 So, what we will do is before we go to derivation of the momentum equation which is anyway quite involved is going to take some time, we have not taken up any questions since morning. So, we would like to start with questions some question answer sessions for now. So, we go to YC college, Nagpur. So, any little question about the Prandtl number criteria which is cited in the morning session, actually you cited when the Prandtl number is greater than 1, then there is a very less diffusion of the heat, but as we see in the chart the Prandtl number for the gases is very low, but the Prandtl number for oil is more than 1, first still we are preparing the oil cooled engine or the gas cooled engine, why this is so? The question asked is the Prandtl number of oil is much larger than that of Prandtl number of air. So, why is still oil is preferred over air in oil thermal boundary layer thickness is smaller compared to that of air? So, that itself is answering directly. Number one actually first of all why are we using oil is not from boundary layer thickness because its specific heat is very high. So, its heat carrying capacity is very high that is the reason why we use oil. So, it is not directly related to boundary layer thicknesses. So, it is related to specific heat, its heat carrying capacity if I take for a given mass flow rate MCP delta T is very high in case of oil compared to that of air compared to that of even water. So, CP of air is 1050 and CP of oils will be more than water which is water CP is around 4180, I am sure CP of oil is much more than 4180. Please go ahead. With reference to the previous discussion I will make one comment sir. Please correct me if I am wrong. If there is we have to we have to select certain fluid for the specific application then the criteria of the Prandtl number will not be sufficient. We have to take the specific heat and the other thermo physical properties will account. The question asked is if we have to choose the fluid for any specific application for maximizing the heat transfer actually it is little premature but still I will go ahead and answer this question. See actually if we see irrespective of the Prandtl number we are going to see let us say it is flow over a flat plate. So, Nusselt number is going to be a function of Reynolds number and Prandtl number. Now that we have defined Reynolds and Prandtl I think we can we can appreciate this Reynolds and Prandtl quite easily. But the function the functional form that is the Nusselt being dependent on Reynolds and Prandtl that functional form will be different in case of Prandtl numbers less than 1 and Prandtl numbers greater than 1. So, using those Prandtl numbers we will compute the Nusselt number in turn we will compute the heat transfer coefficient whichever is having the higher heat transfer coefficient is the one which we are going to choose. So, directly we cannot say based on CP I am going to use the oil or based on density I am going to go for air or water or oil it is not so. It is based on Reynolds Prandtl using which I will take the heat transfer compute the heat transfer coefficient and whichever fluid is fetching me higher heat transfer coefficient I am going to choose that fluid. So, but still I think we will understand this in great detail tomorrow when we compute the Nusselt number distributions for flat plate case and pipe flow case we will appreciate this problem in greater detail. For now this much should suffice PSG college Coimbatore. Yes, this is with respect to the problem on the fins some two days back. So, we occurred there for individual fins the Q and un-fins also and we added both in and multiplied by the number of fins which has been given as 200. But number of un-fins will be 201 that we have left. Yeah, actually the question as these for finned problem there is for finned portion and the un-finned portion we have multiplied by 200. Yeah, one of the participants one of the professors is saying that for fins it will be 200 and for un-fin it will be less than 199 it will be less than 200. Yes, professor you are right we have just neglected that because the area of the un-finned portion is very small, but yes you are right I have to un-finned portion I have to take 199 and fins I have to take 200 you are right professor. Thank you. Go for it. Sir, I have a question regarding the fetch. So, how it come possible to have a temperature gradient if y equal to 0 that means no slip condition in which T s equal to T s of the surface is equal to a temperature of the fluid that is at the rest condition and in contact with the surface. Your question is when I say no temperature jump condition the fluid in contact with the surface is at T s same as the surface temperature right then how is the possible for us to have dT by dy at the wall which is greater than 0 is that your question. Yeah, yeah of course. So, actually what we are saying is the heat is going to be transferred from this fluid to the next layer of fluid which is going to be at a lower temperature correct the surface fluid layer will be at T s, but the subsequent layer there is going to be a slight difference in temperature. So, that difference is what is manifested as dT by dy at the wall what we are saying is what you are saying is correct velocity is very easy to understand because one of the quantities is 0 other is a finite velocity ok next layer of fluid is at a finite velocity here also it is the same thing you have one fluid which is at one layer of fluid which is at T s next layer of fluid which is at a temperature if you are heating the fluid next layer of the fluid is at a lower temperature. So, there