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This is a deceptively simple problem in fluid mechanics that I can't quite figure out because it produces a paradox.

Suppose that there is a straw and a small column of water inside the straw, such that the height of the column is significantly smaller than the height of the straw, and that the top of the column is initially at the same height as the top of the straw. Suppose also that the cross sectional area of the straw at the top is $A_1$, and at the bottom it is $A_2$, such that $A_1>A_2$, and that this change in the cross sectional area of the straw occurs somewhere around its midpoint.

Now suppose that the water column accelerates from rest due to the gravitational force until the bottom of the column is at the same height as the bottom of the straw. As the column falls through the midpoint of the straw its cross section changes from $A_1$ to $A_2$. Therefore, its velocity must increase due to the decrease in cross sectional area, as follows from the continuity equation. This means that there is a force acting on the bottom portion of the column as it accelerates through the midpoint, and that this force cannot be the gravitational force because that accelerates the entire column uniformly.

So the question is this: what is the additional force?

Typically this force comes from a pressure differential, but in this case both the top and bottom of the water column are at atmospheric pressure, so there is no pressure differential.

The paradox is that if this force doesn't exist then the continuity equation is violated. If the continuity equation is not violated then a force exists without a pressure differential.

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    $\begingroup$ Would there not be some upwards force on the water from the straw at the point where the straw narrows? It not a question there being extra force on the water past the narrow but rather there being a force opposing gravity on the water above the narrow. $\endgroup$ – M. Enns Apr 22 '16 at 0:37
  • $\begingroup$ The acceleration of each tiny volume of water in the water column increases as it passes through the midpoint. If there was a force opposing gravity at the midpoint then those tiny volumes of water would experience a decrease in acceleration. But in fact the very opposite happens! $\endgroup$ – The Riddler Apr 22 '16 at 0:43
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    $\begingroup$ You are completely neglecting the resistance to flow a restriction actually imposes. Therefore you start from the false premise that $v_2>v_1$ which is certainly not necessarily true, see also Duncan Harris's case for a pinhole. There's no paradox here. Continuity doesn't apply here in the narrow sense of the understanding. $\endgroup$ – Gert Apr 22 '16 at 1:06
  • $\begingroup$ Yes, I believe Duncan Harris got it right, although it is still definitely the case that $v_2 > v_1$ and that the continuity equation still holds. $\endgroup$ – The Riddler Apr 22 '16 at 1:29
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There is a pressure differential.

When the falling water hits the partial obstruction in the middle of the straw, the pressure around the edges of the straw increases. This increased pressure accelerates the water into the smaller passage below (and also tries to decelerate the water coming from above).

This effect is not consistent over the period in which the water passes through the midpoint. By the time the top of the water column reaches the midpoint, the pressure at the constriction has fallen back to atmospheric pressure, because there is no more water crashing down against the walls of the straw.

If you don't see why the pressure increases when the column reaches the midpoint, consider the extreme case in which the straw is reduced to a pinhole. Then the water will smash into the constriction and be brought to a halt by the increased pressure where the bottom of the column hit the obstruction (the pressure is contained by the straw on the sides and bottom, and by the inertia of the water from above). This same effect will be at play with any less drastic reduction of cross section.

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  • $\begingroup$ I believe you successfully resolved the paradox, so thank you very much for that. However, I am still unsure how one would calculate this pressure differential. $\endgroup$ – The Riddler Apr 22 '16 at 1:54
  • $\begingroup$ The behavior of the water column depends on the exact shape and dimensions of the straw. If there is an abrupt constriction, then turbulence and shock waves will be produced and you are in for a very demanding numerical simulation. If the straw has a very gradual taper and simple laminar flow is preserved, then you can set up and solve a system of equations to find the pressure as a function of height and time. $\endgroup$ – Duncan Harris Apr 22 '16 at 2:03

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