 So, now for little bit more physics or you know fundamentals, internal flow lends itself to lot of discussion actually, it is very, very conceptually what you say stimulating, I think that is the right word because you think we think we have understood many things but we have not understood honestly. So, today I understood many things when he taught external flow. So, same thing I, internal flow unfortunately again as it is taught in the undergraduate curriculum is just correlations unfortunately and that is in fact how far away from reality that you are that is what we are doing. Internal flow can be taught very beautifully when you again couple fluid mechanics with heat transfer. Now, as I said you cannot separate you are like the two wheels of the train you have to have them together. So, always always always I have to keep that in mind. So, just whatever if I have a doubt fluid mechanics will tell me oh this is the reason for it I can explain many things because of that anyway. So, internal flow refers to any flow through a confined environment, confined geometry, circular pipe, rectangular pipe, triangular any such geometry and so all of us are familiar with this concept of average or mean velocity. So, when we when we give problems or when we have seen textbook problems oil flows through a pipe with an average velocity with a velocity of 2 meter per second it is essentially the average velocity which is given to us. Everybody knows that there is a velocity profile there is a distribution associated with respect to r we do not we do not worry about center line velocity or velocity at r by 2 or anything we just give the average velocity. So, this slide essentially tells you how to calculate the average velocity this average velocity has no physical meaning except that as he has rightly said it is our engineering approximation, but it is not an approximation it is just a number which is obtained very nicely by one fundamental equation which is conservation of mass. So, I am equating the mass flow rate calculated by means of this average velocity to the actual mass flow rate which I will get by using the velocity profile. So, rho cross sectional area times vm which is the average velocity books will call it u bar v bar u m v m whatever it is it is just the mean or average velocity that is equal to rho v dA this v is the local velocity. So, conventionally we are used to doing this to help students understand I will also do that we take a circular pipe and we say for a fluid element elemental ring of radius at radius r and thickness dr the area of this ring is nothing but dA c is equal to 2 pi r dr and velocity at that for that ring is r comma x question. So, I will say dm dm is essentially dm dot is essentially rho dA c times v of x correct m dot is rho A v. So, I am writing the mass flow rate through the differential element and when I integrate this over the entire cross sectional area it will give me the mass flow rate through the cross section of the pipe that I am setting equal to rho v r comma x 2 pi r dr equal to some average mean pseudo whatever you want to call some value times rho times cross sectional area of the pipe is given by pi r square. And when I do the maths I will get v mean is nothing but 2 by r square integral 0 to r p of r comma x r dr. So, this is all given there and this is essentially the average velocity. So, if I know the velocity distribution then I can calculate compute evaluate the mean or average velocity which is what is given to an engineer. I need a velocity of this our Reynolds number is based on this average velocity. So, though there is a velocity distribution we do not care about it we do care about it, but for calculating Reynolds number etcetera we use this average value. And the profiles are rightly drawn here the idealized profile is shown like this is this correct physically it is not correct right because it violates no slip condition. So, this is an idealized profile this is the actual velocity profile and when students draw it also if we say these are some common mistakes they make they will draw the vertical line somewhere here to show it cannot be in a logically it has to be much smaller than the value of v max common mistakes which students make. So, I think that concept of mass balance if it is ingrained such mistakes will not happen. Now, if there is a mean or average velocity then there should be a mean or average temperature ok. So, temperature profile yesterday when we did boundary layer we told about the temperature distribution. We said if surface temperature is greater than the fluid temperature let us just revisit this for internal flow I will have this is x is equal to 0 and this is the center line of the pipe r equal to 0 r equal to r y coordinate is measured from the wall of the pipe r is measured from the center line upward fluid is coming in with a velocity u infinity sorry u mean and T infinity is the free stream temperature ok. So, we all know hydrodynamic boundary layer let me just draw it for the sake of completeness if a student has successfully completed fluid mechanics he should be able to draw this for us right. What do we segregate the pipe into now this is the hydrodynamic boundary layer delta of x. This region which is inside the boundary layer is where you have viscous effects this part which is outside is the inviscid core. This inviscid core progressively decreases because you have to think in three dimension the boundary layer develops all around the circumference and ultimately at some location it will merge completely and this we will call as the hydrodynamic entrance length or entry length all of us are familiar with this. If I have to draw the velocity profile at any of this location it will be something like this something like this and then a straight line in the inviscid core why