is going to be a gradient between the first layer and the second layer and that is what is going to cause the heat transfer which is given by minus k dT by dy at the wall we are doing it at the wall because it is we are doing it for that elemental for that element in contact with the wall that is all ok. Sir second question can we say at a lower Prandtl number conductive heat transfer through the liquid is greater than the convective heat transfer. The question is for lower Prandtl number can we say conductive heat transfer is greater than the convective heat transfer the answer is as follows irrespective of the Prandtl number the concept that the heat transfer within the boundary layer does not change what we are saying within the boundary layer the conductive heat transfer that is minus k del T by del y at y equal to 0 is always going to be equal to by what we got by Newton's law of cooling that is h into T s minus T infinity irrespective of the Prandtl number this concept is not going to change because h into T s minus T infinity has to be equated with the heat transfer by conduction within the boundary layer. So, this statement has no relevance with respect to only a particular Prandtl number or Reynolds number for any fluid for any Reynolds number for any Prandtl number conduction is equal to convection within the boundary layer. In fact, the definition of h does not involve anything like Prandtl number at all it is only the thermal conductivity of the fluid which is coming in by virtue of Fourier's law. Now, I understood your question why you are saying heat transfer coefficient is going to be higher for higher Prandtl number is it because of lower k no you we cannot look at k itself correct we have to look at the temperature gradient for lower Prandtl numbers my temperature gradient would manifest itself in one particular fashion for higher Prandtl numbers my temperature gradient would manifest itself in a different fashion. So, how del T by del y at y equal to 0 is going to pop up none of us can predict that we have to measure or compute it is not easy just by seeing k or C p or mu we cannot tell that. So, from the definition of the Prandtl number the molar diffusion at lower Prandtl number molar diffusion is lower as compared to heat transfer diffusion. Yeah. Molecular diffusion. Momentum. Momentum diffusion. That means the what will be the mode of heat transfer at the lower value of Prandtl number. See that is what I am trying to see see who is causing this heat transfer who is causing this heat transfer actually let us go back to fluid mechanics. The answer can be answered by the question asked is Prandtl number is defined as momentum diffusion to. Heat diffusion. Heat diffusion. So, for lower Prandtl number the expected thing is heat diffusion is less it cannot be answered just directly like that. See who is responsible for the heat diffusion again the velocity gradients again the molecular momentum diffusion and also remember in the morning I said the momentum diffusion is can be characterized by velocity gradients and also the turbulent stresses turbulent stresses. So, how the turbulent stresses are going to be there are again dependent on the Reynolds number. So, that is what we are saying we cannot directly tell for lower Prandtl number lower heat transfer for higher Prandtl number higher heat transfer no it is decided by my stresses that is the turbulent shear stresses and the laminar shear stresses. But how whom whom on they are dependent on they are dependent on the Reynolds number. So, it is all it is all mixed group effect it is not individuals effect just by directly Prandtl number we cannot say yeah V I T Pune. Average convection heat transfer coefficient and local convection heat transfer coefficient is this concepts are applicable for flat plate solar collector also. Ok the question asked is the concept of average heat transfer and concept of local heat transfer coefficient is it applicable for solar collector plate also. I would go ahead and make a very generalized statement it is applicable everywhere. The heat transfer coefficient are not going to be same for any application whether it is flat plate or solar collector plate or flow around a cylinder or flow in an air flow around an aerofoil or a gas turbine blade cooling passage or a circular pipe. Why do I make that statement heat transfer coefficient is dependent on temperature gradient who is deciding this temperature gradient my velocity profile my velocity turbulent stresses and these stresses and these gradients are going to be different in different location. So, naturally my temperature gradients are going to be different at different location. So, naturally I should be having the heat transfer coefficient different at different location. So, given any application heat transfer coefficient will not be uniform, but in most of the papers you might have seen that heat transfer coefficient is constant even in flat plate case. Why because he might have taken the flat plate very thick plate if you take a thick plate and put one thermocouple there actually who is averaging we are not averaging the thick plate is averaging and the thermocouple is measuring that average temperature using that average temperature I am getting the average heat transfer coefficient essentially averaging is not done by computer or calculation, but it is done by the copper plate. But on the other hand instead of copper plate if I take a thin metal foil let us say stainless steel metal foil which is as thin as 0.05 mm then I can if I lay thermocouples all over the place I will get the local heat transfer