I'm in the Brave and Booth with Isaac Cruz and they've got some interesting products for mobile audio I guess would be one way to put it. Absolutely. Well, this guy here is our BRB Pro It's a speaker that we announced yesterday here at the show and this BRB Pro allows you to attach modular accessories to your speaker to improve your speaker listening experience or your experience in the outdoor space So right now we're in a booth that has a kayak and we got motorcycles and bicycles So this is sort of the out-and-about kind of thing, right? Absolutely. Out and about. One of our modular accessories that we do have is an action-sports mount that allow you to attach your speaker to your action-sports camera accessories out there Companies like GoPro have these types of accessories so you can attach your speaker while you're riding your bike Listen to music, attach your accessory to your kayak or your outdoors And be able to carry music out there with you wherever you go. All right Let's play some music because this was kind of interesting the way these stack Yeah, so that you can stack one speaker on top of another speaker and you get both speakers working, right? Absolutely. All right, let's see if we got enough sound. Can you get we'll get I'll get in here So this is one speaker now he's gonna attach it to a second one Okay, that might be loud enough I Think that got the idea across you got the idea so all of our accessories that we have for the BRB Pro or IPX7 rated and IPX7 rated means what again waterproof Everything's a hundred percent waterproof So if you're on your boat on your deck and your speaker falls off into the water You're gonna be fine if you're using an accessory on your speaker your speaker accessories aren't gonna get fried or Malfunction or anything like that So now you've got another one of these BRBs, but it's got something that folds out on top What is that? Well, this here is an accessory that'll give you endless music. You'll have music that never ends This is essentially a solar panel what you can do is you can listen to your speaker during the daytime using your solar panel It gives you enough juice and your internal battery to play the speaker and when the Sun goes down Use your internal 2,000 milliamp battery to carry you through the night So you're gonna get about 15 hours at least playtime with the internal battery by itself But if you're using the solar panel by the day, you're not gonna really tap into much that never be in silence all day long Okay, so I've been out all day. I'm playing on my bike. I've got my speaker going I got my solar panel now my battery's dead on my iPhone. So what do you do about that? Oh, man Well, the cool thing about all of our Braven speakers at least for the rugged speakers is they have built-in One amp USB charging Some of our larger speakers have larger output like a 2.1 amp so you could charge an iPad The smaller speaker doesn't but you can at least charge your smartphone to keep the music going You know for whatever type of house part of your entertainment You're doing at your home or or why you're hiking But now Isaac keeps saying rugged and I just got the full feel for it because when he picked up the speaker Which is fairly heavy because it's got a big battery in it He picked it up by the solar panel itself. So these things are really right there.  Now this this particular sample is a handmade sample. So it's not NP quality But still it's very rugged the solar panel is fully intact And and you're not gonna really have any any types of issues This is actually the speaker for me because I drop everything. Yes. All right. This is a great speaker Yeah, this is very cool. So what kind of price point are you looking at for these products? The speaker by itself is 14999 not bad at all at all and all the accessories that we're gonna sell alongside with the speaker when it launches They're gonna be $40 ish and below very very good. It's working people find more about the BRB Check out braven.com. That's b-r-a-v-e-n.com. Yes. Thank you very much Isaac. Thank you


Coach, you guys were ahead 7-5 midway through the third quarter. What changed that allowed them to come back and take this one? Well, they played pretty tough. I give them a lot of credit. The out-ground ball, despite 14, that was the biggest part of the game. They got every ground ball, especially in the face-offs, all the 50-50 balls. They got, and that's the biggest difference. We tried to do a couple things there in the fourth quarter to get back in the game. They got a couple in the end, so it's a little bit closer than the final score. But we got to make plays, and I don't think we did. I thought our offense, we dropped too many passes, went quicker, we weren't as patient as we needed to be. In the first half, we played at our speed. In the second half, they slowed it down, and we couldn't get any time or on or anything like that. And they played really slow and methodical and broke us down. And we just played way too much defense. And I thought our goalie actually did a good job making some saves. And we just made some mistakes, and that's what this tournament's about. I mean, we played great teams week in and week out, and York had a much easier game against Nazareth than us. And that thing that might have been a little bit of a factor there coming in the fourth quarter. But we still have no excuse. We still got to play and make things happen. And I'll give them a lot of credit. They play tough. Yeah, Coach, on defense, we looked a little fatigued. You think the back-to-back games had anything to do with that? Well, things that we couldn't get any ground balls in the fourth quarter. And they had the ball the whole time. And we couldn't get any timers on. And then when you play slow like that, we thought the timer on, we're going to be a lot quicker like they were in the other seven games. And they just weren't in that game. But it is what it is. We have to adjust. We got to do a better job. And so we played a lot of defense in that fourth quarter. We probably only played three or four minutes of offense.  We'll be better. We'll learn from it. We don't have a lot of time to think about it. We've got to get ready for toss, but that's our sixth-ranked opponent in a row. And they're all really good. And when you play a bad quarter, you end up losing to good opponents. And so we had a bad fourth quarter. Coach, you were pretty happy with the face-offs at half when you talked to Greg. What changed there in the second half as they brought Sasso back in? And how do you improve that moving forward? I don't think he did anything. I think they out-scrapped us.  