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Current time:0:00Total duration:11:06

Viscosity and Poiseuille flow

Video transcript

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 and just keep moving with a constant speed let's say there's no air resistance no resistive forces at all now what I'm going to do I'm going to stick a fluid let's say water and I'm going to 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 fluids 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 topmost 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 resists the motion but it's worse than that because if this topmost layer gets pulled this way then the layer right below it gets pulled by the topmost 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 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 and the fluid is going to exert a force by Newton's third law to the left on the lid and I'm going to call this a viscous force I'm going to 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 going to be dragging the larger the viscous force that makes sense so it's this area here and something else that depends on is the speed with which you drag the lid so the faster I pull the lid well the faster I'm going to be pulling this water the bigger that force which means the bigger the viscous force so it's 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 Aida is going to 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 resists flow and so coefficient of viscosity so to give you an idea honey or corn syrup would have a large viscosity water would have a smaller viscosity coefficient and gases 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 ADA what would we get we 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 forces and Newtons distances 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 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 ADA the coefficient of viscosity is a pressure times a time Pascal seconds but some people use the unit pause and one plus is equal to one tenth of a Pascal second or in other words ten pause and it's abbreviated capital P is one Pascal second and so you'll often hear this unit pause as a measure of viscosity so what are some real-life values for the viscosity well the viscosity of water at zero degrees Celsius is and I'm not talking about ice but water that's actually at zero degrees but not frozen yet is about one point eight but one point eight million seconds and another way to say that look at mili Pascal seconds that would be a center ascent applause CP because applause is already one tenth of a Pascal second number one pause is one tenth and so essenti pause is really a mili Pascal second or water at 20 degrees Celsius is one point zero mili Pascal seconds or scent applause 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 of fluid typically gets which you know because if you start your car and it's too cold outside that car is not going to want to start that oil inside is going to 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 milliseconds or sent applause and then engine oil can have viscosities in the hundreds around 200 cent applause and then gases gases would have viscosities that are even less air as a viscosity of around 0.018 Santa Paws 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 if the coefficient of viscosity does depend on the speed with which the fluid is flowing or the speed with which you pull 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'd be a non-newtonian fluid now you might be thinking well this is kind of stupid I mean how many cases are we going to have where you're trying to pull a lid over a box you probably not never try 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 this is a constant of the fluid now anywhere that this fluid is flowing now that you've measured it carefully you can determine what kind of flow rate you would get so imagine let's get rid of all this 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 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 cubed 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 the formula is called kwazii's law and it says this it says the volume that will flow I'm is dependent on delta P times pi times R to the fourth divided by a2 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 PA's a is law so this Delta P is referring to the pressure differential so it would be pressure on this side we'll call it P one will be pressure on this side P two if those are the same this flow is not going to be flowing very long there's to 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 p1 minus p2 the more volume that's going to flow per time and that makes sense and then PI is the geometric factor R to the fourth this R is referring to the radius of the tube so that's R and then eight and eight oh we know ADA is the viscosity so this is the viscosity of the fluid and the volume flow rates inversely proportional to the viscosity because of 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 fluids got to flow through the smaller the volume flow rate so this is called Plaza's law it's useful in 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 8 is 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 fluids stay in their Lane basically they do not cross over 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 Plaza's law gives you the volume that flows through a pipe per time