is it a straight line in the inviscid core will it be the same as the free stream velocity it will be greater why will it be greater mass balance. So, if the velocity initially or inlet velocity I will just draw it here for want of space is this average velocity V m shown by a line of length so much this mass flow rate is rho times cross sectional area of the pipe times V m that has to be balanced by this whole area because in the boundary layer the velocity distribution is such that the velocity is smaller than V m this magnitude of the velocity in the inviscid core has to be larger than the velocity of V m it is like a cone which is extruded how many of you are with me on this concept velocity at alaswamy velocity in the inviscid core will be larger than the average or mean velocity within clear everybody because see these are things which a student always has a doubt why why why and books will not explain because this is obvious what is to explain these things when once I come to a region which is beyond the hydrodynamic entry region what do we call this flow in this part fully developed everybody familiar with this term fully developed flow what do you mean by fully developed flow we answer this question yesterday also fully developed flow hydrodynamically fully developed so if I take the velocity distribution here I am drawing a profile the V max here will it be smaller than V m or larger than V m larger than V m for the same reason because in these portions the mass flow rate velocity is smaller than V m this is my pseudo V m this velocity distribution whatever I have drawn paraboloid three dimensional paraboloid or figure obtained by rotation of this paraboloid it is called as a paraboloid this paraboloid will have the same form at any x location downstream downstream of the portion where the boundary layers are met that means in the hydrodynamic entry region all this activity will happen beyond that if I take a velocity profile at x 1 and I measure the velocity profile at some other downstream location it should come out to be identical if not then the flow has not yet developed fully. So, this is fully developed flow. So, I think this is also there here a little there is nothing which will not be there in a set of slides the slides are so complete thanks to it. So, velocity boundary layer velocity profile this is the inviscid core the inviscid core region shrinks and this is the hydrodynamically fully developed velocity distribution and we have to write this d by dx of V is equal to 0 that means the there is no change in the velocity profile downstream from this location. Such a region we will call as a hydrodynamically fully developed region. Now, let us come to temperature life becomes a little difficult as we saw yesterday we had already in fluid mechanics we had two velocities local velocity u and u infinity in case of external flow. Now, when I did internal flow I have a local velocity and a mean velocity temperature external flow I had t surface t infinity t local here in internal flow I have if I have a heated pipe I will have a surface temperature T s the fluid will come in with some initial in a temperature which is t infinity after that the t infinity has no value. So, then there is a local temperature and then one very very important very very useful measured temperature which we call as mixing cup temperature. So, we say water coming out at 80 degree centigrade is 80 degree centigrade is essentially if you take water flowing from that heater or geyser or whatever take it in a cup stir it well put in a thermometer it will show 80 degree centigrade that is what it means it is called as a mixing cup temperature it is not a magic what it is just saying that it is a well mixed homogenized mixture where all the temperature variations across the radius is nullified or it is the energy thermodynamic equilibrium that is what it means. So, this mixing cup temperature is playing the same role as your average or mean velocity. So, this temperature is what we use in most engineering applications you say that hot oil at 100 degree centigrade is used to heat cold water heat exchanger classic problem which are there what is 100 degree centigrade when the when it is flowing through the pipe we know that the temperature is going to change along the radius. So, this 100 degree centigrade refers to the mean temperature properties are evaluated at this temperature calculations are done based on this temperature m c p delta t temperature at inlet at outlet we take only the average value what is this average or mixing cup value that we will see here and I will draw the diagram again but it is drawn here very nicely this is the temperature profile for a case where T s surface temperature is greater than the fluid temperature that means heat is coming from the wall into the fluid fluid is getting heated this profile it is very hard for me to get temperature at every location. So, I will change this temperature to this dashed line which we will call as P m some constraint has to be there otherwise there is no constraint then there is no fun in life. So, we say just as mass flow rate was conserved your energy content has to be equated. So, all I am doing is energy of that fluid element. So, this profile fluid will have a different energy in this region as opposed to this region because its temperature is different. So, I am integrating the mass flow rate for every elemental ring mass flow rate times c p times the temperature at that element m dot c p times the local temperature that is the energy content of that fluid element ring not clear anybody who is not clear in this please. I imagine always he used very right word ring. So, imagine so many stream tubes you know each one is coming each stream tube is coming with a definite temperature. So, if I average the energy