And we got out-scrapped there.  He was sick. And we missed another one of our D-Mitties who had the flu. So we had two guys out that played for us a lot. But that's not an excuse. They were out. And we have to do better.  And we have to do things a little bit better there. And that's what we have to coach them up better and do things. But they played a lot of motion when they got in the role. They did what you're exposed to do. And so then we just couldn't get the breaks we needed. Yeah, coach. And obviously, Kyle Donoffery had a great game. He's definitely a special player. He's playing great all year. What do we got to do on all offense? What do you think you have to do to get other guys involved to get other guys to kind of get the same kind of impacts for being their guy and creating all things? I think our attack's got to play much better. I think they're not doing the job they need to be. They've got to rotate more. They've got to move more. We're just not getting the production or the things out of Watson. He's doing a good job. But we need Wade. And we need JT to do much better things when you get to get Tyler and they're more. So they've got to play better. I think our midfielders are doing a really good job. I think our attack's got to get much better. Coach, you mentioned Tufts Tuesday night. But before we go to that, just putting some finishing touches on the Mustang Classic in general. I know you would have been much happier coming out of it with two wins. But the event itself, another huge success, another great couple of days here at Stevens. Yeah, I think it's great. I mean, it's great for Divisory 3 Cross. We're happy with all the games. They all went pretty well for the most part. They're all pretty close for the most part. And I think it really did a great light shining the light on Division 3 lacrosse, a ton of publicity. We love that. I think all the coaches are coming. And I think they were all ecstatic about it. And so they're the ones that really make it happen. And all the fans that come. We had great attendance all weekend. It was really, really great. And we can't be happier how it went. So I mean, for that part, it couldn't be more exciting. Dickinson had a great show. They're toughness today after a terrible loss. Yesterday they come back and beat RIT. And I thought Amherst Pondspec, well, Lynchburg showed. Even though they didn't win any games, I think they played what showed really well. But this is what the tournament's about. You've got six ranked teams, seven ranked teams. And you don't show up for a quarter against a good team. You lose. And that's kind of what happened today. Coach, I appreciate you taking the time. We'll see you on Tuesday night. I appreciate it.


I found out that today we get to see a brand new intro and I could not be more excited but I'm gonna kind of miss we are It was the very first one in one piece and it's a banger But again, I am very excited to see the new songs and what I know people say that one piece is really known for its intros But nonetheless, it's still kind of cool to see something fresh and new the ladies and gentlemen welcome back to the channel My name is Jack the Buster Ice Shinobi today We have episodes 48 and 49 of our watch through now in the last two buggy made his grand return much to my surprise I thought the man was dead and he's now teamed up with this woman who still won't even reveal her name the show's kind of teasing it a little bit and they're off to find Luffy for this fat bounty while Luffy and go are going to be soon be at Logetown where Gold Roger actually died I think previous King of the Pirates the one that the treasure that everybody's looking for nonetheless We're gonna hop right into it if you knew this channel drop a sub hit that people bell Smith he future episodes double fat like on this one. Let's just hop right into it Okay, kind of got a lot to impress with how good we are is a lot of characters We got a little narration. Okay, some new people Bellamere shanks So he's Gold Roger. I keep blanking on his name even though it said his name in we are at every time at the beginning of every episode Okay, this is kind of wholesome. I sense aesthetic somewhere Yeah, I just thought about this Sanji would be freaking cracked at DDR actually The leave and wonderland Oh new characters That's how it kind of how it teased the like buggy and like a bunch of people in the first intro It teased us a lot with with shanks being back But I don't know how at the forefront he'll be all scenes we've seen okay kind of covering what we've covered up until now Very wholesome. I don't think it quite matches. We are but I mean also it's brand new So it's gonna need to take time to grow on me We made it look down This is the final destination spot beforehand So wholesome new blades, dude. He's not gonna buy like cheap ones the what It's the number one tourist destination spot Execution plank. I just want to know more about him and like how they captured him. He's like the king of the pirates, dude I love how much Luffy is like fanboying though. All right show time I have a feeling that buggy and that woman are going to eventually they're gonna meet them at oh My great googly moogly was she actually going clothes shopping? So how much does this cost? Oh my Ingredients bro, what oh No Love love pirates. Oh no, dude. They're already there Oh, that's kind of cool. There's got to be some good stuff in there some hidden gems. He's like perfect Is the panda yes, I got another one, baby The favorite gun Okay, dude, you're just a scammer. Oh, that's straight up like a toy water gun It looks like and of course who stops going to fall for it. How much is he buying man?  Oh, they're already here, buddy 30 million dude. That's so much Yes, very evil villain. Oh my he's kind of ripped. Was he part of like capturing gold roger?  I think he was from the first intro. He was in we are Bro, what is that blade? He's sure I'm just going out there. He's gonna do it himself Hmm He's a lot of confidence. I believe it's gonna be a big clash between the marines buggy and her and oh wait What was that name to takashi to shaggy? Man, I wonder if that's her But she's a pirate.  Oh, he got angles like our long Oh That's cute.   