Essentials of Fluid Mechanics

A fluid. Okay. The definition, first of all, of a fluid. The official definition in Chapter 1. It's a substance that deforms continuously when acted upon by a shearing stress of any matter. How many people here have had ME218 strength materials? Almost everybody, of course. Take a piece of copper plate. Copper plate. And apply a force F on it. I don't know, an inch thick, 6x4, something like that. Apply 400 pounds on it. The top of it. This is fixed with a base. When you apply that force to it, and you know any 2x18 for instance, There's the plate. Now, again, here's the brown. Now I take a layer of water. I apply a force on it. You know, it doesn't look like that when I'm done applying a force to that plate on top of water. Think what would happen if you applied a force to that plate on top of a waterway. I applied 10 pounds. Would the plate ever stop? Of course not. You apply 10 pounds, this is a room flooded with water this high. I'm walking in the water on my knees. I put a styrofoam plate on top of it. push on two pounds two months my finger is it going to stop me from pushing ever no and the key word is continuously this will deform a little bit then it's going to stop but a fluid will deform continuously That's the fluid. Air, 10-way oil, JP for jet fuel, you name it, these things we live with, water, they're fluids. Now, there's different kinds of fluids though. We're all used to fluid and water. I'll be working there. It's really, really nice. What's in this cup went in that cup. Now, fill that cup there with cold air. Try and pour that cold air in that cup over there. It won't do it. It won't do it. It'll spread out when it comes out. It'll go all apart in a big plume. Why? Because the molecules are not attached to each other as strongly as they are in that liquid. Are they both fluids? Sure. Of course they are. Put air in here. Push on a plate and air. It was told me. Yeah, they're... They're both fluids. Liquids and gases. But not solids. Not solids. Well, maybe not quite. Maybe not quite. I think you can't off the toothpaste. Turn upside down. Does it run out? Really? No. This has been the car for a few hours. It doesn't run any temperatures. No? What makes it come out? I've got to press on the tube. Okay, so it takes an initial stress, shearing stress, to move it, to get it moving. Once it's moving, it moves pretty good. But it takes that initial shearing stress. It could be, you know... Some very thick things like some epoxies, you know, you can check it out. They don't come out until you squeeze a tube or a plunger. So, yeah, those things are kind of like fluid, but we won't discuss those in this class. We're not going to worry about, you know, toothpaste and epoxy and things. We'll worry about water, air, oil, things like that. Helium, freon, you name it. So that's a big difference in that. We're also going to look at two sets of units. You know, a lot of textbooks now, they are almost always, almost all in SI. Which is so much easier than the British system. So much easier. But unfortunately, the real world doesn't go that way. So our textbook, I forget the percentage, but it has quite a few, I'll call it English, it's British gravitation, quite a few problems, whether they're example problems or homework in English, and then a lot of SI. But you graduating have to be conversant in two languages, otherwise you're not going to show off very well. You have to know two languages, forward and backwards, really, really well. The easy one is SI. The hard one is English engineering. For instance... How many centimeters in a meter? Oh, I'm sorry. I'm just guessing, okay? How many millimeters in a meter? I'll tell you, about a thousand. How many cubic centimeters in a liter? I'll tell you. It's a thousand. Do you get the point? Everything in SI... Where'd you go there and fill that word, dude? Oh, okay, thanks. Everything, every conversion factor... Okay, we'll tell you later. One joule divided by one second. What is it? Joules per second. Uh-huh. Uh-huh. One. Yeah, one. Is it one? Yeah, one. OK. One. Everything in version SI starts with a 1 times 10 will pop up. You can't get any easier than that. Now, you tell me this. How many pints in a cork? How many quarts in a gallon? How many gallons in a barrel? Ho, ho, ho. Nothing starts with one. Nothing. Want to memorize that stuff? No, no, no. You don't memorize this right here. I don't know, but I've got this thing in my pocket. Yeah, I'll tell you in a minute, okay? Gee, that's okay, but I mean, if you've got a gallon in a barrel, blah, No, it's awful. It's awful. Oh, power. How many foot pounds per second is a horse's power? Well, you know, how many watts in a kilowatt? Well, I'll tell you, how about a thousand? One times ten to the third. So, oh yeah, I mean, don't you wish the whole world was SI? Of course we all do, but it's not. And you live in a country that's, you live in the worst country for it. The country that still has both unit systems. So you have to be really, really, really good in both of them. That's why my point is, that's why when we select a textbook, we make sure the textbook has, if we can find it, dual unit systems in it. Because we know the more you practice, the better you get. The problem is most people work problems in SI where they need it. practice in English engineering. Okay, well that's either here or there. There are problems here. There'll be problems on homework of both. There'll be problems on tests of both. You can use your smart phone for conversions. There's conversions of the inside back cover too. SI to greer's gravitational. So good choice, but there will be both kind of problems. for homework and exams. Okay, let's just briefly go over the two unit systems. Let's do SI first. Okay, force is in newtons. Our mass is in kilograms. And of course, length is in meters. Okay, let's do British gravitational. Force is in pounds. Mass is in slugs. Length is in feet. And they both have seconds for time. There's a third unit system that has force in pounds force, mass in pounds mass. But this is the British gravitational. You don't put a subscript on this guy because you know if it's in this system and it says pounds, it's got to be force. So you can skip putting the subscript F on pounds. Unfortunately, you don't deal with pounds under mass, you deal with slugs. These are not little creepy crawly slimey things on the floor. These aren't stale slugs. A slug is a unit of mass. How big is it? Anybody know? how many pounds mass are in a slug? Let's figure out the acceleration. I think it should be 32.2, right? There you go. 32.2 pounds mass in one slug. That means a slug is more massive than a pound mass. A slug is more massive. Okay, so those are the two unit systems. When you work problem for homework or exams, that's what you use there. Okay, now let's take a look at, let's see, pressure. Let's look at some other. Oh yeah, here, here. Okay, a Q-pad with Nixon thermal. Rho, density, mass per volume. If it's in, Si, kilograms per cubic meter. If it's in British computational mass, slugs per cubic foot. Specific weight. Gamma. Weight per volume. SI. Weight. Weight's a force. Newtons per cubic meter. Free gravitational force. Pounds per cubic foot. Specific gravity. Okay, gamma of the fluid in question under gamma of water at a certain temperature, like I think it's 4 degrees, yeah, 4 degrees C. R is 39 degrees F. So it's a ratio. A ratio is dimensionless. So it's a dimensionless number, specific gravity. Gamma is the specific weight of the fluid you're studying divided by gamma of water at a certain temperature. An ideal gas, our perfect gas law, P over rho equals R times T. P and T are absolute