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  • today would do in Reynolds number.

  • Famous Number, named after Reynolds talks about fluid mechanics to do with Naevia Stakes equations.

  • But it's not one number.

  • It's not like pie.

  • It's not.

  • No, no, this is.

  • This is like the superhero version of pie, because pious fix pie has, like one superpower like circles and things.

  • This is Reynolds number.

  • This guy can Maur from change to be whatever value it wants to be, but depending on what sort of size it is, whether it's really big or really small, you get different behavior with your fluid.

  • So it's more like a label that you give to a particular situation.

  • You have a fluid moving in, some set up, and that's like High Reynolds number.

  • It does one thing, but if you have the same fluid moving in a slightly different set up where it's now got Low Reynolds number distantly in completely different, like choosing your superhero from the Avengers.

  • If it needs to be invisible, it could be invisible.

  • If it needs to reverse time, it can reverse time.

  • It can do whatever it wants.

  • Starting point Naevia Stokes equations.

  • So these guys model the motion.

  • The flow of every fluid on Earth in the universe.

  • In fact, we have these equations.

  • They come from sort of universal laws of physics.

  • We have masses conserved.

  • We have Newton's second law.

  • We get our equations so we'll start by writing those guys down.

  • We talked about these guys before.

  • We did the full sort of what everything means, but we've got the top one on the little guy, which is conservation of Mass.

  • And then we have Newton's second law.

  • I've made another slight simplification from the standard Naevia Stokes and I've said there are nobody forces.

  • Normally there's a big plus f here.

  • We're gonna pretend there's not.

  • That's the simple case.

  • So here's an obvious takes equations.

  • Now we want to get the Reynolds number, so we need to define what the Reynolds number.

  • Yes, we label it with R E, and it's going to be our density, wrote times a lens scale L a Times a velocity You divided by viscosity.

  • Tom.

  • Why do we even need a Reynolds number?

  • Thes never Your Stokes equations, which you've explained to us before.

  • Serviceable things for figuring out how fluids work.

  • What was missing in the first place.

  • here.

  • So Equation one is nice and simple.

  • So So we actually, that guy doesn't change with the Reynolds number discussion, but equation to we've got, like, three bits, so we've got a different car.

  • So we've got this guy is acceleration because it's a density time to change in velocity.

  • This guy's a pressure on, then this guy is viscosity.

  • So viscosity is like an internal friction, your layers of fluid sliding over one another and interacting.

  • We got three bits on.

  • The point is in different situations.

  • Some of these bits will be more important than others.

  • Practical example is, if viscosity was important.

  • When you go fast, you could clean your car by driving really quickly what the air would just blast off.

  • Exactly.

  • So if you have those little bits of model of your car, you just really, really fast The air would actually washing up because the friction in the air would grab onto the dirt particles.

  • Clean your car for you.

  • So disgusting doesn't matter if you're going fast.

  • Turns out as we're going to see because of the Reynolds number.

  • In some situations, they're all equally important.

  • But in certain situations like the extreme cases, you can get that one of them doesn't actually matter, or one of them is so important that dominates the other two.

  • And then you've got a nice, simple our equation, which we can then solve because as we know, we can't always solve Naevia steak.

  • So if we can get our hands on this Reynolds somewhere, allows us to simplify them on, actually solve them and get somewhere.

  • So the Reynolds number is what apportions different waiting's to the different fact.

  • Exactly, exactly way.

  • Can I have a really big Reynolds number or really small stuff?

  • But that's sort of the standard thing we do in maths that's consider if it's really big or it's really small.

  • And just so I've got some idea of what numbers we're talking about.

  • We're talking like billions and negative numbers like Tell us where the S.

  • O.

  • The resident has to be positive because it is a density times of length, times of velocity over a viscosity.

  • These were all positive physical things you can measure, and you plugged those values in.

  • You'll get a number so it could be a small as sort of back down towards zero can't be zero point just above zero, or it could go as big as you want.

  • But generally we're talking sort of 10 to 1000 most everyday situations.

  • If it's over 1000 its turbulent.

  • So turbulence is this chaotic, crazy motion of fluids crashing into each other.

  • So we have two cases.

  • The Reynolds number.

  • We've got big Reynolds number.

  • So we say it's not just greater than one.

  • It's much greater than one so sort of 1000 above turbulence on your equations.

  • So we've got rid now of all of this.

  • Sort of these guys like them.

  • You're on the road.

  • They sort of their in their rentals number now.

  • So they're gone.

  • And so what?

  • That means you can start to compare your terms and you end up with this guy.