Oh, no This is so awkward. He didn't look at the the bounty. She does why he doesn't know What he looks like and of course this other guy wants to be king of the pirates too She's getting info Oh, but when he stepped in oh He's kind of reshaped this place Oh shoot Oh, no, no wonder he's able to keep charge. Oh and it has to do with smoke. It's like part of his power Uh, he lacks critical info. Oh, no, he doesn't But aren't loopy and them already there are they like park on another port or another port? Oh, it's a little awkward He doesn't know So clueless Oh, yeah Why did that shock him so much? Oh, he's actually leading him I'm like, how loopy doesn't question it at all. Like no way this kid could be maybe he realized it as soon as he said that He realized he's like, wait, is this is who it is. I knew they'd be expensive Bring it a percent interest Hey, you gotta go unprepared They've got some blades.  Or is it just a perspective? This girl's like, this is perfect timing Oh, oh she has a blade This boy's gonna be like, this is my time, but girl Bro, she was she was uh baiting them in or they just messed with the wrong person. Oh no Wait, how did she fall Just tripped Dude, her face kind of looks like Oh, I literally Ha The plot thickens. I don't think she's alive. I think she actually did die back in the dizzy, but still it is weird What is this? He's still following the smoke. He's leading him astray Bar, oh he had a bar Dude, what is this music and why are there all the wanted posters all destroyed? I have a feeling we're gonna get some good lore here.   That's what i'm saying though. He might are Oh, I thought he was gonna talk about rogers The eternal pirate He's fanboying so hard Eric down It's too many aims Dude, how did they capture gold roger then that is insane. I want to meet the person that did You can tell luffy anything he'll believe it Uh, you're looking at someone else that doesn't feel the grand line, sir Luffy's excited. I would argue that the rest of the crew. Maybe not usopp is like not excited but not that scared either Yeah, this is wasn't all oh shoot. Oh shoot. That's the same guy. Oh, so roger used to just hang here What a beast A little stroll That's quite the mustache Dude this bartender used to be built He was giving him advice So Man's drunk ship driving he wasn't even shocked Luffy's about to explode. Yeah, how do we get to that point? It feels like they skip over that so hard So inherently his death did good for the pirate world though since it set off the great pirate era This town is more under control now, too He's gonna see like a resemblance of gold roger in in luffy Deja vu deja vu, baby But the difference is luffy does not in as intimidating looking as gold roger Oh, that's so wholesome The navy guy, what's the name smoke he he led luffy here, too He doesn't drink milk That's a that's a sick title Eternal king He like brought spirit and hope back to that old man. Oh Oh, maybe we're gonna find out why he led him here This guy confuses me Oh, maybe the smoke didn't lead him here he just got lost Today is a special day. He can move. Oh shoot. Why is today a special day? Dude, he can use the smoke to move stuff.  No Or was he just there? Oh, this is like the anniversary. He said it's a special day Hey, yes, this is what I want to see at least maybe more backstory to this But still it wasn't the beginning of every single episode for a reason clearly Oh, so he was young. It's the kid when it happened. It said 22 years ago God, what a unit.  What a shot. That's so clean King of great pride What badass last words dude god like who put him in these shackles Dang man just like that that is ridiculously high up unnecessarily might I add That's where it ends They didn't behead him. They just stabbed him like through the heart that is even more sad I feel like beheading is a quick death and that's where it began Dude that one guy in the crowd inherently started it all I know inherently gold roger did but still that is ruthless He just left him and then died A new one he's gonna see it captain smoker. Okay. I wasn't that far off.  Oh no. He's straight up booking it to him. Stop these swat-lo's Oh Yet my question remains who captured this man this man who like one shot someone who like killed hundreds I feel like it just wants to build more and more and more which it makes sense This is kind of a lot of the first war we've gotten aside from the first intro and it's so intriguing to me I just want to learn more about this world and the overarching schemes because like they said gold roger Is the one that started it all his death set off this era that we're in now That's kind of dark I want to see the last sky he saw the last buildings he looked at dude I know there's like a four kids version of one piece where they censor stuff. How do they censor smoker? Isn't his whole thing smoked? He got two cigars hanging out of his mouth. Do they need to censor cigars? He knows He's Quite a bit of confidence. We're gonna see a 1v1 They're like what what is he doing? I want to see how he uses this thing on his back Yeah This hair is really clean-looking he's like i'm getting arrested He's so like just like one's track mind Is that he's gonna be like okay? Oh, this is gonna be this is gonna be a sick five I can tell already if he has a double fruit powers He sees it as like a Something in his way. Yeah, it's just another objective. He has to take care of Now how the question is will it be easy or not? He just showed his hand too The cigars were like floating out of his mouth. Is he really stuck? Oh luffy stop embarrassing me brother Okay, he's fast all we know is he I guess he's fast and he uses his smoke Oh, he does seem kind of outclassed here actually although I think luffy isn't expecting much out of him And he's like you're not worthy you're not you're not him Oh, I guess the sheer like just look of gold roger was intimidating luffy doesn't have that But he doesn't need to have that he just needs the attitude True good attitude this thinking creatively now he's probably gonna hit like a random civilian This might be the escape he needs though low key unless he falls in the ocean and he's just gonna die Mademoiselle Oh, no, oh wait, that's not her. There's someone else wearing the exact same outfit Oh shoot, she's got a spiked bat What is that in the sky? Wait, that was the yeah from buggy's crew. Oh shoot. They chillin Oh man, it's going down soon Oh, I thought he was about to land there him How far Well, if nami saw him it would have been like a flashback to when they first met Is luffy ever gonna stop just freaking flying across towns like this I feel like every fight with him. He ends up being sent like 500 miles away Yeah, and he had their person would have died and then passing out. I I presume no, they know where he is now too Somehow bro, you know, yeah, you you broke Oh, that's kind of mean Please calm down Oh Okay, is she gonna make him like work or something?  