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Viscosity and Poiseuille flow

Video transcript

- [Voiceover] Check out this empty box. Now, check out this lid that I can put on the empty box. Let's say there is no resistance so that if I gave this lid a little nudge, it just keep moving with the constant speed. Let's say there's no air resistance. No resist to forces at all. Now what I'm gonna do, I'm gonna stick a fluid. Let's say water. And I'm gonna fill it to the brim all the way to the top. This thing was overflowing. This goes all the way to the top. Now what happens is you take this lid, you go to slide it across again. You give it a little nudge, it doesn't keep going, it slows down and stops. You give it another nudge, it slows down and stops. The fact that this fluid's in here now is resisting the motion that's because it's viscous. What do I mean by that? I mean that when this lid was moving across the top, the fact that this fluid was in contact with the lid caused this top most layer to start moving with the same speed as the lid. There's adhesive forces between this fluid and the lid on an atomic and molecular level, this fluid gets pulled with it. And so it resist the motion but it's worse than that because if this top most layer gets pulled this way then the layer right below it gets pulled by the top most layer and the second most layer pulls this third most layer and this keeps going down the line, and what you get is a velocity gradient is the fancy name for it. Which just means that this velocity of this fluid gets smaller and smaller. Once you get down to the bottom well, now the fluid is in contact with this surface at the bottom and this surface at the bottom is not moving. This fluid is at the lowest most point doesn't move at all. All right, so that's why this lid slowed down when we gave it a nudge, it was dragging that fluid along with it. And if it exerts a force to the right on the fluid, then the fluid is gonna exert a force by doing its third law to the left on the lid and I'm gonna call this a viscous force. I'm gonna call it Fv. So, what does this depend on? What does this viscous force depend on? One thing it depends on is the area and not the whole area of the lid. It's just the area of the lid that's actually in contact with the fluid. So if you imagine the dimensions of this box would only expand here. So it's only that part of the lid that's actually in contact with the fluid. So that area and it's proportional to that area. The bigger that area, the more fluid you're gonna be dragging, the larger the viscous force that makes sense so it's this area here. And something else that it depends on is the speed with which you drag the lid. So the faster I pull the lid, well, the faster I'm gonna be pulling this water, the bigger that force which means the bigger the viscous force, which was proportional to the speed as well. It's inversely proportional to the depth of the fluid. I'll call that D. And then it depends on one more thing. It depends on the viscosity of the fluid. Maybe the most important factor in this whole discussion. Eta is gonna be called the viscosity of the fluid or the coefficient of viscosity. And what this number tells you is how viscous, how thick essentially the fluid is. How much it resist flow. And so, coefficient of viscosity. So to give you an idea honey or corn syrup would have a large viscosity. Water would have that smaller viscosity coefficient and gasses would have a coefficient of viscosity even less. So what are the units of this coefficient of viscosity? Well, if we solved. If we were to solve for the eta, what would we get? We'd get force divided by area. So this would be force divided by area, and multiplied by the distance divided by the speed. What units do these have? Force is newtons, distance is in meters, area is in meters squared, speed is in meters per second. So I bring that second up top because it was divided in the denominator. So it goes up top. And what am I left with? Meters cancels meters and I'm left with the units of viscosity as being newtons per meter squared times a second, but a newton per meter squared is a pascal so this is pascal seconds. So these units are a little strange but the units of eta, the coefficient of viscosity is a pressure times a time, pascal seconds. But some people use the unit poise and one poise is equal to 1/10 of a pascal second. Or in other words 10 poise and it's abbreviated capital P is one pascal second. And so, you'll often hear this unit poise as a measure of viscosity. So what are some real life values for the viscosity? Well, the viscosity of water at zero degree Celsius is, and I'm not talking about ice but water that's actually at zero degrees but not frozen yet is about 1.8. But 1.8 millipascal seconds. And another way to say that, look at millipascal seconds, that would be a centi, a centipoise, cP. Because a poise is