blurrscreen said:
But imagine gravity pushing down on the fluid, thus applying a pressure to it which, according to pascal's law is equally transmitted at all points. Then why do points at different heights in the fluid have different pressures.
The answer to this is that gravity does not apply pressure, it is a force with a very different nature than pressure. Gravity is what is known as a "body force", which means, if you think in terms of time rate of change of momentum, gravity acts to remove momentum from every gram of material at a fixed rate. Since it in some sense "acts on the body of the system", since it removes momentum per gram, it is a body force. What this means is, you should think of it as source or sink of momentum that does not need to move any momentum through any boundaries, it just plain deposits or removes it (depending on the sign you are imagining) at every point in the body, like magic if you will.
Pressure is quite different, because it is a surface force. The key difference is that when you think about how pressure adds or removes momentum from something, it always has to pass that momentum through the surface of that something, it cannot just make momentum appear like magic the way gravity does.
In the problem you are thinking about, like water in a pool, all the surfaces we need to think about are horizontal surfaces, and the only direction we need to worry about is vertical. This simplifies the situation, we don't need to worry about the fact that pressure is isotropic, we can just look at what it is doing with momentum in the up and down directions, and that suffices to see what is going on in a pool.
So imagine any horizontal surface inside the water in a pool, and ask, what kind of momentum is passing through that surface? The answer is, pressure is causing upward momentum to pass upward through that surface, and an equal amount of downward momentum to pass downward (that's the "isotropic" aspect). Let's call this equal rate of momentum crossing the surface in either direction P
1A/2 (the notation is motivated by the fact that P is pressure and A is area, but it doesn't matter, it's just a number I'm talking about). It doesn't matter how it is doing that, this is just what pressure does, but if you want to know how, realize it is the action of all the little water molecules that are moving up and down across that surface all the time-- an upward molecule is obviously going to transport upward momentum upward across the surface, and a downward molecule is going to transport downward momentum downward across the surface. Now your question is, if an equal amount of upward momentum is moving upward as downward momentum is moving downward, why does anyone care?
Good question, and it has a good answer. Next imagine a second surface that is a bit deeper down below that first one. Again the same amount of upward momentum is crossing upward as downward momentum that is crossing downward, but the numerical value of this rate of crossing of momentum is different from before, now call it P
2A/2, and you will not be surprised to hear that P
1 < P
2, since I mentioned that P is really pressure here.
Now for the reason you care about these P: think about an imaginary box that is bounded above by the first surface, and below by the second surface, and think about the rate that momentum is entering this box. There are 4 terms you have to add up, two from the top surface and two from the bottom. The top surface has P
1A/2 rate of downward momentum coming down into it, so that's a net accumulation of downward momentum. It has the same rate of upward momentum leaving it upward, but here's the reason they don't cancel-- upward momentum has the opposite sign from downward momentum!
So having upward momentum leave has the same effect as having downward momentum enter. Thus the two terms add up to a total of P
1A, not to zero. In short, there is downward momentum entering the imaginary box through its upper surface, which is a downward force on the box, due to the P at its upper boundary.
Play the same game at the lower boundary, and you will find that P
2A is the rate that upward momentum is entering from below, but that's the same as -P
2A rate of downward momentum, since momentum is a vector. So the top and bottom momentum fluxes are the things that might cancel out, not the momentum passing through a given surface into our imaginary box. But if there's gravity, that body force that is adding downward momentum to the mass in the interior of our box, then we can only get force balance if pressure is removing downward momentum (or adding upward momentum, it's the same thing), and that's what the surface momentum fluxes that pressure controls is doing. That's why P
1A < P
2A, and that's why pressure matters, even though it is locally isotropic. It matters because it is different in different places, and Pascal's law says that once you have set up those differences, if you exert additional pressure (like adding weight to the top, for example), it can't affect the
differences in pressure, because those were already what they needed to be to move a net flux of momentum around as needed to balance gravity.