He's gonna work for it. You want a job? To his face Is he actually gonna stay here and like work? I I know he does feel bad. What do you mean you bought a lot? There's all the luffy boasters. Did she steal that or she actually buy it? I guess she has an absurd amount of money. Oh, no way. She left the money Wait, how's she buying this stuff? Bro the three That's the only way to be most productive Do you have everything Dude, he's he's gonna get even more flashbacks to uh, what's her name? I'm blinking. Oh, he was hired Yeah, you're gonna arrest him Are these the same guys that arrested gold roger? Oh Hey, did he leave a note? I mean to be fair, it was in self-defense. Oh some money. Oh, I'm guessing it's like Most of the rest of whatever he has unfortunately Gwena, thank you sorrow. I appreciate the monologue. Yeah looks like her and as a swordsman and as a talented swordsman Oh, he's going to the shop. Did he oh, did he steal from those dudes? 200 years Oh, he totally stole from them. Oh, maybe not. He just wants the barter Like he could sharpen them. He just wants three Is he gonna ask for a trade? There's no way zoro ever gives this blade up. Does he mean is it like I mean, I know it's a really nice blade, but still he knows it I He thinks he's an amateur Okay, dude, you can't do this. Nope. Sorry. This is just shameless Yeah, he's not gonna sell it to you buddy. Look how desperate he is. He's showing his whole hand 800k This is awkward Who could it have been? Wait, is she being sarcastic? She's serious? Wadoichi Monji, it's an- wait, did they say? Yeah, they never said a name in the flashback. Did they? She's revealing that it's a legendary blade 21 who was the wazamono? 10 million oh my gosh suing from For telling the truth Wait, her blade is sick. Shigure? Oh, she is kind of clumsy. Ooh Wait, how many were there 21? I think he knew it was worth something, but he didn't know exactly how much Yeah, witch bounty hunter. She's just a little bit behind on things. Yeah, despicable. There's none good in the world That's a perspective She's gonna gather all of them 21. Okay. Oh was the monos. The real was them? Okay. That's a lot of blades Wadoichi Monji We have names all of them Oh Red assessor is Didn't he tell Zoro he could pick any two? Oh Oh my gosh Dude that is clean the music It's curse Oh interesting, but they're still rare I like how she was giving the lore on it though. That's cool. She's gonna take it That's his that's his type of style like someone destined to die with it, bro He's willing to take that risk All right, dude That's that's cold, baby.  You can pick any other. Oh, does he have another in mind?  Oh, wow Redemption art for my man's my boy. He's back with three Hell yeah He's in he's in the comfort zone now Family treasure Trusting a dream to another kind of like what zef did misogy Is he just gonna like waltz back? Oh shoot. They're on the prowl. They're like a colt walking around Oh, man, they're getting close too and luvi's not gonna care that they're looking for him.  I'm not even as disappointed in luvi Actually, I'm not disappointed in buggy either. I should expect it from the both of them Dude luvi's got everyone on him bro.  Of course they are He knows his like his attitude Yeah Now she's gonna know She actually didn't I think she actually was clueless about it. Oh no, this is spiraling so fast God dude smoker is a unit despite saying luvi wasn't worthy of being king of the pirates It's like he saw that spirit in him though a ton An absurd amount of lore in these past couple. I need to make I need to make a statement real quick though one sec So back here. I have blades I don't usually read comments on youtube videos So I probably won't for this one either because I know you guys like to partially spoil things even when I don't ask But if you missed it on on my non reaction channel jack the bus, I'll have a link down below I always do I did an amazon stream where I basically let my chat pick any anime items from amazon And I bought them and unboxed them on stream and one of the items was three of zoro's blades Now I think what was it called this kind of name this episode I think that's this one the white one is like his og Is this just straight up like black and it's the white, you know like outer But then these two is this the red one he got this episode because it has like the any Is this the cursed blade the uneven like pattern and then this one This one has like spikes kind of like as the pattern, but I think this is actually the cursed blade I'm not really sure again. I'm not asking for confirmation I just wanted to kind of point out why these are back there But with that being said once again big fat lore dumb these past two episodes both with zoro's new blades And learning a lot about gold roger, which I just I love I eat that stuff up I eat up just any world building lore that we can get but it seems like in the next two We're gonna see a big showdown between literally everyone versus luffy And if we can get nami and and and usop and zoro back in the picture and usop Usop hasn't got much screen time these past two episodes.  I feel like he's just gonna come out with some insane Oh thing he bought and it's gonna actually make a difference just one prediction out of many But then the last has been it for episodes 48 and 49. We're actually at episode 50 next episode for one piece We're 50d, baby. They're very exciting stuff. But thank you guys for watching I hope you did enjoy drop a sub if you're new and with that being said I hope you guys have a wonderful rest of your day as per usual and I'll catch you in the next one Peace peace