coefficient. So, the point is local remains local no matter what is the application that is the answer for this question. What about plate area temperature is ok what about area of plate see the question is temperature is ok what about the area of the plate the question is if the temperature is varying everywhere where is the question of area coming into picture. Small area or large area there will be variation there is variation. So, if you take small area over which you have put the thermocouple I said that is what I said if you put a thermocouple putting multiple thermocouples does not mean anything the plates thermal conductivity means a lot if I why did I take the example of stainless steel foil because two things there stainless steel is made of lower thermal conductivity than that of copper and stainless steel I said thin metal foil that means it ensures that averaging is not being done. So, it is not about area professor it is about averaging it is about again temperature gradient. I hope you are convinced. Sir there is one more question ok while we studied different motions of the particle we have seen that particles get stretched. Yes. Is it the particle or small mass of the fluid gets stretched and second question is how and why particles will be stretched. The question asked is how the fluid particle is having a certain fluid mass or not that is part number one and part number two how is the fluid particle getting stretched. See first question yes when I say fluid particle we have not stated one thing I want to state this very clearly whatever we are studying in this fluid mechanics we are studying actually continuum mechanics that is the we say that whatever effect is there if I take whatever cubes we are drawing all over the place today for conservation of mass or momentum that cube is a small cube we can imagine that as 1 mm by 1 mm by 1 mm. So, if I imagine a 1 mm by 1 mm by 1 mm cube definitely it is containing fluid mass may be of approximately I think my number might be wrong that is it is of the order of 25 into 10 to the power of 6 number of molecules would be there. When I say a pressure within that infinitesimal controlled volume I am actually saying that I am actually saying that it is the net effect of the various molecules which are colliding with each other and that is the net pressure. This is what we have studied even in thermodynamics. So, the answer for your first question is that the small infinitesimal fluid particle fluid particle is a small infinitesimal volume and it is consisting of a mass of a fluid as much as it can get filled based on its density based on its density within that fluid volume number 1, number 2 who is stretching it who is stretching it how does it flow how does it flow it cannot flow on its own it has to be there has to be some pumping power even if there is a head that head has to be generated either by potential head or by a pump. So, when I am pumping when I am pumping the fluid particles that pumping power is pushing because of that pushing nature there is stretching or angular deformation or translation or rotation this is what is happening. So, who is stretching my pump is stretching while it is flowing is that ok professor. Hello sir my question is related to somewhat general query in the conduction that is I want to have the explanation regarding two terms that is equivalent thermal conductivity and the variable thermal conductivity equivalent thermal conductivity. Ok the question asked is I want to have a definition of equivalent thermal conductivity and variable thermal conductivity no problem confusing question what we mean by first I will take variable thermal conductivity what I mean by variable thermal conductivity is there are n number of questions asking thermal conductivity varies with the temperature or we took k equal to k naught plus a t where a can be positive or negative that is variable thermal conductivity that is the thermal conductivity varying with the temperature. Now, let us come to equivalent thermal conductivity equivalent thermal conductivity concept is usually introduced like this in fact we answered for all eyes for example we have material a material b we mixed it and we came up with the plausible explanation I do not say right explanation plausible explanation as k equal to k a k b upon k a plus k b this can be told this can be considered as equivalent thermal conductivity is that ok. Sir there is one experiment in the heat pipe that we have to find the equivalent thermal conductivity of the heat pipe ok this is an average experiment see the question I ok now I understand in heat pipe you have something called equivalent thermal conductivity so without details I will not be able to answer this question I would implore upon you to put up the details of your experimental setup and this equivalent thermal conductivity definitely we will come back to you ok then yeah. Sir there are two ingredients for the acceleration unlike a material flow one is one that one that varies one that varies with the time and other varies with the space that means the relative position of the particle with reference to some data. So, traditionally we define acceleration as something that varies with when the velocity varies with time so I just cannot comprehend how we can call that second component as acceleration is that physically possible to call that second as acceleration ok. So, the question asked is it is very difficult to physically feel this is regarding substantial derivative or the total derivative in which we have acceleration or the change in the velocity with respect to time and also change in the velocity