of each stream tube and imagine a temperature that is going to be the bulk fluid temperature or the mixing cup temperature. Mixing cup temperature is so said because as he said that is how we measure also mixing cup temperature. We put it in a cup and we mix it then only I will get the mixing cup temperature. So, imagine always stream tubes different stream tubes each uncaring different energies. If I equate these energies and imagine a temperature that temperature is my mixing cup temperature. Right enthalpy essentially c p t is enthalpy ok. So, for benefit of those whom it is not very clear I have tried to draw it see this is the cross section of the pipe all of us appreciate that there is a velocity profile. So, this I have taken a fluid element ring at radius r and thickness d r. So, imagine it is like an extrusion just the ring which is coming out ok madam. So, it is like a pipe of a certain thickness which is being extruded the fluid element is of that thickness and it is coming out that fluid element will have a velocity which is given by the local velocity associated at that position u of r comma x. Similarly, it will have a temperature associated by virtue of a temperature profile its temperature will not be T s it will not be this temperature it will be some local temperature given by the magnitude of this length ok. So, now what I am saying is energy content for that fluid element d e is nothing but d e dot because it is flowing it will be mass flow rate of the fluid through that element times c p times p of r comma x mass flow rate of the fluid times the enthalpy associated with that fluid elemental ring. Enthalpy I am writing in terms of c p times p for an incompressible fluid T essentially why because we are interested in the temperature distribution we are not concerned with enthalpy distribution we are concerned with temperature distribution. So, this is T comma r x this m dot as we did in velocity it is going to be rho u of r comma x that is where I am marrying fluid mechanics with heat transfer. So, this is the bride this is the groom ok you cannot write it u of r comma x times d a c this is the mass flow rate this multiplied by c p multiplied by p of r comma x this is the infinitesimal control volume energy flow of energy this ring extends from r is equal to 0 to r is equal to capital R total energy is obtained by integration this is over the entire cross sectional area a c will be written as 2 pi r d r and the integral will go from 0 to capital R. Energy conservation tells me whatever energy I do by this method integral should be the same as energy content if I took that entire fluid into a cup put in a temperature I mean mixed it and put in a thermometer I will get a temperature which we call as the average or mean or mixing cup temperature. So, I will say this is also equal to m dot c t times t mean or t mixing cup and this m dot now again is not based on local values this will be our rho a which is pi r squared and v mean or u m sorry I have written u m the nomenclature is v m it means the average velocity mass flow rate evaluated using the average velocity times c p times the so called mixing cup temperature whose magnitude I have to find out. So, if I do the maths I will just get a very simple dirty looking integral equation which will be nothing but t mean is equal to 1 over rho pi r squared c p times u m anything I saw please point out integral 0 to R let us be slow here times rho times 2 pi r times c p times u of r comma x times t of r comma x times d r every term is included incompressible rho is cancelling of pi pi cancels of c p c p cancels of I am left with 1 sorry 2 over r squared u m or v m I am sorry I am sticking with u m integral 0 to r u of r comma x t of r comma x r d r this is your mean mixing cup average temperature let us just go back and write our average or mean velocity and average or mean velocity was a very very nice and simple equation just 2 by r square integral 0 to r u of r comma x r d r look at these 2 equations has not life become very complicated suddenly product of 2 distributions both of which are functions of r comma x that has to be integrated with respect to r mathematically yes you can do it but conceptually because of the flow of the fluid and the fact that it has a different velocity at every location temperature also is changing at every and this mixing cup temperature or average temperature is what is specified in our problems most of the type okay and we have to appreciate that this mixing cup of average temperature is something which is given for convenience but despite it being something something which is generated for convenience it is so useful that it has become indispensable okay so cutting all this we go back to the slides I have used u m it is v m here it is the same thing so this expression you do not have to ask the student to memorize this okay this comes from fundamental I do not remember this I am sure he also does not remember this but given a velocity and temperature profile student should be able to get an expression for the average or mean temperature we will do it when we do the derivation for Nusselt number for constant okay any questions on this average temperature deliberately we have gone slow but if it is clear I think it is not just to get a set of properties I will explain that in a minute when I evaluate heat transfer coefficient H okay we are we are going little 5 6 slides more there are faster than what we are how is Q what is Q double prime H is equal to I mean Q is equal to H times Q is equal I will write down for you why why ask why speak in air Q is equal to Q double prime is equal to H times please tell me help me T s minus T infinity okay external okay external flow I will accept this equation because there is only one T infinity okay internal flow where is this T