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?

  • $\begingroup$ 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. $\endgroup$ Jul 1, 2019 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, 2019 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. $\endgroup$ Jul 1, 2019 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, 2019 at 12:46
  • $\begingroup$ @denklo I will add this assumptionm to the post $\endgroup$ Jul 1, 2019 at 13:52

2 Answers 2


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.


Your example demonstrates the mass flow for 3 different pipes. "The equation of continuity for pipes" concerns pipes of different diameter that are connected successively to each other so you have a single long pipe that might be fat for the first half and thin for the second half. If the long pipe is initially full of water, then if you stuff 1 kilogram of water in one end (in one second) then 1 kg of water must come out the other end (in one second), so the mass flow in the fat section must be 1 kg/s and the mass flow in the thin section must also be 1 kg/s. Hence the continuity equation. (Assume the water is incompressible, which it is near enough.)

The equation of continuity for pipes is best illustrated by the last of your 3 diagrams. The mass (or volume) of the water going into the glass at the end in one second is equal to the mass or volume entering the same pipe on the left in one second and is the same in the fat sections, as it is in the thin section, so it is independent of the diameter of the pipe in any given section.


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