with respect to space. See in traditionally in plus 2 at least I was introduced to acceleration in plus 2 where in which I was told that v equal to u plus a t that is a equal to v minus u by t that is what I learnt. So, for me also acceleration always was getting accelerated that is getting picked up my car is getting accelerated when I put on my accelerator I think even the world accelerator comes from that. So, but however what we mean by acceleration is increase of velocity with respect to time or with respect to space it does not tell. So, that is what we mean. So, acceleration means it is increase of velocity professor took up the example of nozzle even in steady flow there is increase of velocity, but not with respect to time because it is steady flow, but with respect to space. So, all that I think we need to comprehend or we need to appreciate here is that acceleration mean increase of velocity with reference to what we have not told, but in plus 2 we have been taught usually acceleration as velocity increase of velocity with respect to time, but all that I would say is that acceleration can be with velocity increase with respect to space or time. Just let us go back to the whiteboard where we had drawn the nozzle see if you try to draw the stream line let us take there are 5 particles and you are tracking the 5 particles the stream line represents the relative position of each of them. You will see the stream lines are far apart here, but when they come to the exit of the nozzle they come very close to each other. So, imagine you have room in a room like you as your 5 of you sitting in a large room you have you can sit wherever you want, but if you are going to get out through a small door all 5 of you at the same time you are going to come very much closer to each other that is what is this concept of change in position relative to one another. So, that is causing an acceleration in a fluid particle n i t 3 chi over to n i t 3 chi. With reference to the transient heat conduction you said the long infinite cylinder could be considered as a short cylinder and flat channel I do not understand what this the short means how do you consider that as a short cylinder and a flat. No the question asked is a long cylinder and a long cylinder can be considered as a short cylinder and a flat plate that is the question that is the question asked, but the answer is actually the question seems incorrect because a short cylinder can be considered as a combination of a long cylinder and a plane wall it is not the other way. A short cylinder is formed by the intersection of a long cylinder and a plane wall which is cutting through the cylinder. So, what we are saying is the two second dimensional effect of this shortness in height comes in when we take into account this whatever you say product solution of this long cylinder and the plane wall which is in the y direction. So, it is not that a long cylinder is not composed of a short cylinder and plane wall that is not correct the correct thing correct question should be short cylinder how is it a combination of long cylinder and plane wall. Any other questions from here? Sir, in journal purpose they are referring in the internal flow when we are inserting some turbulators we are disturbing the we want to disturb the thermal boundary layer why we want to disturb the thermal boundary layer when we are disturbing the thermal boundary layer there is increase in the heat transfer, but when we are inserting that any turbulator there is decrease in increase in pressure drop. The question asked by one of the professors is that when we put the turbulators in the pipe actually we are breaking the thermal boundary layer because of which there is increase in the heat transfer coefficient and this also results in increase in pressure drop it is little premature actually I am going to deal with this in convective heat transfer in great detail. However, let me go ahead and answer because I have introduced already laminar sub layer what we are doing when we put the rib turbulators or any turbulators or any enhancers is that in the morning lecture I had introduced laminar sub layer, buffer layer and turbulent boundary layer. What we are saying is that when we put this rib enhancers or rib turbulator we are going to break this laminar sub layer such that my boundary layer becomes completely turbulent and we all know that turbulent shear stresses are orders of magnitudes higher than the shear stresses because of laminar that is because of the velocity gradient. So, my heat transfer coefficients will go up significantly so that is the reason why we put the rib turbulator so, but at the same time pressure drop is increasing because my shear stress is also increasing that is because of minus rho u prime v prime shear stress is increasing means resistance for the flow is increasing that means pumping power required should be high. So, that is the answer for your question but to get more insight into this as we go along actually we will be dealing this in greater detail in internal flows this much would suffice for now for this question. Jabalpur college any questions please. Yes sir I want to know only three layers in turbulent section not more than three why. So, the question asked is in turbulent boundary layer we have classified three layer structure that is laminar sub layer, buffer layer and turbulent boundary layer why only three why not four why not two why not one ok that is a valid question. In fact, historically if we see historically actually there are only two layers that is one was laminar sub layer and the other one was turbulent boundary