infinity for you T infinity is gone once the fluid has come inside this confined environment which is at either uniform surface temperature or uniform wall heat flux there is no concept of T infinity anymore T infinity history so what do we deal with we have to it plays a similar role of TM but very very important TM is not a constant why because if I take this pipe I will just go to 2 slides later it is gone if I take the profile at I will we will explain all this little later I am just showing a temperature profile where the fluid is hotter than the wall and heat is going to go out a temperature distribution here versus another temperature distribution here will not be the same because heat is being lost by the fluid to the ambient because there is heat transfer because the Q is a finite number this if I keep T s fixed TM will have to change this profile will become a little bulged in words it will come in words okay unlike T infinity which was a constant all through the flow I did not bother the air flows over a plate at a velocity of 30 meter per second and T infinity of 25 degree centigrade at 25 remain 25 we did not change it here at every location my TM is going to change and that produces additional complexity because I have to track the variation of TM at every point so my heat transfer coefficient definition therefore would be Q double prime is equal to sorry H is equal to Q double prime divided by T surface minus T am I clear local value okay if Q s Q double prime is fixed this is constant these two are changing if this is fixed and these two are changing whatever it is H is going to be a local quantity if neither of these are fixed life becomes even more difficult okay is everybody with me on this part it is not to scare or you know you know say how how great this subject it is not for that it is just to emphasize that with just one additional aspect of the flow that the flow is now instead of external it has become internal you have a all together new dimension associated with the problem you have to appreciate that there is a mean fluid temperature which necessarily has to be carried with you and it is not a constant unlike the infinite okay so I did not know this when I studied heat transfer okay so in concept of hydraulic diameter all of us are familiar with it essentially has come from the idea that you like to deal with circular pipe geometry so circular pipe geometry is something where I get the hydraulic diameter essentially equal to the diameter of the pipe the definition historically has come from that aspect so 4 times flow area divided by the wetted perimeter and I think this is not a big deal now just as we had a hydrodynamic entrance length that means where the velocity profile was changing where there was a inviscid and the viscous region similarly we have what we call as a thermal entrance length thermal entrance length is nothing but the region where the boundary layer is still developing just as the hydrodynamic boundary layer developed this is the thermal boundary layer it is developing till the point where the boundary layers intersect we will call this as a thermal entrance length or thermal entrance region inside that region the let us say the fluid is entering at some temperature T i and the surface is maintained cooler than the fluid so heat has to go out from the fluid so a typical temperature profile would be like this so there will be a temperature distribution which goes from a higher value to a lower value as we move outward correspondingly here and there will be a central portion where the boundary layer is not there more important aspect here is what happens in the so called thermally fully developed region I use the word so called because it is not very convincing for me to use the word thermally fully developed why because when I had hydrodynamic boundary layer and we said fully developed flow I just had one very nice equation d by dx of velocity equal to 0 what does it mean velocity profile is identical or d by dx of the velocity is equal to 0 means irrespective of what x I am doing the v is identical at every point so if I say at r equal to r by 2 if I take the velocity at this cross section versus another cross section it is the same physical magnitude of velocity the profiles are identical so I can superimpose all the profiles and there will be no change in the profile nothing will be outward or nothing will go in it will be identical in shape and size am I clear on this now when I come to thermal things are not so nice why because few minutes ago we argued for this case where heat is going out from the pipe what is happening let me say surface temperature is kept constant this length will be constant because this is an indicator of the surface temperature if heat is going out the net energy content of the fluid is going to decrease with increasing length is everybody with me on this if the net energy content is decreasing this area represents the net energy content integral that is what we did when we did mean temperature definition so if this is anchor and the area has to decrease this curve will have to come in what it has to sink in what and when it is changing T s is fixed locally at every point temperature is changing if you are not able to appreciate it I will just draw it for you here and then I hope you will appreciate I have this pipe I will take the same case where T s is smaller than T fluid at all points this was one profile T s at say X 1 at another profile at X 2 I am going to have X is equal to X 2 this is at X is equal to X 2 and this is my center line this is velocity if I draw I just told you velocity if I cut the profile and keep one over the other you will not know what is the difference it is identical so whatever be the velocity at X 1 X 2 profile will be sitting one over the other this is the