hello all um today i'm gonna give you a lecture on chapter eight three we'll be talking about fluids in motion okay so here we're going to talk a little bit about um the motion of fluids and how we can predict speeds and pressures and things along those lines so this uh section we'll talk about those particular topics the first thing that i'd like to do is i'd like to define the term ideal fluid an ideal fluid is a fluid that meets very specific so that we can use it effectively in determining uh theoretical flows of fluids and then we can compare how actual fluids flow compared to those ideal situations it's a way to help us understand the process in a kind of a pure concept think about it as like the idea of you know removing gravity or friction so this is the kind of stuff that we're going to be talking about so an ideal fluid is a fluid that is not compressible that means that if you were to put it in a cylinder and you were to squeeze on that fluid you wouldn't be able to make its volume any smaller so it would be one of those things where as you push on the fluid the fluid would squeeze out into some other area an ideal fluid is something that is called non-viscous that means it has no internal friction so there's that idea of no friction that we keep talking about it would flow very very freely and easily the ideal fluid exhibits laminar flow and i will talk a minute about what laminar flow is but basically it's a very smooth flow without any turbulence and then we're going to talk about the velocity density and pressure are going to be the same at every point in the fluid and what that might even indicate is that there's no impurities or things like that inside this fluid so the ideal fluid is something that we use as a comparison so make sure that you have those particular characteristics written down when we talk about fluid flow types a minute ago i talked about laminar flow and laminar flow is a very smooth type of flow and you can see here on the picture on the right hand side this area right here where we have nice flow lines that are compressed together but there's no turbulences or disturbances in those areas that is an example of laminar flow it's very very smooth turbulent flow on the other side is just that it is turbulent and what we see is with some of the things that characterize turbulent flow are eddy currents and these eddy currents are spiraling sections you see these at the back here where the fluid comes in and flows around the back and we get these little currents in the back there that end up creating air disturbances and these eddy currents can actually be quite powerful and you've probably noticed them if you've ever been passed by a semi tractor trailer or something along those lines and you feel uh number one there's a change in pressure but at the back of the truck you'll feel your car kind of buffeted around a little bit like if you're driving behind the semi you'll feel your car is kind of bouncing or being jostled and it's being jostled by those eddy currents those those turbulent areas of airflow and we talked about an ideal fluid having no viscosity and what this truck talks about like i said is that internal friction of fluid fluids that typically have high viscosity and we'll typically think of things like syrup molasses and other types of things that typically have a relatively high sugar content things that have low viscosity would be things like water alcohol like like a rubbing alcohol things along those lines they tend to be relatively low on the viscosity they will flow relatively easily they don't tend to stick or roll one thing that you could kind of think of is that things that have higher viscosity would tend to make mounds of fluid things along those lines so we'd you know if you put a drop of syrup down it kind of holds its shape like that on the surface whereas if you have something with low viscosity it would tend to kind of flow over the surface and level itself out and then so this is what we would see with something that has low viscosity so we'll quite frequently see that many cases like things like water alcohol those really low viscosity fluids will tend to spread out pretty easily on a surface whereas a low viscosity fluid will stay pretty much in the same place when we talk about the motion of fluids one of the things that we talk about is how do fluids move through different types of pipes and setups so in this particular case we're going to take a look at a comparison of what happens to the velocity that's this v right here notice the lower case to the velocity of a fluid as the area changes of the pipe that it's flowing through and we talk about the area you can see here that i've made the end of these little cylindrical shapes red and we're talking about the area that's what we're talking about in order for us to figure out what the volume would be in this particular case we'd have to take the the area the surface area of that pipe and then we'd have to multiply it by the length of the cylinder in that pipe area and this would give us the capital v volume for that area one so the way when we're talking about this type of flow keep in mind that the a up here is an area a surface area like that red disk whereas the v the lower case in this particular equation is a velocity one not a volume one now the idea of this continuity equation is very helpful in like i said determining how fast fluids flow through pipes because this becomes kind of an engineering challenge to figure things out as people are designing things like plumbing systems in homes or sprinkler systems in your yard things along those lines it's really important to understand these basic concepts so this whole idea was derived from this idea of mass conservation and we know that mass is one of these conserved quantities where whatever mass we start with is the mass we end up with it doesn't end up being created or destroyed so if we take a look at mass conservation so m1 equals m2 in this diagram here what we end up seeing is that we can apply the terms that we used for mass earlier where we have rho v 1 is equal to rho v 2. so once again density times the volume is another way that we can express our terms in mass so you see this idea of mass conservation and the equations that we've been using so far become very very important now to get to this continuity equation we already talked that volume can actually be rewritten as a delta x so i'm going to kind of change this up a little bit we're going to go to row 1 area delta x is equal to row 2 area delta x now if we go back to our motion equations okay see like i said this stuff like keeps cycling back on itself ladies and gentlemen but if we go back to our motion equations we'll remember that delta x is equal to the velocity of the object times time so what we're going to do is we're going to do another quick substitution in here density of the fluid it so density of the fluid area 1 velocity times time is equal to density two area two velocity delta t now if we take a look at this equation we're saying wow that gets to be kind of ugly but a couple of things we want to take a look at this equation and see if we can simplify it down first of all we know that the fluid that's contained in the pipe is the same fluid so the fluid that is at area a is the same fluid that's at area b so what that means is the densities are the same so they end up canceling each other out the next thing that we notice is that the time that it takes for our fluid to travel delta x1 and delta x2 that time is going to be exactly the same so that means that those two terms can cancel out and what that ends up leaving us with is this continuity equation that we're looking at above where the area of position one and the velocity