layer, but when they did the Karoo fitting with the shear stresses whatever they measured and then they plotted there was continuous there was sudden discontinuity when they ended up in the buffer layer that is in the buffer layer both the laminar shear stresses and the turbulent shear stresses are of equal magnitude. To make the Karoo smooth they introduce this concept of buffer layer. In fact, if I remember correctly this was introduced by professor von Cormann it is called as von Cormann's mixing layer concept buffer layer concept. So, it was professor von Cormann who introduced this intermediate layer that is the buffer layer. So, that answers you why only three layers with these three layers we are able to model the complete boundary layer that is the reason we are sticking to three boundary layer why not more than three because we do not need more than three to model this that is why we are stopping at three, but people make simplifying assumptions and go ahead on three also, but we know that if we take three we can smoothen out our shear stress distribution when we move along the y. Okay, VJTI Mumbai any questions? Yesterday I was asked the question regarding the boundary layer. Within the boundary layer the fluid flow is there as per our today's discussion. I have answered your question. I think it takes less. But repeat your slide. Conduction heat transfer within the boundary layer. The question asked is yesterday also one of the professors had asked the question within the boundary layer whether my fluid is still. So, we had said that we had answered today during the class session know the there is flow in the boundary layer, but now the question posed is a very valid question asked by the professor if the fluid is flowing how can you apply conduction equation Fourier's law of conduction, but what are we doing? We are not applying for the complete boundary layer. We are applying just on the layer above the flat plate and at that flat plate there is no slip condition, no temperature just the first layer of fluid. The first layer which has zero velocity. Yes, that is minus k del T by del y at y equal to zero y equal to zero there is no slip condition it is stationary. So, there is no what to say paradox in this it is straight forward it is straight forward. I hope now you would have understood what we mean by conduction heat transfer. So, conduction happens in the first layer of fluid and that heat from that first layer has to go out by convection. So, if you think of this flat plate or the thin layer of fluid E in is by conduction E out is by convection. That is right. So, much about questions we will move on to Navier-Stokes equations now. So, far we have studied what is called as continuity equation. So, I am just going to look at the same equation in little different way let us see what can we or in which way we can look at it that is what professor taught us is that so we have the equation d rho by d T plus rho of del u by del x plus del v by del y plus del w by del z all that I am saying is I am just going to connect this equation with my stretching that is all that is with the fluid dynamics basics what we studied that is what am I trying to say here is that yeah. So, what we are saying is that see we said that volumetric dilation rate that is del u by del x plus del v by del y plus del w by del z is equal to 0 that means net change in the volume is 0. So, when can this occur physically when the density is constant when the density is varying the change there will be change in the volume so that is what it has been restated here in this equation. If the density is constant if the density is constant my del dot v would be equal to 0 this is what we will be using for continuity equation. Moreover we should keep telling our student we always write one dimensional conduction equation rho 1 a 1 v 1 equal to rho 2 a 2 v 2 in fact this is also coming from this equation the only thing is that this is one dimensional conduction equation sorry continuity equation. So, another thing is that I just want to give another way of looking at this total derivative I have one standard example which I always quote I just want to touch that and then go if I write a total derivative of temperature that is d t by d t that is I will have to write that as del t by del t plus u del t by del x because total derivative does not only tell about acceleration I just want to say that this total derivative concept can be used even for temperature that is the reason why I am writing this although this is little early, but it is ok because this is the point where we have to understand the total derivative concept of the substantial derivative concept plus v del t by del y plus w del t by del z. What does this mean del t by del t means temperature variation of temperature with time and this variation of temperature with space by virtue of moving by virtue of moving because of velocity u v and w. Professor Anderson in his fluid aerodynamics textbook he states this example this is what he gives I am just copying his example and telling you restating it same, but I like this example that is why I am restating it see let us say I am entering a room I am entering a room let us say this is the room where in which it is fully ac I am entering from outside to inside when I am entering means what I am having a velocity that is let us say u let us not worry about v and let us not worry about v and w only u is there. So, if I am moving with velocity u, but I am going to undergo a temperature gradient why because I am entering from a hot air environment to cooler environment that is this u del t by del x