already 1/10 of a pascal second. Remember one poise is 1/10. And so a centipoise is really a millipascal second or water at 20 degrees Celsius is 1.0 millipascal seconds or centipoise. Now you can see there's a huge dependency on temperature. The viscosity is highly dependent on temperature. The colder it gets, the more viscous a fluid typically gets which you know, because if you start your car and it's too cold outside, that car is not gonna want to start. That oil inside's gonna be more viscous than it's prepared for, and that engine might not start very easily. Blood typically has a viscosity between three to four millipascal seconds or centipoise. And then engine oil can have viscosities in the hundreds. Around 200 centipoise. And then gasses, gasses would have viscosities that are even less. Air has a viscosity of around 0.018 centipoise. Now it's important to note, if a fluid follows this rule for the viscous force and the coefficient of viscosity does not depend on the speed with which this fluid is flowing or with which you pull this lid over the top does not depend on that. Then we call this a Newtonian Fluid. Then it's a Newtonian Fluid. But if the coefficient of viscosity does depend on the speed with which the fluid is flowing or the speed with which you pulled this lid, then it would be a non-Newtonian fluid. So if this coefficient of viscosity stays the same regardless of what the speed is, it's a Newtonian fluid. If that's not the case, it would be a non-Newtonian fluid. Now, you might be thinking well, this is kind of stupid. I mean, how many cases are we gonna have where you're trying to pull a lid over a box, you've probably never tried to do that in real life. But this is just a handy way to determine the viscosity. Once you know the viscosity you can apply this number. This is a constant of the fluid. Now, anywhere that this fluid is flowing now that you measured it carefully, you could determine what kind of flow rate you would get. So imagine, let's get rid of all these stuff here. So you had a stationary tube or a pipe and there was a fluid flowing through it, maybe it's a vein or a vessel and it's blood flowing through it. Anyways, now it's stationary though. Both the top and the bottom are stationary so that means the fluid near the top and the fluid near the bottom aren't really moving but it's the fluid in the middle that's moving fastest and then slightly less fast, and you get this somewhat parabolic type velocity gradient where it gets bigger and bigger and then it gets smaller and smaller, and so, the velocity profile might look something like this. And, if we wanted to know how much volume of fluid, how many meters cube of fluid passed by a certain point per time, we can figure that out. There's a formula for this. The volume per time, the meters cube per time. Now the formula is called Poiseuille's Law. And it says this. It says the volume that will flow per time is dependent on delta P times pi, times R to the fourth, divided by eight eta, times L. Now this is a crazy equation. Let's break this thing down and see what it's really talking about. So here's Poiseuille's Law. So, this delta P is referring to the pressure differential. So it will be pressure on this side. We'll call it P one. There'll be pressure on this side, P two. If those were the same, this fluid's not gonna be flowing very long. There's gonna be a difference. If we want the fluid to flow to the right, P one has to be bigger than P two, and the greater the difference, the greater this difference P one minus P two, the more volume that's gonna flow per time and that makes sense. And then pi is a geometric factor. R to the fourth. This R is referring to the radius of the tube. So that's R. And then eight and eta we know. Eta is the viscosity. So this is the viscosity of the fluid. And the volume flow rate's inversely proportional to the viscosity because the more viscous the fluid, the more it resists flowing, and the less meters cube you'll get per second. And it's inversely proportional also to the length of the tube. The more tube this fluid's got to flow through, the smaller the volume flow rate. So this is called Poiseuille's Law. It's useful on a lot of medical and engineering applications. For whenever you want to determine the volume flow rate, now you've got to be careful. We're assuming this is a Newtonian fluid, that means eta's not a function of the speed of the fluid. We're also assuming you have nice streamlined laminar flow. So laminar flow means these layers of fluid stay in their lane basically. They do not crossover. Once you start getting this, you'll start getting turbulence and once you get turbulence, you'll need a much more complicated equation to describe the dynamics of this. So we're assuming no turbulence and a nice Newtonian fluid. And if that's the case, Poiseuille's law gives you the volume that flows through a pipe per time.