Hi and welcome to the session. Let's discuss the following question. The question says integrate the following functions sin inverse root x minus cos inverse root x by sin inverse root x plus cos inverse root x. Let's now begin with the solution. In this question we have to find integral of sin inverse root x minus cos inverse root x by sin inverse root x plus cos inverse root x with respect to x. First of all we should know that sin inverse root x plus cos inverse root x is equal to pi by 2. And sin inverse root x plus cos inverse root x equals to pi by 2 implies cos inverse root x is equal to pi by 2 minus sin inverse root x. By substituting pi by 2 in place of sin inverse root x plus cos inverse root x and pi by 2 minus sin inverse root x in place of cos inverse root x, we get integral sin inverse root x minus pi by 2 plus sin inverse root x by pi by 2 dx. This is equal to 2 pi pi into integral 2 sin inverse root x minus pi by 2 dx. This is equal to 4 pi pi into integral sin inverse root x dx minus integral dx. Now let i1 is equal to integral of sin inverse root x with respect to x. Now put root x equals to t. Now root x equals to t implies x is equal to t square and this implies dx is equal to 2t into dt. So by substituting t in place of root x and 2t dt in place of dx, we get i1 as integral into sin inverse t into dt. This is equal to 2 into integral of sin inverse t into t with respect to t. Now here we have product of two functions and we know that if u and v are two functions of t, then integral of uv with respect to t is equal to u into integral of v with respect to t minus integral derivative of u with respect to t into integral of v with respect to t whole with respect to t. We choose the first function as the function which comes first in the word iLate. Here we have algebraic function and trigonometric inverse function. According to iLate, we choose first function as inverse trigonometric function that means sin inverse t. So using this rule, this integral is equal to sin inverse t into integral of t with respect to t minus derivative of sin inverse t into integral of t with respect to t whole dt. This is equal to 2 into sin inverse t. We know that integral of x to the power n with respect to x is x to the power n plus 1 pi n plus 1 plus c. So using this formula integral of t with respect to t is t square by 2 minus integral derivative of sin inverse t is 1 by square root of 1 minus t square into integral of t dt is t square by 2 whole dt. Now this is equal to t square sin inverse t minus 2 into 1 by 2 into integral t square pi square root of 1 minus t square with respect to t. And this is equal to t square sin inverse t minus integral t square pi square root of 1 minus t square dt. This is equal to t square sin inverse integral we are now going to add and subtract 1 in the numerator. So we have t square minus 1 plus 1 by square root of 1 minus t square dt. This is equal to t square sin inverse integral t square minus 1 plus integral 1 by plus t taking minus common from the numerator we get plus plus integral. So using this formula of these integrals I am going to replace t by root of 1 by 2 sin inverse integral t to sin inverse root x plus root x into square root of 1 minus x by root of 1 minus t square dt. So by substituting values, we get 2 can be written as 2x minus 1 by 2 into sin inverse root x root 1 minus x by 2 plus c1 minus integral dx is x plus c2 is equal to minus 1 by sin inverse root x plus into root inverse root x 2 into root x into root 1 minus square root x minus 1 by 2 into sin inverse root x minus 1 by 2 into sin inverse root x


So could you tell us a bit more about Bushmaster? Bushmaster, he is very driven. His purpose in life is to avenge the wrongs that have happened to his people on a personal family level, but as his people as a group. So he's kind of revolutionary with some misguided methods. So there's a lot of big fight scenes in the series, but the one you come face to face with Luke and take him out for the first time, what was it like filming that? Super cool, man. We were like right on 116th in Harlem, it's like right down the street from my sister's old apartment, so some of my stomping grounds. And how long does it take to train for these sort of big fight scenes? I mean I think it's different for everyone, but we had some fights and choreography training like once or twice a week, we'd take videos of it, I'd take it home and study it, get it locked in my muscle memory and then come back and then execute. Any accidents or quite plain sailing for you guys? We had a couple close calls, we had one really bad close call. But I pulled my hamstring and rehearsed it one time, but we just worked around it. We made it work. A trap door fell, yeah, and it just missed me, it was a big metal door. Any funny moments from filming? Always bring my little Bluetooth speaker with me and I took it and put on some music between takes and everybody got into a big dance session and these songs and raunchy songs for about a good ten minutes, that was great. Anyone in particular got good moves? Alfrey Wooder, say hello to Alfrey, she would be getting down. So Marvel crossovers, if you could bring any other Marvel characters into the world of Luke Cage, who would you like to bring? I would just link the X-Men, I would just, you know, because there's a difference between Netflix television and cinematic, I would just link the worlds, you know what I mean? Luke Cage would be in there fighting Thanos, you know, with the rest of the crew. How would Bushmaster fare against Thanos, do you think? Without the glove, I think he'd have a chance. With the glove, like, who has a chance? No, no, he's just, nobody. What was it like having Lucy Liu direct the first episode? It was the best intro ever because she's super exciting, exuberant personality, and then, you know, your first day, you got the jitters because you haven't, like, popped it yet. But after, like, my third take, she just made me totally comfortable and set the tone for the rest of the shoot, so it was awesome. And why is it that people should watch this new series? I think all the criticisms that you guys had about the first season were taken, heated. I think you'll be really excited to see where we go with it. Definitely up to stakes.