velocity profile r, X at all X values. In fact I should not even be writing r, X it is just v of r whereas here I can just looking at this tell you that if I move to X 3 which is further downstream and this T s being fixed I will get a new curve which is going inwards so what does this mean I cannot define thermally fully developed based on my temperature profile alone I cannot define thermally fully developed so what we say okay let us make life a little bit make life a little bit complicated we will say we have this ratio T we have this ratio T s minus T divided by T s minus T m this gradient of this ratio with respect to X is invariant with respect to X what it means T s is fixed let us say we deal with T s being fixed T m is changing with respect to X local T is also changing with respect to X what we are saying let us understand for T s is constant that is easy so this term is constant this is changing this is sorry sorry very very sorry see this gradient is invariant with respect to X how did I come up with this for a case with T s is equal to constant that means this term first term in the numerator and denominator is fixed constant say wall is maintained at 100 degree centigrade T m is going to change with every X T local is obviously going to change T local being I mean changing is what is causing T m to change all I am saying is this variation the difference between the surface temperature and the local fluid temperature to surface temperature minus the so called mixing cup temperature this non-dimensional theta non-dimensional temperature is invariant with respect to X location temperatures are varying T s is fixed just for example case T s is fixed I am fixing T s for ease of understanding the situation T s is fixed T local everybody appreciates is going to vary because the fluid is either losing heat in this case the fluid is losing heat I have drawn the velocity profile where I have shown that for the same point let us take this point this at this location this fluid will have a temperature which is given by this at X 1 it would have had a temperature which is given by this so for the local temperature is going to change with X mean temperature is also going to change with X just for simplicity we will say surface temperature is constant what we are saying is this non-dimensional temperature which is nothing but T s minus T which is the temperature difference between the surface and the local value to the temperature difference between the surface and the so called mixing cup temperature which plays the role of some kind of a arbitrator this is not independent it is directly related to T remember your equation for T m it was a big integral which had u times T dr r times u of r x times T of r x times dr so this is not some magic this is directly related to the temperature it is in fact related also to the velocity all we are saying is this change is such that it is invariant with respect to X so even though the temperature profile shape changes with respect to X at every location what is constant is that this difference so if this differs if let us say for example this is 100 and at a local at a given location 100 minus say 80 and this is 100 minus 60 at another location the ratio 100 minus 80 is 20 divided by 40 which is half that ratio will still be half at another location very crudely I am explaining with numbers it is not correct to explain with numbers what we are trying to say is a non-dimensional temperature profile which is given by this is invariant with respect to X we have forgotten the actual velocity profile actual temperature profile we have introduced a non-dimensional velocity non-dimensional temperature profile which encompasses within it the velocity distribution and the temperature distribution right here velocity distribution is also here so this is what we call as so called thermally fully developed this has been the definition for thermally fully developed condition temperature is varying agreed mean temperature is varying local temperature is varying seemingly on front of it flow will never be thermally fully developed that means d t by d x is definitely not equal to 0 d t by d x is definitely not equal to 0 unlike you had d v by d x that is ruled out so what do we do how do we have something similar conceptually so instead of having a dimensional value dimensional temperature they say local temperature is feeding is is influenced by what it is influenced by the surface condition okay fine so I somehow bring T s also some reference temperature has to be there so this is a reference temperature is what T infinity is not possible anymore so T m but this T m itself is influenced by the local temperature so I bring up this kind of a non-dimensional temperature profile which will be invariant with respect to x this is called as thermally fully developed I am sure I have lost 50 percent of it open to questions. The question is always very difficult whereas we will also face the same problem when you take up we will introduce this because only layer thickness for fully developed for very easy to understand so how how did one know that I have to take this ratio that's why professor Arun went at great lengths to make it understand because in no textbooks they will define why is it defined like that I mean they will not make us understand they will simply throw this and we are supposed to absorb it so this is the best logic I think what professor Arun has taught us is the best logic one can give both T is varying with x T m is also varying with x so both the variations I am trying to capture and then try to make it invariant then try to make it invariant that's the only possible option one can think of. What we are trying to say is probably that may be wrong is that the nature of variation of T and T m obviously it should be similar it cannot