values. Absolute pressure, absolute temperature. We'll mention absolute pressure in just a little bit. You know absolute temperature at 273 degrees C, degrees K. Add 460 degrees F, degrees R. Good day. It's in the book. This is for gases. So the good news is, when we're studying this class, pretty much all the gases behave like that. As long as you're around ambient conditions, air, hydrogen, oxygen, hydrogen, behave like that. Viscosity. All these properties are in the back of the book. Page after page after page of properties. The graphs of physical properties of fluid. Appendix B. I'm giving you, on both exams in the final, copies of these out of the back of the book. You'll get a package of properties, probably eight or nine pages long. You'll have these same properties on the exam. So learn how to read them. That's the key point. I'll pick them up after exam one. Your name's on. I'll keep them. Before exam two, I'll pass them out. You'll use them for the test. Pass them back to me. I'll keep them. The final exam, I'll pass them off to you. You use them. Pass them back to me when you're done. So they'll be your package. You can mark them up with a pen, pencil, whatever you want to do. Mark them up. You'll get it back for each exam, including the final. Okay, now let's get back to our definition of mu. Mu, this is a big property. It's different in a thermal capacity. What means is that? We'll look after it. This is sharing stress. Tau is sharing stress. Now, du, dy, this is called a velocity gradient. So if we take a simple case, again, think that plate on water, and here's water. Maybe the velocity looks like this. This would be capital B velocity, and then this is typical of how the velocity might look between the plate. Let's make it linear, make it like easy right now. We're going to do two. Exactly, day one. So this is called a linear velocity gradient because it's a straight line. Velocity versus the height. The height is y. The velocity is lowercase u. u is a function of y. Differentiate that. That's called the velocity gradient. You take the velocity gradient. Like in a bearing, a bearing, one surface is rotating, one is stationary, stationary, moving. You've got 10-way oil, that's mu, multiply, where do you get mu oil? Back in the book, in the appendix, what's the temperature? I don't know, 120 degrees Fahrenheit. Go back to the book, find mu, put it in here, multiply it by du dy. which is V divided by the separation distance. And that's going to give you the shearing stress that the fluid creates. Okay. So viscosity is a property which is useful to determine shearing stresses. Okay. And then there's another property, which is nu, letter nu, which is mu over rho. This is called kinematic viscosity. So there are two possible viscosities. One is a real thing. The real thing is this one up here. This is really the viscosity. If someone just says viscosity to you, then you assume they mean this one. But that's why they're also called absolute and kinematic, just to be certain that someone knows what you're giving them. If you say viscosity, though, the default definition is here. Kinematic viscosity is this value divided by the density of growth. This thing comes in handy. We'll see later. It appears the ratio mu divided by rho appears in some equations. So it's given a special name called kinematic viscosity. Okay, the units are the unisons. Okay, if you don't memorize those things, and I hate to, if you don't memorize them, then figure out what they are. What is this guy? Force divided by area. Okay, so force divided by area. The area of, let's just say, length squared. What's this guy? Velocity, length over time. What's this guy? Length. Okay. Cancel, cancel. So figure it out that mu is F, this guy's a denominator, this guy's a denominator, put him up here, FT over L squared. If it's new in a natural viscosity, divide that guy by mass divided by length cubed. Density, mass divided by length cubed. We go back here. Viscosity. English units. What are they? Pound second per foot squared. Look at it. Pound...over there. Pound second per foot squared. Yeah. I don't memorize that stuff. I derive when I have to. When I have to. So, there's no reason to memorize it, because you know you can derive it. Where does it come from? The basic governing equation. Right there. Okay, now let's go on to our last two properties. The next one, by the way, before we forget, let's just go ahead and put the scrap up here. This is Shearing's Dress Tile there. This is a G-E-P-Y velocity gradient, sometimes called a break of shear-ease strain, but the velocity gradient is sufficient. If this is air, this is water, this is oil. Oh yeah, we know, intuitively, we know oil is very viscous. It will give you very high shear stress for a certain value of dUdy. Given a value of dUdy, air is not very viscous. Water is in the middle of the road. Oils are very viscous. Notice they are linear too, which is very important. They are linear. Those tools are called Newtonian. There are many which are not linear which we won't study in ME 311. Toothpaste, no it's not. Latex paint, no it's not. When you put that paint on a brush, put that brush in water, pull it out of there, all the water runs off. You'd be running for the wall to get up on the wall before right off your paintbrush. No, no. You put that in there in latex paint, take it out of there, turn the brush. It stays on the brush fibers until you get to the wall, then you either roll it on or you paint it on. Yeah, it's different. It's not linear. It's not linear. The graphs from the book, by the way. Quicksand. No, not linear. The more you try and get out, the harder it is for you. You've got to move really, really slowly. Don't try and pull your foot out, because you're going deeper. It's all this graph tells a whole story. I'll just show you a couple. Some go like this, some go like that. Here's quicksand, here's latex paint. There's toothpaste and maybe stuff like that, but that's just for your own sidelight right now. What you're worried about are the three things us M.E.s and C.E.s look at most in life. Arrows and M.E.s, arrows and C.E.s, mostly. heavy engines and things like that you know so we're not going to worry about those other exotic ones we're going to work up things like these three right here they're all going to be Newtonian fluids in me three of them okay let's go to one more Surface tension. Before I forget, of course, we need videotape. I'm trying to hear Matt. Matt, you're in the student, right? What year are you in? A lot. Did you start here? Did you start here? Yeah, I started here. Oh, did you? Okay, okay. But you're taking senior classes now, right? Yes. It was close. Yeah, very close. It should be my last quarter. Okay, good. So he'll be in every class meeting, video taking, and lectures. Okay, surface tension. The symbol for that is sigma. And sigma is the force per unit length. So sigma has dimensions of force per unit length. Okay, here's a reservoir. I put this tube in there, invert it. The liquid will rise up in that reservoir something like this. You might have seen a mercury barometer on the wall. And the mercury rises up that glass too, and you can read off what the mercury level is. The guy, night in the news says, okay, the weather forecast, a cold front's coming out from Canada, and the barometer's dropping, and it's right now 30.05 inches of mercury. You say, what do you mean? There's mercury dropping out of the sky on people's heads? No, no, no. Ask any of your friends at night, say to them, what does he mean when he says the pressure is 30 inches of mercury? Most folks don't have the slightest of course. They don't ask me their tire pressure. Don't tell me 32 