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The equation of continuity for pipes

$$\frac{\Delta m_{1}}{\Delta t} =\frac{\Delta m_{2}}{\Delta t}$$

states that mass flow rate inside a pipe doesn't depend on pipe diameter. I'm confused on this being valid even in the following scenarios:

In these three different systems, an open tap creates an ideal fluid like magic that keeps filling the larger/taller tank in order to maintain constant pressure.
A pipe depart from each tank at the same level.

Given that the end of the pipes are closed at time zero, and we open them simultaneously when the systems have their tank and pipe completely filled, will the glasses at the end of the pipes fill up all at the same time?

(Assume that the width of each pipe is much smaller than the height of the tank.)

Or in more scientific terms:
Will mass flow rate be the same in these pipes filled with an ideal fluid?

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  • $\begingroup$ The mass flow rate depends on the diameter of the exit pipe. it is the velocity of the fluid in the exit pipe that does not depend on diameter. $\endgroup$ – Chet Miller Jul 1 '19 at 11:47
  • $\begingroup$ $\Delta m_1 = \Delta m_2$ is the same as $$\int_{t_0}^{t_1} \dot m_{\text{in}} dt = \int_{t_0}^{t_1} \dot m_{\text{out}} dt.$$ The are just the differential and integral forms of the same equation. $\endgroup$ – alephzero Jul 1 '19 at 11:59
  • $\begingroup$ @Nat Yes I think the equation refer to the scenario you depicted ( one pipe feeding into another of different diameter). I'm just trying to understand how this relate to these systems scenarios if it does. Moreover, maybe an answer just describing what happens in the three different systems would be enlighting. $\endgroup$ – Gabriele Scarlatti Jul 1 '19 at 12:01
  • $\begingroup$ Please take a look at Hagen-Poiseulle equation. I think you may actually solve the problem with some minor modelling assumtion, namely that the width of your pipes is much smaller that the height of the container. The pressure-difference $\Delta P $ is then simply the difference between ambient pressure and the pressure due to the water in the container. $\endgroup$ – denklo Jul 1 '19 at 12:46
  • $\begingroup$ @denklo I will add this assumptionm to the post $\endgroup$ – Gabriele Scarlatti Jul 1 '19 at 13:52
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Mass flow rate is the number of molecules per second passing a cross-section of the pipe. If it's different in different places in the pipe, that means molecules are piling up somewhere. That can't go on very long.

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