As I sat in the theater waiting for Ghostbusters Frozen Empire to start on a Thursday afternoon, alone, I couldn't help but think to myself, man, it was so fun years ago as a kid going to movies, being hyped up for films months in advance, getting my ticket for that midnight showing, but here I am now, a grown man with two kids who have zero interest to go to a free movie with their dad in the afternoon to a property that's been around for decades because they don't give a single crap about it no matter how hard I try to make them. Let's rant about kids these days and whether or not the movie industry can handle them or sustain itself as they grow older. Before I begin, if you like movie criticism, conversation, rants, roasts, and live streams, I post them every single week on the channel. I do movies here, it's in the name, it's all I do and I have a good time and I like to think I'm chatting with you as a regular friend, someone that we share a common goal which is to express our opinions on movies and hopefully we get a bunch of good ones every year that we can be excited about and share with each other. Alright, with that being said, I really wish my kids have that someday as well, but I fear because of all the tech and all the exciting things to look at 24-7 movies are just kind of been there, done that for them before they even were there or did that very much. Now I think in more of an exception to the rule because I have raised my kids on movies, I know I'm on a movie channel so you watching probably do the same, but a lot of regular folks out there that watch sports and do all sorts of other fun things that I really don't care to do, they don't put their kids in front of movies and watch them with them. That's like a thing that doesn't really happen anymore, at least not where they're fully engaged. They might be sitting on their phone or doing the dishes or something while their kids just they're alone for the most part and I find that kind of sad, but everybody has their own way of parenting and doing things. Regardless, my kids are getting older now. I have a 15-year old and I have an almost 12-year-old boy, a girl and a boy, and I raise them on like Goonies, Sandlot, Star Wars, pretty much everything I grew up with. We dip into older stuff as they got older, you know, my daughter actually loves the shiny and it's one of her favorite movies. One of my son's favorite movies is The Thing, so I think I'm doing a decent job as a parent because both those movies I think are freaking awesome and I didn't really push them into liking those. I just wanted them to see movies that are kind of classics and there's plenty more to show them over the years. That's for damn sure. A week or so back we watched 300. Now that Connor is getting a little older, I thought, all right, now's a fine time. Let's watch 300. It's still easily my favorite Zack Snyder movie, but we had a great time watching that film and it just brought me back to when I actually saw that in theaters and what a freaking amazing experience it was. I saw it with my wife. She was of course captivated by all the rippling strong alpha males. That was cool. That was fine, CGI abs and all, but I was just, I thought the story was great. I thought the way that Snyder presented the film was so different, so unique. It was such an experience that could only be really appreciated on the big screen. It was going to always lose a little bit of its luster at home and the same goes for a lot of films like even the recent Dune movies. Seeing that in the big screen theater, that's going to be just so much better than at home. Even if you have a home theater system, I just don't think it can match the sheer overwhelming scale of a big screen theater in IMAX with the great seats and with an awesome crowd. That is unmatched. Now, of course, my theater is not that great, but I can imagine it's even better for some other people out there. This hasn't really been a thing that I've thought about a lot until recently. When my kids were younger, I didn't expect them to get hyped up for films or to know when movies came out. They rely on the parent for that. So I would take them to the Kung Fu pandas and the How to Train Your Dragons and all those things. And they were great. Kids loved them and we would go home and we would watch something later on a weekend. We watched a movie or two and it was just a fun bonding experience.  My kids will obviously sit down and watch a movie and have a great time with it, but it's different now. They don't really seek them out like I think you and I probably do. I know I do. I like to search for movies. I like being excited about what's coming up, the pipe, even though a lot of it's pretty bad now. There's still good films being made and coming out. I'm excited for a couple. I guess it just kind of blows my mind that a new Ghostbusters film comes out. My kids have seen the previous entries and they don't have any real interest in going. They didn't ask me when it's hitting theaters. They haven't even really followed up about what I thought about the movie. They did see the trailer. They just don't really, really care at all. Indiana Jones was the same way. No interest in going to see that. There's just every movie I can think of actually outside of the Taylor Swift live concert for my daughter.  And that's kind of a bummer, I guess. It's kind of a bummer because it's just different. You know, everything's different than it was when we grew up and watching that kind of unfold and play out is a sad reality, I guess, that you have to hit. Like back in my day, I used to bike to the theater and buy a ticket for $4.50 and a pop for $3 and now a soda I think is like $7 for a small at Regal.  So basically you're giving me poison.  And I'm supposed to give you $7.  This is insane. This highway robbery. It's just a different world. And I think movies and studios have really shot themselves in the foot when it comes to younger audiences because they have oversaturated the market. There isn't a day that goes by where there isn't 20 different new things available on Amazon Prime, Disney Plus, the Coq, Paramount Plus, Apple Plus, Bonus Plus, Minus Plus, Plus Plus, like everything's a freaking math equation. That's another thing. Kids don't like doing math for the most part.  There are a ton of factors into what's happening, but I do genuinely have some concerns about what's going to happen with the movie theater industry once these kids start to grow up.  And I think some of the reason is there's no real reason for them to go out to the theaters anymore. It costs a ton of money. It's typically a pretty mediocre or miserable experience because the movies aren't very good. They're all rehashes of films I grew up with.  Remaking Roadhouse? That's for me, I guess, but none of us wanted it either. We liked Swayzee. Even though I didn't mind the new Roadhouse, I still was annoyed that they used the title when it really didn't have anything to do with the original or like Twisters. Or the fact that there are five Indiana Jones movies and four Ghostbusters movies, five if you count the reboot, it's just overkill and it's nostalgia. Beetlejuice 2 coming out, Beetlejuice, Beetlejuice. My kids have seen Beetlejuice. They still don't care about another one. They're like, great, it was what? The 80s that came out? What would he do? It was a fine movie. We move on. I've been trying for several weeks now to get them to watch the new Mission Impossible movie because it's on Paramount. I've also been trying to get them to watch John Wick 4. My son and I did a marathon of the three of them within a week. We watched all of them and now he doesn't care anymore. He's like, there's another one, Dad. The last one was really long. If you think I'm too harsh on film, you should talk to a kid sometime. They will tell you exactly how they feel and man is it refreshing. That movie's trash. I don't have time for that. No, it's too long, blah, blah, blah. It's so fun to talk to them about some of these things. I thought for sure Connor was going to love 300. My daughter loved it, 10 out of 10 for her. My son gave it like a six and a half, seven. He's like, it was too long. I didn't like the bad guy and I didn't like that Leonidus died. Those were his critiques and I thought, they're all kind of fair. That's fine. They're all opinions at the end of the day. I wasn't expecting it, but fair enough. And I think that's the biggest thing for a lot of them now. If it's not a huge event like a Taylor Swift concert or something that you can kind of get on social media with, such as Barbie doing the barbenheimer thing, then they just don't really care that much because it's a time commitment to drive several minutes, maybe a half hour to the local theater, wait for the trailers for 20 minutes to end, watch a two hour, sometimes longer movie, and then go back home.  There's just a lot of ways for them to occupy their time. They could be having a FaceTime with someone while they're playing around Fortnite and they have their computer up with it.  And I know I'm not alone with this. I know there's kids all over that just are in the same boat. They don't know when movies come out. They're not really interested in seeing the movie in theaters. They might want to see it, but they can wait. They can wait. When it comes out, I'll see it. When you present it to me, not make me have to go find it. That's great.  So we are looking at things in a different way. Kids are seeing things in a different light than we did. We didn't have much. And we had way more than our previous predecessors. But when I grew up, it was, okay, we have a big movie coming out in a couple months. The original twister I remember was coming out around opener fishing. Every year around opener fishing, when I used to do that with my family, there would always be a big movie that would come out. And I didn't like fishing, but I loved the camaraderie with the friends, with the guys. And I liked going up north to a different environment, and we would always go to a movie after a couple days of fishing. We'd always celebrate with the film. And one year, I saw Mad Max Fury Road with the guys. One year, I saw one of the Spider-Man movies. I think it was Spider-Man 2. Another year, it was Volcano. Not good, but it was a spectacle. Twister. I mean, these are the movies you live for. You're excited about.  Again, not for the fishing, but for that moment in time when I'm in the theater, ready to watch something blow me away. And I think that's another thing that kids just don't have anymore, because it's all been done to death. Watching a tornado rip through a house has been done. It's been done very well, as a matter of fact. So doing it again, not quite holding the same weight, even if the special effects might have been updated, or there's a little bit more, you know, 3D at play. It doesn't matter. It all still looks kind of the same. Watching a volcano erupt downtown LA. That's cool. That's different. Watching Spider-Man for the first time on the big screen go between building, saving people and webs. This was so amazing. And now it's so easy to do.  And it's all become so incredibly mediocre. The oversaturation of it all. And it's not just superheroes. That's across the board with cinema. It's all so played out now. And I think younger generations already know it. And since the studio and the industry is catering to my generation, with these reboots and these nostalgic hash grabs and sequels and remakes, they're alienating the younger audience. And then what do they have for them? Well, they try to jump on their trends.  There's trans people in this one. Is that getting you to come out? Because you're seeing an influencer on YouTube shouted out? Or is that maybe not enough? Spoiler, it's not enough. It's not enough. They're not that desperate to see it. This is a rant video. I don't have a solution. I don't have a proposition. I'm just telling you how I see it. It's anecdotal, I guess to an extent. But we are seeing the box office numbers drop a lot. And yes, a lot of it has to do with the amount of money they're pumping into these films. But I think it's also because they know in the back of their mind, they know that if they're not spending $200 to $300 million on these super bombastic larger than life films, people are not going to go see them in the theaters. Because Fast and the Furious 25 does not have a smart script. It does not have good acting. It has spectacle. But you've already blown up everything.  Well, what's past space? The multiverse? Does Vin Diesel fight himself in the multiverse next? Probably. We'll see multiple versions of Paul Walker now. It's been a long day without you, my friend. Driving on the beach side, Tokyo drifting into our hearts all over again. But it's going to be superficial. No one's going to buy it. All right, that was a good rant, I think. Let me know your thoughts, though. Am I wrong? Are kids just as into movies now as they've ever been before? Or are you noticing it, too? Are you seeing it, too? That 15-year-old demographic not quite hitting the streets and hitting the theaters like they used to, they're sitting home on a Friday night, they're streaming something from the comfort of their own couch. You might see the parents drag their little ones out to the theater and I just pity them at the end of the day because, yeah, good luck. You've got a four-year-old kid who's already crying before she sits down.  Not going to happen.  Okay, let me know your thoughts. Please like the video if you had a good time. Subscribe if you haven't. Again, I post tons of movie content, commentary, all sorts of stuff every week. We'd love to have you stick around. Take care.