pounds in my tires. Do they really mean 32 pounds of air into your tire? No, no, they're taking shortcuts. They mean to say 32 psi in my tire. Well, if you put mercury in here, fill the tube with mercury, invert it, come off there, the mercury rises up. At this point right here, mercury does this. Around the glass perimeter, where the glass and mercury are in contact, the mercury is being pulled down. That's surface tension. Take water. Here's a tube of water. Do the same thing. Doesn't work quite as well. That, the water being pulled up by the glass, it seems like. So it's a property, it's a property of the fluid and the temperature. So that's why it's 4th per unit length. What length is this length? It's the perimeter of the glass tube, the perimeter, pi times d. So that's a phenomenon that occurs quite often, that influences, that you should be aware of. What's this thing called surface tension? Try and put, if you do it, if you do it real carefully, a dime, a dime on water. If you really, really... really careful you can put it on water it won't sink it won't say something's hold pull it up around the perimeter so let's pull it up that's the water doing that and that's the surface tension in the water so So those are all part of the concept of surface tension. Okay, I think that pretty much covers all of our important properties in there. So let's take a look then at pressure. All right. Now, let's go back over here. Hey, this thing's on here. Let's talk about pressure. One of the big concepts in our first fluids course, what do you call pressure in any 218 string? What's stress? Force per unit area. What's pressure? Force per unit area. Right, right. Different, but just different. Those guys are solids, these guys are liquids. Okay, let's start off by absolute pressure. We'll put them all down. Gauge pressure. Okay, absolute. Referenced to zero pressure. Reference to local atmospheric pressure. We'll use our standard pressure. 14.7 BSI and I think for our one of the just use 101kPa. 101.3 something something something. 101kPa. So these are the standard pressure values. We're not going to use the value of the moment which is Here in the field off sea level, these guys are at sea level and I don't know, 34 degrees latitude, something like that. Standard boundaries. We'll use those. Okay. Gauge pressure. Reference to local atmospheric pressure. The word local means where you are when you make the measurement. Local means where you are when you make the measurement. If you're in Pomona, Cal Poly Pomona, it's the pressure here. That's the local atmospheric pressure. If you're in Denver, Colorado, it's the pressure in Denver, Colorado at that time of day, that day of the year. So that's reference to where you are. Okay, now let's talk about these guys. Gauge pressure can be positive or negative. These guys are always positive because they're reference to zero. No such thing as a negative absolute pressure. So here we're vacuum pressure and some gauges will show you positive pressures and vacuum pressures. That's what they mean by that. So let's draw a little picture here that shows these pressures. These pressures mean pressures. I'll put on the board the values of some conversion factors too for you. Let's start off with zero pressure down here. Zero absolute. And let's say that this is the local atmospheric pressure. And let's call that 101kPa. And let's call the pressure up here Pa. If I measure... PA from the local atmospheric pressure, reference to local atmospheric pressure, that's this line right here. And PA is going to be, this is given, 200 kPa. This pressure is 101 plus 200 subscale. 301. If the pressure is down here... And let's say that it's a minus 50 kPa gauge vacuum. Then the absolute pressure always is measured from the bottom down up to here. 101 minus 50, 51. 1K today, and that's absolute. In the textbook, I think he gives us a big hint of what the pressures are here. This picture is in the book, too, by the way. Yeah, it's on page 50. Okay, here's what he says in italics. In this textbook, pressures will be assumed to be gain pressures unless they're specifically designated as absolute. In this book, for a whole month, in an exam, for a problem. The pressures are assumed to be gauged unless it specifically says absolute. So if I get your problem on exam and it says the pressure is 50 kPa, that means you assume that's gauged. Unless I put that. Then it's absolute. So that's the default rule. The default rule is pressures that just are in kPa assume that they're gauge, and thus they're absolute. There will be a current abs after. Or the problem will say the absolute pressure is 50 kPa. Okay. In... English engineering or gravitational, bridge gravitational. If someone tells you pressure of the air in your tire, make it easy, is 30 psi. How do they measure? Probably a tire pressure gauge, I would suspect, a tire pressure gauge. How does a tire pressure gauge work? I want a typical example. There's a spring in this thing and the stem pops up and you read it. Or there's a circular gauge you put on your tire and you read it. Take the spring where you want it. What's the pressure in your tire doing to that thing? Oh, it's compressing the spring as it pushes it up. What's it working against? What's outside that pencil-like device? The atmospheric pressure. So what's it measuring the pressure with respect to? 30 psi means that's 30 psi gauge and we don't say 30 psi gauge we say my tire pressure 30 psi G that's standard practice psi G means gauge If somebody says, well then what's the absolute pressure of the air in my tires? It goes. There it is. You're up here now at 30. 30. Above the global atmosphere. 30. Global atmosphere. I think I erased it. Yeah, there it is. 14.7. 30 plus 14.7. 44.7 PSIA means absolute. That's just the way most engineers work. They use PSIG for gauge pressure, PSIA for absolute pressure. But in the world of SI, in the world of SI, you don't say You don't say this. There's no such thing as a K-Pag. It's no. There's no K-Pag in the book. So what do you do? You have a rule. Textbook rule says, if I don't put anything after KPA, you assume that it's a gauge. If it's absolute, I'll put ABS after it. Just so on the exam, you'll say, Professor Bill, what do you mean by that KPA? I'll say, go back to the default, what it would say. You don't want to get confused on an exam, a timed exam. It makes stuff miserable. Okay, so that's the pressure now. And the guy that says, what do you want your tires at, 35 pounds? Say, no, I want 35 PSI. Because they take a shortcut. They just say pounds. They really mean PSI, of course. Okay, now let's take a look at some problems similar to Homer. Let's see which one I like to have here. I'll take this one first because this is... 177. 