All right everybody quiet down quiet down. Thank you for coming. Thank you. Thank you for showing up. I appreciate it. Very you sly dog. Thank you for showing up. We're gonna get started here. I'd appreciate it if you would all thank you. Thank you. Perfect. Perfect. All right ladies and gentlemen, I appreciate you all for coming and I asked you to please stay until the very end for a very special surprise by a very special guest. Trust me. When I started this channel over four years ago, I founded it on the name I Shinobi with no knowledge whatsoever on the places it would go. I never expected it to become part of my full-time job or me to even fall in love with anime the way that I did. And throughout those four years, I Shinobi branched out to simply everything I started enjoying far beyond the show about ninjas that I just really liked. With the many long-lasting memories by my side, I have made the difficult decision to end I Shinobi. And change the name to Jack the Bus Reacts. Dokes aside, I have a big vision for this channel and my brand is a whole. I may not want to do the same thing forever, despite how much I enjoy it right now. Opening myself up for music reactions, trailers, anything, while reviews, gameplay, basically everything non-reaction is on Jack the Bus, widening my audience to those even outside the anime community and beyond. And this both simple and big change is that first step. I'm honestly hoping that this will also bring more clarity to my online persona as a whole, really synergize the channels and as much as I hold the name I Shinobi near and dear to my heart, after quite literally years and years of thinking on this, it's time to let it go. The Jack the Bus brand is in full swing now, baby! All right, all right, hang on, hang on, hang on. And to bring in this big change, I would like to invite our special guest to bring us home. Uh, drum roll please? I've finally done it! After all of this time! Come here. The more I look at this, it's just like two big balls. I hope you're happy. The Gekkomoriya cosplay is real! Anyways, thank you guys for your patience. I'm back for the foreseeable future. I will also be live right now as you're watching this video on my Twitch channel in this cosplay. So come hang and celebrate! But if not, I will see you tomorrow where we are off to the New World!


In her book, Insight, organizational psychologist Dr. Tasha Urik discusses a topic that is vital to both our professional and personal lives, self-awareness. We often discuss self-awareness as it relates to individuals, and so does Dr. Urik, but she also discusses the importance of self-aware teams. One crucial element for self-aware teams is a leader who can communicate their principles clearly and act upon those principles. Psychologists call this authentic leadership. Research shows that when leaders are open and genuine about their own performance and standards, it encourages other team members to be honest too. When authentic leadership is paired with an atmosphere of trust, where people feel safe to ask for help and are encouraged to admit failures, this enables the entire team to develop their collective self-awareness. Lead well this week.


In our previous lecture, we started with a brief introduction to lubrication theory. So we will have a brief recapitulation of the physical situation, what we are trying to address. So let us say that there is a passage where the lateral length scale is much less than the axial length scale and let us consider this example where the top wall moves with a velocity relative to the bottom wall and that velocity may be arbitrary with the velocity components along x and y. So the situation is 2 dimensional and one of the important hallmarks of the particular arrangement is that the existence of a physical small parameter. The small parameter is in this particular case, the ratio of the transverse length scale to the axial length scale. So with this arrangement, let us try to see that how we can physically describe this arrangement through governing equations. Let us come to the board and try to explain that. So the physical situation, let us draw a schematic that you have 2 walls and you set up your x and y axis like this, the gap h is a function of position and time, you have an arbitrary velocity of the top wall relative to the bottom wall. The axial length is lc and the scale that we consider in the transverse direction. So I mean because h is continuously varying, you may consider the scale to be any local height provided that the variation in the height is not strong. That means there is a weak variation or slow variation of height as we move along x. If there is a slow variation in height, there is not much difference between this height, this height, this height and this height. So any particular height you can choose as the characteristic scale. Just as an example, if this height is Ho, then we can consider that as a characteristic length scale. I mean if this is the maximum height, then if the maximum height by lc is small, then all the other heights are also much smaller than the corresponding axial length scales. So with this particular arrangement and we have to keep in mind also that the top wall moves with a velocity v. And as we discussed yesterday that we will indicate the dimensional quantities with prime and non-dimensional quantities without prime. So that we will consider here and we will begin with the continuity equation. So let us assume that it is an incompressible flow. So the continuity equation for 2-dimensional incompressible flow. So this is x prime and this is y prime in terms of dimensional coordinate. Dimensionless there will be x and y. Let us describe the orders of magnitude of various terms. So what is the order of magnitude of this? Let us say that the characteristic scale is uc, characteristic velocity scale is uc. Now what is uc? We do not know because it depends on the physics of the problem that what is the characteristic length scale. If the characteristic length scale is, sorry what is the characteristic velocity scale? Characteristic length scale is described by the geometry. But characteristic velocity scale is described by the physics of the problem. So if the top plate velocity is the dominating physical influence then its velocity along x will be the characteristic scale. However if there is some body force which is dominating towards dictating what is the velocity then there will be a velocity scale dependent on that body force. We will see that how that can be obtained. So at this stage uc is totally arbitrary. It depends on the boundary condition or the physics of the problem for us to decide that what is the appropriate choice of uc, okay. So this scales as uc by lc where you do not know a priori what is uc. This scales as say this is vc by h0. So as we have discussed earlier that these 2 terms they must have the same scale so that the net effect is 0. So that means vc by h0 is you can say is of the order of uc by lc. That means vc is of the order of uc into h0 by lc. H0 by lc we denote by epsilon. So vc is equal to epsilon into uc. So that if epsilon is small that means there is one dominant flow direction. That is also another hallmark of the lubrication theory that you have a dominant flow direction because the dominant flow direction is dictated by u and not by v. However you cannot trivially neglect the momentum transport along y because of may be possibility of existence of a pressure gradient may be due to body force. Some body force may be there and