h2 norm of u0 times a factor like this. There are two terms. The first term has a factor, y, but here is the gradient of psi integrate in the whole omega. Second factor has a square root of y, if you're small. So this actually is worse, but you integrate actually in the neighborhood of the boundary here. So that is this lemma here. We'll see that the second theorem follows easily from this lemma, once you have this lemma down there. All right, so how do you do it? To see the second theorem, you simply take your test function to be w, y. That would be a function in h1, 0 of omega. So you take psi to be w, y, and use the ellipticity to bond this from below by the l2 norm of the gradient of w, y, l2 norm squared here. And then you simply replace. You'll see this, you can actually replace the second term by the first. Well, first, bond is two terms by square root of epsilon times l2 norm of the gradient of psi. And so that will give you this estimate. You can do better than this, and now that's down in the lecture notes. And so this is a very rough use of this estimate, but there are other things you can do which actually produce a better estimate. But I just want to get the idea across using a simple case here. So now let's see how do we prove this lemma. So there are some calculations. I'm going to have to show you on the board. Well, actually, I don't think I have time. But it's pretty straightforward. You take the double y option and you take the gradient. There are three terms in double y option. You take the gradient of each term. And this is the first term, second term. In the third term is a product. So the gradient by product rule will generate two terms. And the last term, because the derivative goes to the second factor, your y option is still there. But you lost your y option on the first term because when you take the derivative of chi of x over y option, you're going to have an y option in the denominator which cancels it out. So the last term will be, it's a good term. You already have your y option in place. So for the rest, we're going to have to deal with how do we deal with the second term, the first three terms here. So we're going to simply put this term a hat, the gradient of u0, up there. And then we'll be replacing the gradient of u0 by the COVA function multiplied to the smooth in of the gradient of u0. So you generate a difference. You generate a difference here. And when you combine all the remaining terms, you end up with a b, b of x over y. The b is the function, the matrix. We introduced earlier, which has these two properties. One is that the mean is 0. Another is the divergence of each column is 0. So now let's simply multiply the gradient of a test function to the both sides and integrate it on omega. And then we use the fact that both u0 and u0 are solutions of the Dirichlet problem. So these two integrals are the same. Because the right-hand side, the fourth term, f, is the same for both boundary, both Dirichlet problems. So this is equal to 0. I mean, this is all the same. So you end up with these three terms here. So I want to claim that these first two terms, you're going to have your epsilon in place, or maybe square root of epsilon. Well, how do we know that? Well, look at the first term here. Remember, eta epsilon is supported in omega by vanish near the boundary. So in the interior, I mean, away from the boundary by, say, 4 times epsilon, 1 minus eta epsilon is 0. So this term, this integral, only taking place in the boundary layer of omega. And with a boundary layer, if you use a boundary layer estimate, you can increase the derivative. You're going to generate an epsilon in place. But the price you pay is that you're going to have to go to the second derivative of u0. So that's fine. For this term, here you have a smoothing. Here you have a function. And because of the second lemma, the difference between a function and smoothing can be estimated. With epsilon, again, the price you pay, you have to go to a derivative of the gradient of the u. But that, we can afford with that. There's a second derivative. The last term already has an epsilon in place. So the problem is, how do you deal with this third term, which you don't see any epsilon here? How do we know you're going to have epsilon come up, or square root of epsilon come up? So this come up with the property that we're going to have for the corrector. So here's the place. We use the symmetry of the flux corrector. So the calculation goes like this. So you have bij of x over epsilon s over epsilon du0 dxj and d psi dxi times eta epsilon. So we're going to write this as epsilon, take derivative of the flux corrector kij of x over epsilon, then s epsilon du0 dxj and d psi dxi times the cutoff. So that's one of the properties of the flux corrector here. We're taking the derivative k summed. When you take a derivative, there will be an epsilon on the denominator. So we need the epsilon here to cancel it out there. So the next thing you do is that you're going to use the product rule. You want to throw this guy here into the derivative, pi kij of x over epsilon and d phi dxi s du0 dxj. And then you have, of course, this is not correct because we have to product rule. We have to generate another term, which is in a form of epsilon phi kij of x over epsilon d square psi derivative of force here dxk dxi du0 dxj and eta epsilon. And because of skew symmetry, if I interchange i and the j, it's the same thing. But this guy has a minus sign come up. So this last term actually is 0. So that's the calculation showing on the screen there. OK, so you see how this factor epsilon come up. Yes. I don't have an epsilon. No, third term. Oh, fourth. Yeah, yeah. Gradient affects the eta epsilon? Yes. Wouldn't that give you a 1? Yeah, very good point. So if you take a derivative of a gradient, you're going to have an epsilon denominator. But the gradient of the eta epsilon support near the boundary. So that will give you a back square rule of epsilon. Very good point. Thank you. Yeah. So that's all you need to finish the proof. So you see how this flux correct is used in the process? We use the definition of the effective matrix and the correct. Of course, you know you're going to use it somewhere. Otherwise, you can just replace the effective matrix by some other matrix. It cannot be true. OK? All right, so that's the proof of theorem true, the convergence rate in H1. And now let's come back to the first theorem, which we want to show that the convergence in L2 with a power epsilon here. So first of all, the convergence here. So the way to do this is through a duality argument. So we're going to solve a Dirichlet problem with a zero boundary data. But in the right-hand side, we choose an arbitrary function in state 0 infinity of omega. OK? And then we introduce the sort of analog of a double epsilon for this problem. So the v epsilon minus v0, which is the homogenized problem, minus this epsilon times the corrector for L epsilon star and the same kind of stuff. But we do this for the operator L epsilon star, the adjoint of L epsilon. OK? So the chi star is the corrector for A star, for the adjoint of A. OK? So let's test this W epsilon without the derivative against this function g, which is arbitrary. So we replace this g by the divergence of this. So by definition, you have this one. You write out v epsilon. You're going to have three terms come up. And we're going to have to estimate each term, 1, 2, and 3. OK? Again, the goal is to generate a power epsilon. We do not like a square root. We want to do better in L2. We already have a square root in H1. OK, so the first term is of this form. So you have a integral of A times the grain of W epsilon times the grain of R epsilon. Both of these generate the square root of epsilon. So putting together, you have a power 1 in place. So that's the first term. It's fine. OK? Here, I will need some smoothness condition for omega because I need to have H2 estimated for L0 star for this last step. And that requires, say, for instance, C11 will be suffice. OK? OK, so the second term, you have this integral of A, grain of W epsilon times the grain of V epsilon. Here, we're going to simply use the main lemma. And you treat V0 as a test function. OK? So V0 is in H1 0. So we can take it as a test function. You generate two terms. The first term gave you power epsilon. The second term, you have a square root of epsilon by its integrate on the boundary layer. If you increase the derivative to 2, then you have another epsilon from the boundary layer estimate. So go get back to power 1 again. OK? So finally, the third term is, again, we use the main lemma. And you're going to have to estimate, treat this as a test function, the whole thing as a test function. You have to estimate L2 norm, L2 norm on the boundary. And it turns out it works out just perfectly fine. OK? So putting this together, we have this estimate for any function g in C0 infinity of omega. So by duality, you have L2 norm of W epsilon is bounded by epsilon times the L2 norm of U0. And so that's the W epsilon. There's a third term. But third term actually is already in good term. And so just throw it to the right-hand side. That's what we have. So that's a proof of theorem, too. In the lecture notes, there are more elaborate estimate involved non-tangential maximum function on the leapshift domain with symmetry conditions. So I don't want to show you that it takes too much time. You can read them. But you already see the main idea I have for this simple setting. OK? OK. So actually, I have a theorem here. Suppose you actually have a symmetry condition. Again, I do not need smoothness bounded measurable. It will be fine. You can actually do better than you can actually generate a PQ. Well, P and Q are related by this equation. This is interesting because this relation is the right scale. It's the right scale. Because you can scale both sides, and the constant will remain invariant. And also for h1, you can actually get a more precise estimate in the right-hand side. Instead of u0 h2 norm, you can get h1 norm of the boundary data plus some LQ norm of the right-hand side. So I want to spend the last 10 minutes, maybe five, talk about one application of this second estimate in the context of a boundary value problem and a radic estimate where some of you knows really, really well here. So before I