at position one is equal to the area at position 2 times the velocity at position 2. and it gives us this really nice relationship that helps us further explain how fluids move through a pipe here's kind of an example and what i did here is i said if we assumed that we knew the velocity at position 2 would be or the flow rate would be 10 meters cubed per second we could use that information to help us figure out what the velocities would be at different points um in the uh pipe at area one so if when i show here two different things is that we can see that if we have the flow rate at 10 meters cubed per second and we have a 2 meter squared area we can very easily calculate our velocity to be 5 meters per second in this example 1. if we look at example 2 we change the area we make that area larger we can see very quickly that the velocity goes down quite a bit so we get the velocity on on example two would be two meters per second instead of the example of the velocity in the first example which would be five meters per second you know me that i like to always show ways that this actually applies in the real world so this is an example of the continuity equation in real life we know that as water leaves the faucet velocity increases as the water falls due to gravity we know that because the water has mass it's going to be affected by the acceleration due to gravity so we know that as the speed of the fluid increases we should end up getting a smaller area of water flow so if we look up here at the top right near the faucet where the water is moving its slowest we can see that the surface area that's taken up at that point is relatively large and we can calculate surface area simply by taking that diameter across cutting in half and doing the pi r squared equation but then as the fluid picks up speed and falls you'll notice that the cross section of the stream gets much much smaller so this goes right to the idea of these two equations that i just showed you above that as the surface area gets bigger the velocity gets smaller so we see that example right here in the everyday life of water at your faucet and we would notice that if we could see more of the picture eventually this water stream breaks up into droplets that have different separations as well and it kind of continues on with this idea of flow separation another area that this section talks about is the pressure in the fluids as they relate to the speed of the fluids and this one is kind of an interesting one i'm not going to spend a whole lot of time going through the mathematics on it because unlike the one that we just did a minute ago this is more intricate and i don't want to spend a whole lot of time digging into the mathematics of it what i just showed you above with that continuity equation in all reality was relatively straightforward mathematics down here it gets a little more complicated so i'm going to ask you to just memorize things and in this case using this picture to help you memorize them makes it quite a bit easier so here what we have is we have a pipe with a relatively large cross-sectional area and that tube is shrinking down to a smaller cross-sectional area and as this leaf is flowing through the tube we see that as the cross-sectional area gets smaller the velocity increases which holds entirely true with our example up here so as the tube gets smaller we see a much higher velocity of the fluid the part that's a little interesting about that is how do we get to that point so we know that if the two if the um leaf is flowing flash faster in the constricted area of the tube that means at some point in the game there had to be an acceleration and that acceleration is going to happen in the range where the pipe is constricting itself so this is where we get our acceleration you can see here on the diagram that it's showing an acceleration right there with that particular vector so that acceleration vector means that in order for us to have an acceleration we have to go back to our old friend f net equals m a because we're dealing with the mass of a leaf that mass is going to stay the same it's the same leaf in the tube so if we are experiencing acceleration that means we're getting a net force and you can see that the acceleration is pointing to the right so therefore the net force also has to be pointing to the right and what we see is that when we apply that net force and if we remember from the previous lecture where we talked about the buoyant force is that the net force is pointing in the direction of the greater force so in this particular case the pressure behind the leaf is going to be greater than the pressure in front of the leaf so therefore the pressure in area one is greater than the pressure in area two so very important thing that i want you to take from this is basically bernoulli's principle that the pressure in a fluid decreases as a fluid's velocity increases so the way to there's a couple different ways to you can use this diagram to memorize the concept and you can also use bernoulli's principle so bernoulli's principle the pressure in a fluid decreases as the fluid's velocity increases now a lot of times um people talk about bernoulli's principle being the way that uh is a thing that causes airplanes to fly and it's partially true and i kind of crossed out that not entirely true because at one point a couple years back somebody had an issue with the terminology of not entirely true but it is partially true airplanes actually fly on two basic principles number one um you see that we have a big jet engine on the back of this airplane and that jet engine is going to push the airplane through the air and as it pushes the airplane through the air you'll notice that the wing if we draw a chord line in here the wing is actually angled relative to the body of the airplane and this angle is called the angle of attack and this angle of attack is one of the major ways that airplanes also fly that as the air as the airplane is being pushed through the air you can see that the air on the bottom here hits the wing bounces off and changes direction okay and this is going to apply a force of its own and then you see that because of the shape of the wing the air is forced to accelerate and travel across the top of the wing at a higher rate so the air up here is moving at a faster rate than the air on the bottom side of the wing so we just learned that if we have fast moving uh fluids in this case that the air we're going to see that the top of the wing is going to have a lower pressure so in when an airplane flies we actually get two characteristics that cause this airplane to fly one is the air being forced off the bottom of the wing as a result of the angle of attack and the other way is that we get a net force called lift as a result of the lower pressure on the top portion of the wing because the air going across the top of the wing is moving at a much faster rate than the air going across the bottom of the wing so this is not something that's uh it talked about a little bit in the book i kind of added quite a bit to the description of the book so that we can kind of understand it a little bit better but that is basically one of the ways that airplanes fly a couple of different characteristics that cause it to stay aloft so here we are at the end of this particular section um please make sure that you take some time to work on the section reviews and work on some webassigns as we go through we're going to know that the coming up next tuesday we're going to have i'm going to give out a an example of a practice final exam there's going to be two days of studying after that followed up by the final exam that will be given on friday at the end of next week thanks for your time have a great day