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.]
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:
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