We humans have the unique ability to interact with the world around us using our upper limbs. Precise motor maps created in the brain enable us to move specific muscle groups in discrete ways needed to play this music. Sensory information from our fingers also allows us to explore the world through touch. But how do you connect the brain with peripheral receptors and muscles? How do nerves get to where they need to be? Our upper limbs evolve to position our hands in space for use as motor and sensory tools. This requires precise muscle movements and precise control over those muscles. But how does such an intricate tool develop and what are its components? We're now going to explore the evolution and development of the brachial plexus, the nerve bundle that makes this unique human experience possible. During development, the precursors of the musculoskeletal system, the somites migrate into what will be a limb. Each somite is associated with its own spinal nerve. Muscles are organized into compartments that share similar functionality. For example, here in the upper limb we have a flexor and an extensor compartment. Parts of somites can migrate into more than one of these muscle compartments and they take their nerves with them. But look at this, there's only one nerve in each compartment. So that means that each one of these peripheral nerves contains fibers from more than one spinal nerve. If this mixing and matching of fibers does not occur within the brachial plexus, the complexity of our movements would never develop. Let's take a look at the formation of this plexus in more detail. Here we have spinal nerves C5 to T1 emerging from the spinal cord. The anterior ramai of these spinal nerves or roots will form the brachial plexus. We will now recombine these roots into trunks, divisions, cords, and finally peripheral nerve branches. Let's do this step by step in this diagram. First, let's take the upper two ramai from C5 and C6 and combine them into a superior trunk. C7 forms its own middle trunk. C8 and T1 combine to form the inferior trunk. Each of these trunks then divides to give rise to an anterior and posterior division. These divisions recombine to form the three cords. The medial cord is simply an extension of the anterior division of the inferior trunk. Notice it contains fibers only from C8 and T1. The lateral cord is formed by the anterior divisions of both the upper and middle trunk merging, and you'll notice that it contains fibers from C5, C6, and C7. And look at this. The posterior cord contains fibers from all spinal nerves because it is a combination of all posterior divisions. The terminal nerves for the upper limb are a result of one last recombination of these cords. The muscular cutaneous nerve emerges directly from the lateral cord. It carries fibers from C5, C6, and C7. It will innervate the flexor compartment of the arm. The median nerve is the result of branches from both the lateral and medial cord combining. It contains fibers from C6, C7, C8, and T1. It will go on to innervate the flexors of the forearm and parts of the hand. The ulnar nerve emerges from the medial cord. It contains fibers from C8 and T1, and interestingly from C7 joining here from the lateral cord. It will go on to innervate the intrinsic muscles of the hand. The radial nerve is an extension of the posterior cord. It contains all spinal levels involved in the brachial plexus. It innervates all extensors of the upper limb. The auxiliary nerve branches off the posterior cord as well and innervates muscles around the shoulder. As you can see, the brachial plexus allows for fibers from various spinal levels to recombine into each one of these peripheral nerves associated with the different specific muscle compartments of the arm. Now let's take a look at this in the anatomy lab. This is a dissection showing the right brachial plexus. This is medial, this is lateral, this is superior, and this is inferior. The brachial plexus forms posterior to the axillary artery, so I'm going to remove the artery so that we can clearly see the various parts. The roots of the brachial plexus are clearly visible here. C5, C6, C7, C8, and T1. C5 and 6 join to form the superior trunk here, and C7 becomes the middle trunk. C8 and T1 join to become the inferior trunk. Each of these trunks divides into an anterior and posterior division that then recombine to form three cords around the axillary artery. The medial cord is here, the lateral cord here, and the posterior cord is posterior to the artery. From these three cords arise the major peripheral nerves, musculocutaneous, median, and ulnar. You'll notice these together with the medial and lateral cords form an M over the anterior aspect of the axillary artery. The musculocutaneous nerve innervates the flexor compartment of the arm. The median nerve innervates mainly the flexor compartment of the forearm. The ulnar nerve innervates mainly intrinsic muscles of the hand. The posterior cord is posterior to the artery, so we have to move structures laterally to see the nerves that arise from the posterior cord. Here is the axillary nerve and the radial nerve, which are the major branches of the posterior cord. The radial nerve innervates the extensor compartments of both the arm and the forearm. It's this whole amazing evolution of the control of our upper limb that allows this violinist to create such beautiful music.


This is great, this cat is like... Alright, you stand on the edge of a bathtub It looks like there's a little bit of water in there There is a little bit of water, what's that? But not a lot It's like an inch or two of water I'm a cat?