that is why we call this as convective term convective term, but what is this while entering let us say all of a sudden I have entered, but all of a sudden someone has hit me with an ice cube or a hot ball. So, I am undergoing going to undergo temperature change with because of hitting instantaneously I just before I realized someone has hit me with the ice ball. So, that is del t del t. So, it is not always the total derivative is about acceleration total derivative can be temperature or total derivative can be with the density also. So, it is a it is a concept all that it says that any parameter is going to change not only with time, but also with space that is all it is telling. In case of fluid flow this is very important because with see the derivative concept itself you have u v w if there is no flow then I have I am coming back to only transient that is del by del t. So, with this I think I will move on to what is called as navier stokes equation that is the conservation of momentum. So, this is again same thing I will reuse the same thing that is m dot in what did we use conservation of energy e dot in minus e dot out plus e dot g is equal to e dot st. Instead of that here what we are saying is force is equal to mass into acceleration that is there is a net rate of increase of momentum and there is some momentum entering and there is some momentum getting out. So, the conservation of momentum what does it say force equal to mass into acceleration that is what we apply it to system, but if I have to apply it to control volume I have to apply Reynolds transport theorem. So, this is the change in control volume and this is the change in the control surface that is rate of increase of x momentum I am writing it only for momentum in the x direction x momentum and rate at which momentum is entering in the x direction and the rate at which the x momentum is leaving in the x direction this has to be balanced by whom rate of change of momentum has to be balanced by whom it has to be balanced by the forces applied which forces the forces applied in the x direction that is the sum of the x component forces applied in the control volume, but now the question is what are the types of the forces which are applied. So, there are two types of forces one is surface forces another one is body forces who is creating these surface forces these surface forces are created by normal stresses shear stresses and pressure. So, what is shear stress just now in the morning we have found that shear stress is the one which is caused because of angular deformation angular deformation there is delta alpha and delta beta that is the angular deformation. Now what is normal stress normal stress is because of stretching it is because of stretching or compressing. So, that is normal stress that is because it is acting normal to the surface when can it act normal to the surface when it is either stretching or compressing. So, that is normal stress and pressure we all understand, but always there is a confusion there is a normal stress and there is a pressure also what is this pressure business that is the question always we use we always myself and professor Arun after the class we always come back with two three students who ask us this question the answer is or the way we understand this is like this what is pressure what is pressure when does when to understand pressure I always ask the question back to the student when is pressure zero when do you say pressure is zero when it is perfect vacuum when it is perfect vacuum pressure is zero. What is perfect vacuum how do you define perfect vacuum or how do you realize perfect vacuum when there is no molecule when there is no molecule then only we call that as perfect vacuum that means when there is no molecule then there is no question of collision taking place when there are plenty of molecules they hit each other then there is a collision because of this intermolecular collision there is pressure. So, that is pressure, but then what is normal stress normal stress is because of stretching or compressing either del u by del x del v by del y or del w by del z. So, that is because of the velocity gradients remember pressure is because of the intermolecular collisions. So, this is a very important concept very important concept which we need to differentiate between normal stresses and pressure. So, I think this is this is very important I hope you have understood. So, professor Arun wants to pitch in and clarify little bit more. See in fluid mechanics pressure concept was introduced in statics itself. So, non-moving fluid also can have pressure, but hydro static pressure that is why the word static is there hydro static non-moving pressure. So, we have to keep that in mind irrespective of whether the fluid is stationary or in motion there is a pressure associated with it only when there is a velocity component associated which causes this kind of a normal which causes a stretching or elongation in a particular direction there is a normal stress both of these are force per unit area only both of these act normal perpendicular to the surface, but the cause is different all stresses will go to 0 in a stationary fluid except pressure force. So, keep this in mind and I think all students will have these doubts. Yes, you will be you will be asked by your student these doubts these are normal questions. So, next is this is as already professor has told this is force per unit area these surface forces are force per unit area. Now body forces are the forces per unit volume what are the body forces gravity force centrifugal force and Coriolis force these are called body