Interpreting Bernoulli Equation

[cj]

Homework Statement

The photo below shows a stream of water in steady flow from a kitchen faucet. At the faucet, the diameter of the stream is 0.960 cm. The stream fills a 0.125cm³ container in 16.3 s. Find the diameter of the stream 13.0 cm below the opening of the faucet.

Homework Equations

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The Attempt at a Solution

The official solution says to take P₁ = P₂. Why is this? Isn't the pressure at the bottom of the column greater than the pressure at the top? I'm not quite grasping an explanation that says "the pressure at the top and at the bottom of the stream is the same: atmospheric pressure." Wouldn't a pressure gauge inside the stream near the top register a different value than one near the bottom?

Thank you!

 

[Chestermiller]

The official solution says to take P₁ = P₂. Why is this? Isn't the pressure at the bottom of the column greater than the pressure at the top? I'm not quite grasping an explanation that says "the pressure at the top and at the bottom of the stream is the same: atmospheric pressure." Wouldn't a pressure gauge inside the stream near the top register a different value than one near the bottom?

No. It would actually be very close to atmospheric at both locations. Certainly at the surface of the stream, the pressure is atmospheric at both locations, since it is contact with the air. There is very slow flow in the radial direction compared to the axial direction, so the pressure gradient in the radial direction is going to be very close to zero (from radial momentum balance considerations). So the pressure in the middle of the stream is essentially equal to the pressure at the surface (i.e., atmospheric).

Chet