get to that, let me briefly mention that the same argument can carry over for Neumann. And that's one of the good things of using cutoff. The boundary data actually does not enter into the play. So we do a cutoff on the other boundary. You can do the same proof for Dirichlet and Neumann boundary data here. So you have the same estimate. So going back to this h1 estimate, I want to mention how this will relate to some kind of radic estimate. I call this large radic estimate in homogenization. So if I write this, one of the estimates, the first one there, so that is what we're claiming is that suppose you have a solution to a Dirichlet problem with a right hand side 0, boundary data f. The second, the first line, so what we claim here is that if you integrate the gradient near the boundary layer, so this is less or equal to constant times epsilon. And this is bounded by a constant times epsilon times the tangential derivative of the boundary data f. So that's the first equation. The second inequality related to the boundary to the Neumann problem here. OK. So why do we call this large-scale radic estimate? By the way, again, here, this is true for Liebstein's domain. A is elliptic, periodic, and symmetric, but we do not need any smoothness condition on a coefficient. Just bounded measurable. We'll give you this one here. So why this is related to the radic estimate? So I want to bring up the small-scale radic estimate. So let's say you have a domain omega. You pick up a point on the boundary. So this is a local radic estimate. And we claim that if you integrate on this piece of boundary, say with the radius of epsilon, say this point is p here, the gradient u epsilon is a solution. d sigma, this is bounded by a constant on a bigger ball, so the gradient of f, plus a constant over epsilon times a volume integral in a ball centered at the p of radius 2 epsilon intersect omega. So here, I'm assuming l epsilon of u epsilon equal to 0 here. So this is true if I assume A is further, continuous, and elliptic. And I do not need periodicity. This is a small-scale estimate because you can scale. It's suffice to prove this for epsilon equal to 1. And then scale and blow up to scale epsilon. And this is true. So we do not need periodicity.  This is a volume integral. These are surface integrals. So this is, again, it's a small-scale estimate. We do not need periodicity. And as you can see, the large-scale here, if I combine this two-scale estimate, small-scale and a large-scale estimate, this estimate allow you to control this volume term by the first term. So we have completely separate two scales. In the small-scale case, we only need the operator to be elliptic and the Hewler continues. In the large-scale case, we only need the operator to be elliptic and periodic. But if you have all three, you can combine them to generate a four-scale radic estimate. So this is actually was approved by Kennegan and myself back in 2011 using a integrand by parts. So here, we present a different proof here. So you've got a four-scale that the boundary, the four-grading is controlled by the tangential and also by the co-normal derivative, with a constant independent of epsilon. And the coefficient assumed to be elliptic, periodic, symmetric, and also Hewler continues. Omega is elliptic. So with this, you can solve the L2 boundary value problem using the method of layer potential. And this is also done in the same paper. Because once you have the radic identity, then you can prove using layer potential to show the estimate for the non-tangential maximum function. So going back, how this is done? Well, it's a simple consequence of this what we have for the estimate in H1, this line of estimate here, square root of epsilon times the H1 norm on the boundary. And now, if I just look at what happened near the boundary, in the boundary layer, the third term, because of the cutoff, drops out. So you simply have the derivative of u epsilon minus the derivative of u0. And so you have this term here, controlled by the derivative of u0 in the boundary layer. Then because u0 solve a constant coefficient, when we do have non-tangential maximum function estimate. So the volume integral in the boundary layer can be controlled by the boundary data. Because the boundary layer has a sickness epsilon, this generates the square root of epsilon, which just matched the second term. And that's all I want to say. Thank you. Any questions? Yes. Yes, one of my former student, Chang Xu, has written several papers on the subject. So his most recent paper actually deals with layer potentials for operators with low order terms. You probably can find on the archive, his last name is Xu, first name spelled as Q-I-A-N-G. So he can deal with the first order and zero order, operator with first order and zero order terms. All right, thank you.


Welcome to the Pyramid Insider. I'm Tyler Patner. Today we're going to be taking a look at the brand new Seneca Eagle Claw Carbine in 25 caliber. So of course the Eagle Claw is an import under the Seneca line for Air Venturi bringing this in here and kind of dropped on us a little bit last minute here in 2019 as an end-of-the-year new product launch. Wasn't announced at Shacho and actually the Eagle Claw has been on the market in other parts of the world in different variations and stuff like that for almost two years now. So really excited to see this finally come to the US because it has a lot of intriguing features that we're going to go over here in just a minute, but before we do don't forget to like comment and subscribe. We appreciate it very much. Now let's check out the Eagle Claw in more detail. So starting at the front of the Eagle Claw. Now remember looking in line with the rest of the Seneca line, this is really the next evolution of the Sumatra, which is also a lever action. But unlike the Sumatra, the Eagle Claw is fully shrouded and actually does have baffles here in the end of the shroud. So it makes it much more backyard friendly than the Sumatra ever was, which was a huge gripe with that gun. Underneath the full length shroud you have a 17 inch barrel here in the carbine version. This is available in 22 and 25 caliber. It's also available in a full length version with a little bit longer barrel and a little bit longer overall length. Like I said, 17 inch barrel, you are looking at a 41 and a half inch overall length here in the carbine version. Now as we come back on the shroud, you'll notice that we do have a nice barrel band here that goes around both the cylinder and the shroud. And that leads us down to the quick disconnect fitting here on the end, which is very nice and easy to use. Comes with a nice cap here that just snaps on right over it. Very easy, simple system, love the QD on there, nice and exposed, easy to get to, no problems filling this gun. It does fill the 200 bar, 2900 psi. And thankfully there is a gauge on the underside, which is color coded. You got that 200 bar marking where it stops the green, goes to red. So very easy to read gauge, very nice overall presentation. 360 cc cylinder and it's rated for about 20 shots on high power. I say high power as well because we do have an adjustable power wheel very similar to what we had on the Sumatra here underneath. So we'll test it for you at a couple different locations to give you guys some idea. But as you can see here, you have a ton of adjustment range. So anywhere in between that, you know, almost a thousand down to 500 range should be achievable right with this wheel here. And of course, as you adjust it, it's going to change your shot count as well. We'll cover all that once we head out to the range. And coming up to the breach here, we do have a full length Picatinny rail that actually splits the breach here. So a nice thing about this, we'll talk about the magazines here in just a second. But your magazine doesn't sit up higher than your breach. So you don't have to worry about scope clearance or anything like that with the saddle of the scope. If you do want to go over the breach there, which is really nice. But just got a hawk air max compact of the six to 24 version mounted here should compliment the rifle pretty nicely. One thing I do want to point out about the breach here, and I'll actually go ahead and open that up. Don't know if you guys can see it there, but there's this little screw just in here. So what this does is actually kind of matches up with these slots on the bottom of the magazine and allows you to have kind of a stopper position for the mag when you load it in. So when you load it in, it can't go any further through the breach. And what that screw does is you can actually flip it to the other side if you are maybe a left hand to shooter, and you prefer to load it from the other side. So as it sits right now, so I can get around a side wheel or something like that on my scope, I can load it here from the right hand to the left hand side very easily. And that screw stops it. So adjustable if you want it. And talking about the magazines, these are plastic magazines, which is kind of a departure from the Sumatra, but a little higher capacity, which I like a lot. So eight rounds as opposed to six, and you do get two of them with the rifle. So very nice there. And these are very basic spring loaded magazines.  