177. This book, by the way, why do we use this textbook? It is the most popular Blue and Connection textbook in the United States. It's the most popular blue and connection textbook in the United States. That's why. And the reason why, it's very bold and very good. Okay. 177. You have 178 and 180. They're all the same kind of problem. They're viscosity problems. So we don't need this. Okay, I'll read it for you. It's a snow sled. The sled shown in the picture slides along a thin horizontal layer of water between the ice and the runners. So here's the sled. Here's ice. I'm going to expand it up. Here's water. Here's the runner on the sled. A blade, that's a skate blade. The horizontal force that the water puts on the runners is equal to 1.2 pounds. The sled's speed is 50 feet per second. The sled, the runner, the blade, runs on water, not ice. Water. The total area of both the runners in contact with the water is 0.08 square feet. The viscosity of water, if they didn't give it to you, it's in the back. Viscosity of water, 3.5, so this is mu. They didn't say kinematic, they just said viscosity, assume it's absolute, 3.5 times 10 to the minus 5. Pounds seconds per foot squared. Determine the thickness of the water layer under the runnage. Deal. Fine deal. Keep reading. Assume a linear velocity distribution in the water layer. Assume a linear velocity distribution in the water layer. So, come on. V is the velocity at the top of the water layer. It goes down to zero at the ice. velocity at a solid surface is assumed to be at rest the water it's called the no slip condition right here this is called no slip means the water doesn't slip along the ice the velocity of the water at the surface of the ice is assumed to be zero we'll make that assumption pretty much throughout the whole course in those conditions okay so I recently I raised our equation so we have our towel equal view the you be why Okay, our shear stress we know is force over area. Mu, du, dy. We're measuring y from here up like this. Okay, force is 50 pounds force. 50 pounds. Oh, 1.2 pounds. Which reminds me, if I put something on the board which doesn't make any sense to you, it looks like I made a mistake, let me know right away. I don't want to have to copy the mistake on the paper and come back and make a class meeting and change it. I don't mind. I've got to correct it right away. And you know, I'm not perfect. I make mistakes. So just let me know and we'll change it on the board right away if something's wrong. Okay, 1.2, we're talking about area, zero, zero, zero, eight. Equal mu, okay, there's mu right there, 3.5, 10 to the minus 5. Pound, second, or foot squared. Multiply by du dy. Don't forget it's a linear profile. So you know lucky you. du dy, if it's linear, equals delta u over delta y. Delta u at the top, b at the bottom, zero. Delta y at the top, d at the bottom, zero. There it is. du dy is delta u over delta y equal b over d. b, 50. d, that would be b. Check it out. Better come out right here. Okay, seconds divided by seconds. Feet, feet squared, feet squared here. Feet, feet gone. The right hand side, pounds per foot squared. Left hand side, pounds per foot squared. Good, we're good to go. Solve for D. It'll be a G. 7.7 to the 10 to the minus 4. One ten-thousandth of an inch. One ten-thousandth of an inch. That's the amount of liquid water between the blade surface and the ice on King's Coffee Bean. Not a lot. It doesn't take a lot. It doesn't take a lot. But there is a layer of water under there. Okay, I did that just to kind of show it to you. Did you give me a signal? I'm sorry about that. No, I got it. Okay, thanks. I want to show you how you set up the homework problems. And I'll move it again when you pass your first homework statement. But, do that. The first thing you do is you draw a sketch, if it's appropriate. I mean, we're engineers. We love sketches. If you're a mathematician, you love... equations and you paint sketches. You want the theoretical solution. If you're an engineer, you love pictures and you'll do the math because you have to do the math. You don't love the math, maybe, but you do it because you have to do it. You do your toolbox. So, most engineers think more clearly if they have a picture. On an exam, I guarantee that if you try and sketch something on an exam, whether it's a free body diagram, which is essential, maybe 214, What it does to you is it builds a time delay in your brain. You don't start writing down and putting numbers in something. And you read the problem, when you sketch it, your mind is working in the background, you're not, and it makes life a lot easier if you do it that way. Because the way you get flustered and frustrated is try and start putting numbers on that handheld calculator right away. Oh! That's just a story. If you're unsure of yourself, that puts you down, way down there. What you do is you build about a six... second time delay in there so your mind starts to think about that problem and one of the best ways to do it is to draw a picture you got a picture your mind starts to think it's amazing engineering thought process so anyway if it's appropriate If it's appropriate, draw a sketch. Once you draw the sketch, I don't care if the problem has it listed there in paragraph form. If you put these guys down and look at the picture, you start to see things. When you write the equation down, you start to see things. What do I know? What don't I know? I know it. I know it. I know it. I know it. I don't know it. Your mind starts to think like that. So much of engineering is to have an approach to solve a problem. We're not backyard, you know, in your garage making something. You know, that's, you know, we're not doing that in your garage. We engineers are trained to think correctly. It's pretty fascinating. Pretty fascinating. Okay, next thing you do, put the equation down in symbolic terms. I don't want to see any numbers in there until you write the equation down so I know you're using the right equation. Then you put the numbers in. Then you put the unis by each number. I'm repeating things I know have been beating your head for three, four, five years, but I'm going to do it one more time. Put the unis in, cancel them out, like this, and end up with what you want. All those steps. are important steps. If you do them in the right order, you may be amazed how things kind of fall out. There might be three or four possible equations. You might write down three equations. You look at the equations, you look at what's over here, and you say, No, that's not going to work. One equation, two unknowns. No, that's not going to work. One equation, three unknowns. Yeah, he's going to work. So you don't always know. You don't always know which equation might work at the start. Sometimes you do, sometimes you don't. But if you don't, write down the equations you think might work and stare at them. Stare at them a while. Maybe something will jump out at you. Whether it's homework or an exam. So, you're going to have to think about what you're going to Now, let's say something else. You don't get ready for an exam by copying Solutions Matty. No, no, no. I mean, you know, tell Kershaw, you know, did you read how to throw the curveball before the game? He said, are you kidding me? No, I don't do stuff like that. I threw it myself 50 times on the sidelines. I developed a new pitch, a slider. How? No, no, I practiced it. I threw it a thousand times on the sideline. You don't get good until you. suffer through some homework and say I just don't get this I went through that I get so mad sometimes I take my fist I go I don't know why I'm not getting this at home you might be better you know nice music on touching my hand but I still got mad I got so frustrated and clustered but I work my way through it and boy you're better for it you're better for the pain the way you get ready for exams is you're not surprised and how are you not surprised You look at all the problems I work in class, you read the example problems in the textbook, and you go over homework. Don't read their solutions because you won't be ready for the exam. Oh, sure, you've seen the solutions. But on an exam, you won't be ready. You've got to practice yourself and put yourself in a tough spot, in a real tough spot. Okay, we have finished chapter one. Here are your assignments for the next two class meetings. Oh, by the way, my office hours, I can make other times, Monday, Tuesday, and Wednesday. I'm here all three days. If those office hours don't match with yours, let me know. We can meet another time of day, that's fine. Alright, we'll see you there on Wednesday.