also I mean you may have some sort of motion along y because the plate is being pulled or pushed vertically upwards or downwards that might be a possibility. So either the kinematics or the kinetics may decide that there may be some significance in the momentum transport along y although the velocity component along y is one is significantly less as compared to the velocity component along x okay. So this is what the continuity equation gives then let us write the x momentum equation put dash in all the parameters just to indicate that they are dimensional parameters. Find out we will now non-dimensionalize various terms. So we will write u is equal to u dash by uc, v is equal to v dash by vc, x equal to x dash by lc, y is equal to y dash by h0, p is equal to p dash by pc, fx is equal to fx dash by fxc. I am trying to derive a very general formulation where whatever body force may be there is present and so the normalization factor for the body force depends on what physical it is. Now certain scales we have already discussed like uc, vc, lc, h0 but we have not discussed about what is pc, what could be fxc. What could be fxc depends on what is actually acting as a body force along x. So in a general treatment it is difficult to prescribe what could be fxc but it is possible to prescribe what is pc and we will try now to find out what is an appropriate pressure scale that is one of the key parameters in the lubrication theory. So now let us one thing we have not yet done the time t is equal to t dash by tc. So there are different possibilities of choosing the time scale like effective time scale, diffusive time scale, forcing based time scale we have seen that these are there. Now here an appropriate choice would be an advection based time scale. There is no forcing here and the advective time scale will naturally have this term of similar effect as that of these terms. So if these terms are not important, this term will not be important. So the advection time scale so what is tc then tc is lc by uc okay. Now let us write the various terms in a dimensional form and certain non-dimensional parameters will come out of that.  Next we will make some simplifications. So tc is lc by uc so this becomes uc square by lc alright uc square by lc this is uc square by lc now vc by h0 is same as uc by lc from the continuity equation. So this also becomes uc square by lc so the left hand side let us write the next step you have rho uc square by lc as common into del u del t plus u del u del x plus v del u del y is equal to minus pc by lc del p del x plus mu uc by h0 square next maybe let us utilize the top part of the board. When we are writing these equations what appears to be a small parameter to me is the size of the board because I mean whatever you write it does not fit either in the left or in the right or in the top in the bottom so that is at least one small parameter. Now let us continue with this equation in the top little bit down we write so that it is covered by the camera. So what we do now is let us try to find out what is the appropriate pressure scale what is pc okay so we have to keep in mind that the pressure scale see the dominant pressure gradient term first of all forget about the pressure scale first of all what we do is that we multiply both sides by h0 square by mu uc okay this h0 so basically to get rid of this we multiply both sides by h0 square by mu uc so rho uc square by lc into h0 square by mu uc. Now let us look into various terms see one has to keep in mind that the order of magnitude of various terms depend on how the terms are non-dimensionalized for example see there is a key difference between normalizing with an appropriate scale and any non-dimensionalization for example in the problem that we have we have we have been discussing you could non-dimensionalize y dash by lc as well right because lc is also a length but that is not a proper normalization that is not a proper scaling based normalization since y dash it is range is from 0 to h0 not 0 to lc. So if you normalize a variable with its range then the dimensionless parameters are constrained to be within 0 to 1 in terms of order when the dimensionless parameters in terms of order are constrained to be within 0 and 1 then all their derivatives are also constrained to be within the order of 0 to 1. So that means you can safely say that this is of the order of 1 if this is of the order of 1 you can now quickly convert to a conclusion that this is of the order of epsilon, epsilon square sorry not epsilon of the order of epsilon square. So because this is of the order of epsilon square we can conclude that this term is asymptotically smaller than this term. So whenever you are considering an asymptotic expansion formally this term will not come out to be important but from physical arguments you can tell it without doing any asymptotic expansion into h0 square by mu uc right into h0 square by mu uc. The next observation is that in a situation where this term is of the order of 1 and there is a competition between viscous force and force due to pressure gradient and appropriate choice of the pressure scale should be such that this is of the order of 1. So that you will get a term like just dp dx. So this gives a choice for PC. So what is PC? So PC by LC is equal to mu uc by h0 square. So that means PC is equal to mu uc into LC by h0 square right. So this we can write as multiply both numerator and denominator by LC and h0 by LC is epsilon. So this is mu uc by epsilon square LC. Now what is this term? Let us calculate this, this is rho uc into LC by mu into h0 by LC square. 1 uc gets cancelled so 1 uc into LC by mu into h0 by LC square. So what is this? This is nothing but the Reynolds number, Reynolds number based on LC and this is epsilon. So this is epsilon square finally the last term the multiplying factor what is this? In place of h0 you can write LC square by mu uc epsilon, epsilon square. So the final equation in terms of epsilon is Reynolds number into epsilon square del u del t plus u del u del x plus v del u del y is equal to minus del p del x plus epsilon square del 2 u del x2 plus del 2 u del y2 plus fx fxc into LC square by mu uc epsilon square. Which one? Last one yes yes right. So this will be this will be LC square and this will be epsilon square right. Oh sorry sorry yes epsilon square LC square right. So this is epsilon square will be in the numerator right epsilon square LC square. So now can you tell from this equation just by looking without making any mathematical treatment or anything that which terms have potential importance relative to