and welcome back or welcome to my name is Yoshida and this is Norse Cove I decided to start including my name once Nika was like I can't my plan of baby Nika and over at Nika's creations she's like I can't pronounce your name up I didn't chop it up so it's Yoshida I am Norse Cove just for those who are new to my channel Norse Cove is named after my grandmother she passed away a few years ago and she taught me a lot of things and all those things like to incorporate so I figured I just make a hub Norse Cove but it turned into planning so I just took my grandmother's name and turned it into all of this and it turned into planning and it will hub a lot of things that I do and like to do such as cooking things like that but for now it's planning and stationary and that's why I named my store Norse Cove but I am Yoshida I'll try to put the pronunciation across the screen for you guys the I is the E so it sounds like the E but anyway if you're new to my channel I'm glad that you stopped by I hope that you will stay a while and if you decide to stay a while don't forget to subscribe button also hit the notification bell so that you are notified whenever I upload new content so let's get into this planning today so I'm using my classic this is what the cover looks like I've added yeah I added my home thing here so I haven't done anything to this planner I just because I wasn't going to use it so I'm not gonna ramble today but I'm going into the wrong wrong and it's simple and it's functional that's what I need for everything to be functional we got a little blue over here any another blue over here nope it balances out so this is my pretty much catch all I love this planner I do like I said it's just always a lot of green so I always have to kind of coincide and make things green so this was last week and it worked thank you all so much for watching and I will see you in my next video again if you're new to my channel don't forget to subscribe button also hit the notification bell so it's your notified whenever I upload a video thank you for watching


Hi, I'm Daniel Donato and today we are here at Roberts Western World and we are playing with some Black Star Amps This one in particular is the club 40 Mach 2 Kentucky special. I'm not from Kentucky and we're not in Kentucky But it sure sounds good


So that is the big picture of today energy equation. So energy equation whatever we have been applying whether it was momentum or mass whatever was applied to system we are applying it for control volume by using my E dot in minus E dot out plus E dot st is equal to sorry E dot generated is equal to E dot st. So similarly here also the same thing we are going to do we are taking a small control volume and I am taking the change in the internal energy is equal to the heat transfer to the system and the work done on the system. Now if I take that E change in the energy what is the change in the energy if I do the budget always we do this budget what is that here we are saying what is that we are saying there is net rate of energy increase in the control volume something E is getting in and something E is getting out. So how do I if I take again a cuboid of delta x delta y delta z we know this business quite well so I am not writing this on this. So what is that which is entering mass into energy. So rho u dy dz is mass flow rate into energy what is getting out same Taylor series expansion I do not think I need to write it again so we have been using left hand write this. So by now we are adapting this so rho e u dy dz plus del rho e u by del x into 1. Similarly I can write in the y direction I am not consciously writing this equation for z direction because it makes my equation quite huge so I want to keep it little minimal so I have put it only for x and y direction if I take in the y direction I have rho v dx dz which is the mass flow rate in the y direction and E similarly I can apply for exit. So now if I rate of energy increase within the control volume is del rho e by del t into dx dy dz getting in and getting out if I cancel out this will get cancelled out with this and this term I am left out with del plus plus is missing here plus is missing here plus del rho e u upon del x plus del rho e v by del y. So what did we take care d e we took care now we have so we will do some algebra here same term I am writing rho e plus rho e copy paste problem whatever problem was there here same problem because of copy paste. So now if I expand this if I expand this I get rho del e by del t plus e del rho by del t blah blah blah blah blah is that okay. So now if I rearrange that if I rearrange that what is this what is this continuity equation this is my yes some problem second line okay there is a problem first rho del e by del t plus e del rho by del t is that okay plus e del rho u by del x plus rho u del e by del x I am just taking two parameters keeping one as constant and differentiating the other second term is that okay. So now if I expand that so this is my continuity equation so this I can throw it out I am left out with rho del e by del t plus rho u del e by del x plus rho v del e by del y if I take out rho what is this this is the total derivative of e so I have rho d e by dt okay. So this energy is having the internal energy and also the finite.  So that is about the so let us keep let us keep this in the back of our mind this is my d e remember now heat transfer dq I have to get to heat transfer. So this heat transfer is straight forward this heat transfer is by conduction we have done this already this what we have done here is already familiar to us because this we have done in heat diffusion so what am I going to end up with k del squared t by del x squared plus k del squared t by del y squared I have not done that because to do that I have to make the assumption that k is independent of x and y is that okay. So what is the term left out now energy is over heat transfer is over work done on this system who is doing work on my system who is doing work on my system pressure forces normal stresses and the shear stress is that right. So now let us take the rate of the work done by pressure is that okay I know I am going fast but these are all now very familiar because we have done this several times in momentum equation all that I am doing is work done. So what is work done if I take the pressure what is the what is the work done by pressure force force into velocity force into velocity force is what in the x direction p into dy dz into velocity here it is gaining direction because of the velocity okay. So p dy dz into velocity what is getting out if I apply Taylor series expansion what will I get the same term plus del up by del x into dx dy dz similarly I can do it in the y direction. So what is left out if I do the balance this this this this term will get cancelled out and I am left out with minus del up by del x plus del vp by del y into only is that okay what is this this is the work done by pressure. So same thing I will have to do for so first let me take normal stresses normal stresses will be same as pressure only thing is that the direction is different. So if I I will do the same thing sigma xx sigma xx sigma yy sigma yy I am going to end up del u by del x sigma xx plus del v by del y sigma is that okay. Now shear stress same thing for shear stress shear stress here is sigma xy u dx dz okay. And here it is taking negative sign because why is it negative u is in the opposite axis okay. So now here you take apply Taylor series expansion you have sigma xx plus del sigma xy or instead of writing this it might be appropriate to write this as actually consistently that is what in the mail professor Arun has sent today. So instead of this we could we can write notation wise what is that I should write minus sigma xy u dx dz right okay in the top it will not now plus what it should be del by del by del del by del y del y of u sigma xy dx sorry dy dx dz expansion of this is what is written here you are getting me I have skipped one step and gone here because my feeling was we are 2 I have skipped this whatever I have written and expanded that here if you expand that you will get this I have skipped this because we are now familiar. So if I expand all this in x and y direction so I get del v sigma yx by del x plus del u sigma xy by del y dx dy dz is there always so now we budgeted work done by pressure work done by normal stress work done by shear stress. So we have to now add all of them if you do that if you do that so rate of increase in E rate of heat transfer rate of work done by surface and body forces of course body force also does work. So what is the work done by a body force effects into rho u mass into velocity into body force is that right. So now this is the whole lot of equation I get is that right. So let me take stock of each term what is this term I will randomly pick someone yeah Baluswamy quite young you look that is why I have caught you yeah. So what is this term this is energy this is internal and kinetic energy. So what are these two terms heat transfer by conduction what is this term work done by pressure force what is this term work done by normal stress x x y y work done by what work done by body force in the x direction work done by body force in the y direction z direction is there imagine it would become still in the air I have cut it short okay. So now we have to do some rearrangement why are we again I come back to my favorite question why are we doing what are we doing and why are we doing what are we doing we are trying to derive energy equation why are we deriving energy equation okay to answer this we derived momentum equation and conservation of mass why did we derive this to get the velocity distribution why do I need velocity distribution why do I need the velocity distribution I have not even come to heat transfer as on its own why do I need velocity