You load another one, you keep on going until it's full really easy. And we have a cross bolt style safety, which is actually engaged right now. I'm going to disengage it to show you guys how to de-cock this gun before we talk about the lever action. The trigger does have an adjustment to it for the release point. So basically that release weight is adjustable, haven't touched it out of the box, we'll get it on the scale and see how it does. But to de-cock the rifle once it's in the cock position, you're just going to want to hold the lever open. And then go ahead, pull the trigger, you kind of feel it come back under spring tension, then move your hand out of the way and close it up. And you're all good to go. Very easy to do. Just make sure you're keeping your hands clear of this area here. Talking about that lever action, you guys will notice it's got some nice shape to it, really contours nicely with the grip of the rifle. One thing for me personally, I could not cycle the Sumatra like this. Obviously you saw me just do it there. I can do it with the Eagle Claw with ease. I don't know what air venturi has done with this, but this is so easy and smooth to cycle. It is really a dream and allows you to get shots off so quick. This brings lever action love right into the equation for me. I'm digging on this lever action system. It's super awesome and really for me a highlight of this Eagle Claw. Last but certainly not least, the stock, a very nice Indonesian walnut. You have checkering on both the fore end here and the grip area. The cheek piece, you got to raise cheek piece for those right handed shooters out there which feels very nice, brings the gun into the shoulder and onto the cheek really nicely. For you lefties out there, don't feel bad. It is completely kind of not raised over on the right hand side of the stock, so you really shouldn't have a problem using this gun if you are a lefty. Of course, we've round out at a rubber butt pad at the back. Overall, the fit finish feel of this gun is great. I'm digging it. Just a matter of how it's going to perform on the range because they say big power and if it's got accuracy to back it up, this thing is definitely going to be one to watch. Let's go ahead out to the range and see what it can do. Loading the Eagle Claw magazine is really simple. Now, this is the backside of the mag here. You don't want to load it from the backside. The pellets will actually fall out of the back.  So very simple. You just drop those skirts in, kind of give them a little push. They'll drop on down and then you just rotate. It clicks into place and locks it. You don't have to hold it under tension or anything. You do that for all eight rounds and you are good to go. Probably my favorite feature on the Eagle Claw is this lever action. Just how much smoother it is than the Sumatra that came before it. This thing is incredibly fast with follow-up shots for you hunters out there. If you need that quick follow-up, this is absolutely going to deliver it at blistering speeds. Check this out. Just that fast. I mean, it's probably a second. Just have to reacquire the target and you're good to go. This thing is so smooth, so easy to operate. Absolutely some lever action love. So going straight out to 45 yards here with the Eagle Claw gun in this price point should absolutely be able to perform at that distance. Now, while this isn't a bench gun by any means, this is a hunting rig. So we're looking for one inch or better accuracy. That's going to do a squirrel's head every time at 45, 50 yards. And this thing's got the power for far more than that. So let's see how it did here. The Seneca 43.2 grain pointed pellets. You got a 1.3 inch group here with 7 out of 8 and 1 inch. Definitely can do better than that. Let's see what we got. The King Heavies JSBs, of course, 6 out of 8 and about three quarters of an inch. If you take these two away here at the bottom, 1.2 inches total. So still not that great. But moving up to the King Heavy, Mark II is a pure 1 inch group for 8 shots, a full mag there. A little bit of a big vertical dispersion could have been some velocity change. Gun's not regulated. So as you shoot through the air consumption, you're going to get a little bit of up and down there, especially at distance. But the best of the day, the standard JSB Kings 25.4 grains, I'm a little surprised by this because it's going to be pushing them very fast on this high power setting, three quarters of an inch for all 8 shots, sending about 4 or 5 right through the center of that group. They're absolutely phenomenal. This is going to get the job done on game every time for you. Good accuracy. Out of the Eagle Claw, let's put it over the chronograph and see how many shots we're getting. Starting off with the high power setting from 200 bar down to about 140 bar. The 25.4 JSBs are putting out an average of around 975 feet per second. And that's really over those 16 shots in that sweet spot we're seeing. And that sweet spot just so you guys know, considering that under 30 feet per second since the gun's unregulated. So 27 feet per second is our extreme spread for those 16 shots. And that's about a 53 and a half foot pound average, which is a ton of power. Dropping the gun down to medium power. And we'll show you on the power wheel where that is here in just a second. From 200 bar to 140 bar again, we're getting a couple more shots here. But really, we're still looking at that same sweet spot, about 16 good shots there within 30 feet per second. Actually, the extreme spread is identical to our high power string at 27 feet per second. And we're actually getting an average that is three feet per second higher than our high power setting at 978 feet per second. So you're still looking at the same 53 and a half foot pound average. Dropping the gun down to the lowest power setting, you're going to get a ton of shots. Now, whether or not it's actually going to be useful for you at just a roughly 380 feet per second, maybe not. But looking at about eight foot pounds of energy, and you're going to be able to do around 90 shots over the course of an entire string. Now, we're just showing you about 32 good shots here that we took, but it only used up about 20 bars. So you're getting really, really great air consumption. Very consistent results as well. Super low standard deviation. But again, very little energy, just eight foot pounds of energy out of the 25 fours on the lowest power setting. Wanted to take a second now that you've seen the chronograph results and show you what those power wheel settings we were using were. So what you're seeing right now is our highest power setting. That's that red dot. That's where the wheel stops. Okay, so that is ultra high power. That's where we were supposed to be getting our best energy about 55 foot pounds with those 25 fours. And we're getting about 16 good shots within 30 feet per second. Now moving down here, this black dot is where I was on for medium power. So this is about half of the revolution adjustment. Interesting thing we saw was that we're getting about the same energy, actually a slightly higher max velocity, but roughly the same number of shots, maybe a couple more here and there. So you're still looking at about 55 foot pounds, little over two magazines where the shots a little over 16 there. We had to go all the way down to this red dot, which is the next adjustment point down. And this is Mark's one full revolution from high power before we really started to see a difference. And here we're actually looking at about 960 feet per second at the highest. So again, all with the 25 four JSP is just a few foot pounds less than our max. Now this is where things get interesting because each notch down to the bottom here is actually represents a pretty significant jump in power. So this first notch right there was producing about 900 feet per second. So again, another incremental adjustment downwards, but still a ton of energy there with those 25 fours. This adjustment point here marks about 600 feet per second with the 25 fours translating to just over right around 20 foot pounds of energy. And then right here was where we tested. So the last position with our green dot there was our low power setting and that's about 380 feet per second. So really you're getting the bulk of your adjustment within that last segment from the red dot down to the green dot, which is really interesting to me. So it also almost makes the entire revolution of this wheel useless. But that said, still does have a considerable power range. But just wanted to show you guys in a little bit more detail after diving deep on this, because again, those chronograph results were weird to me. I've actually confirmed this function on a 22 caliber one of these as well. So that kind of seems to be the consistent pattern for the Seneca Eagle claw power adjustment. Before we close out today's video on the Seneca Eagle claw, don't forget to hit that like button, subscribe to the channel if you like the content and want to see more and give us a comment down below and let us know if you're feeling some lever action love. So the Seneca Eagle claw a lot of power in a very nice sleek good looking package. I'm really a big fan of this gun, three quarters of an inch at 45 yards, especially for a gun that is this powerful and specifically aimed at a hunting application, certainly will get the job done. The lever action is absolutely stellar, whatever they have changed to make this lever action as smooth as it is and as seamless and easy to use kudos to the guys over at AirVentury. This is awesome, digging on this a lot. I would love to see more lever actions come out onto the market. Magazines worked well despite being plastic, so no gripes from me there. The power adjuster a little bit weird, but we're getting huge power 60 foot pounds out of this carbine configuration. And that's with your heavier 33 to the 43 grain, you know, Seneca pellets, a true power monster right here. Not a ton of shots, but plenty for hunting for small game, medium game. This thing is going to be lethal. And in my opinion, despite the numbers that we got on our decibel meter, I do think this gun's backyard friendly. Of course, the longer shroud, the full length version is going to do a little bit better than this carbine. So keep that in mind if you're interested in the Seneca Eagle claw. But overall, guys, a thumbs up for me too. In fact, really digging on this Eagle claw. I think you guys will like it as well, especially you hunters out there. That's all I've got to say about it. For the insider, I'm Tyler Patner. We'll see you guys at the next one. I really want to like try the Schwarzenegger, like, yeah, but the things, the things, the things too heavy and I'm too small, you know what I mean? I don't want to know.


Alan Hamill and Suzanne Sommer's relationship is a testament to power of love, resilience, and ability to overcome life's challenges together. Alan Hamill, born on June 30th, 1936, is a Canadian TV personality, producer, and entertainer. Suzanne Sommer is born October 16, 1946, was an American actress, author, businesswoman and more. These two individuals, despite their different backgrounds and nationalities, found themselves drawn to each other in a way that transcended borders and any differences they had in their lives. The pair's paths first crossed in the late 1960s when they met on the side of the TV game show Anniversary Game.


