A bit of fluid mechanics from scratch not from scratch
I’ve been reading various things about fluid mechanics so that I can think about microfluidics. But I haven’t really been studying it like I used to study a math area (instead just kinda getting various impressions). I’d like to think it through a bit “from scratch”, though it’s very much not actually from scratch of course. This is written as close to real-time thinking as possible (though of course much slower than without writing), over a few hours, with the intent of getting a “thinking trace” because that seems interesting.
[In all diagrams, pretend the top is open to the air, and ignore differences in air pressure.]
1 Static pressure gradients??
If I ask myself what I’m intuitively confused about, I’m like, hold on—it makes intuitive sense that pressure gradient would cause the fluid to accelerate. But can’t you have pressure gradients in a static situation?? E.g. if you have a tank of water, there’s a pressure gradient, where the lower water has more pressure:
But it’s just water sitting in a tank and it’s obviously not moving, let alone accelerating. But there’s a pressure gradient, so why isn’t it accelerating? Presumably it has to do with gravity? But like, the gradient is pointing downward, so my synesthetic unconscious thinks that the acceleration is supposed to be downward, and also the bottom is the most pressurized and pressure feels intuitively related to going fast, so the bottom should be the fast part.
Ok duh, there is indeed a force from the pressure gradient; it’s pointing upward, because of course high pressure pushes stuff away so the force (which I suppose is the negated gradient) points in the direction of decreasing pressure, which is straight upward. This force is exactly counterbalanced by the force of gravity. The gradient is constant throughout the water, dependent only on gravity and the density of water.
2 Spouts and acceleration
Now suppose we have a tank with a spout:
The water is shooting out of the spout. When does it accelerate? It seems like the full spout head, on the far right, is full of water shooting out. But if that’s the case, since water is incompressible / not stretchable or something, that means that the whole horizontal pipe has its contents flowing at the same rate. Because if the water were accelerating while in the tube [[narrator: he’s assuming that the water at different vertical positions, at a given horizontal position, is moving with the same vector]], then you would have more water leaving the horizontal tube than entering it.
So that means that the water is already accelerated to full speed before entering the horizontal pipe… How / where? Something doesn’t make geometric sense…
…Oh, we can’t model this as a bunch of simple (straight, say) slices with constant velocity on a given slice. That just won’t work. Let’s imagine a different scenario:
And let’s presume that the velocity at a given height is constant. Well, we know that the velocities are always strictly vertical (and downward). Further, to conserve mass, and given incompressibility, we’ve got to have that the velocity is inversely proportional to the width, simple as that.
What about with a simpler tank?
Something seems really weird. The water coming out should be going pretty fast. But if the slices of water at each height have constant velocity, this makes no sense? Because the tank is much wider than the spout. And also it is constant width, so therefore the water has constant velocity. So the water is going slowly down, but then suddenly it zooms out of the narrow spout at the last infinitesimal moment?? When did it accelerate??
And furthermore, the tank is wide and the spout is small. So the water reaching the bottom of the tank, say in the corner, has to get over the spout really fast?? How??
Makes no sense. I think I can imagine what ought to happen:
Where the corners are stagnant, or now that I’ve made the diagram I guess maybe little vortices or something, I don’t fucking know lol.
Anyway the point is that the case where a fixed flow-orthogonal slice of a pipe / channel always has a constant velocity, is a very special case. …Well actually, I don’t know how special… Is it determined between two points? Like if I tell you the width of the top and bottom of this:
Can you then deduce the full curve? Let’s come back to this…
3 Height paradox thingy
Ok so I have a serious problem. As everyone learns in kindergarten, the speed at which water comes out of a spout in a tank depends on the height:
But something is weird. Come with me for a sec. Suppose we have a spout at the bottom:
So the water comes out at some rate, fine. And that rate depends on how tall the tank was, yeah. But now suppose that we add more pipe, at spout-width, to the bottom:
Ok, so, it’s the same as before, but the outlet of the spout is now significantly deeper / lower. So the speed of the water should be higher, right?
Ok, but if the water is faster at the bottom of the long spout… We could view the top part of this system as an exact copy of the short-spout version. At the interface between the tank bottom and the pipe-spout, the velocity of the water should be the same as in the no-pipe version, right? But that means the water inside the pipe is accelerating inside the pipe:
But that’s… not… possible. Water is incompressible and mass is conserved. The water at the bottom of the spout cannot be going faster than at the top!
I’m not confident about what’s going on. I think all my reasoning is pretty solid. Unless it’s like, the water inside the pipe-spout is somehow pulling or pushing on the water in the tank differently than if there were no pipe? But that doesn’t sound right? Because it’s falling away anyway…
My best guess is that the above reasoning is correct, and what actually happens is like this:
In other words, the water does indeed accelerate, and it also gets narrower; the pipe is partly filled with air.
4 Deducing the equivelocity curve?
Does this solve the problem stated a little while ago? In other words, do we know what shape a pipe would have to be in order to start with a given width, end with another given width, and have constant velocity on each flow-orthogonal slice?