the other terms or which terms are potentially less significant as compared to the other terms. See here you have certain terms which are of the order of epsilon square and epsilon being small order of epsilon square will make it significantly smaller as compared to order of 1 right. So this term and this term you may well argue that well this term has also epsilon square but that is a dangerous conclusion to talk about because you really do not know what is the scale of fx fxc okay. So it might so happen that adjustment with the scale of the fxc will not eventually make it of the order of epsilon square although elusively epsilon square is happening to be there in the numerator. So we cannot make any conclusion of the relative dominance of this force or not because simply we do not know what is fxc. Not only that we do not know what is uc. So uc depends on the physics of the problem as we discussed. So we do not know what is fxc we do not know what is uc. So just by looking into this term although it contains epsilon square that does not mean that eventually when all the expressions are substituted finally it will be also of the order of epsilon square. Therefore we cannot conclude that this term is trivially negligible but we can conclude that these two terms are trivially negligible. That means if you make an expansion if you write an expansion u equal to u0 plus epsilon u1 plus epsilon square u2 this is called as an asymptotic expansion.  You may well argue that if we make this asymptotic expansion how will we ensure that this expansion is justified. This expansion is justified only if there are 2 possible I mean 2 important constraints that are obeyed. One epsilon is small that we have ensured the other is that this parameter u0 u1 u2 these are constrained to be within 1 and that is ensured by considering appropriate scales for normalization and that is why we are actually doing it in a dimensionless form otherwise we could work on this with a dimensional form as well but the dimensionless form gives us this possibility that we are dealing with individual parameters which are of the order of 1. Dimensionally if we are handling these parameters that we cannot ensure. Dimensionlessly we can ensure by choosing suitable scaling parameters okay. So now if you substitute you will see just let us do it mentally because if we I mean this board does not give me enough space of writing all these expansions so let us just do it mentally and from that we can figure it out.  So when you do that you will see that the dominant terms of the order of 0 will be what of the order of epsilon to the power 0 that is what I am calling of the order of 0 of the order of 0 means of the order of epsilon to the power 0. So of the order of epsilon to the power 0 which terms will remain. This term will definitely not be there because there is a epsilon square multiplier. So you will get 0 is equal to minus del p0 del x plus del square u0 del x square and the body force u it is better to retain because you do not know what is the scale what is the appropriate scale. So plus fx fxc epsilon square lc square by mu uc y square right del y square if no body force is present then the last term is not important. So then we are left with an equation which is del p0 del x is equal to del square u0 del y square that is the leading order equation and in lubrication theory we do all the calculations using the leading order equations only because that gives the dominant effect. Now there may be an interesting yes no no no that is what I told that it is this is illusion you do not know what is uc you do not know what is fxc you do not know that what is the net effect. So in the leading order the body force may be dominant okay. So we should have a provision for keeping it and if the body force is not dominant in the leading order that has to be that has to be considered from the physics of the problem not from anything else then you drop this term. Now a very interesting point before we move on to the y momentum equation. The interesting point is that out of pressure and viscous stress which is more dominant in lubrication theory let us see. So when you are writing the pressure okay the viscous stress the viscous stress is what viscous stress is of the order of like mu del u del y right. So mu uc by h0 right that is the viscous stress. So h0 is what epsilon into lc right. So for a given lc the viscous stress scales with 1 by epsilon. The pressure scales with 1 by epsilon square right. So for small epsilon the pressure is what is dominating and not the viscous stress right. So see this I mean these are I mean certain things which are intuitive and certain things which come out of the smallness of the parameter epsilon. So that is one very important observation. So that is why in lubrication theory at the end to calculate the force we use the pressure distribution only. I mean people may argue that there are viscous stresses which are present then why do you just calculate the force only on the basis of pressure. The reason is that in the leading order the pressure is what is dominating being scaling with 1 by epsilon square whereas the viscous stress scales with 1 by epsilon. So to sum up the discussion on the x momentum equation we can see that the x momentum equation boils down to this. We will come to the boundary conditions of this equation later on but now let us move on to the y momentum equation. y momentum del v del t plus u del v del x plus v del v del y is equal to minus del p del y plus we will add a dash to all the terms this is the dimensional form. Next what we will do we will use the suitable scaling parameters to non-dimensionalize this. So as usual we have v is equal to v dash by vc u equal to u dash by uc then x equal to x dash by xc y is equal to y dash x is lc right y is equal to y dash by h naught t is equal to t dash by tc p equal to p dash by pc and f y is equal to f y dash by f yc. One very important point is that the pressure scale whatever is established to the x momentum equation same pressure scale has to be considered for the y momentum equation pressure is just a scalar. So its scale will not change as you go from one direction to the other okay. So let us quickly write the various terms in their dimensionless forms from our previous experience of writing it for the x momentum equation we can do similar thing for the y momentum equation. So rho see how do you do these things quickly just it is very important to learn how to do these things quickly without getting into like writing all the derivatives and all. See this term will be u scale into v scale by x scale right. So and by using the continuity equation and the advective time scale all the terms will have the same coefficient at the end.  