Gweithio'r gweithio yw hwn yn ystafellio'r ffaith, ond mae'n fewn i'n meddwl i'ch thangos eich gwasanaeth, fewn o'r ffasol gweithio'n ffaint a'r hyn. One of my students, Peter Rudiatt Gould, ganddwch gyda ffildwr yn y gynnig o'r Lleidau Marseill ilywyr, y gweithio'r gweithio, ac yn lluniau yn y ddechrau. Felly hanfyddiw'r gweithio'r gweithio'r barod, y dreidiau, fewn i'r ardal Llywodraeth yn cael y gweithio, a mae'n fwyeg o'r llwysiau yn ymdod o'r hwn yng nghymru. Mae'n gorfodd diwethaf, i bwybliol, yn gwneud o gwybod yn ymddi'r llwysiau, a ffodd yfynodd o'r twymu yn ymddi'r llwysiau. Dyn ni'n rhaid i'r ffordd a'r ffordd, mae hyn yn dwylo'n weithio'r gwrthau. Efallai, mae'n gwybod o'r twymu'n gwisig yw'r thiru angen i gweithio'r gweithio fel gyd, neu'r gweithio'r rhan o'r ffawr. Bydd yw'r cyfnod haf wrth ddweud y grwpeth sy'n golygell yw'r ffaith cyllid, sy'n hollwch ymlaen yn unifau. Felly, y gael eich bobl. Y galat pol yn 2010 yw'r 40% am ystafell ymlaen i'r gweithio bydd ymlaen yn ei wneud erbyn o'r hwn yn ei gwneud ymlaen i gael 10,000 oed. Mae rhywbeth sy'n hollwch eich gobl sy'n hollwch i'w hyfforddiadau, drws gweithio i gynnwys, sy'n deall i'n gwybod i gyd, a phobl i gyd yn ymgyrchol, a'r iawn i'r unifyd. Pactpunigini, mae gweithio'r ddechrau o'r ddaf yn cyllunio o gymryd, bydd yn ddechrau i gyd yng Ngheilio'n gwybod a'r product o'r proles ymlaen, a'r creau o'r bydau a'r goleidau o'r gwybod i gyd, a'r ddau'n gweithio gyd yn gwybod i gyd, a amddân rŵlwg sy'n cysgu estosydd. Cyn oes un bwysig o'r system ffathol i Umhylch yn unrhyw dim, yn ffarth cymdeithas nhw bredd 공wg wahanol و gynllun gyda'r yn un bwysig o'r ffathol i Umhylch o homogwpen Melanesiolaeth. yn y ffathol i Umhylch, ddod o'r clymau a ddannag, o'r clymau a ddannag o gyfysg, o'r ddannag o bod azernig o arno, y mhenau i Umhylch fel y bywones. ..finden when we really need to understand the psychology involved. My colleagues and I have been studying this problem for quite a number of years... ..and I'm tempted to share with you the many interesting facts this has generated... ...but our experience isn't going to get us on the same page quickly. So instead, I'm going to tell you a bit of a story... ..about my trials Tri musicalistic and anthropology levels. And I'm going to slip a few facts in along the way. My story begins here in Papua New Guinea, where I went to live for two years... Gun y mal gan gailedydd ar alon roedd gyrfa человека yng Nghymru gan gan costechoedd o'r bodici, ein modiau'r cy expl о fy hwn. Rwy'n meddwl iawn i fydd festive i arbennig ac mf Dalfawr anfril, yna all Teis geleu nifer. Ty wedi arwb publisharnio'r adael i'r meddlisteb yng Nghymru, will soon return from the dead bearing all the wonders of Western technology. They'll flatten the rainforest and produce magically overnight a vast city of high rise buildings, the likes of which people had only ever heard stories about. Before this could happen, however, the ancestors needed to be convinced that the living were ready for this miracle to occur. It said that the ancestors, though invisible, are all around Getting all around at any given moment, keeping a very close watch on peoples coming and goings, they are gratified when people perform the correct rituals of the tradition and offended when they transgress against the norms and rules and morals of the thing. Now one other thing I should say is that these ancestors when they return are expected to take the form of ethnically European people so when I a young white Mae'r cymdeithas wedi gwneud am ddod yn cael ei wneud hynny, ymddangos ei gweithio'r gweithio'r gweithio a'r interpreter. Roeddwn ni'n rhoi'r gweithio sy'n gweld i'r gweithio'r community, byddwn ni'n cynnig i'r gwneud am y llwyddiadau gwaith o'r gwag gweithio'r cyfnodau, wedi'u llwyddiadau gweithio'r ffordd, ddaeth, yma'r dweudiau. Also, during my stay, I came to observe the development of a whole set of new rituals that were very intense emotionally and had never been performed before. They included very euphoric things like dancing and singing and a mass marriage and things of that kind, but they also involved some very dysphoric, very unpleasant rituals, and these seem to be especially bonding for people. Eventually, as they waited for their ancestors to return and consumed through large feasts, all of their livestock and garden produce, they ended up facing the spectre of starvation and had to go back to their ordinary lives and rebuild their shattered economy. Now you might think that the failure of prophecy here would have dented enthusiasm for the cult, but not at all. This cult has been going for more than 50 years. It's as strong today as it's ever been. And in fact, nobody's enthusiasm was at all dented by this experience of the failure of prophecy. In fact, if anything, people were more tightly bonded together than ever before. Now, I wanted to then see to the extent to which some of these things I'd observed in Papua New Guinea generalised to other cultural groups. So, my colleagues and I began to gather together a huge amount of information about the world's rituals extracted from an amazing resource called the Human Relations Area Files, which is a vast storehouse of ethnographic writings. I did say I was going to slip in a few facts. Here's an example. What we found is that if you analyse the rituals of the world, they do fall into the kinds of categories that I observed in Papua New Guinea. There's a sort of inverse correlation between frequency and emotional arousals and, in fact, a real clustering of rituals around the high frequency, but rather boring end of the continuum, rather like the laying out of offerings to the ancestors in those temples that unite large populations and a clustering of rituals around the lower frequency but more intense end of the spectrum that are much more bonding. This all seemed to confirm our hypotheses, but we also learned something that really surprised us, that as rituals become more frequent, agricultural intensity increases. In other words, as societies become more reliant on farming, as they become larger and more complex, the rituals become more frequent, rather less intense emotionally, and the really rare intense rituals, like the ones of the Splinter Group that I described, become less common. In fact, they're often perceived as a threat in larger societies to the unity of the state or the nation or what have you. Now, to test these theories, to see whether they really tell us about the way ritual and social complexity and group size etc evolve, we constructed a really large database of global historical materials called Seishat, named after the ancient Egyptian goddess of writing. This is a really ambitious endeavour. We have already 150,000 complex data points in the data bank, and it will enable us to test a wide variety of theories about the nature and origins of social complexity, but it will also crucially allow us to be able to make predictions about global trends in ways that we think will be a great value to policy makers. We also test some of these theories experimentally, so the idea that intense emotional experiences are especially bonding for the group is something that we've been studying for a long time, and our evidence does suggest that dysphoric experiences are the most bonding of all. One of the reasons we think this is the case is that dysphoric experiences prompt a great deal of reflection, and that is used to build the sense of the personal self, the stuff that makes you, you, and that makes me, me. When you share those sorts of things with a group, it's incredibly bonding. We've been studying the psychology of these sorts of patterns of group bonding among tribal warriors, victims of terrorist atrocities, religious groups, even football fans, and actually it turns out that if your football team loses regularly and you suffer a lot as a consequence, you're much more bonded than if your team consistently wins, so you should join the bad teams really. This also helps us to understand certain otherwise very puzzling religious doctrines. Why have a religion which a god senses only sun to be horribly tortured and killed for a sin that occurred many generations before he was born or any of them at the time? It sort of makes no sense in terms of factuality, but in terms of the psychology involved it makes perfect sense. This is an incredibly intensely bonding doctrine for Christians both with one another and with their Messiah figure, and I think this is also the spirit in which we need to understand these cargo cults. They're not fundamentally about the facts of where technology comes from. They're about the shared dysforic experience of colonised peoples, indigenous peoples in the face of oppressive colonial regimes, and their efforts to unify themselves in response to that. Fats don't unite us, but rituals and shared experiences and emotions do. An interesting feature of this is that it doesn't seem to matter whether these things are actually shared. What matters is the conviction of sharedness, and this actually may have some important policy implications. Violent groups are very often united, unified by the conviction of shared dysforic experience, so the idea that other members of your group share your sufferings. But what if we could develop interventions that challenge those shared assumptions, those convictions in the sharedness of the dysforia? My colleagues and I, as part of a European Research Council grant, are attempting to do exactly that, and if we're successful this may actually be one of those occasions where it may not actually be such a bad thing that facts don't unify us. Thank you.