profile what is the engineering interest of writing these equations okay what do I do with the boundary layer thickness what is my engineering yes shear stress why am I interested in shear stress I need to find the pressure drop then only I can decide my pumping capacity if I have to decide to how much pumping power that is whether I have to put a 0.5 HP pump or a 2 HP pump to pump from here to Puna let us say if I have to decide that who will decide that the pressure drop will decide that shear stress will decide how will I get that shear stress until and unless I have the velocity profiles I cannot get those gradient I cannot get the so engineering interest drives us to derive these equations on the same note okay to emphasize I want to emphasize I like this in fact this up this I have learned from Professor Bajans book he always reminds us what is the engineering interest of all these equations so the engineering interest of momentum equation and conservation of mass equation is what is the engineering interest of this equation to get the velocity profiles u v w and pressure once you get that you can from this you can compute the gradients right that will okay gradients let us write that gradients of velocity that will fetch me shear stress if you have to relate now itself what is shear stress v non-dimensionalized shear stress in terms of kin friction factor cf equal to tow all upon half rho u square u squared can be u infinity squared in external flow u average squared in case of internal flow to remind you f equal to 4cf f is Darcy friction factor cf is kin friction factor or fanning friction factor so why do I need this shear stress why do I need this shear stress for as ma'am told us it is pumping power the key for all of this is this we need to emphasize this to the students otherwise they will not understand why are we doing this equation what is going on they will get lost okay that is why I said for every derivation in between I have me I have tuned my mind you see I did not even plan I landed up asking question what am I doing because we tend to get lost in the derivation we should not get lost ourselves we should stop and ask ourselves it is a digression the same question we will ask yes how will I decide in fact the boundary layer concept itself came by Prandtl because he was asked to decide the blower capacity for blowing on one drying for a flat plate energy equation what am I doing what am I doing with the why am I deriving energy equation to get what temperature distribution what do I do with this temperature distributions what do I do with this temperature distributions what did you write for velocity after velocity what did you write velocity gradient here you are interested in temperature gradients why you want to get heat transfer coefficient let me remind you what is tau wall for fluid mechanics that is heat transfer coefficient that is heat transfer coefficient for heat transfer tau wall is the resistance offered by my solid surface for the flow to happen h is the resistance for the heat transfer to happen you can think analogically very straight forward manner no confusion on that score okay so h what is h you remember what was the h definition professor Arun had written very nicely minus k del t by del y at y equal to 0 upon t s minus t infinity I cannot get h until h definition is not q double dash equal to h into t wall minus t infinity h definition is this he told us yesterday so I will get from the temperature gradients I will get h which we usually write in terms of Nusselt number like I wrote tau wall in terms of cf or f okay now why do I need h why do I need h now you can answer heat flux or q t wall why do I say heat flux or t wall if you know heat flux you can compute what would be the wall temperature it is going to attain if the wall temperature attained is going to be more than the metallurgical limit in your heat exchanger your wall temperature turns out to be let us say 1000 degree Celsius you cannot so you have to either increase your Reynolds number so that your h goes up so that the wall temperature comes down so that is why so why we are doing we should not lose sight of this is pumping power or you can on the same note you can call it as maybe what can we call heat transfer rates maybe I think we will leave it like that okay so that is the that is the whole idea of doing this energy equation fine so that was good thing so this he this is done by Weijan before deriving he does this okay okay so now this was our energy equation now let us do some rearrangement why I reminded this equation do I see anywhere t no so I have to reduce this equation so that I need to t is there but still in e I have to write in terms of t so everything has to be energy has to be put in terms of t so momentum equation is given by this we have derived it yesterday Cauchy's equation it was called it is not I have not put for sigma xx I have put I have not put the velocity gradients this is the this is Cauchy's equation okay so now if I take this was my this is my rho du dt if I divide it by rho and multiply by u please listen to me carefully I am going to go fast I am not going to take you along with me until you make a conscious effort to be with me I am not going to take an extra strain because this is all algebra there is no concept involved so that is why I am going to go fast so if I multiply by u by rho throughout this is what I end up with so u du dt I can write as du squared by 2 and this is my right hand side so on the same note I can write v dv squared that is dv squared by 2 what is this y momentum equation so why am I doing this if I add this I will get u squared by 2 plus v squared by 2 and the right hand side is that okay now this is my equation b and this is my equation a so why am I trying to do that is I want to get rid of this u squared plus v squared by 2 if I subtract b from a b from a a b is nothing but u squared plus v squared by 2 on the left hand side so I get a minus b d by dt equal to blah blah blah is that okay huge equation but it is not it is not scaring us because we know what is each term so now d e by dt now I will go ahead and substitute for sigma xx sigma yy sigma xy I will substitute these equations okay if I substitute that and call this whole term as phi okay it is called as viscous dissipation I will discuss the importance of that little later but if I substitute for all this for sigma xx sigma yy sigma xy which are there here and call that as phi that phi turns out to be this what is this it is all what is this it is all gradients what is del u by del x stretching del v by del y stretching in y direction so what is del u by del x sorry no that is right this is right this is also stretching what is this del u by del y angular deformation angular deformation okay and shear stress of course agreed so now if I write this now I know that internal energy h equal to e plus we know h equal to e plus pv we write left and right so instead of writing v specific volume I have written it as p by rho why am I doing this because once I get in terms of enthalpy I can write enthalpy in terms of cp into t then I can get my left hand side also in terms of temperature all over the place there is temperature that is the exercises I am doing so nothing great I am doing so dh by dt equal to if I take total derivative dE by dt plus dt by rho by dt so let me expand d of p by rho by dt equal to 1 by rho dp dt minus p by rho square d rho by dt I have just differentiate kept key constant and rho constant so what is this d rho by dt what is d rho by dt what was my continuity equation d rho by dt plus rho del dot v equal to 0 so if I have to get d rho by dt that d rho by dt equal to minus rho del dot v so that is what I have reminded you and written for d rho by dt as rho del dot v which is here 2D I am writing so del u by del x plus del v by del y that is equal to 1 upon rho dp by dt plus p by rho blah blah so now let me substitute this here so what do I get what do I get what am I done can anyone explain me what I have done here in the next step because I tend to go fast I will catch someone can you tell me what did I do in the next step what am I trying to do in the first step I figured out what is dp by rho by dt why did I figure out dp by rho by dt what did I do I did what did I tell in the previous equation let us go slow then what did we write in the previous equation what did we say I am finding e in the left hand side I want to replace this e as h minus p by rho is that right but I have to take a total derivative of h minus p by rho first I have tried to find out the total derivative of p by rho that is what I get I am sure all of you have got this but now what am I doing in the next step I have done nothing I have written I have written the same equation which was there here rewritten the same equation I have done nothing now my next step would be for d e by dt I will write dh by dt minus d of p by rho by dt so that is dh by dt but minus dp by rho by dt becomes plus if I push it to right hand side and that is nothing but 1 by rho dp by dt plus p by rho del dot p. So this is the conduction term this is this term and this is the viscous dissipation term that is 5 by rho because instead of writing this big thing all the time I have written 5 by rho plus this 1 by rho dp by dt plus p by rho del u by del x plus del v by del y.  