force because they are acting all over the volume that is. So, gravity force I think all of us can understand easily and centrifugal force also we can understand whenever there is rotation or whenever there is we are going across a band there is going to be centrifugal forces even when there is a flow in a band 90 degree band it is going to experience centrifugal force. Then what is this Coriolis force the Coriolis force is that force let us say we have a pipe and in a pipe there is a flow taking place at the same time this flow is rotating that is orthogonally rotating that is this pipe is there this my hand which is there towards you is from inward to outward there is a flow from inward to outward there is a flow that is from myself to yourself there is a flow and this is my pipe and this pipe is rotating orthogonally it is rotating like this continuously it is rotating. So, in that case the fluid particle which is moving which is having linear velocity and also rotational velocity is going to experience an additional force called Coriolis force. So, this is the thumb rule what we use right hand thumb rule that is this is the rotational direction this is the fluid direction this is the Coriolis acceleration. So, the Coriolis force would be opposite to that. So, that is what is Coriolis force and this is acting all over the place that is complete volume. So, that is why we are we are using this as a body force this is the Coriolis force. So, you have gravity force Coriolis force centrifugal forces you can have electromagnetic force also that is also a body force one of the participants was asking on the other day how does thermal conductivity vary with a body force which is electromagnetic force electromagnetic force is also a body force with each body force my nature of my conservation of momentum equations or the Navier-Stokes equations are going to change each body force taking it affect effect will be one phd. So, that is how we can look at the body force what we study in the class is we do not consider these Coriolis forces and centrifugal forces we just put it as body force, but we neglect them that is how we handle for getting the closed form solutions in the class, but if you have to account any of these body forces that we that will be involved exercise even if it is laminar flow. So, now coming to conservation of momentum what I will do is I will just state this so that is let us say I have a cuboid let us say I have a cuboid let me draw this I know it is little involved, but let us try to draw this before we we cannot complete momentum equation, but let us draw this and then sign off let us say this is my q again infinitesimally small volume I have delta x delta y delta z delta x delta y delta z. So, what is the momentum in the x direction on this surface delta z delta y what is that mass into velocity that is the momentum which is entry what is the mass what is mass mass flow rate rho a v so rho is rho area velocity is velocity in the x direction we have stated that u and area is delta y delta z delta y delta z. So, this is the mass flow rate and momentum to get the momentum I should be taking mass flow rate into velocity that is u. So, this gives the momentum which is entering into the control volume in the x direction this is what I we had stated in the when we made the statement that is the rate at which the momentum enters, but here we are worried about x direction this is the x momentum direction. Now, what is the momentum getting out the momentum getting out is the momentum getting out is again Taylor series expansion. So, the same thing f of x I will get f of x plus del f x by del x. So, that is u rho u delta y delta z rho u delta y delta z plus of del of rho u delta y delta z plus of del of rho u square upon delta x delta x delta y delta z. Here please read this as delta z I am not able to complete that delta z. Similarly, this is straight forward there is no problem in this, but in the y direction there is little confusion always. So, we have to be little careful. So, you might be wondering we are talking about x momentum, but I am writing in y direction, but what we are saying is there is mass flow rate let me put what is the mass flow rate in the y direction is rho v into area that is delta x into delta z. This mass flow rate can have momentum in the x direction by virtue of x velocity that is u this is the x momentum carried by the mass flow rate in the y direction because of the velocity in the x direction. So, now it is straight forward having understood this it is like a jet being deflected because of velocity in the x direction. So, I will have the getting out mass flow momentum in the y direction is rho v delta x delta z into u plus del by del y of rho u v delta y delta x delta z. I think two long equations you are all getting tired it is fine. So, we will continue with this derivation as we at 2 o clock again. So, professor Arun wants to pitch in and tell something. This can you just go to the diagram yeah this your your diagram. This fluid mass coming in the y direction having a momentum in the x direction is analogous to you wanting to cross a road and there are bunch of people walking along the road. So, when you are trying to cross the road you are not going to get a free path to cross you will be pushed along with the people and then only you will be able to cross. So, that is the carrying you are carried physically by the people who are pushing you and in the process you end up crossing the road. So, you are displaced from that particular x coordinate because of this motion of the people. So, I think we will end this session. Thank you.