Let us move on to the next topic which is one of these additional topics and this is what we call introduction to compressible flow, I am Balchandar Puranik, technically this is slightly different from your standard thermodynamics business I will say. Compressible flow as many of you would know is really part of fluid mechanics, however this is a special branch of fluid mechanics in the sense that it is a combination of fluid mechanics principles and thermodynamics principles and therefore in some situations, in some courses, some places some of this material is included in the thermodynamics course especially toward the end. In IIT as far as I know some instructors included in the thermodynamics course toward the end of their course some instructors do not and typically in fluid mechanics the standard undergraduate fluid mechanics class that we have at present does not include the compressible flow material. So, in that sense in case your fluid mechanics class does not have compressible flow included maybe it makes sense to include it in the thermodynamics part if it can be permitted. As somebody had said I think two days back in the initial session obviously there are places where a complete semester long course on compressible flow exists and in that case you can do a far more detailed but what we decided is that at least let us look at some of these basic ideas and in case there is any requirement in the final workshop we can expand these and include those. So, it is really the choices yours at the end of this I suppose you can decide whether some of this material can be included in the final workshop or not. So, what I have done is I have just put together a bunch of slides for two reasons one is that the time is about one hour or probably even less now. So, I thought let me just give you a brief overview of what we are going to do if this material is going to be included in the final workshop and this is typically what we include in a thermodynamics course. So, let me just go through this. So, what we will go through is a discussion on the following topics. So, conservation of momentum speed of sound in normal shock analysis and finally a discussion on steady quasi-1D isentropic flow analysis. We will discuss each of these in as much detail as time permits. But right at the beginning when I said that compressible flow is actually part of fluid mechanics and uses sufficient amount of thermodynamics ideas. The first point here will emphasize that the conservation of momentum. So, conservation of momentum is something that you normally do not see at all in thermodynamics. It is a concept that we normally utilize from fluid mechanics. However, when it comes to compressible flow analysis it is going to be required. So, I do not know really when at your institution when students go through a thermodynamics course whether they have completed a fluid mechanics class or they are simultaneously doing it or I do not know exactly what the situation is. But here for our purpose what I am going to do is I am just going to talk about what is the overall principle of conservation of momentum as employed or applied to an open system or what we will call in fluid mechanics a control volume rather than getting into the detailed mathematics of the conservation of momentum principle. I will write it in plain English so that hopefully we can understand what is going on there and then utilize it directly for a couple of situations immediately afterwards namely the determination of speed of sound and for the normal shock analysis. And finally, when we will come to the steady quasi-1D asymptotic analysis we will try to understand what that means. So, here is something this is a slide which I am sort of recycling from the CFD workshop that we had run several months back if anyone of you was part of that workshop the first part of that workshop included fluid mechanics as was required for CFD and then the later part was CFD. So, this was one of the slides which I had used in that fluid mechanics part to simply convey what is meant by conservation of momentum when we employed for an open system or equivalently what we call a control volume in fluid mechanics. I do not really expect that you will write all this down. What I am planning to do is for whatever it is worth I will put this set of slides on model so that in case you want to go back and download it and look through it you can do that. At present my opinion is that it is sufficient if you just look at the screen and simply go through whatever is written out as I discuss it. Of course you are free to write it if you want but there is no need. So like I said in fluid mechanics we call this a control volume I think in thermodynamics you will call it a an open system and a balance statement just like what you would write for a mass balance or an energy balance for an open system we write it for momentum. Now it is not that obvious actually when it comes to quantities such as momentum what this balance statement is going to be but try to couch it in the same form as what the mass balance statement you are aware of and what the energy balance statement that you are aware of. When we are talking about control volume or open system we are necessarily talking about fluid flow coming in and going out of the control volume and as the fluid flow comes in and goes out along with it it brings in certain rate of momentum and takes out a certain rate of momentum and those are the two first terms on the right hand side of this equation. Finally it is something like an account balance statement as you can think of and for a control volume or this open system what we will say is that the rate of change of momentum that is contained within the control volume is simply going to be the rate of inflow minus the rate of outflow plus in case there is a source term available within the control volume because of which there is a chance of increase of momentum within the control volume. This is exactly the same the overall statement is exactly the same as what you would write for a mass balance statement when it comes to control volume or open system. There you would write rate of change of mass contained within the control volume or open system is equal to the rate of inflow of mass minus the rate of outflow of mass plus the rate of increase of mass if there is a source. So now let me just take a minute to say that what we mean by rate of increase due to a source as far as mass is concerned. So in case you are dealing with a situation where your open system or control volume contains nuclear type of reactions where mass is actually created or annihilated through nuclear reactions using the famous Einstein's law. Then you would require a rate of source to be included for the mass balance. Normally in standard engineering applications we never are dealing with such situations and when we write the mass balance for an open system or a control volume customarily you will realize that the last term which is the rate of increase of let us say mass due to a source is never really brought into picture. We simply say that the rate of change of mass within the control volume is simply going to be equal to the rate of inflow minus the rate of outflow of mass that is it. That is the standard mass balance statement. So in the same way you can come up with a balance statement for control volume momentum situation as well and that is what I have written out here. So the only thing that is worthwhile pointing out in the momentum balance statement is what do we visualize and what can we convey to the students as a source term for momentum. And typically the easiest way to convey that is that what we are talking about here is an open system. So that there is a mass flow in and there is a mass flow out. So from instant to instant the mass content within the open system or control volume is going to change. That is fine but at any given instant if whatever mass content exists inside the control volume if there is a net force acting on the mass content or the material that is contained instantaneously within the control volume that is going to be treated as essentially a source for momentum. Because Newton's second law of motion tells us that if there is a net force acting on a certain fixed material its momentum is going to change. It will experience a rate of change of momentum and that is the way one can come up with a balance statement and that is what I have written here. The other thing which I would like to point out immediately is that just like what you have in mass balance statement if you are dealing with a steady flow situation there is no rate of change of mass within the control volume in which case what we say is that whatever is coming in is whatever is going out. I cannot resist it here but this is a typical situation where in class I will say that many times this is a situation with the students. You consider your head as the control volume and whatever information is coming in if it goes out at the same rate nothing is getting accumulated and that is not a good thing. So your head should not be a steady state system is what I jokingly point out. But yeah if we are dealing with steady flow situation there will not be any rate of change of mass momentum energy whichever one that you want to talk about within the control volume and in that sense you can simplify this balance statement. Now those who have background in fluid mechanics and those who have gone through fluid mechanics particularly can relate these terms and relate to these terms and can put these in mathematical form. I am not going to do that here the reason is because we are going to immediately employ this for a fairly straight forward situation where we have a one dimensional flow and we will see exactly how this statement is going to be employed. So then moving on it turns out that the speed of sound is one of the most important parameters that one has to deal with when it comes to compressible flow analysis. So what it really implies or entails is that we are talking about some sort of a wave propagation through the fluid medium and sound is just one type of wave. So what I have tried to point out here is that in general when it comes to compressible flow analysis we may have to deal with different types of wave propagations. So one type of wave is what many of you would know something called a shock wave. Some other people will be also aware those who have background in compressible flow analysis is that there is some other type of wave possible which is called as an expansion wave and so on. So what we mean by wave motion is essentially that let us say you introduce some disturbance at some location. For example right now when I am talking I am continuously introducing some sort of a pressure pulse or a energy pulse at this location and this increases the local molecular activity at this location. So what happens is that when you impart a certain amount of energy at location it will go into increasing the molecular activity at that location. Likewise what ends up happening is that these higher energetic molecules will move this energy pulse outward by bringing out more collisions away more and more away from the location where it all started and therefore the physical picture associated with wave propagation is that the molecular collisions is essentially responsible in transferring this pressure pulse from one location to the next location and so on. So that is what I have pointed out in the couple of slides, so couple of points on the slide at the top. While the molecular picture is fairly well known for these wave propagation we do not really have to bother with dealing with that when it comes to a course like thermodynamics or even fluid mechanics. What we end up using is a standard model where we say that let us represent this wave with what we call a wave front across which the fluid properties are going to change. So you can imagine and you can try to point out to the students that sound wave is something that you can visualize or represent using a model like a wave front across which there is a change in the properties from P to P plus delta P let us say pressure likewise temperature likewise density likewise velocity and then this wave front is moving with a certain speed which I am calling C here for a general wave propagation. So what I have shown is in the picture a standard one dimensional representation of a wave propagation picture where the wave is shown as a wave front which is moving with a speed of C in general. The region of fluid into which it is moving has properties pressure and temperature density etc. given by P rho T and just behind the wave front the properties have been changed by amounts delta P delta T delta rho etc. And if it turns out that these changes are infinitesimally small then that particular wave front is what we call a sound wave front. And this has actually experimental backing in the sense that people have measured what is involved in sound wave and it turns out that the sound waves as we know fairly well approximate what I have put it inside the box namely it can be actually shown that the sound waves do form wave fronts across which the fluid properties will change infinitesimally. So we will have P to P plus delta P where delta P is very very small. So this is what typically we would like to begin when we want to talk about what is meant by speed of sound and the information that you want to convey to the students. So formally we want to start with a reasonably general picture of wave propagation and bring about that molecular collision idea. Eventually when it comes to the analysis we say that the experimental evidence suggests that it can be modeled as a wave front across which the conditions will change by certain amount. If it so happens that those changes are infinitesimally small then we essentially have by definition what is called as a sound wave. So with this background let me just go through the formal derivation which is doable in this case to determine the speed of sound. So now the previous slide was sort of a general description of wave propagation. Now I am specializing to speed of sound and most authors prefer calling that speed of sound with the letter A. Again it is no hard and fast tool but typically at least in fluid mechanics you will see that people prefer using the letter A. C has also been used but A is something that I am using. So what we have here is this wave front traveling in air. We are necessarily restricting ourselves through dimensional situation. The wave front is traveling with a speed of A in two conditions which are P, T, rho, pressure, temperature, density etc. And just behind the wave front the conditions are changed to P plus delta P, T plus delta T and so on where again since we are talking about sound wave the changes are necessarily very very small. So infinitesimal changes.  