Now what is pc just tell me from your notes what is pc mu by epsilon square lc. So the next step what we will do we will multiply both sides by h naught square by mu vc right just the same way as we did for the x momentum equation. So it becomes rho uc vc by lc into h naught square by mu vc. Now let us just simplify this this term. So rho uc vc is uc into epsilon. So uc square by epsilon lc anyway one vc gets cancelled out right. So we do not have to care about this vc this vc gets cancelled out with this. So rho uc into epsilon square lc by mu. So that is equal to what Reynolds number into epsilon square alright this is then let us come to this term oh we have not written del p del y right del p del y was missing here. So this term mu uc one h naught gets cancelled one h naught by epsilon is okay mu also gets cancelled uc by vc into h naught by lc into 1 by epsilon square right. So uc by vc gets cancelled with h naught by epsilon. So this becomes 1 by epsilon square and the last term if you want it you can simplify it it will not make much of a difference so far as what we are doing currently because we have not assumed any body force but just for the sake of completeness this h naught square is lc square into epsilon square and vc is lc into epsilon sorry uc into epsilon. So the last term becomes fy fyc into lc epsilon by mu sorry lc epsilon square by mu uc lc square epsilon lc square epsilon right there is a epsilon in the denominator so fy fyc lc square epsilon by mu uc. Now you multiply both sides by epsilon square okay let us do it at the top you multiply both sides by epsilon square the reason is 1 by epsilon square has come here so just to bring it in the left hand side so you have Reynolds number into epsilon to the power 4 del v del t plus u del v del x plus v del v del y is equal to minus del p del y plus epsilon 4 del 2 v del x 2 plus del 2 v del y 2 epsilon square del 2 v del y 2 plus fy fyc lc square epsilon cube by mu uc.  If you do that what will come out as a dominant term forget about the body force keep it separate what will come out as a dominant term only the pressure gradient term. So the consequence of this if you neglect the effect of this term is simply this one because other terms are of the order of epsilon square epsilon 4 like that body force term of course what is the order we do not know but we are not keeping the body force in per view for this particular discussion. Now there is one subtlety here we are considering that this term is of the order of epsilon 4 so this is small but what if the Reynolds number is large but fortunately we are dealing with microfluidic problems where the typical Reynolds number is small. So we are dealing with low Reynolds number hydrodynamics so there is no chance of blowing this term up despite the smallness of epsilon. So the assumption that we are making are very much consistent with microfluidics. So to sum it up let us write our x momentum and y momentum equations. So you have 0 is equal to minus del p0 del x plus del 2 u0 del x2 sorry del y2 and 0 is equal to minus del p0 del y if you are not considering body force that means that because this del p0 del y is equal to 0 that means p0 is not a function of y so this you could as well write as dp0 dx. So your governing equation actually boils down to this one in absence of body forces remember this what are the boundary conditions.  So to make one of these as 0 instead of giving both as non-zero components what we do is that we apply a transformation of coordinates where you are using a coordinate system which is translating with respect to a velocity u star prime along the x direction and we fit that coordinate system with the bottom plate. So with respect to this this will become 0 one of the components will become 0. Now if this be the case then what are the boundary conditions this is y equal to 0 this is y equal to h these are all these h is dimensionless h. So this if you want to put this as dimensional parameters then just get rid of the dash that will make dimensionless parameters get rid of the dash. So these are dimensionless parameters in terms of which the governing equation is written. So we will write the boundary conditions also in terms of dimensionless parameters at y equal to 0 what is the boundary condition u0 equal to minus u star right that is the no slip boundary condition relative to this moving reference frame which is drawn by the green color and y equal to h what is u0 0. See the description of see one of the boundary conditions we have made 0 by using this transformation and this does not depend on what is v at that boundary. See this is what is the very critical thing that whenever we have made this transformation we have not cared about what is v we have made the transformation with the translation along x nothing to do with y. Why because in the leading order equation v has never appeared in the leading order equation the net effect of the momentum equation along x momentum equation along y as well as the continuity equation the continuity equation we have forgotten its role but its important role is to set up the scales for u and v the relation between vc and uc that is set up by the continuity equation. So, the smallness of the parameter epsilon is established through the continuity equation now. So, you can see there is no v0 which is appearing so it does not matter what is v of the top boundary so far as description of this boundary conditions are concerned. So, once we get this equation along with this boundary conditions next we will make an attempt to solve this equation to get u0 and from that establish a governing equation for the pressure distribution. We will see that how do we establish a governing equation for pressure distribution for that we have to couple this equation with the continuity equation. We have not considered the continuity equation as yet because we have used only the x momentum and the y momentum. We have never used the continuity equation to close this system and as we have understood that to obtain a pressure distribution we must close the momentum equations in conjunction with the continuity equation. So, we will bring the continuity equation into the picture and couple that with this equation to get a pressure distribution and hence a force distribution on one of the plates say the top plate that we will take up in the next lecture. Thank you very much.


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