be different we do not intuitively it should not be different so that logic probably has been used to come up with this. And it works and it works people have found that Nusselt numbers after developing region even although we here we see here professor always emphasize that for a minute let us take it is T s now it becomes all the more difficult when I take a different boundary condition constant heat flux it's very difficult now will you convince the student when you take constant heat flux that it will work it's not possible all that you will have to say that it works it works it indeed works how do you say that it works because you are measured the heat transfer coefficients irrespective of the boundary condition what you have imposed and then you indeed find that the heat transfer coefficient in the developing region it is constant that's why it works in the developed region the developing region it is decreasing and in the developed region it becomes constant so that is why you say that it works okay so that is that this is the only plausible explanation one can offer okay sir I have one question is there any stuff to work in developing region people are working people are doing the research based on the developing developing region there are already empirical curl not empirical we have standard relations derived one can derive that's what is if I'm right great problem Venu or sake can correct me in the developing region it's called as greats problem g r a e t z so that's developing region also there are closed form solutions but what are we actually getting from the study on developing in an heat exchanger in an heat exchanger you take all correlations all correlations which correlations you use developing correlations you use or fully developed correlations you use and I will get a feel how much length I should keep so that I am always in the fully developed I don't know that I have answered you I didn't answer you your question if I am understanding correctly why at all I should be worried about heat transfer coefficients in the developing region is that right maybe we can consider like that we can consider like that okay no problem my question is in general not not to depart heat transfer okay as I see the papers of researchers they are now focusing more on the developing region flows is it yeah my question but I thought that developing as far as yeah developing region already closed form solutions then what else is left out to study I don't know if configuration changes if you put it is to tape or if you put some other configuration inside yes then it becomes difficult things become more tedious when you scale down the object scaling down I am not going to deal with I will deal with you alone scaling down over cup of coffee but not here because things will change yes here we are taking TS is lesser than TI if suppose reverse case is there no heat loss will be there will the profile will same in that case heat gain will be there by the fluid right okay yeah yeah but if the fully developed reason here we are getting different profiles due to the heat loss but in that case what I will I will draw it for you in fact I always teach it the other way but it doesn't matter is this the profile will just change that's all so let us say we are in the developed region so called developed region I am making it really small so it is like this purposely I am making it to okay this is x1 and purposely this is made fat okay yes no this will that I would like to ask will it change if it is constant heat flux coming in no heat loss suppose because heat is coming in this is coming in yeah this case is heat coming in okay let let us not take uniform wall okay so question is whether it is for heat flux heat also is constant heat flux okay so in that case no heat loss will be there any case heat loss is not there heat gain will be there if constant heat flux is coming in then what let me just go back to this no what is your question I did not know I kept saying TS is constant just for sake of understanding this concept that's what I emphasized yes that's what yes it is now it is hard to visualize now I cannot visualize now I cannot visualize it is hard to visualize that's why I said it works so what we are saying is the rate of I mean the change in the numerator and the change in the denominator happens identically at all axis that is what I am saying uniform heat flux condition uniform heat thickness will change correct so the mean that is in fact that is why this T and TM are coming into picture to take care of the bulging or the contraction of the profile because if the locally T is changing TM is also going to transpondingly change whoever has come up with this is done a wonderful job actually one more question here if the war calculator is fixed constant the velocity profile will also change because the fluid will get heated correct it will not have same profile now here what we have discussed that the velocity nature of the profile is fixed I never brought in velocity profile at all no you know your point is velocity profile will change because my temperature profile is changing no no no no no no no no no no no what are you saying velocity profile we assume that in nature is fixed when it is fully developed hydro dynamically fully developed if there is heating of fluid then there may be change in the volume of the fluid there may be change in the viscosity of the fluid and so it may affect the nature of the velocity profile so no actually it does not work that way in the developed region in the fully developed region velocity profile does not get affected because of the boundary condition because of heating if I have to put it in whether it is constant wall temperature or constant heat flux no it doesn't change I will tell you why why historically how are we solving this