The conditions in front of the sound wave are given by P, T, rho and the one behind given by the change amount. For the analysis what we do is we take the coordinate system which is ourselves let us say and attach it to the moving wave front itself. So then the bottom picture simply shows that the coordinate system has been attached to the moving wave front. In that case the wave becomes steady and what you will see is that the fluid which is ahead of the wave is approaching the wave front with a speed of A and will leave with a change speed which I am calling A minus some amount delta V. This is what formally mechanics people will call a Galilean transformation where all that I am doing is that I am making a coordinate system change to make a situation so called that was unsteady to what we will call a steady situation. So when we are sitting on the sound wave front you will see that you are not moving anywhere you are always there but the flow ahead of it is approaching you with a speed of A and leaving with a change of delta V. Now what we do is we come up with an open system or a control volume which is essentially surrounding the sound wave front just about hugging the sound wave front you can say and in that sense this control volume or the open system will be a one inlet one outlet open system. So the inlet is with respect to the flow that is coming into the sound wave front and the outlet is the part which is where you see that the flow is leaving the sound wave front. Once this is done what you can go ahead and do is you employ your standard conservation principle. So the first conservation principle that you can employ for this open system is conservation of mass which is shown on the left hand side. Since the coordinate system within which we are performing this analysis makes the flow steady when you are sitting on the sound wave front you will essentially see that there is no accumulation of mass within that control volume. So whatever mass flow rate that is coming in is the same that is going out and that is what the first line there says that m dot is simply the mass flow rate which is coming into this control volume or open system is the same that is leaving. What we do is since this is a one dimensional picture we perform the analysis on a per unit area which is normal to the direction of propagation. So you can imagine a normal sorry a unit area along the sound wave front and in that sense we perform the analysis. Since we are assuming essentially a uniform one dimensional flow picture the mass flow rate in on a per unit area basis is simply going to be equal to the density times the velocity with which the flow is coming into the control volume and with respect to our control sorry coordinate system that is simply going to be rho multiplied by A which is now seen as the flow velocity coming to the sound wave front. And mass flow rate that is going out is simply rho which is now changed by an amount delta rho times whatever is the change in the flow velocity which is simply going to be A times delta rho. Usually when you come to such calculations there is a little bit of algebra involved. So in this case for example what one will be able to do is you can open up those two brackets on the right hand side and perform the calculations. When you do that you will see that that rho times A on the left hand side and rho times A on the right hand side will drop out. In addition when it comes to the product of delta rho multiplied by delta V what we will claim is that we are necessarily working with the situation where each of these quantities delta rho delta V delta T delta P are very small. So their products are going to be even smaller. So those products delta rho times delta V etcetera will be neglected in comparison with the other terms. This is a simplification and it becomes more and more accurate as your delta rho and delta V are smaller and smaller. And by definition since we are dealing with a infinitesimal change in each of these properties it makes sense to do that. When you perform such simplification you will see that whatever I have boxed out there is the final form of the conservation of mass statement employed for this particular control volume or open system. On the right hand side I am writing the momentum balance.  So all that I am now going to do is I am going to come up with expressions for each of these terms and put it up there for that particular control volume. Since we are dealing with a steady flow situation the left hand side which is the rate of change of momentum within that control volume is going to be 0 and that is what you will see here. The equation which you see here p dot is something that I am using as a rate of momentum that is flowing out minus the rate of momentum that is coming into this control volume is simply going to be equal to the net force acting on the control volume. And the rate of momentum coming in or going out is simply going to be the mass flow rate multiplied by the corresponding velocity component. So the mass flow rates are already there for example if you look at m dot out it is that rho plus delta rho times a minus delta v that is the mass flow rate on a per unit area basis. You multiply that by again the velocity of the flow which is in that case a minus delta v and you will have the rate of momentum leaving this control volume and that is what the first term is likewise rho multiplied by a multiplied by a will be the rate of momentum that is coming in. And in this case we are basically neglecting any viscous forces altogether which are acting on the control volume. This is a fairly useful approximation to make and in particular you want to point out to the students that in general viscous forces will be present in fluid flow situations. However, if it turns out that the domain that you have selected for analysis if it is sufficiently far from solid walls etcetera the effect of viscosity in general can be neglected. So the only terms contributing to the net force acting on the control volume turn out to be the pressures which are acting on to the control volume. So if you see the pressure on one side is p on the other side it is p plus delta p and the signs have been taken into account such that I am essentially using the direction pointing to the left as the positive direction for the flow in that sense the signs have been taken into account. Again you perform a simplification of terms drop out the higher order terms as what we say and the equation at the bottom is something that is the left out part of the simplified momentum value. And what is then you can do is right here this term rho multiplied by delta v is something that will appear in the simplified momentum equation which you can eliminate using that conservation of mass statement. You can see that rho times delta v is available as A times delta rho. So you can replace that and if you simplified I am going slightly faster understand but I am just trying to give an overview of what is involved in such analysis. If it turns out that we are going to include these topics in the final workshop I will do a far detailed job of working out the derivations and pointing out how things are going. But right now hopefully I want to just give an overview. Finally what turns out after the simplification is that the square of the speed of sound turns out to be the ratio of the change of pressure across that wave front divided by the change of density across that wave front. Let me point out something which some authors prefer to do in addition to what we have done. It is up to you whether this approach is something that is appealing to you or you would like to do something else when we want to argue that the process across this wave front that is happening is essentially isentropic. So let us see how we can do it in a couple of different manners. At least some authors prefer to do it in this fashion. So in addition to that conservation of mass and momentum balance for the control volume one more conservation equation that you can come up with is the energy balance and I think after having gone through the open systems analysis earlier you would recall the so called one dimensional steady flow energy equation. We can employ that steady flow energy equation in one dimension for this control volume. You can note that there is no external heat transfer into or out of this control volume neither there is any sort of work transfer going in or out of this control volume in which case the neglecting of course changes in the elevation because we are talking about essentially one level where everything is happening. You can bring about the one dimensional steady flow energy equation to say that it is simply the specific enthalpy times the inflow velocity squared over 2 is going to be equal to the specific enthalpy on the outlet side plus the velocity at the outlet side square over 2. So this is simply the form of the one dimensional steady flow energy equation that will reduce to in this particular situation. Going ahead and neglecting higher order terms which will be in this case delta v times delta v or square of delta v what you can come up with is a reduced form of the one dimensional steady flow energy equation which is that the change in the specific enthalpy across this wave front is going to be simply equal to the speed A times delta v. These are fairly straightforward to work out. This algebra is really extremely straightforward. So I am skipping these right now. If required we will look at it later in the main workshop in detail but right now this is one. So this is some sort of an intermediate result that one can obtain from the energy equation employment. Finally you use your so called TDS relation from the thermodynamics that you have gone through and starting with that you are able to simplify step by step using the two previous expressions that we have obtained namely that the change in the specific enthalpy across this wave front is equal to A times delta v and A square is equal to delta p over delta rho. If these two are used in a sequential manner which I have outlined here on the bottom part of the slide what you can actually show is that the change in the entropy in the specific entropy in this particular case across the sound wave front can be shown to be equal to 0. In other words what we can argue in this case is that the process that is occurring across the sound wave front is essentially isentropic. So this is one way of doing it and as I said some authors especially from the compressible flow gas dynamic side prefer to do it in this fashion. Some others will simply say that let us go back to the control volume. Here is my control volume. What do I have? I have a control volume where there is no external heat transfer that is one good thing and the properties across the wave front or from the inlet of the control volume to the outlet of the control volume are changing by infinitesimally small amount. So simply by looking at these two factors that there is no external heat transfer and the properties are changing only infinitesimally. Right there we can argue that the process is essentially isentropic that is one way of looking at it. One way or the other finally when we want to close this discussion we realize that the sound wave propagation can be approximated as an isentropic process and therefore formally when we want to write this speed of sound we want to write it as the change in the pressure with respect to the change in the density all occurring at constant entropy and formally the mathematical way to write it is partial differential of pressure with respect to density at constant entropy. So this is what we will normally see as the expression written out for the speed of sound that the square of the speed of sound is given as the partial derivative of pressure with respect to density at constant entropy and hopefully I have tried to argue why we can say that the process which is involved across the sound wave front is isentropic. So that subscript s here is added to remind us that the process across the sound wave front is essentially isentropic. We can go ahead and include one more relation from your previous knowledge and that is isentropic relation which simply relates the pressure and density through that isentropic law which is that the pressure is equal to a constant times density raise to gamma. I am using gamma as the ratio of the specific heats Cp over Cv. I think some authors use k as well I am preferring to use gamma. So gamma here is the ratio of specific heat at constant pressure to that at constant