Almost everybody, of course. Take a piece of copper plate. Copper plate. And apply a force F on it. I don't know, an inch thick, 6x4, something like that.

Apply 400 pounds on it. The top of it. This is fixed with a base.

When you apply that force to it, and you know any 2x18 for instance, There's the plate. Now, again, here's the brown. Now I take a layer of water. I apply a force on it. You know, it doesn't look like that when I'm done applying a force to that plate on top of water.

Think what would happen if you applied a force to that plate on top of a waterway. I applied 10 pounds. Would the plate ever stop?

Of course not. You apply 10 pounds, this is a room flooded with water this high. I'm walking in the water on my knees. I put a styrofoam plate on top of it.

push on two pounds two months my finger is it going to stop me from pushing ever no and the key word is continuously this will deform a little bit then it's going to stop but a fluid will deform continuously That's the fluid. Air, 10-way oil, JP for jet fuel, you name it, these things we live with, water, they're fluids. Now, there's different kinds of fluids though.

We're all used to fluid and water. I'll be working there. It's really, really nice. What's in this cup went in that cup. Now, fill that cup there with cold air.

Try and pour that cold air in that cup over there. It won't do it. It won't do it.

It'll spread out when it comes out. It'll go all apart in a big plume. Why?

Because the molecules are not attached to each other as strongly as they are in that liquid. Are they both fluids? Sure.

Of course they are. Put air in here. Push on a plate and air.

It was told me. Yeah, they're... They're both fluids. Liquids and gases. But not solids.

Not solids. Well, maybe not quite. Maybe not quite. I think you can't off the toothpaste.

Turn upside down. Does it run out? Really?

No. This has been the car for a few hours. It doesn't run any temperatures.

No? What makes it come out? I've got to press on the tube. Okay, so it takes an initial stress, shearing stress, to move it, to get it moving. Once it's moving, it moves pretty good.

But it takes that initial shearing stress. It could be, you know... Some very thick things like some epoxies, you know, you can check it out. They don't come out until you squeeze a tube or a plunger. So, yeah, those things are kind of like fluid, but we won't discuss those in this class.

We're not going to worry about, you know, toothpaste and epoxy and things. We'll worry about water, air, oil, things like that. Helium, freon, you name it. So that's a big difference in that.

We're also going to look at two sets of units. You know, a lot of textbooks now, they are almost always, almost all in SI. Which is so much easier than the British system. So much easier. But unfortunately, the real world doesn't go that way.

So our textbook, I forget the percentage, but it has quite a few, I'll call it English, it's British gravitation, quite a few problems, whether they're example problems or homework in English, and then a lot of SI. But you graduating have to be conversant in two languages, otherwise you're not going to show off very well. You have to know two languages, forward and backwards, really, really well.

The easy one is SI. The hard one is English engineering. For instance...

How many centimeters in a meter? Oh, I'm sorry. I'm just guessing, okay?

How many millimeters in a meter? I'll tell you, about a thousand. How many cubic centimeters in a liter?

I'll tell you. It's a thousand. Do you get the point? Everything in SI... Where'd you go there and fill that word, dude?

Oh, okay, thanks. Everything, every conversion factor... Okay, we'll tell you later.

One joule divided by one second. What is it? Joules per second.

Uh-huh. Uh-huh. One.

Yeah, one. Is it one? Yeah, one. OK. One.

Everything in version SI starts with a 1 times 10 will pop up. You can't get any easier than that. Now, you tell me this.

How many pints in a cork? How many quarts in a gallon? How many gallons in a barrel? Ho, ho, ho.

Nothing starts with one. Nothing. Want to memorize that stuff?

No, no, no. You don't memorize this right here. I don't know, but I've got this thing in my pocket. Yeah, I'll tell you in a minute, okay? Gee, that's okay, but I mean, if you've got a gallon in a barrel, blah, No, it's awful.

It's awful. Oh, power. How many foot pounds per second is a horse's power?

Well, you know, how many watts in a kilowatt? Well, I'll tell you, how about a thousand? One times ten to the third. So, oh yeah, I mean, don't you wish the whole world was SI?

Of course we all do, but it's not. And you live in a country that's, you live in the worst country for it. The country that still has both unit systems.

So you have to be really, really, really good in both of them. That's why my point is, that's why when we select a textbook, we make sure the textbook has, if we can find it, dual unit systems in it. Because we know the more you practice, the better you get.

The problem is most people work problems in SI where they need it. practice in English engineering. Okay, well that's either here or there.

There are problems here. There'll be problems on homework of both. There'll be problems on tests of both. You can use your smart phone for conversions.

There's conversions of the inside back cover too. SI to greer's gravitational. So good choice, but there will be both kind of problems.

for homework and exams. Okay, let's just briefly go over the two unit systems. Let's do SI first. Okay, force is in newtons.

Our mass is in kilograms. And of course, length is in meters. Okay, let's do British gravitational. Force is in pounds. Mass is in slugs.

Length is in feet. And they both have seconds for time. There's a third unit system that has force in pounds force, mass in pounds mass. But this is the British gravitational. You don't put a subscript on this guy because you know if it's in this system and it says pounds, it's got to be force.

So you can skip putting the subscript F on pounds. Unfortunately, you don't deal with pounds under mass, you deal with slugs. These are not little creepy crawly slimey things on the floor. These aren't stale slugs. A slug is a unit of mass.

How big is it? Anybody know? how many pounds mass are in a slug?

Let's figure out the acceleration. I think it should be 32.2, right? There you go. 32.2 pounds mass in one slug. That means a slug is more massive than a pound mass.

A slug is more massive. Okay, so those are the two unit systems. When you work problem for homework or exams, that's what you use there.

Okay, now let's take a look at, let's see, pressure. Let's look at some other. Oh yeah, here, here. Okay, a Q-pad with Nixon thermal.

Rho, density, mass per volume. If it's in, Si, kilograms per cubic meter. If it's in British computational mass, slugs per cubic foot.

Specific weight. Gamma. Weight per volume. SI. Weight.

Weight's a force. Newtons per cubic meter. Free gravitational force. Pounds per cubic foot.

Specific gravity. Okay, gamma of the fluid in question under gamma of water at a certain temperature, like I think it's 4 degrees, yeah, 4 degrees C. R is 39 degrees F. So it's a ratio. A ratio is dimensionless.

So it's a dimensionless number, specific gravity. Gamma is the specific weight of the fluid you're studying divided by gamma of water at a certain temperature. An ideal gas, our perfect gas law, P over rho equals R times T.

P and T are absolute values. Absolute pressure, absolute temperature. We'll mention absolute pressure in just a little bit. You know absolute temperature at 273 degrees C, degrees K.

