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There are two discs in the pipe but not at the same time. You can consider conditions with X but without Y, then with Y and without X.

I am pretty sure that disc X will feel less force from fluid than disc Y does. I am looking for its reason.

Total pressures (static + dynamic) are the same for both conditions, because it's the same flow. In R1 area, dynamic pressure is low, but static is high. In R2 area, dynamic is high, but static is low, but in conclusion, total pressure is the same.

If total pressure is the same but force on X and Y is not the same, then how to justify this observation?

Does it mean that discs feel only dynamic pressure, not total? When I go for momentum equations, I find that disc Y must feel more force than X does since flow has higher velocity in R2 than in R1, which means higher momentum.

So, in this case, discs feel only dynamic pressure, I think.

What's your comment?

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  • $\begingroup$ My comment? Check out D'Alembert's Paradox. $\endgroup$
    – Solar Mike
    Mar 27, 2023 at 8:23
  • $\begingroup$ "Force on a disc" implies that the flow is slowed down by pushing against the disc? In this case, the dynamic pressures will reduce, and the static pressures will remain, and the forces will be as you expect? $\endgroup$ Mar 27, 2023 at 8:26
  • $\begingroup$ No, I mean total force trying to push discs to right. Which one faces more force from fluid? Which one starts to drag to right firs? $\endgroup$
    – Jawel7
    Mar 27, 2023 at 8:49
  • $\begingroup$ Ok, so if the fluid isn't being stopped by the disc, then the static pressure will be present on both sides of the disc - cancelling itself out! $\endgroup$ Mar 27, 2023 at 8:55
  • $\begingroup$ But dynamic pressure will be only on left side pushing them to right? $\endgroup$
    – Jawel7
    Mar 27, 2023 at 9:38

1 Answer 1

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As sketched here, the discs block the entire pipe. Flow would be zero. From your description, that is not what you have in mind. I think you mean that there are some leak openings left, for flow to go through?

When the opening area is small relative to the pipe area, then (perhaps surprisingly), neither the static nor the dynamic pressure in the main pipe is directly relevant for the answer. The relevant parameter is the dynamic pressure in the open leakage area, where the velocity is much higher than in the main flow. That's where the main losses are generated.

Roughly put, and for turbulent flow only, the pressure drop across the disc will k * 1/2 rho (Q/A_L)² Where rho is density, Q is volume flow rate, A_L is the open area of the leak path trough the disc. k depends on details, but will be typically between 1.5 and 3. You can use k=2 as a good starting point.

If both discs have the same A_L, then they will generate the same pressure drop (deltaP) across them. Since force is deltaP times disc area, the larger disc would see a larger force on it.

If the leak area is proportional to the disc area, then the pressure drop across the larger disc is smaller as Q/A_L is smaller. This effect is quadratic, so the effect is a lower force on the larger disc.

In between , there is the situation where the leak area is proportional to the diameter of the disc instead. For example, if there is a constant-width gap around both discs. In that case, a larger disc area and a smaller pressure drop would roughly cancel out. Both discs would have a similar force on them.

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