volume. This note that this relation p equal to constant times rho raise to gamma is applicable only for isentropic flow. So if you go ahead and perform differentiation of this particular relation you can actually find out this dp d rho as we will say and substitute it back here and if you simplify that again this analysis we can do later in detail in the main workshop. I am simply pointing out what you can do here. If you do that then you will see that the dp d rho at constant entropy can be equivalently written as simply gamma times p over rho and then finally using ideal gas equation of state p over rho can be replaced in terms of the specific gas constant r times t and therefore finally we can arrive at probably the expression which many of you are definitely familiar for the speed of sound that is the square of the speed of sound is given by gamma which is the ratio of specific heat at constant pressure to that at constant volume times r which is the specific gas constant times t which is the temperature of the fluid medium into which this sound wave is propagating and that is the square of the speed of sound. So here you know normally what I do is once you have arrived at this you can say that let us estimate the speed of sound in standard conditions in air for example. So for air we have gamma is equal to 1.4 in standard conditions r is about 287 joules per kg Kelvin 287, 289 and let us assume a temperature of say 27 degree Celsius for the purpose of calculation which is 300 Kelvin. If you put it all in here you can show to the students that the speed of sound is roughly about 350 meters per second in these conditions. So this is something that is the first part that we want to do in these compressible flow discussions. It turns out that this speed of sound is one of the most important parameters that you will always come across within this portion of compressible flow and that is what we normally begin with. So just to summarize quickly what we have done we have gone ahead and utilized three conservation principles. The additional one that normally thermodynamics does not see is these momentum balances but that is something that we are borrowing it from fluid mechanics and then the conservation of mass, the conservation of momentum and finally as an additional exercise we do this conservation of energy as a steady flow energy equation for the control volume and arrive at certain important results one of which is the estimation of speed of sound. So is that roughly acceptable in the sense that has it been followed what I am doing? I know that I am going fast. Then let me move on and then define the other parameter which is of importance in compressible flow and that is something that we call the Mach number and that is simply the ratio of flow velocity to the sound speed which we have just calculated. And if you look at the fluid mechanics side of it normally we like to provide some sort of a physical meaning to many of these ratios. Those who are familiar with fluid mechanics will recall that the Reynolds number has a certain interpretation in terms of ratio of inertial forces to viscous forces and so on. Likewise there is a nice little interpretation that we can associated with Mach number and to do that although I have not done it here I will do that in the main workshop if it is taken up. We look at the square of the Mach number. So when we look at the square of the Mach number on the right hand side you will deal with v squared over a squared and with certain manipulations including the expressions for the specific heats and the gammas and so on. You can actually come up with an interpretation that the square of the Mach number can be used as a measure of the ratio of the bulk kinetic energy to the random thermodynamic energy within the flow. Why this is important is because in compressible flow typically eventually you will find out that the bulk kinetic energy contained within the flow on a specific basis which will be simply v squared over 2. m times v squared over 2 is what we will call the kinetic energy. So v squared over 2 is simply on a per mass basis, unit mass basis. That if you compare with the thermodynamic internal energy which can be roughly classified as cv times t let us say. You will see that when it comes to compressible flow that bulk kinetic energy is not negligible in comparison with the thermodynamic internal energy and that is what one of the most important facts is about compressible flow. If you want to point out to the students that what is difference, what is different between compressible flow and incompressible flow, one factor that you can point out is that the bulk kinetic energy incompressible flow is not negligible in comparison with the random thermodynamic energy. In case of incompressible flow which normally we are familiar with at least in mechanical engineering, the bulk kinetic energy usually is very, very small compared to the thermodynamic internal energy and that is why many times flows in mechanical engineering you will see can be safely approximated as isothermal flows. There is no change in the temperature within the flow unless there is an external heat transfer happening. But in compressible flow that cannot be done even though there is no external heat transfer occurring because the bulk kinetic energy is sufficient as a fraction of the internal energy. There is actually an interchange between those two happening and you will see that because of which the temperature within the flow will change from location to location even though there is no external heat transfer in or out of the flow. So that is something that I normally like to point out to the students. To be honest with you this is more on the fluid mechanics side than a thermodynamic side but something that you may want to keep in mind. Once we define the Mach number then we can say that if it is less than 1 we will classify the flow as subsonic flow. If it is equal to 1 we will call that as a sonic flow where the flow velocity is exactly equal to the speed of sound. That is v equal to a right here. And if it turns out that the flow velocity is higher than the speed of sound we call that situation a supersonic flow. There are many more slides obviously I knew that I will not be able to complete it but let me just quickly walk through what we can expect in the later part. Here you know rather than going through the isentropic flow discussion now I thought that while we are discussing the conservation equations for a control volume that is surrounding a wave front which we just did for a sound wave. One thing that you can do is immediately you can talk about what is a shock analysis. Fundamentally there is nothing different between what we did for the sound wave analysis and what one can do for shock analysis. It is exactly the same. The picture is the same. The only difference is that the fluid properties now are not changing by infinitesimally small amounts across the wave front but they are changing by finite amount. So a normal shock is something that I will just go through very fast. It is essentially a wave front across which fluid properties will change by finite amounts as opposed to infinitesimally small amounts in case of sound wave. So some people will want to classify sound wave as a special case of a shock wave where the changes across the wave front are infinitesimally small which is a special case of any change. Physical measurements have shown that the stream wise extent of the shock is very very small. Typically it is about fractions of micrometers and therefore for a primary analysis as what we would do in case of a typical undergraduate course we would like to treat this shock front as essentially a discontinuity. In that sense that sound wave front which we talked about can be considered as a weak discontinuity. So weak is supposed to convey to us that the changes that are occurring across the sound wave front are infinitesimally small that is about it. So here we have again the same picture. There is a upstream flow which is coming to the shock or approaching the shock with properties p1, t1, rho1, v1 and the mark number m1 and across the shock things have gotten changed to pressure 2, temperature 2, density 2, velocity 2 and mark number 2. So the formal purpose or objective of a normal shock analysis is that given all these upstream quantities can we determine the downstream quantities without really bothering about what is exactly happening through this shock front. So what we do is we again come up with that control volume which surrounds the shock front and we simply look at what is coming in and what is going out and simply relate those through the standard conservation equations which have already been discussed. So if you did that what you will see is the first equation where my cursor is pointing right now is your mass balance equation for that control volume again written on a per unit area normal to the flow direction. The second equation is that momentum equation which we had discussed earlier when it comes to a normal shock the momentum equation is written in the form that is written on the board right now. And finally the third equation is your 1D steady flow energy equation again noting that there is no external heat transfer and work transfer in or out of this control volume. And immediately I am writing the specific enthalpies in terms of the CTs and the temperatures. Again as was being discussed with in the previous session there is a certain datum that we define and let that datum be sufficiently low so that we do not bother about getting into any negative or whatever quantities. We are going to essentially look at only the changes with respect to that datum and therefore I can safely write that H is simply Cp times T and so on. Finally the closure relations for this set of equations as we would like to point out is provided by the fact that the ideal gas equation of state is valid everywhere including the front of the shock and the back of the shock. And therefore P equal to rho RT is applicable both in conditions 1 as well as say and conditions 2. So that is it this is really the normal shock analysis if you want to simply say that all that you need to do is employ the conservation equations of mass momentum and energy across that control volume and use the closure relations of P equal to rho RT to relate the properties behind the shock to the properties ahead of the shock the shock analysis is over. The remaining part is only algebra manipulations of these equations to couch them in certain form which I have not outlined here but we will look at it in detail if it is to be included in the final workshop. There are couple of different ways of doing it and it turns out to be slightly more tedious than the sound wave analysis but it is fine it is purely algebra is nothing more. So sometimes students get dissolution that you know what is going on. Every time I would like to point out to them that we are doing nothing but simply manipulating the same set of governing equations that we are familiar with. So that you need not lose the sight of what you are up to finally there is a little bit of mathematics and manipulations involved but finally we are simply looking at the governing equations and looking at how those can be transformed in a manner which we can use. So just for the sake of this purpose I have written that algebraic manipulations you can do and one way of obtaining the solution is in terms of what is called as the Rankine-Ugonio relations which are nothing but the governing equations represented in a special manner where we are representing everything as ratios of properties now. So we represent the density ratio, the velocity ratio, the pressure ratio and the temperature ratio across the shock. The solution of this problem turns out to be in the form of a single parameter which is that p2 over p1. So you can see that here rho2 over rho1 which is same as v1 over v2 through the mass balance equation can be written using the expression that is shown here which involves this p2 over p1 ratio. Similarly the t2 over t1 ratio can be written in terms of an expression which involves p2 over p1 and therefore to summarize what I will say is that this Rankine-Ugonio relations are essentially the normal shock solution in terms of the parameter p2 over p1 which has a special name in the gas dynamics or compressible flow terminology it is simply called as the shock strength. So far you point out that everything is algebra nothing more nothing less. Finally some physics has to be brought in and we say that what is it that makes the shock shock in some sense. So then you say that we go back to our control volume and we realize that conditions are changing from 1 to 2 across the shock no heat transfer no work transfer. So the only way this process is going to be permitted is by employing our second law of thermodynamics which must state that entropy in conditions 2 has to be equal or greater than condition 1. So finally you come up with that statement that the requirement of the second law for this shock process to be feasible or possible is that s2 minus s1 has to be greater than or equal to 0 and with much more algebra which again I am not doing here you can show that this s2 minus s1 can be greater than or equal to 0 only if that shock pressure ratio or the shock strength is greater than or equal to 1. So only if this condition is satisfied then you can show that we are essentially in a position to call a situation shock and to conclude this we then say that the pressure must increase likewise you show that the temperature and density must increase across the shock and then using the continuity equation or the mass balance equation you can show that