Add 460 degrees F, degrees R. Good day. It's in the book.

This is for gases. So the good news is, when we're studying this class, pretty much all the gases behave like that. As long as you're around ambient conditions, air, hydrogen, oxygen, hydrogen, behave like that. Viscosity.

All these properties are in the back of the book. Page after page after page of properties. The graphs of physical properties of fluid. Appendix B. I'm giving you, on both exams in the final, copies of these out of the back of the book. You'll get a package of properties, probably eight or nine pages long.

You'll have these same properties on the exam. So learn how to read them. That's the key point. I'll pick them up after exam one.

Your name's on. I'll keep them. Before exam two, I'll pass them out. You'll use them for the test.

Pass them back to me. I'll keep them. The final exam, I'll pass them off to you. You use them. Pass them back to me when you're done.

So they'll be your package. You can mark them up with a pen, pencil, whatever you want to do. Mark them up. You'll get it back for each exam, including the final. Okay, now let's get back to our definition of mu.

Mu, this is a big property. It's different in a thermal capacity. What means is that? We'll look after it. This is sharing stress.

Tau is sharing stress. Now, du, dy, this is called a velocity gradient. So if we take a simple case, again, think that plate on water, and here's water. Maybe the velocity looks like this. This would be capital B velocity, and then this is typical of how the velocity might look between the plate.

Let's make it linear, make it like easy right now. We're going to do two. Exactly, day one. So this is called a linear velocity gradient because it's a straight line. Velocity versus the height.

The height is y. The velocity is lowercase u. u is a function of y.

Differentiate that. That's called the velocity gradient. You take the velocity gradient.

Like in a bearing, a bearing, one surface is rotating, one is stationary, stationary, moving. You've got 10-way oil, that's mu, multiply, where do you get mu oil? Back in the book, in the appendix, what's the temperature? I don't know, 120 degrees Fahrenheit. Go back to the book, find mu, put it in here, multiply it by du dy.

which is V divided by the separation distance. And that's going to give you the shearing stress that the fluid creates. Okay.

So viscosity is a property which is useful to determine shearing stresses. Okay. And then there's another property, which is nu, letter nu, which is mu over rho. This is called kinematic viscosity.

So there are two possible viscosities. One is a real thing. The real thing is this one up here. This is really the viscosity.

If someone just says viscosity to you, then you assume they mean this one. But that's why they're also called absolute and kinematic, just to be certain that someone knows what you're giving them. If you say viscosity, though, the default definition is here. Kinematic viscosity is this value divided by the density of growth. This thing comes in handy.

We'll see later. It appears the ratio mu divided by rho appears in some equations. So it's given a special name called kinematic viscosity.

Okay, the units are the unisons. Okay, if you don't memorize those things, and I hate to, if you don't memorize them, then figure out what they are. What is this guy? Force divided by area. Okay, so force divided by area.

The area of, let's just say, length squared. What's this guy? Velocity, length over time.

What's this guy? Length. Okay. Cancel, cancel.

So figure it out that mu is F, this guy's a denominator, this guy's a denominator, put him up here, FT over L squared. If it's new in a natural viscosity, divide that guy by mass divided by length cubed. Density, mass divided by length cubed. We go back here. Viscosity.

English units. What are they? Pound second per foot squared. Look at it.

Pound...over there. Pound second per foot squared. Yeah. I don't memorize that stuff.

I derive when I have to. When I have to. So, there's no reason to memorize it, because you know you can derive it.

Where does it come from? The basic governing equation. Right there.

Okay, now let's go on to our last two properties. The next one, by the way, before we forget, let's just go ahead and put the scrap up here. This is Shearing's Dress Tile there.

This is a G-E-P-Y velocity gradient, sometimes called a break of shear-ease strain, but the velocity gradient is sufficient. If this is air, this is water, this is oil. Oh yeah, we know, intuitively, we know oil is very viscous.

It will give you very high shear stress for a certain value of dUdy. Given a value of dUdy, air is not very viscous. Water is in the middle of the road. Oils are very viscous.

Notice they are linear too, which is very important. They are linear. Those tools are called Newtonian. There are many which are not linear which we won't study in ME 311. Toothpaste, no it's not.

Latex paint, no it's not. When you put that paint on a brush, put that brush in water, pull it out of there, all the water runs off. You'd be running for the wall to get up on the wall before right off your paintbrush.

No, no. You put that in there in latex paint, take it out of there, turn the brush. It stays on the brush fibers until you get to the wall, then you either roll it on or you paint it on. Yeah, it's different.

It's not linear. It's not linear. The graphs from the book, by the way. Quicksand. No, not linear.

The more you try and get out, the harder it is for you. You've got to move really, really slowly. Don't try and pull your foot out, because you're going deeper.

It's all this graph tells a whole story. I'll just show you a couple. Some go like this, some go like that.

Here's quicksand, here's latex paint. There's toothpaste and maybe stuff like that, but that's just for your own sidelight right now. What you're worried about are the three things us M.E.s and C.E.s look at most in life.

Arrows and M.E.s, arrows and C.E.s, mostly. heavy engines and things like that you know so we're not going to worry about those other exotic ones we're going to work up things like these three right here they're all going to be Newtonian fluids in me three of them okay let's go to one more Surface tension. Before I forget, of course, we need videotape. I'm trying to hear Matt. Matt, you're in the student, right?

What year are you in? A lot. Did you start here? Did you start here?

Yeah, I started here. Oh, did you? Okay, okay.

But you're taking senior classes now, right? Yes. It was close. Yeah, very close.

It should be my last quarter. Okay, good. So he'll be in every class meeting, video taking, and lectures.

Okay, surface tension. The symbol for that is sigma. And sigma is the force per unit length.

So sigma has dimensions of force per unit length. Okay, here's a reservoir. I put this tube in there, invert it.

The liquid will rise up in that reservoir something like this. You might have seen a mercury barometer on the wall. And the mercury rises up that glass too, and you can read off what the mercury level is.

The guy, night in the news says, okay, the weather forecast, a cold front's coming out from Canada, and the barometer's dropping, and it's right now 30.05 inches of mercury. You say, what do you mean? There's mercury dropping out of the sky on people's heads?

No, no, no. Ask any of your friends at night, say to them, what does he mean when he says the pressure is 30 inches of mercury? Most folks don't have the slightest of course. They don't ask me their tire pressure.