So dh by dt is equal to k by rho del square t plus 1 by rho dp by dt plus 5 by rho we have not achieved what we are supposed to achieve what we are supposed to achieve now write for h cp t so if I take cp t and take cp is constant so I get rho cp dt dt rho cp dt dt plus k del square t plus dp by dt plus 5. Let us write this equation come on at least final equation we are supposed to write so what do I get rho cp rho cp dt dt is equal to k into del square t by del x square plus del square t by del y square plus dp by dt plus 5 okay plus 5 what will be let us expand the first term on the left hand side rho cp into can anyone expand this we have done this n number of times yesterday yeah del t by del small t plus u into del t by del x very good plus v into by del y actually plus w by w del t by del z but I am not taking w so that is why that is not there is equal to k into del square t by del x square plus del square t by del y square plus I would like to keep dp by dt as it is dp by dt plus 5 what is okay now let us take each one let us take each one what is the left hand side term what is the left hand side term convective term it is a material derivative of my temperature so this is the temperature variation for unsteady that is transient temperature distribution this is unsteady and this is the temperature variation by virtue of by virtue of velocity that is why it is called as convection term that is why it is called as convection term okay what is this diffusion or what we call as conduction okay so usually in all the equations in the undergraduate level for the energy equation we would not be writing these two terms in next couple of slides I will be saying why okay for now let us call this as this is called as pressure term people also call this as noise term noise term next is viscous dissipation term viscous dissipation term why is it called viscous dissipation if you go back what is it consisting of all velocity gradients followed by viscosity so it is the viscous dissipation these two terms are important only for high speed flows I will show you through non-dimensionalization why it is so you have these two terms we would take so I think we are now clear of each term now let us non-dimensionalize yes now we will use principle of similarity what is principle of similarity this is my energy equation this is my energy equation I am going to non-dimensionalize every length term it can be x or y by some characteristic length l okay and u star by some characteristic velocity if it is internal flow it will be average if it is external flow it will be u infinity okay and pressure term by rho u square rho v square and t by t minus ts upon t infinity minus t if I non-dimensionalize this each term each term okay so if I non-dimensionalize this each term I will end up I will not go through the algebra okay it is too lengthy it is just substituting that is to tell you what will I get for let us say now let us say if I take one of the terms rho cp u del t by del x let us do now all of you do now so that you know when you go back home how to do it so rho cp there is its rho cp only what should I replace u with u star equal to u by v so u will be getting replaced by u star into t so similarly t star was t minus ts upon t infinity minus ts so what will be del t by del x it will be del t star into t infinity minus ts upon del x star upon l is that okay so like that I have done for like that I have done for each and every term it is not okay some problem is the mungesh upon l why did I get x star I have replaced x I have replaced by x star into l l is constant that is why I have taken it out is that okay yeah today you are in comfort zone you are asking everything I am happy about it okay so now if I non-dimensional this is what the non-dimensional parameter will look like let us write this equation let us write this come on if we non-dimensionalize what is that I get u star del t star by del x star plus v star del t star by del y star equal to 1 upon r e p r into del squared t by del squared t by what do you expect del x star squared plus del squared t by t star you are right del squared t star by del del y star squared plus e c which is called as occurred number occurred number many of you have used the book occurred and Drake I saw in your syllabus that occurred only is this occurred okay so occurred into u star del p star by del x star plus v star del p star by del y star plus of plus of phi star into occurred by Reynolds number okay what is occurred number let us write each term what is Reynolds number rho v l by mu we know this what is parental number I am just reminding mu c p by k I would not like to remember like this rather I would like to remember as mu by alpha for reasons very obvious now after professor around spending so much time on importance of Prandtl number and occurred number is v squared upon c p into t infinity minus t s okay now now let us stop here what is the left hand side term we said in this equation what does it represent what does it represent sir what does the left hand side of this equation represent it is pressure term convection term what it is it is convection now what is the second term representing that is this term representing this is diffusion or the conduction but what are the non-dimensional numbers associated with the conduction term Reynolds number and now that means if I vary Reynolds number and parental number this term also becomes important okay these two terms now next is what is this occurred number what is this term by the way pressure term this is the pressure term and this is the viscous dissipation term this is also having occurred what is occurred number occurred number is v squared upon c p into t infinity minus t s now let us calculate occurred number for a velocity of 10 meters per second if the velocity is 10 meters per second and c p of air is 1050 okay and typically our temperature differences are of the order of when I am heating or cooling let us say where of the order of 10 degree Celsius t infinity and t s what is occurred number reducing to 10 squared upon 1050 into 10 it will be of the order of 0.01 or 10 to the power of minus 3 yeah 1000 upon 100 upon 1000 into 10 let me take like that 1050 let me make it as 1000 so what do I get not 10 to the power of minus 3 it is 10 to the power of minus 2 so what does it what does it mean this is just to show what does this mean because I said that I we usually do not take these two terms what one is the pressure term another one is the viscous dissipation these two terms in the energy energy equation we do not take these two terms why can now anyone answer this question very small for low speed flows subsonic flows incompressible flows it becomes only important in compressible flows that is high speed flow you take a Mach number of let us say 1 or 2 if you take a Mach number of 2 what will be the velocity Mach number 2 means velocity of sound in air is 330 meters per second roughly 330 into 2 is 660 okay let us take 1000 let us take 1000 velocity because Mach 4 is not at all a wrong Mach because when when when my special is landing back what is its velocity what will be Mach 7 Mach 8 it easily goes so 1000 is not an number thrown out of the world it is very much applicable to the real life world for high speed flows if I take 1000 square 1000 square divided by 1000 in fact T infinity minus T s also will be very high okay it will be very high as high as 500 to 600 okay you can compute how can you compute what will be the temperature you can compute no t0 equal to or h0 equal to h plus v squared by 2 so t0 equal to t plus v square divided by 2 cp okay so anyway I have just got digress the point is the well the temperature will be very high and this v square divided by cp into T infinity minus T s will be significantly low I cannot afford to neglect any more these two terms which is the pressure term why it is called noise term perhaps now you can understand why it is called noise term because it creates noise and noise is created at high speed okay flame you are taking a blowing torch sorry flame torch oxy acetylene welding torch let us say it is a very high speed velocity so their noise term is very important for all incompressible flows this term is not these two terms that is the pressure term and the viscous dissipation term are not important I hope I have answered your question v squared by cp delta t where does it come from kinetic energy okay what what is denominator heat supplied coming here actually I have given the I am showing the equation so that you can get the clue who is generating that temperature when it is coming that that high speed flows high speed flows the kinetic energy gets transformed into thermal energy thermal energy okay I take this example all the time Kalpana Chawla died because that one of the tiles flew off and the because of thermal energy or the friction the wall temperatures shot up and it caught fire okay so that is that is this so it is how much of this kinetic energy is getting transformed into thermal energy is being represented by that is the that is the physical significance of occurred yeah one we said in the numerator kinetic energy in the denominator that is thermal energy now you can know dissipation of heat to kinetic energy right smaller so v squared by 2 is smaller compared to cp delta t frictional heating maybe we can say frictional heating is so that is how we need to understand this it becomes very important if you are working in high speed force these two terms are they will create completely they will change things completely in fact h definition also will change let us not get into that so that is that is the importance of the energy equation I think I think we have understood this yes yes and we are adding the macro size dimensions but if suppose we are working with the micro and mini channels type of dimensions should we ignore these two terms their rarefaction comes into picture their rarefaction yes there we have to there we have to include this even if it is less velocity less velocity because why I have to include there when I say rarefaction what does it mean density variations come come into picture density variations last two terms are coming because of what let me let me clarify let me clarify I am clear now how to reach you this pressure term and the viscous dissipation term came into picture in high speed force high speed force it is very