the flow must decrease in its velocity. So this is one way of performing the normal shock analysis the same analysis rather than using that parameter p2 over p1 can be performed using an upstream mach number as the parameter. So everything can be couched all entire solution that we went through can be couched in terms of m1 which is the mach number of the upstream flow and again when you put in the condition that the second law requirement is that s2 minus s1 has to be greater than or equal to 0 for this process to be feasible you will come up to the conclusion that upstream mach number has to be supersonic. So the condition that you come up with then we will work out this in detail later is that m1 must be greater than or equal to 1 corresponding to which there is an associated conclusion that the mach number behind the shock has to be less than 1. So therefore then to conclude again the upstream flow with respect to the shock has to be supersonic and the downstream flow with respect to the shock has to be subsonic. I will just use three more minutes although I do not think I am going to be able to complete it but just to point out what else we are going to plan to do in the main workshop and that is this so called steady quasi-1D isentropic flow. So quasi-1D in plain English is supposed to mean almost 1D which basically is supposed to mean that we are talking about a flow inside a duct whose cross sectional area is varying very very slowly as you move along the stream wise direction. So that at any particular cross section the two dimensional effects are fairly negligible. That is all that means by the word quasi-1D which is almost 1D and we will look at adiabatic and frictionless flow which makes everything isentropic. Adiabatic and frictionless which essentially means that there are no irreversibilities involved within the flow will make the situation isentropic. In which case what we end up doing is we perform the same analysis everything is the same again and again in fluid mechanics nothing is different. All the time what you do is you select a certain control volume and you employ the conservation laws as what we have seen already the mass balance, the momentum balance, the energy balance and then come up with some more manipulations to see how things work out. So here in this particular case what we are going to do is we will identify a little section within your duct which is varying in cross sectional area. The length of the section is dx which is elemental length and things change from p, t, rho, v, m etc to p plus delta p, p plus delta t and so on across this length and area also changes from a to a plus dA. So now what I will do is I will just point out what is done here I am not going to get into the details because there is a little bit of algebra involved again. We employ the mass balance which is on the left, we employ the momentum balance which is on the right, we obtain certain intermediate relations which you combine to bring the so called very well known area velocity relation based on which then you can point out what happens in case of supersonic flow versus a subsonic flow. So if you are dealing with a subsonic flow which means that the m is less than 1 what you will see is that if you decrease the cross sectional area of the duct correspondingly there has to be an increase in the flow velocity as shown by this area velocity relation. On the other hand if you are dealing with supersonic flow it turns out that if you decrease the cross sectional area the velocity will also decrease. So that is something that is worth pointing out from this area velocity relation and further there are a few more additional things that you say that in case there has to be a Mach number 1 to be achieved it will be achieved only at one area minimum within that converging diverging duct and that area minimum is something what we are going to call as the throat. One thing that I like to point out at this point to the students though is that just because you have a throat does not necessarily mean that you are going to get m equal to 1. It depends on the operating conditions. However if m is equal to 1 is to be happening it has to occur at that area minimum. That is something that is worthwhile pointing out. Many times students go out with this concept that as long as there is an area minimum there is going to be a Mach number of 1. That is not true only if certain conditions exist Mach number of 1 will occur at that area minimum. On the other hand because physically we have an area minimum we will always have a maximum or a minimum velocity depending on whether you are dealing with a subsonic flow or a supersonic flow. But m need not be necessarily equal to 1 just because you have a throat. That is something that is worth pointing out to the students. And then there are more terminologies like the stagnation conditions and so on which I will skip right now. Finally what we are going to wrap up this is with a discussion on the operating characteristics of nozzles and here what I personally want to do and like to do in the class is first discuss the operating principle of a purely convergent nozzle. So that we have only a continuously decreasing area of cross section ending up with the so called throat let us say there is no further part to the nozzle. It is essential because with respect to that purely convergent nozzle I think you can reasonably well explain the phenomenon of what we call choking. Many times what I have found is that if you do not do the discussion on purely convergent nozzle and go directly to the converging diverging nozzle sometimes students have difficulty in figuring out exactly what is happening. On the other hand if you do the discussion on purely convergent nozzle first you can then say that well now I will simply add one more part which is a divergent part and make it a CD nozzle or a convergent divergent nozzle and it turns out to be at least in my experience that the explanation becomes slightly better I think and the students also appreciate it little bit more. So I am sorry I have used up almost one hour and again I am sorry that I have not been able to go through enough details here but roughly I thought that I have given I hope so what we will be incorporating in case this topic is to be included in the final workshop. So I think that is where I will stop. Thank you for your attention.


So I was thinking. I know it happens once in a while about this here boat and I recently took it on its first real voyage and couple things happened that I can improve on one is I massively overloaded it like way more weight than I was you know Originally intending to put in the boat, but that's how much stuff was there and I have to presume this is gonna happen again So this is one of the things that would be nice to fix is more buoyant And then the other thing I just haven't addressed it yet, which is that the solar panels are a little bit too low voltage to charge the battery Like directly effectively. I mean it charges the most of the way But it won't charge them all the way and it kind of doesn't slow it bottom line is Higher voltage solar panel would better or something. I mean there's there's various different solutions I Have an idea that will solve both of those things and it's really easy to say and not that easy to do All right, let me just barf it out Cut the boat in half extend it like four or five feet like a meter and a half and put an extra solar panel in the front and The extra solar panel in the front would be one of the glass ones that have which are heavier, but I could just do one and I would put one on that has a high enough voltage to charge the batteries and then they'd use all these four existing solar panels to Direct power the motor and then the one in the front to charge the batteries And then the extra length in the boat would make it more buoyant for more passengers more weight and whatever and The solar panel with the higher voltage to charge the battery. Oh, it's gonna be a big chore if I do that I'm thinking about it though. It really would make the boat pretty awesome So If I cut it, you know right about here Then I could get some sheet something and you know Just match it right to the hull after the spread out and then put the fiberglass and it wouldn't really be that hard to do Just I have to take the whole boat out of the water and kind of dismantle everything to make it light enough to do that The actual doing if it wouldn't be that hard I Think I have to do it And that would basically double the amount of passenger space up here Oh And then this solar panel will come up to here be a really long boat then whatever I Could just slap an extra solar panel on here and it'd come up like tip right up to the front of the boats But that that I don't want to do that I definitely need to add some length and if I add length it's not really gonna add much You know water resistance to the boat movies look at these bottles God, what is this? This is not from us. This is just People of the world stop throwing your bottles in the water seriously Hmm. I guess I haven't cleaned up my boat making mess yet. Oh That would have been so much easier to do at the beginning Of course, I didn't know what I know now at the beginning All right, I'm not doing it today or anything, but I think one of these days coming coming up I'm just gonna yank the boat out of the water Cut it Need to get some fiberglass I'm gonna think about that for a few more days. There may be a better idea Just didn't think of yet All right, I'm gonna go back to working on this car And I'll have more buoyant and longer so I can put an extra solar panel on it and so it can hold a lot more cargo space and have more passengers and Go over here exactly what I'm Looking for I'm gonna have to entertain this notion now. All right, let's think about What if I took the motor these solar panels and all that stuff and put it on this boat? It would have plenty of room and buoyancy to put another extra couple of solar panels No problem and Well, what I was thinking about doing with this boat anyway was Okay, so these are the same Hull shape as my pedal boat my my old pedal boat, which is amazing has been super reliable. We use all the time The pedal boat just has shorter ones, but this this is longer pontoons So I could conceivably let's see I could take the top off here Take the top off here. I can put a pedal station here one there and two on the other side There could be four pedal spots So the reason I was thinking about making this huge pedal boat There's also the reason that I want to have more cargo space or one of the reasons I want to have more cargo space in the Electric solar boat is because I have some more friends here now like some people came down and They're cool and they're staying. They've been here for a few months already And they're they're not leaving.  I actually have a couple people to go with me so It would be nice to have a little bit more space And this boat wouldn't go to waste or anything for sure I I'd probably just turn it into just a pedal boat if I took all that stuff off or maybe I'd put another motor I don't know. I don't know. It would definitely get get use out of it still But okay, let's just say hypothetically take all the stuff put it on there Put four pedal stations and I'd probably just do like paddle wheels in the middle kind of like on my old Pedal boat because those are so reliable. It's just you know a thing of pedals a stick across and pedals and paddles Attached. There's like hardly anything that can go wrong with it So what would I have to do to do that? Flip the boat over get all these barnacles off. Maybe I don't know who basically Re-everything the bottom of the boat, which really isn't very hard. It's just scrape those off Maybe put some resin on there and re-sand it just make sure it's all smooth on the bottom then You know cut out the pedal stations make the pedals Put a put the roof on here and I already have roof stuff for this boat It's like over there on the island Because this used to be a solar boat Then I would just make this my go-into-town boat and Yeah, I could handle a lot more Lot more cargo and we could all go. You know, I wouldn't even need to flip this over to To get to the bottoms because it's a cataract. I just need to lift up one side Do take care of that side and go lift up the other side and take care of that side There's not much to do basically scrape off the stuff for a bit of sanding Maybe put roll a bit of resin on it


Welcome back to AT&T training camp, the Falcons back at it. Hey everybody, I'm Dave Archer. In front of an enthusiastic throng of fans, a tremendous amount of fans came out today. Really cool day, literally and figuratively, high competition today among the defensive backs, wide receivers, certainly along the offensive and defensive lines. And how about the quarterback? A lot of conversation around Marcus Marriota and Desmond Ritter both performed extremely well today. The thing that stuck out to me was ball placement. I mean, the ball was impeccably placed everywhere. A lot of contested catches by Drake London and Kyle Pitts and all these guys, but tremendous defensive plays by AJ Terrell and others, contested throws. Those two guys were impeccable in their accuracy and where they placed the ball. Is there a competition? There's always competition. I remember talking to Joe Montana and he was going into his sixth year in the National Football League. And I said, Joe, what's the number one thing you're worried about going into camp? He says, winning my job. That's the way it is for every NFL player.  So is there competition at the quarterback position? Yeah, there is, but there is everywhere here in Falka Training Camp.