Don't tell me 32 pounds in my tires. Do they really mean 32 pounds of air into your tire? No, no, they're taking shortcuts. They mean to say 32 psi in my tire. Well, if you put mercury in here, fill the tube with mercury, invert it, come off there, the mercury rises up.

At this point right here, mercury does this. Around the glass perimeter, where the glass and mercury are in contact, the mercury is being pulled down. That's surface tension. Take water. Here's a tube of water.

Do the same thing. Doesn't work quite as well. That, the water being pulled up by the glass, it seems like. So it's a property, it's a property of the fluid and the temperature. So that's why it's 4th per unit length.

What length is this length? It's the perimeter of the glass tube, the perimeter, pi times d. So that's a phenomenon that occurs quite often, that influences, that you should be aware of. What's this thing called surface tension?

Try and put, if you do it, if you do it real carefully, a dime, a dime on water. If you really, really... really careful you can put it on water it won't sink it won't say something's hold pull it up around the perimeter so let's pull it up that's the water doing that and that's the surface tension in the water so So those are all part of the concept of surface tension.

Okay, I think that pretty much covers all of our important properties in there. So let's take a look then at pressure. All right. Now, let's go back over here. Hey, this thing's on here.

Let's talk about pressure. One of the big concepts in our first fluids course, what do you call pressure in any 218 string? What's stress? Force per unit area. What's pressure?

Force per unit area. Right, right. Different, but just different. Those guys are solids, these guys are liquids. Okay, let's start off by absolute pressure.

We'll put them all down. Gauge pressure. Okay, absolute.

Referenced to zero pressure. Reference to local atmospheric pressure. We'll use our standard pressure. 14.7 BSI and I think for our one of the just use 101kPa.

101.3 something something something. 101kPa. So these are the standard pressure values.

We're not going to use the value of the moment which is Here in the field off sea level, these guys are at sea level and I don't know, 34 degrees latitude, something like that. Standard boundaries. We'll use those.

Okay. Gauge pressure. Reference to local atmospheric pressure.

The word local means where you are when you make the measurement. Local means where you are when you make the measurement. If you're in Pomona, Cal Poly Pomona, it's the pressure here.

That's the local atmospheric pressure. If you're in Denver, Colorado, it's the pressure in Denver, Colorado at that time of day, that day of the year. So that's reference to where you are.

Okay, now let's talk about these guys. Gauge pressure can be positive or negative. These guys are always positive because they're reference to zero.

No such thing as a negative absolute pressure. So here we're vacuum pressure and some gauges will show you positive pressures and vacuum pressures. That's what they mean by that.

So let's draw a little picture here that shows these pressures. These pressures mean pressures. I'll put on the board the values of some conversion factors too for you. Let's start off with zero pressure down here. Zero absolute.

And let's say that this is the local atmospheric pressure. And let's call that 101kPa. And let's call the pressure up here Pa.

If I measure... PA from the local atmospheric pressure, reference to local atmospheric pressure, that's this line right here. And PA is going to be, this is given, 200 kPa.

This pressure is 101 plus 200 subscale. 301. If the pressure is down here... And let's say that it's a minus 50 kPa gauge vacuum. Then the absolute pressure always is measured from the bottom down up to here. 101 minus 50, 51. 1K today, and that's absolute.

In the textbook, I think he gives us a big hint of what the pressures are here. This picture is in the book, too, by the way. Yeah, it's on page 50. Okay, here's what he says in italics. In this textbook, pressures will be assumed to be gain pressures unless they're specifically designated as absolute.

In this book, for a whole month, in an exam, for a problem. The pressures are assumed to be gauged unless it specifically says absolute. So if I get your problem on exam and it says the pressure is 50 kPa, that means you assume that's gauged.

Unless I put that. Then it's absolute. So that's the default rule. The default rule is pressures that just are in kPa assume that they're gauge, and thus they're absolute. There will be a current abs after.

Or the problem will say the absolute pressure is 50 kPa. Okay. In... English engineering or gravitational, bridge gravitational.

If someone tells you pressure of the air in your tire, make it easy, is 30 psi. How do they measure? Probably a tire pressure gauge, I would suspect, a tire pressure gauge.

How does a tire pressure gauge work? I want a typical example. There's a spring in this thing and the stem pops up and you read it. Or there's a circular gauge you put on your tire and you read it. Take the spring where you want it.

What's the pressure in your tire doing to that thing? Oh, it's compressing the spring as it pushes it up. What's it working against?

What's outside that pencil-like device? The atmospheric pressure. So what's it measuring the pressure with respect to? 30 psi means that's 30 psi gauge and we don't say 30 psi gauge we say my tire pressure 30 psi G that's standard practice psi G means gauge If somebody says, well then what's the absolute pressure of the air in my tires?

It goes. There it is. You're up here now at 30. 30. Above the global atmosphere.

30. Global atmosphere. I think I erased it. Yeah, there it is. 14.7. 30 plus 14.7. 44.7 PSIA means absolute.

That's just the way most engineers work. They use PSIG for gauge pressure, PSIA for absolute pressure. But in the world of SI, in the world of SI, you don't say You don't say this.

There's no such thing as a K-Pag. It's no. There's no K-Pag in the book.

So what do you do? You have a rule. Textbook rule says, if I don't put anything after KPA, you assume that it's a gauge. If it's absolute, I'll put ABS after it.

Just so on the exam, you'll say, Professor Bill, what do you mean by that KPA? I'll say, go back to the default, what it would say. You don't want to get confused on an exam, a timed exam.

It makes stuff miserable. Okay, so that's the pressure now. And the guy that says, what do you want your tires at, 35 pounds? Say, no, I want 35 PSI. Because they take a shortcut.

They just say pounds. They really mean PSI, of course. Okay, now let's take a look at some problems similar to Homer.

Let's see which one I like to have here. I'll take this one first because this is... 177. 177. This book, by the way, why do we use this textbook?

It is the most popular Blue and Connection textbook in the United States. It's the most popular blue and connection textbook in the United States. That's why.

And the reason why, it's very bold and very good. Okay. 177. You have 178 and 180. They're all the same kind of problem. They're viscosity problems.

So we don't need this. Okay, I'll read it for you. It's a snow sled. The sled shown in the picture slides along a thin horizontal layer of water between the ice and the runners. So here's the sled.

Here's ice. I'm going to expand it up. Here's water.

Here's the runner on the sled. A blade, that's a skate blade. The horizontal force that the water puts on the runners is equal to 1.2 pounds.

The sled's speed is 50 feet per second. The sled, the runner, the blade, runs on water, not ice. Water. The total area of both the runners in contact with the water is 0.08 square feet. The viscosity of water, if they didn't give it to you, it's in the back.