for internal flows if I take if I take how do I solve how do I solve this problem how do I solve this problem first I write mass momentum and energy equation I take the velocity profile independently whether I heat or not because in my when I non-dimensionalize I solve only Reynolds number so I will keep the fluid fixed and I get the velocity profile now taking this velocity profile taking this velocity profile I am going to put that velocity profile in my energy equation if you remember energy equation at u del t by del x plus v del now taking this velocity profile I put the temperature now what does this mean if my velocity profile were to get affected by my temperature then my momentum equation and the energy equation would have been would have been coupled that is what happens in natural convection that is what happens in natural convection but here these two are independent things velocity profile doesn't get affected at all because of heating or cooling no two ways about it no confusion on the score yes but not the other way around what the question is because of temperature velocity profile is affecting that is not right what you said is right temperature profile is getting affected by my velocity profile because that is there in my bulk fluid temperature right so velocity and temperature they are they are going to go in one way that is velocity comes first and then the temperature but not reverse sir I have a question that just you have said that once you are fixing up once you are finalizing the velocity profile inside the pipe then you take the aid of those profile and guide the solution of temperature yes I think this been a segregated solver what you are solving segregated part ok but the fluid flow and heat transfer is inherent it is inherent to a couple phenomena if it is coupled one has to take care of it is a recurrence formula no no no see in first case you are getting the steady state of the velocity profile then if it were to be coupled we would have coupled it in my equation itself but see whatever we are getting without taking care of the heat in the velocity profile and if you are taking care of the heat transfer in the velocity profile the first question is whether my velocity profile gets affected at all or not forget equations forget equations forget equations whether my velocity profile will get affected or not I think it will get affected no it will not what is that concept what is that concept ok ok no no I think it is ok it is a good question I think the only way to convince you or convince ourselves is that when will my velocity profile change let us go back to my m dot equal to rho a u which of the things which of the things changes means my velocity will change the mass flow rate doesn't change no no no mass flow rate doesn't change please I have to I have to make you think I have the answer in my mind but until I get the answer from you I am not able to reach you when will my velocity please listen to my question carefully whoever are not getting this question when will my velocity profile change as long as my mass flow rate is constant I cannot do anything about it when will my velocity profile change only when my density changes what is the basic assumption in all of this all of this analysis what is the basic assumption in all of this analysis my fluid is incompressible you have landed into compressible fluid that is why you have gotten into problem ok so my density is constant so there is no way that my velocity profile is going to change with temperature ok it took a while to reach you but it's fine no that we will handle as property variations property variations that we will handle it as property variations ok in my m dot viscosity how is it how does it come into my m dot it's not coming it's not temperature is varying my velocity profile has to change means who will have to change no m dot what is m dot m dot has to be conserved no rho a u if my velocity profile has to change means the only way it can change is because of density not because of viscosity not because of viscosity ok that is why Prandtl number we don't bother you are probably thinking of the momentum equation and there is a new therefore you are saying viscosity is that why you are saying no how will velocity vary how will velocity vary how tell me tell me how it has to come obvious to us well I can say m dot equal to rho a v if my density decreases my velocity has to increase but similarly can you convince me if viscosity decreases my mass flow rate either increases or decreases please convince me then only I can answer your question there is no connection there is no connection between viscosity and mass flow rate there is no connection I cannot answer it any other way but all that I can say is m dot equal to rho a u only viscosity has no influence on mass flow rate ok whatever the change has been done according to the temperature in terms of flow properties ultimately it gets affected in Rellon number until it reaches the certain critical Rellon number the I think so velocity profile doesn't change it gets beyond that limit that is for correlations to worry about that is for correlations to worry about we have not reached there that part ok but that's fine we had good discussion it's better to get I don't tell I mean always my student whenever my student or when I get confused I don't become restless at all confusion is the source of understanding if I don't get confused perhaps I am not thinking I am just accepting things as they are it's good to get confused it took for me a while I could not conceive initially how to answer your question as it went along it occurred to me ok this is what you are thinking so it takes a while you have made me think now you have made me think so I think it's better to get confused ultimately we will get the answer eventually