clear right in high speed force what is the character of high speed force high speed is gained because of lot of compressibility that is because of density variation again in rarefaction what is rarefaction I am only working at low pressures but they are also again my density is varying that itself suggests that these two terms are it depends on the fluid it depends on the fluid which fluid you are handling if it is water if it is water these two terms are important no if it is a rarefaction will come into picture in micro channel no sir when we are defining the nodes and number that is the value is less point correct so in the Rayleigh-Fakson effect will not come into picture no I understand what you are trying to say is then whether we are in slip flow or again in if the Knudsen number is very high of course we are not reaching everyone I am trying to reach only this guy but that is okay because he has asked me so Knudsen number is high only rarefaction comes into picture is that right so if Knudsen number if I am operating for high Knudsen number high means how high typically 0.15, 0.2, 0.3 for that case these two terms are indeed important I cannot really get them neglect them but as you said if it is air only Knudsen number comes into picture if I am operating water in a micro channel are density variations there are density variations there no then these two terms are not important so I cannot generalize in one shot and say that for micro channel these two are important it is specific to the oh yeah their velocities are quite less that is the way mass flow rates are of the order of centimeter cube per minutes Mangesh knows well because incidentally one of our student works on earlier he was working on rarefied gas so that is how I am little familiar with that okay so now let us come back to our we will try to non-dimensionalize we non-dimensionalize only the energy equation what are the non-dimensional numbers we arrived at we will forget here afterwards high speed flows we got Reynolds number and Prandtl number similar non-dimensionalization if I do for continuity and momentum equation how did I get this momentum equation how did I get this momentum equation quickly I have not I have not written all the terms if you carefully see there continuity this is steady let me remind you steady two-dimensional incompressible flow let us write those terms steady no I just want to get it registered in their minds steady incompressible two-dimensional flows steady incompressible two-dimensional flows what will be my continuity equation del u by del x plus del v by del y equal to 0 what is my x momentum equation here it is there for those guys who do not remember u steady del u by del t is not there u del u by del x plus v del u by del y equal to minus of 1 by rho or maybe I can okay minus 1 by rho del p by del x plus nu into del squared u by del x squared plus del squared u by del y squared let us stop for a minute here what is what is the scale of u to what scale you would be let me take flow over a flat plate let me take a flat plate and let me take flow over a flat plate which is having a velocity u infinity what will be the scale of u u is having a scale of x direction velocity scale means it is of the order of not exactly two meters it will be of the order of 1.5 1.6 so u is of the order of u infinity what is the order of x the length is l so u infinity upon l is the scale of del u by now these two terms should be there means what will be the scale of v I do not know the scale of v let the scale of v be v only what will be the scale of y what is the scale of y boundary layer thickness delta so what is the scale of v now u infinity delta by okay now let us see what is this let us put the scales of each term scale can anyone okay what am I trying to say we know that u is very high we will be higher or smaller than u v velocity in the way it will be small all of us register the question is how small is what we are asking so it is very easy to see that u is of the order of u infinity v I do not know how small it is how small it is definitely I can see the scale of x and y x is of the order of length of the plate and y is of the order of boundary layer thickness so now v is of the order of it is smaller by delta by l compared to u infinity each so now let us see the term of each scale of each term what will be the scale of left hand first term u infinity into del u what is the scale of that u infinity by del x scale will be l so what about that means I get u infinity squared by l now what is the scale of the next term what is the scale of v u infinity delta by l and what is the scale of u u infinity by delta so delta delta cancels out and I get what is that I get u infinity squared by l what does this mean both are coming u infinity squared by what does this mean both are of the same order I cannot afford to throw anyone both are important both are important now del p by del x del p by del x will let us keep it as it is let us not worry about that okay now what about del p by del x I do not know the scale so I am not bothered about what is the scale of del squared u by del x squared nu into what is the scale of del squared u by del x squared u infinity upon x squared is what about next term nu u infinity by delta squared so are these two how do you compare both of them second term can be second term is very large because delta is very small compared to so I can neglect this is that okay that is what precisely is being done in the this equation I am going to digress little bit and I will get to show you how to get to delta in terms of Reynolds number from this equation okay maybe I will do that when we do the external flows because for that I need to throw del p by del x I am not told fine what happens to the y momentum equation you have not written that I have not written that is right I have not written here y momentum equation x direction v is very small compared to u so it would not have that much weightage compared to x momentum equation so that is why y momentum equation is completely thrown off is that okay we will come to the pressure gradient when we handle external flows for now this is enough okay and consciously I have not put that you are right what you said this right okay so now these are the boundary conditions there are no sleep boundary conditions and constant wall temperature I have taken that is okay now if I non-dimensionalize them the same way what we did here it is easy to non-dimensionalize because the terms are compact so what do I see in the momentum equation Reynolds number what do I see in the energy equation Reynolds and Prandtl what does this mean I will ask you one I will ask you one thing I this time only I got the trick how to reach that Moody's chart what is that you plot in Moody's chart I am coming off but it is fine in Moody's chart what what versus what you plot friction factor versus Reynolds number why do not I plot and why does in Prandtl number coming there why does in Prandtl do I friction factor is plotted against Reynolds number why is it plotted against Reynolds number if I do measurements with Reynolds number with water and if I am doing measurements with water I can take the f for given Re and get the pressure drop let us say I have done that experiment and get the friction do I have to again do it with air or if I am doing again with oil should I be doing again with oil that is I am asking again with the Moody's chart f and Re is only there Pr is not there question is why trick is answer is here answer is here no one I have not reached anyone I am I will wait for some time I will wait for some time what is non-dimensionalized momentum equation comprising of Reynolds number so friction factor is coming from where no no no no no no no no no let us go slow in order to go in the morning in the morning we wrote what did we write in the morning in the morning what did we write I do not want to hurry up what is velocity profile no I have written in the morning I have written from velocity profiles I get gradients and from that I get shear stress and from that I get my pumping so how do I get this velocity gradients by solving my mass and momentum equation so what is the non-dimensional number sitting in my mass and momentum equation Reynolds number that means my friction factor has to be dependent only on Reynolds that is precisely the Moody's chart is plotting f versus Re but then you have to answer me the question should I again can I use that for oil can I use that for air can I use that for water yes it is independent of Prandtl number it is independent of Prandtl number that is what my non-dimensionalization is teaching me okay that is why we do not have Moody's chart for oil again water again air again we do not have that we do not have that even if you want to find out pressure drop pressure drop for any unknown configuration unknown configuration you do it with the simplest possible fluid or blower whatever facilities you have you do with that fluid that is valid for all that is coming from my non-dimensionalized parameter but is it true for is heat transfer coefficient independent so what is my energy equation telling temperature distribution is dependent on Reynolds and Francis that means my H is dependent on Reynolds and so what heat transfer coefficient I measure for air as a fluid is not valid for oil or water if my fluid is changing I have to sit down and do my measurements again that is what this non-dimensionalization is so is that okay did I reach you this is very very important for this work only professor Nusselt got his professorship which is what is embedded in oil textbooks and we read it I think we need to spend ample time with students on this concept on this concept because this is what is very important when they are planning their experiments even if you are running fluid it does not make sense again to calculate pressure