I used the same argument in the proof of equation of continuity to flow of stones. Suppose I drop stones from the upper end of a vertical pipe. I am continuously dropping the stones so that at any instant the pipe is full of stones. The stones clearly have streamlines just like fluid flow because they have only straight line downward motion. So, after applying law of conservation of mass, we get the equation of continuity for the stones. Since the stones are freely falling, so they clearly come out of the pipe with a speed greater than the initial speed. So, the area of lower end of the pipe should be smaller. But this doesn't make sense because all the stones only have downward acceleration throughout their motion. So, there's no way their paths would have curved in the journey so that they come out of a smaller area. So, how's this possible? I've one confusion with the equation of continuity for fluid flow also. If I drop water from a height $(h)$ with initial velocity zero and having some finite initial area of tube. Then, the area of the tube after any finite time will be: $A_2=\frac{A_1v_1}{v_2} = 0.$ So, the water flow will have zero area after any finite time $(t).$ How's this possible?
[Physics] Equation of continuity for stones
fluid dynamics
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Best Answer
The answer to the first question is that the rate of flow involves not just the velocity and cross sectional area, but also density. With stones in a pipe, the area stays the same, so when the velocity rises, the density falls to compensate (the distance between stones get stretched vertically as their velocity rises). With a fluid, we often assume interparticle forces that maintain a fixed density, so as the velocity rises, the cross sectional area is what drops rather than the density. The interparticle forces, absent with stones, make that possible.
The answer to the second question is that the mass continuity equation assumes you have a steady state, i.e., the state of the fluid as a whole looks the same from moment to moment. You can't have that if you want to keep feeding in new fluid at the top yet claim it is dropped with zero speed-- that fluid has to come from somewhere, so must have some kind of motion. In the case of a water faucet, there is horizontal flow that is turning into vertical flow, so just analyze the flow along the pipe direction as the pipe turns from horizontal to vertical. There is never anywhere that has zero speed along the pipe once a steady state appears. And if you only look at the region where the flow is entirely vertical, there is always some nonzero vertical motion, even at the top of the downward flow.