas, the fluid particle at the same location definitely knows that its velocity is not equal to u infinity because it is inside the hydrodynamic boundary layer. So, in in implications from a point of view of solution of problems you will see that where I have a coupling between velocity I am going little bit beyond this your energy equation will have a term like this d t by d x d t by d y so on and so forth in the region beyond the thermal boundary layer temperature is constant. I can simply write the derivative of temperature will be equal to 0. So, certain terms in the big equations that we are going to see later in the day will go to 0 automatically u of course, will be varying does not matter, but temperature remains constant in this part therefore, I can throw away certain terms. Now, if you look at the other case where you have delta t and delta here Prandtl number less than 1 my velocity boundary layer is going to be present only for this part after this the velocity is nothing, but u infinity in this part it is all u infinity it is going to vary only here velocity variation is only in this part temperature of course, is going to vary all through temperature distribution t x comma y is going to be there everywhere. So, my equation u d t by d y or d t by d x which you will see later this u instead of having a variable in this part in this in this region beyond the hydrodynamic boundary layer u becomes equal to u infinity. So, instead of dealing with a variable velocity I will have a constant number associated with it which is a lot easier to tackle mathematically. So, I can simplify the equations because of these concepts of thermal and hydrodynamic boundary layer also of course, they are a good measure of the heat removing capability momentum transfer capability so on and so forth. So, this is some brief history of what professor Prandtl he was born most of the work you know in physics heat transfer fluid mechanics etcetera was done in Europe as you can see. So, Eastern Europe especially Austria Germany Russia so on and so forth and Prandtl we know is indispensable as far as fluid mechanics is concerned because the boundary layer concept has come primarily because of professor Prandtl.