drop if you generated pressure drop for air again to do it with water until I have this concept embedded in my mind so this dimensional similarity is going to tell me when to use which dimensionless is that okay any questions I can take I can stop and take because this is important okay fine yeah yeah I am not going to take any question on this filter I am going to spend ample time on internal flows I am no I am today very disciplined but that does not mean that I am not going to take questions but I am going to take questions but I am going to time it so that my effort to reach you is going to decrease okay that is my point okay so now now what am I doing is this is the non-dimensional same thing whatever I told I am telling that if I am doing for two different configurations whether water or air all that I need to do is Reynolds number I have to keep constant and Prandtl number I have to keep constant for temperature and for velocity I need to keep only the Reynolds number same that is all so now there is another thing what is called as CF so maybe maybe we can skip this okay here I think I cannot afford to skip because CF definition comes here and actually Nusselt number definition comes here okay so non-dimensionalization non-dimensional u star is a function of what see this equation u star is a function of x star y star and Reynolds number yes if I take dp by dx I am not taking dp by dx I am not taking because if it is a flow over a flat plate pressure gradient will be there only in internal flow and pressure gradient will not be there in external flows I will answer why when we reach external flows okay why dp by dx is 0 for external flows it is very easy to show we will show that but if dp by dx is not there then my velocity profile u star is a function of x star y star and REL what is shear stress mu del u by del y at y equal to 0 if I non-dimensionalize this for u and y u star and y star what do I get mu v by L so what is this del u star by del y star from this I can write that it is a function of x star and REL why y star is not there because I am taking it as y star equal to 0 only so you have mu v by L F2 of x star REL now if I non-dimensionalize this C of x equal to tau all upon rho v squared by 2 if I substitute this what do I get here 2 upon rho v L in the numerator mu so that is 2 by Reynolds that means what what is that I get 2 by Reynolds number which is a function of x star and Reynolds that means what C of x is a function of location and this is the inference we did intuitively already I told the Moody's chart example this is Moody's chart C of x that is for internal flow where there is nothing like x star but for flow over a flat plate from location to location my friction factor local is going to change is that understood similarly let us do for temperature T star equal to G x star y star REL PR H is what pressure is defined as K del T del y if I non-dimensionalize that I am going to get K by L del T star by del y star at y star equal to 0 so what is Nusselt number H L by K so if I substitute that here I am going to get del T star by del y star at y star equal to 0 what is this function of what is this function of T star is a function of go back go back to the equation do not have to hurry up T star is a function of Reynolds number Prandtl number and x star y y star is not there again because I am taking del T star by del y star at y star equal to 0 so what does this represent Nusselt number is a function of Reynolds number and location is that understood now so now I guess we know how to utilize non-dimensionalization so that is about the power of non-dimensionalization parameters that is how actually now after understanding this correlation flow how do I know that we have plethora of correlation hundreds of correlation all the correlations are of the form of constant into r e to the power of x p r to the power of some N why do I know that how do I know that I should be taking it that way because in the previous non-dimensionalization I have come to know that my Nusselt is a function of Reynolds and Prandtl we have to emphasize our students that this is what it is that is why we write correlations in this form Dittus-Bolter pressure set that is 0.023 r e to the power of 0.8 p r to the power of either 0.3 or 0.4 depending on heating or cooling how on earth Dittus-Bolter knew that he has to take that form he knew it because it came from Nusselt non-dimensionalization had Nusselt not told that Dittus and Bolter could not could not have come up with that correlation okay yes Mangesh. Sir, can we start the students about this correlation that it is coming from the Buckingham spy theorem. No I would say Buckingham spy theorem is little difficult Buckingham spy theorem I would say it is difficult because it is based on intuition how do you say that my T star is dependent on mu, Cp and K it is see that was hydrodynamics that was hydrodynamics when during time of hydrodynamics when Buckingham's and Reynolds did the analysis their friction factor and Reynolds number when they drew that Nicaragic did those experiments he had no idea of Navier-Stokes equation that was indeed done by the basis of your Buckingham spy theorem only whatever you told that was hydrodynamics that was not called as fluid dynamics after deriving Navier-Stokes equation and after deriving energy equation after Nusselt showed the dimensional similarity only our correlations have flowed in. So I would recommend that we should not show it through Buckingham spy theorem but we should show it through Nusselt's dimensional similarity because it is obvious no no I do not have to make any assumptions to take which parameter and which not parameter into my pitch it is quite difficult for energy equation why because historically it is like that historically it has come like this that is the reason. But sir the methodology is same in both the cases. Yeah but there there is into you know the answer that is why you are able to do it there but here I do not know the answer still I am going to arrive at the same answer that that is the difference between the two. See how to teach a student when you are taking Buckingham's theorem on the right hand side these parameters only I should take if you if he asks you let me ask you why I cannot take K for friction factor in my friction factor dimensional analysis if you ask you why K and Cp are not taken how will you answer it it is not possible with this you can answer that is the power of dimension okay. So I would historically it came we studied it is fine but that is not the approach we are supposed to follow to teach the students it is through writing differential equations and writing and non-dimensionalizing them you attempt this for any of your PhD problem it will work for you it will work for you that is what is called as formulation no any problem first write them in the differential form dimension non-dimensionalize them you get the dimension any problem you take any problem flow through a pipe or droplet spray you take two phase flow you take any problem first thing is write the differential equations non-dimensionalize them you get the dimensionless number then you can plan your experiments to arrive at this type of correlation you change the Reynolds numbers you change the parameter then it is mass production but the thought process is this so there is another interesting thing very very interesting thing but very simple very simple okay now I have what I have done is I have thrown the pressure term if you just see first two terms what is that I have done what is the first equation what is the first equation momentum equation and I have thrown del p by del x because this is valid for a flat place okay again I am postponing why del p by del x equal to 0 pressure is going to answer that next the energy equation in the energy equation I have made Prandtl number equal to 1 actually here R EL into PR was there so I will make PR equal to 1 do you see anything anything magical is there between these two equations those two look if you watch carefully even boundary conditions also will look so the solution of one should be the solution of the other that is what is called as Reynolds analogy so tau s equal to maybe so if you just see that the same thing I am not going to harp on that CFX is a function of extra and R EL Nusselt is a function of extra R EL now PR if it is made one these two functions are going to be same. So CFX into R EL by 2 is equal to so if I write NU by R E into PR NU by R E into PR PR is not there PR is 1 so that is NU by R E into PR is what stanton number so stanton number becomes equal to CFX by 2 what is the beauty of this equation what is the beauty of this equation if you measure CFX you can get the heat transfer distribution in fact I came across a wonderful application of this concept by one of my colleague Professor Menanges he uses reverse for compressible flow over a flat plate in shock tube he measures the heat transfer coefficient and he calculates the frictional drag by stanton from stanton number to CFX and it works for compressible flow also it works okay so it is a very very strong equation please do not relegate this as only mathematical or only bookish it works in real life it works in real life okay for flow over a flat plate whether it is compressible or incompressible it works because measuring pressures fast is difficult but measuring temperatures fast is easy for compressible flow so that is why he measures heat transfer coefficient the point why I took that example is that it is strong it is strong even in pipes even in sub cooled flow we have seen that by and large it works of course with some variations so with this I think I will stop maybe I will not handle this cold burn analogy I will leave it like that because for the positive of time I think I will stop here you can go through this yourself for cold burn analogy.