  • Won over Reynolds.

  • That's good.

  • So if we compare this one too original equation, we have the same terms.

  • We've still got acceleration.

  • We've still got pressure on.

  • We've still got viscosity, but now we've sort of got rid of all the other little bits that were floating around.

  • And now we can compare everything because by introducing the Reynolds number, what we've done is we've non dimensional eyes.

  • The system, which means get rid of the units.

  • If I say I have 10 apples and I have nine but honest then if we consider apples, I have more fruit with my 10 apples than I do with my nine banality.

  • But with my nine bananas, I have more potassium than I do with my apples.

  • The unit's matter.

  • It depends.

  • You can't say I have more apple stand.

  • Been honest.

  • It depends whether I'm talking about fruit or potassium.

  • The units are important.

  • So now, by infusing the Reynolds number, what we've done is a non dimensional ization, and we've got rid of units.

  • So now we can compare acceleration pressure on viscosity.

  • Three very different things with different units.

  • But now they have no units.

  • If the number associated with the acceleration term is bigger than the number associate with pressure, that means acceleration is bigger than pressure.

  • It makes sense to compare them because they have no units anymore.

  • Okay, this is this is the power of the Reynolds number.

  • Really?

  • Does Reynolds number itself have units, then?

  • The Reynolds number does not have units.

  • The Reynolds number is just a number on, we can actually see that.

  • So the Reynolds number itself is density times length, times speed, divided by viscosity.

  • So let's think about the units.

  • That's a real hodgepodge of units.

  • It is so density.

  • That's gonna be kilograms per meter, cubed in three D.

  • So we use the square bracket notation, Tina units for our length.

  • Well, that's just in meters velocity you that's in meters per second and then our viscosity.

  • So this one is possibly less well known.

  • But this guy is kilograms per meter per second.

  • And now if we plug them all together in the red is number formula, we get ari the units.

  • So on the top we've got a density.

  • So it's a cagey and minus three.

  • Then we've got a length That's an M.

  • Then we've got speed and s minus one, and then we divide through by viscosity.

  • And now hopefully we've done this right Kilograms cancel meters meters meters minus three.

  • Gives meters minus one.

  • Cancels second sex.

  • So the rain was number has no unit, so it is just a number.

  • That's why it's called Reynolds number.

  • It does.

  • It's just a number.

  • It's why we can think about what happens when it's big and get this equation.

  • So now the Rennes number is really big.

  • We've got acceleration equals pressure, plus one over a really big number.

  • Times viscosity.

  • So one divided by a really big number is really small.

  • So what's that telling us is that that guy doesn't matter?

  • So in the situation off really large Reynolds numbers, which could be very, very light, very, very high density, very, very long fluids or very, very fast fluids to make any make this thing big, however you have to viscosity doesn't matter.

  • So you're driving your car.

  • You're going really, really fast.

  • That's why the viscosity of the air doesn't clean the dirt from your car because you're in this large Reynolds number and your equations say viscosity doesn't matter, so it doesn't doesn't work.

  • So what if the Reynolds number is now really, really smart?

  • We say it's much less than one.

  • It's not just smaller one.

  • It's much less than what.

  • And so what happens now with our equations is when we substitute this in.

  • We do our non dimensional ization in the same way as before, but now we've got Reynolds number times acceleration equals that pressure term plus the viscosity toe.

  • So now it's slightly different to the one we had.

  • So we've got Reynolds number just a number of times.

  • Acceleration is pressure plus viscosity.

  • You might be wondering, Why is the Reynolds number now?

  • Hear?

  • Why is it on acceleration when before we had it over on viscosity.

  • Now we have a small Reynolds number, and so, in a small Reynolds number flow, we sort of know we don't experiments to say that the pressure term is actually balanced by viscosity, whereas in the big Reynolds number case, the pressure term is balanced by the acceleration so that there is some slight difference in how you non dimension allies.

  • But it still works in the same way it still has units off pressure.

  • You just have this choice between how to balance your pressure.

  • So when the murders were small, we have this guy.

  • So it's Reynolds Times.

  • Acceleration is pressure plus viscosity, but the Renaissance was really, really small, so I've got a tiny number times acceleration.

  • Acceleration doesn't matter, because Reynolds in was really small, so that now goes so this is the term that doesn't matter.

  • In this small Reynolds number regime.

  • So now just we've got pressure on viscosity, and that's it.

  • Our governing equation is just pressured, radiant plus viscosity.

  • My favorite thing I'm going to say in all of fluids is getting this result because