Maybe what we get is like the pipe-spout:
In fact, the shape of this pipe is already determined by the input width! We cannot change it, because the curve is determined by gravitational acceleration. As the water falls, it accelerates. As it accelerates, the pipe must get narrower (total length is inverse to average flux density or whatever).
So we can’t even set the output width.
Well, but this is not fully satisfying. For one thing, we could have just asked, in general, ignoring how this steady state is obtained through forces / pressures, can we just imagine some pipe with some varying width, filled with water which is moving and accelerating according to width, but in a steady state (the way a river holds a steady shape even as water moves and accelerates through it)?
At the moment my guess is yes… In fact, couldn’t we just set up any curves as our pipe?:
I think this would work; as long as you have a pressure difference between one end and the other, the water accelerates in the direction of less pressure. (I just realized a lot of the diagrams above show this incorrectly maybe.)
But, like with the spout at the bottom of a tank, you would very much not get constant velocity on a given slice. Instead you’d get vortices and lacunae and whatnot:
With the pipe-spout at the bottom of the tank, we are fixing the pressures at the top and bottom of the pipe, because we’re using water-depth pressure caused by gravity. If we relax that constraint, we can just imagine that we have some specified pressure differential.
But I’m still not satisfied, because it feels like there’s some constraint / some criterion that specifies the right shape for a pipe…
It can’t actually literally be that the velocity is constant on flow-orthogonal slices. That actually does not make sense because the water is supposed to accelerate along the pipe, and therefore the pipe is getting narrower, and therefore at least some water must move (and indeed accelerate) in a direction with some flow-orthogonal component:
This can’t be constant (or else the slice of water would be moving, on net, in a direction not in the direction of flow…).
We might instead ask for having a constant flow-parallel component of velocity at a given flow-orthogonal slice; or a constant speed.
I’m not really sure what I’m asking for, but it feels like there’s something here. One question would be, for given inlet and outlet widths for a pipe, and given inlet and outlet pressures, what pipe shape gives the fastest steady state flow? (I suppose not all pipe shapes give steady state flows at all, because some would have turbulence—or do you need viscosity or friction or something for turbulence? Which pipes are turbulent or not? Presumably that’s a hard question but maybe there are some nice simple recognizable classes of each.) Something something laminar flow?
Part of why I feel there’s some question here, is that on the one hand, you have a picture like this from earlier:
Where the whole thing considered as a pipe is clearly not good, and the steady state flow (if there is one) is not simple / nice / smooth, because the shape of the pipe is such that it doesn’t elegantly funnel the water laminarly or something into the outlet. But on the other hand, we could have pipes like this:
Here the channel gets quite narrow in a weird sharp way. The flows seem like they would get hampered and not flow smoothly. It feels like, in between getting narrow to quickly and getting narrow not quickly enough, there should be one curve, or maybe some region in curve-space, that makes it so that the flow is nice and smooth and fast and efficient.
Oh wait, I guess both of them get narrow too quickly. It doesn’t really make sense to get narrow too slowly… Well, overall everyone gets narrow on average at the same rate, if we’ve fixed the widths of our inlet and outlet. I guess what I mean is, you get narrow too unevenly, with some high density spikes of narrowingness.
Maybe if the outlet is too differently sized from the inlet, then you might inevitably have problems like this? (I suppose the outlet could also be too big.)
We know that there is at least one good pipe shape, because the pipe could just be straight (equal outlet and inlet) and have no pressure differential (and somehow be in a moving steady state, if we want). I think we know there’s another good pipe shape, which is the gravitational one from earlier with the pipe-spout at the bottom of the tank.
5 Sideways pressure?
One thing that I’m a bit confused about is the sideways pressure from the walls in a pipe that’s narrowing. Or in general, how does the water get pushed into the middle of the pipe as it narrows?
Is there a pressure gradient sideways? There must be; I don’t think we have other available forces. Gravity can be ignored, and what else is there? I guess there’s, like, collision force or something? But isn’t that the same as pressure?
There’s also friction and stickiness and viscosity / strain or something, but I think we’re ignoring those at the moment? Maybe that doesn’t make sense.
Anyway, I think there’s a pressure gradient sideways? Yeah I guess that’s probably right. The walls push inward on the water. But wait… Don’t the walls always push inward on the water?
So isn’t there always pressure on the outside surface of the water? And therefore there’s always a pressure gradient that’s high pressure on the surface and low on the inside? And therefore… the water accelerates towards the center of the pipe?
That makes no sense, right?
Or maybe pressure is constant in a constant-width pipe:
And it’s only non-constant in a non-constant-width pipe:
It would be nice to know these things quantitatively.
6 Generalizing
I guess there’s a few different kinds of questions:
- Statics (water not moving; question of force fields and pressures)
- Constant velocity (water moving but not accelerating)
- Steady states (water moving and accelerating, but constant velocity and acceleration fields; these seem like they should be nice)
- Dynamics (water colliding / roiling… accelerations time-dependent)
I speculate that we can usefully analyze steady states using just pressure fields; the gradient gives the force and acceleration fields; …. and then I guess you need to know how that relates to velocity fields …. seems kinda hard actually.