Dealing with molecules in fluid mechanics (Fluid Dynamics w/ Olivier Cleynen)

If you approach fluid mechanics for the first time, you might be tempted to think that we calculate the movement of fluids based on the movement of molecules. And it totally makes sense. If you think about the question “what is a fluid?” you could really say a fluid is matter and matter is made out of molecules. But this is not true in fluid mechanics, and let me explain why. In fluid mechanics, we work in what we call the “macroscopic scale” yeah and we treat the fluid as a continuum. Tt means, to us, a fluid like air or water is a continuous paste and there is no gap inside.

So there’s no point to us in flow mechanics where we could say “in between 00:46 - those two little pieces of fluid there is no pressure or there is no temperature, there is a void” like there would be in between two molecules. We never do that. To us in fluid mechanics, it’s a continuous paste and so if you want from a mathematical point of view all the properties are continuously differentiable, yeah? So there’s no gap, there’s no stop in the value of pressure inside the fluid, it always has some value that you can differentiate in space. And you might be thinking “well this is not super rigorous, I know for sure that matter is made of molecules and so why not start with the movement of molecules to be able to calculate the movement of the fluid?” okay. Well try this approach, but then you need to think that one empty bottle of air, yeah, so you take a bottle of water like so and you remove the water so you fill it with air, and then you have one empty bottle of air, and how many molecules do you have in there? Well you have approximately two times 10 to the power 22 molecules, ah. And this is the point where you think “well this is a problem”, because each of those molecules is moving on average at a speed of about a thousand kilometers per hour, and your mind goes “huh”, because 2 times 10 to power 22 molecules it’s a lot. Now let’s write it out.

If you want to solve for the 02:06 - movement of those two times 10 to the power 22 molecules then you need to solve times 10 to the power 22 equations of movement; and since all those molecules’ position and acceleration depend on all of the others, then each of those equations has two times ten to the power 22 unknowns. Yes. And you would be busy solving all the equations all the time continuously there, only to land on the average velocity which is ZERO. Because the fluid is not moving inside the bottle. It’s just staying there yeah? And that’s a lot of equations. This is more equations that even the latest iPhone can solve yeah? So we never want to do that in fluid mechanics.

In fluid mechanics we do what we call the 02:53 - continuum assumption; And so we group modules in batches of millions or even billions, yes? And we pretend that within these batch there is no molecule, it’s just a continuous paste. So to me as a fluid dynamicist, air is made out of more air, yeah? There is no molecule inside. And the patch in fluid mechanics, this group of molecules, we call a fluid particle, or sometimes a fluid parcel. And you may be thinking “how big must it be?” Well for ordinary flow like a river flow it could be about one millimeter cube. For a flow where not too much happens, you could take maybe 10 meters by 10 meters by 2 meters so 10 meters cube yeah, in the upper atmosphere.

And so this brings us 03:42 - to this last diagram here which is quite important. In this diagram you see a property this is in this case here velocity, but it could be temperature, we can pretend this is temperature, as you reduce the volume of the measurement volume for that property. Let me just illustrate this. Let’s take, let’s say I’m measuring the temperature of the room in which I stand here. The room has on average maybe 20 degrees C, but it’s an average in space, because some places are hotter maybe near my cheek it’s very hot and near the window it’s pretty cold and so as I narrow down, my the temperature will change. So if I go from the very large volume of the whole room over here and I shrink down the volume, the temperature will decrease until I reach of the temperature of the point at which I’m measuring. And this is the temperature here.

But if I reduce 04:40 - that volume further and further then something really odd happens, which is that the value on average will start to oscillate, in time now. And so the closer I get to the size of molecules inside my volume and the more oscillation I will see in the value of temperature, because suddenly my thermometer measures nothing at all and then comes a molecule and it bumps up the value to a very high value, and then nothing at all and so on so forth. And so we get really odd variations of properties if our measurement volume is too small. In fluid mechanics, we stay clear of that zone here so we never want to enter what we call the microscopic scale, where we have particle physics and have things moving inside. And we stay safe on the macroscopic scale, yeah, so to us the movement of molecules is contained inside the fluid, and we move the fluid! So the way to think of fluid in general in fluid mechanics, is that it’s not marbles, it’s not a bunch of things moving and buzzing around.

Instead it’s 05:53 - like a dough, it’s like continuous paste that expands to occupy all the space that’s given, available to it. .