Viscosity of water, 3.5, so this is mu. They didn't say kinematic, they just said viscosity, assume it's absolute, 3.5 times 10 to the minus 5. Pounds seconds per foot squared. Determine the thickness of the water layer under the runnage. Deal.

Fine deal. Keep reading. Assume a linear velocity distribution in the water layer.

Assume a linear velocity distribution in the water layer. So, come on. V is the velocity at the top of the water layer.

It goes down to zero at the ice. velocity at a solid surface is assumed to be at rest the water it's called the no slip condition right here this is called no slip means the water doesn't slip along the ice the velocity of the water at the surface of the ice is assumed to be zero we'll make that assumption pretty much throughout the whole course in those conditions okay so I recently I raised our equation so we have our towel equal view the you be why Okay, our shear stress we know is force over area. Mu, du, dy. We're measuring y from here up like this. Okay, force is 50 pounds force.

50 pounds. Oh, 1.2 pounds. Which reminds me, if I put something on the board which doesn't make any sense to you, it looks like I made a mistake, let me know right away. I don't want to have to copy the mistake on the paper and come back and make a class meeting and change it. I don't mind.

I've got to correct it right away. And you know, I'm not perfect. I make mistakes.

So just let me know and we'll change it on the board right away if something's wrong. Okay, 1.2, we're talking about area, zero, zero, zero, eight. Equal mu, okay, there's mu right there, 3.5, 10 to the minus 5. Pound, second, or foot squared.

Multiply by du dy. Don't forget it's a linear profile. So you know lucky you.

du dy, if it's linear, equals delta u over delta y. Delta u at the top, b at the bottom, zero. Delta y at the top, d at the bottom, zero. There it is. du dy is delta u over delta y equal b over d. b, 50. d, that would be b.

Check it out. Better come out right here. Okay, seconds divided by seconds.

Feet, feet squared, feet squared here. Feet, feet gone. The right hand side, pounds per foot squared.

Left hand side, pounds per foot squared. Good, we're good to go. Solve for D.

It'll be a G. 7.7 to the 10 to the minus 4. One ten-thousandth of an inch. One ten-thousandth of an inch. That's the amount of liquid water between the blade surface and the ice on King's Coffee Bean.

Not a lot. It doesn't take a lot. It doesn't take a lot.

But there is a layer of water under there. Okay, I did that just to kind of show it to you. Did you give me a signal? I'm sorry about that. No, I got it.

Okay, thanks. I want to show you how you set up the homework problems. And I'll move it again when you pass your first homework statement. But, do that. The first thing you do is you draw a sketch, if it's appropriate.

I mean, we're engineers. We love sketches. If you're a mathematician, you love...

equations and you paint sketches. You want the theoretical solution. If you're an engineer, you love pictures and you'll do the math because you have to do the math. You don't love the math, maybe, but you do it because you have to do it. You do your toolbox.

So, most engineers think more clearly if they have a picture. On an exam, I guarantee that if you try and sketch something on an exam, whether it's a free body diagram, which is essential, maybe 214, What it does to you is it builds a time delay in your brain. You don't start writing down and putting numbers in something. And you read the problem, when you sketch it, your mind is working in the background, you're not, and it makes life a lot easier if you do it that way. Because the way you get flustered and frustrated is try and start putting numbers on that handheld calculator right away.

Oh! That's just a story. If you're unsure of yourself, that puts you down, way down there.

What you do is you build about a six... second time delay in there so your mind starts to think about that problem and one of the best ways to do it is to draw a picture you got a picture your mind starts to think it's amazing engineering thought process so anyway if it's appropriate If it's appropriate, draw a sketch. Once you draw the sketch, I don't care if the problem has it listed there in paragraph form. If you put these guys down and look at the picture, you start to see things.

When you write the equation down, you start to see things. What do I know? What don't I know?

I know it. I know it. I know it.

I know it. I don't know it. Your mind starts to think like that. So much of engineering is to have an approach to solve a problem.

We're not backyard, you know, in your garage making something. You know, that's, you know, we're not doing that in your garage. We engineers are trained to think correctly.

It's pretty fascinating. Pretty fascinating. Okay, next thing you do, put the equation down in symbolic terms.

I don't want to see any numbers in there until you write the equation down so I know you're using the right equation. Then you put the numbers in. Then you put the unis by each number.

I'm repeating things I know have been beating your head for three, four, five years, but I'm going to do it one more time. Put the unis in, cancel them out, like this, and end up with what you want. All those steps. are important steps. If you do them in the right order, you may be amazed how things kind of fall out.

There might be three or four possible equations. You might write down three equations. You look at the equations, you look at what's over here, and you say, No, that's not going to work.

One equation, two unknowns. No, that's not going to work. One equation, three unknowns.

Yeah, he's going to work. So you don't always know. You don't always know which equation might work at the start.

Sometimes you do, sometimes you don't. But if you don't, write down the equations you think might work and stare at them. Stare at them a while. Maybe something will jump out at you.

Whether it's homework or an exam. So, you're going to have to think about what you're going to Now, let's say something else. You don't get ready for an exam by copying Solutions Matty.

No, no, no. I mean, you know, tell Kershaw, you know, did you read how to throw the curveball before the game? He said, are you kidding me? No, I don't do stuff like that.

I threw it myself 50 times on the sidelines. I developed a new pitch, a slider. How?

No, no, I practiced it. I threw it a thousand times on the sideline. You don't get good until you. suffer through some homework and say I just don't get this I went through that I get so mad sometimes I take my fist I go I don't know why I'm not getting this at home you might be better you know nice music on touching my hand but I still got mad I got so frustrated and clustered but I work my way through it and boy you're better for it you're better for the pain the way you get ready for exams is you're not surprised and how are you not surprised You look at all the problems I work in class, you read the example problems in the textbook, and you go over homework. Don't read their solutions because you won't be ready for the exam.

Oh, sure, you've seen the solutions. But on an exam, you won't be ready. You've got to practice yourself and put yourself in a tough spot, in a real tough spot. Okay, we have finished chapter one. Here are your assignments for the next two class meetings.

Oh, by the way, my office hours, I can make other times, Monday, Tuesday, and Wednesday. I'm here all three days. If those office hours don't match with yours, let me know. We can meet another time of day, that's fine. Alright, we'll see you there on Wednesday.

Transcript for:Essentials of Fluid Mechanics

Transcript for:
Essentials of Fluid Mechanics